Università degli studi di Firenze

Università degli studi di Firenze

Facoltà di Scienze Matematiche Fisiche e Naturali Corso di Laurea Magistrale in Matematica

Regularity of solutions for isotropic and anisotropic operators

Regolarità delle soluzioni per una classe di PDEs anisotrope Candidato

Dott. Simone Ciani [email protected].it

Relatore

Prof. Vincenzo Vespri [email protected].it

Correlatore

Prof. Elvira Mascolo [email protected].it

Sessione Ottobre 2018 Anno Accademico 2017/2018

Contents 1 Preliminaries 1.1 1.2 1.3 1.4 1.5

Convex functions and functionals . . . . . . . . . . . . Measurable functions and classical function spaces . . . Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . Parabolic and Elliptic equations . . . . . . . . . . . . . 1.4.1 Physical interpretation and well posed problems The variational approach: nonlinear analysis . . . . . . 1.5.1 The bootstrap argument . . . . . . . . . . . . .

2 Regularity on isotropic growth conditions 2.1 2.2 2.3 2.4

De Giorgi Theorem . . . . . . . . . . . . . . . De Giorgi classes . . . . . . . . . . . . . . . . A Geometric proof by expansion of positivity . The vectorial case: a counterexample . . . . .

3 Regularity on anisotropic growth conditions 3.1 3.2 3.3

. . . .

. . . .

. . . .

. . . .

. . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

Introduction and counterexamples . . . . . . . . . . . . . . . Dierent growths from above and below . . . . . . . . . . . Additional Structure . . . . . . . . . . . . . . . . . . . . . . 3.3.1 The anisotropic Euler-Lagrange equation. . . . . . . . 3.3.2 Regularity of local weak solutions to anisotropic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Boundedness of weak solutions . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

3

3 7 13 22 26 27 40

42

42 59 67 85

93

93 99 117 125

. 129 . 136

The more ambitious plan may have more chances of success

G.Polya

Introduction The object of study of this thesis is the regularity of solutions for partial dierential equations with anisotropic growth conditions. In the claim for a clear homogeneous exposure and for an updating on most recent results on the subject, here have been taken under analysis more than fty scientic papers. The wide and detailed presentation represents an analysis of the whole domain of regularity for anisotropic operators, and it can be considered as a scientic publication for the regularity problem for solutions to anisotropic operators. Technical considerations of last chapter of this work have been developed on the aim of an extension of the recent result [36] of Fatma D. Gamze, Paolo Marcellini and Vincenzo Vespri, for the expansion in positivity in elliptic equations with growth conditions of a particular anisotropy which has a parabolic approach. This result in particular needs no restriction on the distance between maximum exponent and minimum exponent of growth, and a further use of it for more general anisotropic growth conditions would oer a signicant contribute to regularity of solutions to partial dierential equations with anisotropic growth. The thesis observes the following subdivision. In the rst chapter we introduce functional spaces and classical methods in order to approach problems of Calculus of Variations. In its conclusive section we present 19th and 20th Hilbert's problems and the iterative bootstrap argument, in order to obtain C ∞ regularity for the solutions. In the second chapter we present rstly the De Giorgi's theorem, using a constructive approach. In a second section we introduce the De Giorgi classes generalized to quasi-minima of integrals of Calculus of Variations and we demonstrate Harnack's inequalty to the homogeneous case. Lastly we present a geometric proof for the Hölder regularity in the general isotropic case, which shows the method of expansion in positivity of solutions, combining the De Giorgi method with Moser's one, with gures and illustrations. In the third and last chapter we introduce and study anisotropy of growth conditions. Firstly we give counterexamples to regularity and boundedness of solutions, showing that an adjustment of the theory is necessary with respect to the isotropic case. Subsequently we introduce and study the boundedness for solutions to anisotropic equations of type p,q, presenting and demonstrating with detail most relevant results. We present examples and applications in the context of continuum mechanics for the model of Rajagopal and Ruºi£ka 1

[77] on electrorheological uids. Then we introduce operators exhibiting an anisotropy of growth in coordinate axes directions, Sobolev anisotropic spaces and the extension results for Sobolev and Morrey's embeddings. We also present a counterexample for a domain of general shape. Finally, we study with detail the boundedness of solutions for this type of equations, we give exposure to the expansion of positivity technique for a special class of anisotropic PDEs and we establish recent results of Hölder continuity.

2

1

Preliminaries

This section has the goal to x some notation and convention in order to gain readability, and to introduce quickly the reader to the subject. This work is thus self contained. As a matter of notation, we stress the fact that in calculations where appears a constant C , it is supposed by the author that C may change from inequality to inequality. Throughout this chapter we will always assume that U ∈ Rn is an open bounded set, Ω ∈ Rn is an open set and E ∈ Rn is a measurable set.

1.1 Convex functions and functionals The role of convex functions is determinant in Calculus of Variations, as we will shortly see convexity is an essential tool for the development of the theory presented here. It is an useful property to get lower semicontinuity for functionals of the calculus of variation, which, together with other assumptions that we will develop, will imply the existence of solutions for the minimum problem.

Denition 1.1.

Let −∞ ≤ a < b ≤ +∞ and φ : [a, b] → R be a function. Then φ is said to be convex in [a, b] if ∀t0 , t1 ∈ [a, b] and λ ∈ [0, 1] holds

φ((1 − λ)t0 + λt1 ) ≤ (1 − λ)φ(t0 ) + λφ(t1 ) Equivalently φ is convex

(1.1)

i the epigraph

 Lφ(t) := (t, s) ∈ R2 : t ∈ [a, b], s > φ(t) is a convex set. Finally we say that φ is concave if −φ is convex.

Properties of convex functions • Let P {fn }n∈N be convex functions in [a, b], then if αn , n ∈ N the function n∈N αn fn is convex in [a, b] when converging. More on if fn → g for n → +∞ then g is convex too.  • If φi }i∈I is a family of convex functions in [a, b] then φ(t) := supi∈I φi (t) is convex in [a, b]. • Given a convex function f : [a, b] → R, f is continuous and ∀t ∈ (a, b) there exist right and left derivatives (or increment limit) φ0 (t+ ), φ0 (t− ). Moreover when f is convex the derivative f 0 exists almost everywhere (a.e) and such a derivative is an increasing function a.e. 3

• Assume f : R → R is convex, U ⊂ Rn is open and u : U → R ∈ L1 (U ) in the sense of denition 1.5. Then f satises the famous Jensen's inequality Z Z f ( u dx) ≤ f (u)dx (1.2) U

U

Denition 1.2.

A function of n independent variables f : Rn → R is said to be convex when ∀x, y ∈ Rn , 0 ≤ τ ≤ 1,f (τ x + (1 − τ )y) ≤ τ f (x) + (1 − τ )f (y) holds. For such a function and for each x ∈ Rn ∃r ∈ Rn such that holds f (y) ≥ f (x) + r · (y − x) ∀y ∈ Rn : the mapping y → f (x) + r · (y − x) determines the supporting hyperplane to f at x and previous inequality, in analogy to what happens for function of one single variable, says the graph of f lies above each supporting hyperplane. Moreover if f were dierentiable, Df = r and convexity can be interpreted with classic conditions on second derivatives.

Denition 1.3.

Let f ∈ C 2 (Ω) for Ω ∈ Rn open, then f is convex i D f ≥ 0 i.e. if its Hessian matrix is positive denite. The C 2 (U ) function f is said to be uniformly convex if D2 f ≥ αId for some α > 0 i.e. 2

n X

fxi ,xj (x)bi bj ≥ α|b|2 ,

(1.3)

∀x, b ∈ Rn

i,j=1

Uniform convexity will play an important role in the developments of regularity results and in existence of appropriate solution of various variational problems. To see this, we introduce the notion of lower semicontinuity (l.s.c),

Denition 1.4.

Let X be a topological space. A function

1

¯ F :X→R is lower semicontinuous (l.s.c) if ∀t ∈ R the set Ft := {x ∈ X : F (x) > t} is open. Equivalently F is l.s.c. i its epigraph Σ(F ) := {(x, t) ∈ X × R : t ≥ F (x)} is closed. If we add the hypothesis that X satises the rst countability ax¯ is l.s.c. i it is sequentially lower semiconiom2 such a function F : X → R tinuous : this means that ∀{vk }k∈N ⊆ X sequence which converges to some v ∈ X we will have F (v) ≤ lim inf F (vk ) k→∞

1 Where

¯ := {R ∪ {∞}}. We will habitually write ∞ instead of +∞ when sign is clear R

from the context. 2 i.e. if every point x ∈ X has a countable fundamental system of neighborhoods. This is the case for sure when X is a metric space, as a Banach space.

4

Let now V ⊆ X be a subset in which we would like to minimize our functional F . We call minimizing a sequence {xk }k∈N ⊆ V s.t. limk→∞ F (xk ) = inf V F .

Theorem 1.1. (Generalized Weierstrass Theorem) Let V ⊂ X be a subset of ¯ be a l.s.c function. Assume there exists a topological space and let F : X → R a minimizing sequence {xk }k∈N → x0 ∈ V . Then, F (x0 ) is the minimun of F in V . Lower semicontinuity and existence of a minimizing sequence are two rival qualities in the choice of a good topological space X where to minimize functional F : they're usually contrasting requests. The choice of the topology to give to X should be placed at the equilibrium point between these two forces pushing in opposite directions3 .

Denition 1.5. A Caratheodory function is a function F (x, y) : Ω×Rn → R¯ such that

• F (·, y) is measurable ∀y ∈ Rn • F (x, ·) is continuous for almost every x ∈ Ω; It is easy to see that if F (x, y) is a Caratheodory function and y(x) is a measurable function in Ω then the function F (x, y(x)) is measurable in Ω and that if also F (x, y) ∈ L1 (Ω) then F (x, y(x)) is lower semicontinuous in the strong topology of L1 (Ω) by Fatou's Lemma. The role of convexity in this case is to make this property (l.s.c) pass to the weak topology.

Theorem 1.2. Let F (x, y) be a Caratheodory function such that for almost every x ∈ Ω the function F (x, ·) is convex. Then the functional Z F (x, y(x))dx

G(y, Ω) = Ω

is lower semicontinuous in the weak topology of L1 (Ω). Proof. By precedent consideration F is l.s.c. in the strong topology of L1 (Ω)

and as the latter is a metric space, the epigraphic Σ(G) is a strongly closed set. Furthermore convexity of F implies convexity of G which implies convexity of its epigraphic Σ(G). It follows4 that Σ(G) is weakly closed and so that G is l.s.c in the weak topology of L1 (Ω). 3 It

will be useful to prove semicontinuity in the weakest possible topology, where is more likely to nd a minimizing sequence. 4 See théorem III.7 of [8]

5

If we consider F = F (x, u, Du) in U ⊂ Rn a bounded open set with a C 1 boundary, we can reduce hypothesis on convexity of (u, Du) → F (x, u, Du) to the request of F to be convex on the (weak) derivative Du and to be continuous on u. Indeed, by Rellich's compactness theorem (see section 1.1.3) a weakly convergent sequence in W 1,p (U ) has a strongly convergent subsequence in Lp (U ), to which can be applied Fatou's Lemma.

Theorem 1.3. Let Ω be an open set in Rn , let M be a closed set in R and let F (x, u, z) be a function dened in Ω × M × Rn satisfying • f is a Caratheodory function, i.e. measurable in the variable x for every (u, z) ∈ M × Rn and continuous in (u, z) for almost every x ∈ Ω; • F(x,u,z) is convex in z for almost every x ∈ Ω and ∀u ∈ M ; • F (x, u, z) ≥ −c(|z|m + |u|k + g) with k ≥ 1 , p > 1, m < p, g ∈ L1 (Ω).

Then the functional Z F (x, u, Du)dx

G(u, Ω) = Ω

1,1 is lower semicontinuous in the weak topology of Wloc (Ω, M ). 1,1 1,1 In few words, let {uk }k∈N ⊂ Wloc (Ω, M ) converging weakly to a u ∈ Wloc (Ω, M ). α α 1 Then D uk * D u in Lloc (Ω) for α = 0, 1 and we have

Z

Z F (x, u, z)dx ≤ lim inf

Ω

k→∞

F (x, uk , Duk )dx Ω

Finally we can show how can this theorem be useful to prove existence of a minimizer for the functional Z G(u, Ω) := F (x, u, Du)dx (1.4) Ω

By the generalized Wietrass theorem (Theorem 1.1) it will be enough to show that there exists for it a minimizing sequence which converges in the weak 1,1 topology of Wloc (Ω). This can be made, overstepping the diculty given by the fact that this space is not reexive, because a weakly convergent 1,p 1,1 sequence in Wloc (Ω)(p > 1) is also weakly convergent in Wloc (Ω): it will be thus sucient to nd a minimizing sequence which converges weakly in 1,p (Ω) for some p > 1. As the latter is a reexive space, it will be enough Wloc 1,p to nd a minimizing sequence, bounded in Wloc (Ω). There is a very simple situation making each minimizing sequence a bounded one, and it is the following. 6

Denition 1.6.

The functional G of (1.4) is

lim

||u||1,p →∞

coercive if

F (u) = +∞

Remark 1. This condition is surely satised if we assume classical growth conditions as in Theorem 1.3 as F (x, u, z) ≥ δ(|u|p + |z|p )

(1.5)

with δ > 0 and p > 1. Until this moment we have not mentioned other possible conditions (such as boundary ones) to be imposed on the function u(x), and as in general the problem consists of nding the minimum of the functional G among all the functions u satisfying certain properties, this takes us to consider a subset V ⊆ W 1,p (Ω). This subset must be closed in the weak topology of W 1,p (Ω), since we need the limit of a sequence of functions of V to be in V . We conclude this section with an existence theorem by [44] which uses the eect of convexity and coercivity together.

Theorem 1.4. Let Ω ⊆ Rn be open, M ⊂ Rm closed, and let G(u) be a l.s.c. 1,p functional in the weak topology of Wloc (Ω, M ) 5 , p > 1. Let V be a weakly 1,p closed subset of Wloc (Ω, M ), and assume that G is coercive in V. Then G reaches its minimum in V. References and steps foward For simple properties of convex functions see [59], while for their main use in calculus of Variation we refer to [44] (chapter 4) and [30]. Rellich's theorem as well as Sobolev spaces will be more detailed on section 1.3 and its main reference is [2]. Last theorems are exactly what we need to obtain existence and uniqueness for minimizers of calculus of variations as we will see in section 1.4

1.2 Measurable functions and classical function spaces The aim of this subsection is to list some denitions and properties of Lp spaces that will be useful later. As a reference we use the vast literature on Lesbegue integrals, such as [80] , [59], [8] . When speaking of measurable sets and functions we will always refer to 5 We

denote by Lp (Ω, M ) the space of measurable functions f : Ω ⊂ Rn → M ⊂ Rm whose p-module integral is nite, as in Denition 1.7. Regarding their module as a real valued function all denitions can be done analogously , as in [44].

7

Lebesgue measure, and as usual we will not distinguish between functions that dier only on a set of zero measure. This will allow us to work directly in a normed space, as we wil see in Denition 1.7. For a measurable set E ⊆ Rn , we shall denote its measure with |E| . We will denote by C k (E) for k = 0, 1, 2, ... the linear space of functions having continuous derivatives up to and including order k with the conventions that C 0 (E) = C(E) is the space of continuous functions and C ∞ (E) the linear space of innitely dierentiable functions in E, that is the intersection of all C k (E). We will ¯ the linear space of functions in C k (E), whose derivatives indicate with C k (E) up to order k can be extended to continuous functions up to the boundary ¯ made ∂E and lastly with Cok (E) for k = 0, 1, 2... the subspace of C k (E) by functions having compact support contained in E . Given a functional linear space Λ(Ω) made of functions dened on an open set Ω ⊂ Rn , we will write Λloc (Ω) to denote the space of functions belonging to Λ(V ) for each compactly contained V ⊂⊂ Ω. Standard notation about multi-index α = (α1 , α2 , ...., αn ), αi ∈ N is assumed, i.e. if f (x) ∈ C k (E) and |α| = k then

∂ α f (x) D f (x) = ∂xα1 1 ∂xα2 2 . . . ∂xαnn α

With these conventions, the formulae involving partial derivatives of functions of n variables become very compact: for example Taylor's formula may be written as 6

f (x) =

X Dα f (x0 ) (x − x0 )α + Rk (x; x0 ) α!

|α|≤k

where Rk is the rest of order k , whereas the formula for the derivative of a product becomes (for γ ≤ α means γi ≤ αi ∀i = 1, ...n) X α α D (f g) = Dγ (f )Dα−γ (g) γ γ≤α

¯ ) are Banach spaces ∀k ∈ N, with the usual norm The spaces C k (U X ||u||C k := sup Dβ u(x) |β|≤k

x∈U

If D is a domain in Rn , i.e. the closure of a bounded open set, we will denote by C 0,α (D) the space of all Hölder-continuous functions in D, that is 6 Denoting

with α! = α1 !α2 !...αn ! and

α γ




=

8

α! (α−γ)!γ!

being (α − γ) =

Qn

i=1 (αi

− γi ).

continuous functions for which

[u]0,α :=

|u(x) − u(y)| < +∞ |x − y|α x,y∈D,x6=y sup

On a further step we shall denote by C k,α (D) the space of functions k times dierentiable in the interior of D, whose derivatives extend to Höldercontinuous functions in D. The spaces C k,α (D) are Banach spaces if equipped with the norm X  kukk,α := kukC k + Dβ u 0,α |β|=k

Finally if Ω is an open set in Rn , we shall indicate with C k,α (Ω) the linear space of all functins belonging to C k,α (D) for every domain D ⊂ Ω.

Example 1. Sample paths of Brownian motion t → W (t, w) are almost surely 7 uniformly α -Hölder continuous in their interval denition [0, T ] for α < 1/2. This comes as an application by a theorem of Kolmogorov. A detailed and sketchy proof can be found in [32] Example 2. Functions which are locally integrable and whose integrals satisfy an appropriate R growth condition are Hölder continuous. For example, if 1 we let ux,r = |Br | Br (x) u(y)dy ,and u satises Z

|u(y) − ux,r |2 dy ≤ Crn+2α

Br (x)

then u is Hölder continuous with exponent α. This is a result originally due to Sergio Campanato, see for reference ( [47], theorem 3.1 chap 3).

Example 3. We will see later in section 3 that if w ∈ W 1,p (U ) for p > n, an higher exponent than the spatial dimension, then u is Hölder continuous via Morrey's embedding. Remark 2. Let U ⊂ Rn be a bounded open subset and let 0 < α < β ≤ 1 be two Hölder exponents. Then there is an obvious inclusion map of the corresponding Hölder spaces C 0,β (U ) ⊂ C 0,α (U )

which is continuous since by denition we have ∀f ∈ C 0,β (U ) [f ]0,α,U ≤ diam(U )β−α [f ]0,β,U 7 With

exception of a set of probability zero.

9

Moreover, this inclusion is compact: i.e. bounded sets in [·]0,β -norm are relatively compact in the [·]0,α -norm. This is a direct consequence of AscoliArzelà theorem. Indeed, let {un }n∈N be a bounded sequence in C 0,β (U ): as this is automatically equicontinuous, thanks to Ascoli-Arzelà theorem we can assume that {un } → u uniformly as n approaches innity and we can also assume (w.l.o.g.) that the limit is zero, i.e. u = 0. Then |un − u|0,α → 0 as α |un (x) − un (y)|  |un (x) − un (y)|  αβ 1− α 1− α β ≤ [u ] β β |u (x)−u (y)| = n n n 0,β,U (2||un ||) α β |x − y| |x − y|

Denition 1.7.

Let E be a measurable set in Rn by Lebesgue measure, and let 1 ≤ p ≤ ∞. We denote by Lp (E) the space of measurable functions f : E → R such that ( R p1 p , se 1 ≤ p < ∞ |f | dx E ∞ > kf kp,E := ess supx∈E |f (x)|, se p = ∞ where ess supx∈E := inf{t ∈ R : |f −1 (t, +∞)| = 0} is the essential supremum of f over E . The spaces Lp (E) are linear spaces over the eld of real numbers R, and for 1 ≤ p ≤ ∞ with the customary equivalence between functions taking dierent values in a set of zero measure, are Banach spaces. Particular relevance has in the case p = 2 the linear space L2 (E) which is an Hilbert space with the usual scalar product given by the integral of the product of the functions. We will improperly continue to call the elements of these spaces "functions". Moreover for 1 < p < ∞, Lp (E) spaces are reexives, and their dual is isomorphic to Lq (E) with p1 + 1q = 18 . The space L1 (E) has L∞ (E) as its dual, but it is not reexive: indeed for 1 ≤ p < ∞ linear spaces Lp (E) enjoy the property of separability while L∞ (E) does not share with them this quality, which is classically stable by reexivity.

8 Property

of exponents p, q which will be called in what follows conjuguate exponents.

10

Let us now show some inequalities and properties of p-norms and Lp spaces.

 • If f ∈ Lp (E), the set Ft := x ∈ E : |f (x)| > t is measurable since Z Z p |f | dx ≥ |f |p dx ≥ tp |Ft | so that we have|Ft | ≤ t−p kf kpp,E E

Ft

• We recall also the well-know formula Z Z ∞ p tp−1 |Ft |dt |f | dx = p 0

E

• Hölder's inequality: for p > 1, f ∈ Lp (E) and g ∈ Lq (E) with p, q conjugate exponents it gives f g ∈ L1 (E) and Z kf gk1,E =

Z

p

p

 p1 Z

|f | dx

|f (x)g(x)| dx ≤

|g| dx

E

E

q

E

 1q (1.6)

= kf kp,E kgkq,E It is easily understandable that for p = q = 2 this inequality bears also the name of Cauchy-Schwartz inequality, and it remains valid for p = 1 and q = ∞. In the case for which E has nite measure, if 1 ≤ s < r ≤ ∞ we may put f = |u|s , g = 1 and p = r/s in Hölder inequality obtaining 1

1

kuks,E ≤ |E|( s − r ) kukr,E so that if |E| < ∞ and s < r we have Lr (E) ⊂ Ls (E) algebraically and topologically as the metric given by the r-norm is ner.

• Minkowski inequality, which gives us stability for the sum in the structure of linear space on Lp (E) and which says that if 1 ≤ p < ∞ and f, g ∈ Lp (E) then |f + g| also lives in the same space is 1 Z 1 1 Z Z p p p p p p ( |f + g| dx) ≤ ( |f | dx) ( |g| dx)

E

E

(1.7)

E

• For any open set E ⊂ Rn and for any 1 ≤ p < ∞, the space C0∞ is dense in Lp (E), this meaning that ∀f ∈ Lp (E) and for each choice of a positive , ∃g ∈ C0∞ such thatkf − gkp,E <  11

• Interpolation inequality: let 1 ≤ p < q < r such that 1q = pθ + 1−θ r for some 0 < θ < 1. If u ∈ Lp (E) ∩ Lr (E) then u ∈ Lq (E) by an application of Hölder inequality for a good choice of indexes ||u||q ≤ ||u||θp ||u||r1−θ

(1.8)

• Young's inequality |ab| ≤ ap + C()bq , with p, q conjugate exponents 1 < p, q < ∞, a, b,  > 0 and C() = (p)−(q/p) q −1 Finally, we dene a class of functions of very relevant utility in what follows, as they will be the tool we will use to derive most important inequalities just from integral and dierential equations.

Denition 1.8.

Let U ⊂ Rn an open set and let us consider for x0 ∈ U an appropriate radius 0 < r < R, and two concentric balls Br (x0 ) ⊂⊂ BR (x0 ) ⊂⊂ U . A function η : U → [0, 1], η ∈ Co∞ (U ) is called a cut-o function between Br (x0 ) and BR (x0 ) if η ≡ 1 on Br (x0 ), η ≡ 0 in U BR (x0 ).

Proposition 1.5. (Construction of a special cut-o function) It is possible to build a cut-o function η in x0 = 0 ∈ Rn whose derivative is bounded by |D(η)| ≤

C R−r

(1.9)

for a positive constant C and ∀x ∈ Rn .   1

|x| ≤ a Proof. Let f : R → [0, 1] dened by f (x) = a < |x| < b for xed  0 |x| > b a < b , and let's consider its convolution with a mollier9 w . With these assumptions we have that supp(f ∗ w ) ⊆ [−(b + ), b + ] and if |x| < a −  then f ∗ w ≡ 1 while if |x| > b +  then f ∗ w ≡ 0 by simple considerations on supp(f ). More on, as f ∈ L1loc (R) by classic approximation theory of convolutions we have that f ∗ w ∈ C ∞ (R) so that (|x|−b)  a−b

η(x) := f ∗ w (|x|) ∈ Co∞ (R) is a good candidate to be our cut-o function when we chose a = r +  and . Finally as |f | ≤ 1 and Dα (f ∗ w ) = f ∗ Dα (w ) we have b = R −  ,  < R−r 2 9 For

 > 0 a mollier is a Co∞ (Rn ) function w with support contained in B (0) and w (x) dx = 1. A classic exemplar of mollier is w (x) = 1 w( x ) for w(x) = Rn  ( 1 ke |x|2 −1 |x| ≤ 1 . 0 |x| > 1 R

12

Z

α



|D (f ∗ w )(x) ≤

|f (x − y)| |Dα w (y)| dy ≤ 2 max |Dα w | ≤ 10 [−,]

−

1 max w} 2 R just by choosing  = 2{

and this is less than

C R−r

R−r 3

References and further steps For a more detailed information on Sobolev spaces we refer to [8] chapter 4 or [59]. About reexivity we follow [9] on its third chapter. A further step in comprehending this functional spaces is given by the spaces of Morrey and the spaces of Campanato ([44] chapter 2), that will not nd room in this work. The cut-o function we described at the end of this section will be useful in various successive demonstrations as in Theorem 1.24, Proposition 2.1 and in deriving various versions of Cacioppoli's inequality and its applications given further on. We dened the special cut-o function just when considering two concentric balls, but just by modifying lightly its denition it should be peaceful that we are able to construct such cutt-o functions valuables for every boundary-regular open set V ⊂ U .

1.3 Sobolev spaces To extend the class of functions to work with, we dene a summable function which takes the integration by parts role when this is not possible in the classical way. These functions are so important that a whole theory has been developed upon their denition, valuable utility in solving partial dierential equations has been extrapolated from it and its contribute has enriched the beauty of the abstractness of Mathematics.

Denition 1.9.

Let us suppose u, v ∈ L1loc (U ) i.e. u, v ∈ L1 (V ) for every open set V ⊂⊂ U and α multiindex. We say that v is the αth-weak derivative of u, and we will write Dα u = v provided the holding of the integral equality below for every φ ∈ Co∞ (U ) Z Z |α| α vφdx (1.10) uD φdx = (−1) U

U

Remark 3. Thanks to the Fundamental Lemma of Calculus of Variations, a α th-weak derivative v of u is uniquely dened up to a zero measure set. 10 This

bounding to the mollier is referred to last footnote about it.

13

Denition 1.10.

Given m ∈ N, 1 ≤ p ≤ ∞ and E ⊂ Rn a measurable set, we dene the (m, p)-Sobolev norm as a functional || · ||m,p dened as follows,  P 1  p p α || · ||m,p := , 1≤p 0} D(u+ ) = 0 {u ≤ 0}

(1.11)

• if u ∈ W 1,p (U ) and f ∈ C 1 (U ) s.t. f 0 ∈ L∞ (U ) then D(f ◦ u) = f 0 (u)Du

(1.12)

• if U is bounded and ∂U is C 1 , given a bounded open set U ⊂⊂ V then there exists an extension operator E : W 1,p (U ) → W 1,p (Rn ) s.t. for each u ∈ W 1,p (U )E(u) = u a.e. in U, supp(Eu) ⊂ V , and for a

constant C just depending on p,U,V

||Eu||W 1,p (Rn ) ≤ C(p, U, V )||u||W 1,p (U ) 15

As a next step, we discuss the possibility of assigning to a function of W m,p (U ) for 1 ≤ p < ∞ some boundary values along ∂U . The problem is that the Lebesgue measure of ∂U is zero, and so a priori every function of such a space would coincide with the others in that zero measure subset of the domain. That is why we are naturally brought to take into account the notion of Trace Operator, which resolves our problem.

Theorem 1.7. Let us suppose U bounded with a C 1 boundary. Then there exists a bounded linear operator (1.13)

T : W 1,p (U ) → Lp (∂U )

such that T u = u∂U in case u ∈ W 1,p (U ) ∩ C(U¯ ) and ∀u ∈ W 1,p (U ) for a constant C depending only on p and U , ||T u||LP (∂U ) ≤ C(p, u)||u||W 1,p (U )

For a chosen element u, T u is called the trace of u in ∂U . With the help of Trace Operator we are able to describe more easily elements belonging to the linear space Wo1,p (U ). Assuming 1 ≤ p < ∞ and U ⊂ Rn a bounded open set with a C 1 boundary and supposing to have a u ∈ W 1,p (U ), then

u ∈ Wo1,p (U ) if and only if T u = 0 on ∂U Now we turn our attention to the discovery of various Sobolev spaces embeddings into other functional spaces: our main tools are certain inequalities which can be proved for smooth functions and then established for elements of various relevant Sobolev spaces by density with an approximation argument.

Denition 1.12. of p as

For 1 ≤ p < ∞ we dene

the Sobolev conjugate exponent

np n−p

(1.14)

p∗ :=

Remark 5. We remark that p∗ > p, as previous equation gives

1 p∗

=

1 p

−

1 n

Theorem 1.8. (Gagliardo-Nirinberg-Sobolev) Let us assume 1 ≤ p < n. Then there exists a constant C = C(p, n) such that for all compactly supported dierentiable function u ∈ Co1 (Rn ) ||u||Lp∗ (Rn ) ≤ C(p, n)||Du||Lp (Rn ) 16

Remark 6. Take as a counterexample u ≡ 1 to observe that compact support is strongly needed by our assumptions and let us remark that C does not depend at all on the size of supp(u). Theorem 1.9. (Inequalities for W 1,p spaces with 1 ≤ p < ∞). Let U ⊂ Rn be a bounded open set having a C 1 boundary. If u ∈ W 1,p (U ), 1 ≤ ∗ p < ∞ then u ∈ Lp (U ) and the inclusion is continuous, i.e ∃C = C(n, p, U ) s.t. ||u||Lp∗ (U ) ≤ C(p, n, U )||u||W 1,p (U )

(1.15)

Proof. By properties of W 1,p (U ) functions and of the domain U , we know

(Proposition 1.6) that there exists an extension Eu ∈ W 1,p (Rn ) such that Eu = u in U , Eu has compact support and

||Eu||W 1,p (Rn ) ≤ C(p, U, V )||u||W 1,p (U ) . Moreover as Eu has a compact support we know still by properties of W 1,p (U ) functions (Proposition 1.6) that ∃{um }m∈N ⊂ C ∞ (Rn ) s.t. um → u in W 1,p (Rn ). Now, according to Gagliardo-Nirinberg-Sobolev theorem we have the estimate

||um − uk ||Lp∗ (Rn ) ≤ C(p, n)||Dum − Duk ||Lp (Rn ) . So {um }m∈N is a Cauchy sequence in Lp∗ (Rn ) and by completeness we deduce that uk → Eu in Lp∗ (Rn ). Note that theorem of Gagliardo-NirimbergSobolev can also be applied to every singular uk of this sequence, holding the bound ||uk ||Lp∗ (Rn ) ≤ C(p, n)||Duk ||Lp (Rn ) for each k ∈ N and so its limits inherits the bound

||Eu||Lp∗ (Rn ) ≤ C(p, n)||D(Eu)||Lp (Rn ) ≤ ||Eu||W 1,p (Rn ) ≤ C(p, U, V )||u||W 1,p (U ) and this ends the proof as in U the extension and the function are equal a.e.: their Lp∗ (U ) norm is less than the rst term of last inequality and right side of the inequality is dominated by Sobolev norm.

Theorem 1.10. (Poincaré's inequality) Let U ⊂ Rn be a bounded open set, and suppose u ∈ Wo1,p (U ) for some 1 ≤ p < ∞. Then for 1 ≤ s ≤ p∗ we have an estimate of the Ls norm of this function in terms of the norm of its gradient: ||u||Ls (U ) ≤ C(p, s, n, U )||Du||Lp (U )

(1.16)

holds. Strictly speaking, the Lp (U ) norm of Du is equivalent to the Wo1,p (U ) norm of u for u ∈ Wo1,p (U ) within a bounded U . 17

Proof. By the very denition of this Sobolev subspace, there exists a sequence

of functions {um }m∈N ⊂ C ∞ (U ) converging to u in W 1,p (U ) . If we compute the extension of each function um in a way to be zero on Rn −U , we are able to apply to it Gagliardo-Norinberg-Sobolev theorem and to have by approximation the estimate ||u||Lp∗ (U ) ≤ C(p, n)||Du||Lp (U ) . Thanks to hypothesis on U its Lebesgue measure is nite so that if 1 ≤ s ≤ p∗, ||u||Ls (U ) ≤ ||u||Lp (U ) .

Until now we have checked the case 1 ≤ p < ∞. The situation for n < p < ∞ is very dierent, as it is possible to show that each function u ∈ W 1,p (U ) has an Holder-continuous representative: we will call dierent elements of the same equivalence class of a function u ∈ W 1,p (U ) versions, always remembering that they are identied as an unique element on this Banach space (which otherwise would not be equipped with a norm). Next theorem will allow us to identify from now on a function u ∈ W 1,p (U ) when p > n with its continuous version.

Theorem 1.11. (Morrey's inequality for u ∈ W 1,p (U ) functions). Let U ⊂ Rn be a bounded open subset whose boundary is at least C 1 regular. Assume n < p ≤ ∞ and u ∈ W 1,p (U ). Then u has a version (which we will continue to call u ) such that for γ = 1 − np , u ∈ C 0,γ (U¯ ) and ∃ C=C(p,n,U) constant mastering the estimate (1.17)

||u||C 0,γ (U¯ ) ≤ C(p, n, U )||u||W 1,p (U )

Idea of the proof

This is given in an analogous way to the theorem for inequalities for W 1,p spaces, but it uses instead of the Gagliardo-Norinberg-Sobolev inequality Morrey's one which is (1.17) in W 1,p (Rn ) for C 1 functions and whose demonstration relies on the integral inequality Z Z 1 |Du| |u(y) − u(x)|dy ≤ C n−1 dy |B(x, r)| B(x,r) B(x,r) |y − x| found by applying dierentiation under the integral sign. By calculations it's possible to prove that for every x ∈ Rn |u(x)| ≤ C|u|W 1,p (Rn ) so that by arbitrage of x ∈ Rn we can get the powerful estimate supRn |u| ≤ C|u|W 1,p (Rn ) . It's important to mark that previous estimates are possibles R as p > n1 implies p (n−1) p−1 < n which implies the niteness of the integral B(x,1) p dy . (n−1) |x−y|

p−1

We can now collect our information and get a nicer characterization on what happens if quotient np is big or small.

18

Corollary 1.12. (General Sobolev Inequalities depending on np ). Assume U ⊂ Rn be a bounded open set whose boundary has at least C 1 regularity and that u ∈ W k,p (U ).Then, • if k


then the function (and its k − [ np ] − 1 derivative) n is θ- Hölder continuous u ∈ C k−[ p ]−1,θ (U¯ ), θ < 1 ; inclusion getting embedding since ||u||C k−[ np ]−1,θ (U¯ ) ≤ C(k, p, n, θ, U )||u||W k,p (U ) n p

We know now that by the Gagliardo-Nirenberg-Sobolev inequality we ∗ pn have the embedding of W 1,p (U ) into Lp (U ) for 1 ≤ p < n, p∗ = n−p . An essential tool in what follows will be that this embedding is also a compact embedding, i.e. that every bounded sequence in W 1,p (U ) is precompact in ∗ Lp (U ) which strictly speacking means that it has a subsequence converging ∗ in Lp (U ).

Theorem 1.13. (Rellich-Kondrachov compactness theorem) As usual let U ⊂ Rn be a bounded open set with a C 1 boundary , and 1 ≤ p < ∞. Then for each 1 ≤ s < p∗ the embedding W 1,p (U ) ⊂⊂ Ls (U )

is compact. This theorem has an easier version in the case n < p ≤ ∞: the embedding is compact as Ascoli-Arzelà criterion can be applied to continuous versions of a chosen bounded sequence in W 1,p (U ) ,which provide a bounded (by Morrey's inequality) subset of C(U ) whose equicontinuity is given by boundedness of weak derivatives. Thus embedding W 1,p (U ) ⊂ Lp (U ) is compact for all 1 ≤ p ≤ ∞

Remark 7. Observe that the compactness of the embedding Wo1,p (U ) ⊂⊂ Lp (U ) for all 1 ≤ p ≤ ∞ holds without the assumption of the regularity of domain U . Compactness of these embeddings is powerful: just supposing that for p > 1 a sequence {um }m∈N ⊂ W 1,p (U ) converges weakly um * u implies that the sequence is bounded from above and thus by compactness that it has an extract that converges strongly in Ls (U ) for each 1 ≤ s < p∗. By passing to a subsequence we can always suppose that a weakly convergent sequence converges in the overlying Lp space. Moreover this implication has a converse: as for p > 1 W 1,p (U ) is reexive, a bounded sequence has a weakly 19

converging extract by Banach-Alaoglu Theorem. A word more has to be spent in honor of the case p = ∞, with usual assumptions on the domain U we have a precise characterization of W 1,∞ (U ). Next two theorems will lead us to the comprehension of the close link between weak derivatives and strong ones for elements of such a linear space, as we will see that they can be dierentiate in the usual way almost everywhere.

Theorem 1.14. Let U ⊂ Rn be a bounded open set whose boundary is C 1 regular. Then for a function u : U → R we have that u ∈ W 1,∞ (U ) if and only if u is Lipschitz continuous. Corollary 1.15. Suppose now U is any open set. Then u ∈ Wo1,∞ (U ) if and only if u is locally Lipschitz continuous in U. We however have to make a step back for what concerns elements of the space W 1,p (U ) for 1 ≤ p < ∞, as for these elements there is no corresponding characterization. As a limit case, when n < p < ∞ we know that each function U ∈ W 1,p (U ) belongs to C 0,1−n/p (U ), but on the other hand an Holder continuous function with a so small coecient (< 1) need not to belong to any Sobolev space W 1,p (U ) as next example shows.

Counterexample 1. (Holder continuous unbounded functions) Functions of the kind f (x) = |x|β for 0 < β < 1 are Holder continuous in [0,1] with exponent β , but f (x) ∈/ W 1,p ([0, 1]) for p ≥ 2 as the integral of the derivative diverges in 0. 1,p Theorem 1.16. Let u ∈ Wloc (U ) for some n < p ≤ ∞. Then, identifying u as its continuous version, u is dierentiable almost everywhere in U, and its classic gradient equals its weak gradient a.e. Corollary 1.17. If u is a locally Lipschitz continuous function in an open set U, then u is dierentiable a.e. in U. Finally, in order to apply Sobolev space theory to PDEs, we will need an approximation of partial derivative particularly well behaving such as the notion of dierence quotient.

Denition 1.13. Let u : U → R be a locally summable function u ∈ L1loc (U )

and let V ⊂⊂ U be a compactly contained open set. For i = 1, ..., n, x ∈ V, h ∈ R such that 0 < |h| < dist(V, ∂U ) we dene the ith - dierence quotient of u of size h as

u(x + hei ) − u(x) h and the dierence quotient gradient will be Dh u := (D1h u, ..., Dnh u) Dih u(x) :=

20

(1.18)

Proposition 1.18. (Properties of dierence quotients and weak derivatives) • Suppose u ∈ W 1,p (U ) for 1 ≤ p < ∞. Then Dih u ∈ Lp (V ) for each V ⊂⊂ U for which h < dist(V, ∂U ). In formulas we give a more precise

estimate:

∃ C > 0 s.t. ||Dh u||Lp (V ) ≤ C||Du||Lp (U ) • Let u ∈ Lp (U ) for 1 < p < ∞ and suppose there exists a constant K > 0 s.t. for each V ⊂⊂ U satisfying 0 < h < dist(V, ∂U ) we have ||Dh u||Lp (V ) ≤ K . Then the weak derivative Di u exists and satises ||Di u||Lp (V ) ≤ K • Previous assertion is false if p = 1 as L1 (U ) is not reexive. As a counterexample, let U = R, V = (−1, 1) and u = χ(0,∞) . We dene for h > 0, (the argument for h < 0 is analogous)

h

D u(x) =

  0

1 h



0

x ≤ −h if − h ≤ x ≤ 0 if x > 0

(1.19)

It is easy to see that this function has L1 (V ) norm equal to 1. But u does not have a (locally integrable) weak derivative, because this would imply that u is (locally) absolutely continuous. In particular, it would have a continuous representative. For p ∈ (1, ∞) the reexivity of Lp (V ) prevents this kind of behavior. • If v ∈ W 1,p (U ) for some p > 1, then for every k = 1, 2..n the dierence quotient Dkh v converges to vxk strongly in Lp (V ) for each V ⊂⊂ U

compactly contained.

• For u, v ∈ W 1,p (U ) then we have the precious identity of by parts inte-

grating

Z

uDk−h (v)dx

U

Z =− U

21

vDkh (u)dx

(1.20)

References and further steps We mainly follow [2] chap.3, [30] chap 5, [8] chap.8 and [44] in its entire work. Second property of dierence quotients is a powerful tool to prove an improvement of regularity of W 1,1 (U ) functions, as will be done in last Theorem of last section of this chapter and in various other applications as the one of Marcellini [64].

1.4 Parabolic and Elliptic equations Denition 1.14.

For a xed integer k > 1, U ⊂ Rn bounded open set and a given function F of its arguments, an expression of the form

F (Dk u(x), .., Du(x), u(x), x) = 0 for all x ∈ U and for u : U → R the unknown function, is called a k -th order partial dierential equation (PDE). The PDE above, for given functions aα , f is called

• Linear, when having the form X aα (x)Dα u(x) = f (x); |α|≤k

• Semilinear, when having the form X aα (x)Dα u(x) + a0 (Dk−1 u, Dk−2 u, .., Du, u, x) = 0; |α|=k

• Quasilinear, when exhibiting the morphology X aα (Dk−1 u, Dk−2 u, .., Du, u, x)Dα u(x)+a0 (Dk−1 u, Dk−2 u, .., Du, u, x) = 0; |α|=k

• Fully nonlinear, when depending nonlinearly upon the highest derivatives of u

Example 4. (Linear PDEs) Our main example of a linear PDE will be the Laplace equation ∆u =

n X

uxi xi = 0.

i=1

22

A wide theory has been developed around the solutions to this equation, armonic functions, which exhibit a wide amount of properties as mean-value formulas, maximum principle, innite dierentiability with specic bounds for the derivatives, Liouville theorem, Harnack inequality, representation formulas and solvability for the Dirichlet problem, just to cite some ones. Other important linear PDEs are the eigenvalue one, diusion one, Schrödinger's one, Wave equation (and more others) having the morphology −∆u = λu,

ut − ∆u = 0,

iut + ∆u = 0,

utt − ∆u = 0

Example 5. (Nonlinear PDEs) Our main example of a nonlinear PDE will be the p-Laplace equation div(|Du|p−2 Du) = 0.

The p-Laplace operator is dened as p−2

∆p u := div(|Du|

p−4

Du) = |Du|

n n X ∂u ∂u ∂ 2 u o 2 |Du| ∆u + (p − 2) ∂xi ∂xj ∂xi ∂xj i,j=1

Also in this context a wide literature has been brought on: solutions to pLaplace equation, which can be rewritten compactly in terms of the p-Laplace operator as ∆p u = 0 are called p-harmonic functions and satisfy nice properties such as Harnack inequality, maximum principle and in particular the Caccioppoli's inequality, which carries them in a special class of functions11 .Other important nonlinear PDEs are the Eikonal equation, Monge-Ampère one, Hamilton-Jacobi equation (and more others), exhibiting the morphology |Du| = 1,

det(D2 u) = f,

ut + H(Du, x) = 0

Denition 1.15. A linear PDE of the form Lu = f valid in an open bounded set U ⊂ Rn , where u : U → R is the unknown, f : U → R is a given function and L denotes a second-order partial dierential operator having, for given functions ai,j , bi , c, the morphology

Lu = −

n X

ai,j (x)uxi xj +

n X

i,j=1

bi (x)uxi + c(x)u

i=1

is said to be in divergence form if it can be written as

Lu = −

n X

(ai,j (x)uxi )xj +

i,j=1 11 See

n X

bi (x)uxi + c(x)u

i=1

next section for more details on Caccioppoli's inequality.

23

We say that the partial dierential operator L is elliptic if ∃θ > 0 such that n X

ai,j (x)ηi ηj ≥ θ|η|2

i,j=1

for a.e x ∈ U and every η ∈ Rn . This means that the symmetric matrix (ai,j (x))ni,j=1 is positive denite at each x ∈ U and its minimum eigenvalue is still greater or equal to θ, it is referred as the lower modulus of ellipticity, while the upper modulus of ellipticity is the greatest eigenvalue.

Remark 8. If the coecients ai,j are C 1 (U ) functions, then the operator L can be written in both divergence or nondivergence form. When ai,j matrix is the identity and the remaining functions are zero, L = −∆ returns to Laplacian operator. The solutions of an elliptic partial dierential PDE Lu = 0 are similar in many aspects to harmonic functions. Existence and uniqueness of solutions are provided by weak formulation and variational formulation of the PDE, that dene a bounded symmetric bilinear form B : Wo1,2 (U ) × Wo1,2 (U ) → R associated to L; so thanks to Lax-Milgram Theorem existence and uniqueness are achieved. For quasilinear and fully non-linear operators L, ellipticity is dened to extend previous denition and will be referred always as an ellipticiy condition on the growth assumption of the operator L. Denition 1.16.

A subset of the kind UT := U × (0, T ] for some xed time T > 0 is said to be a parabolic domain, it is the natural domain where to solve   ut + Lu = f in UT a problem of the form u = 0 on ∂U × [0, T ] where f : UT → R,   u=g on U × {t = 0} g : U → R are given function, and as usual u : U¯T → R denotes the unknown function. The operator L denotes now for each time t ∈ (0, T ] a second-order partial dierential operator, having the form

Lu := −

n X

ai,j (x, t)uxi xj +

i,j=1

n X

bi (x, t)uxi + c(x, t)u

i=1

for given coecient functions ai,j , bi , c, ∀i = 1, 2, .., n. We say that the ∂ + L is parabolic, if ∃θ > 0 constant such that partial dierential operator ∂t n X

ai,j (x, t)ηi ηj ≥ θ|η|2

i,j=1

for all (x, t) ∈ UT , η ∈ Rn . 24

Remark 9. When matrix ai,j is the identity and the other functions dening ∂ ∂ +L = ∂t +∆ gives the L are zero, the parabolic partial dierential operator ∂t

Heat equation PDE. Also in this case a wide theory has been built on linear parabolic second-order PDEs, comprehensive of existence of weak solutions, regularity results, fundamental solutions and adequate maximum principle and Harnack inequality. For quasilinear and fully non-linear operators L, parabolicity is dened to extend previous denition and will be referred always as parabolic condition on the growth assumption of the operator L, which now is depending also by a time variable.

Denition 1.17.

Regarding both denitions of elliptic and parabolic equations, we say that the equation is degenerate when the upper modulus of ellipticity vanishes and singular when the lower modulus of ellipticity becomes innity.

Example 6. Let us consider the quasilinear, parabolic, partial dierential equation of second order ( 1,p u ∈ L∞ p>1 loc ((0, T ]; Wloc (U )) ut − divA(x, t, u, Du) = B(x, t, u, Du) in UT .

Here we consider T > 0 given, UT := U × (0, T ) and Du = (ux1 , .., uxn ) is the gradient with respect to the space variables; the functions A = (A1 , .., An ) and B are real valued, measurable and regular enough with respect to their arguments, and satisfying for C0 , C > 0 the structure conditions ( C0 |Du|p−2 |Du|2 − C ≤ A(x, t, u, Du) · Du |A(x, t, u, Du)| + |B(x, t, u, Du)| ≤ C(1 + |Du|p−1 ),

In this case the quantity C0 |Du|p−2 is the modulus of ellipticity of the equation. If p > 2, it vanishes whenever |Du| = 0, and the equation is said to be degenerate at those points (x, t) ∈ UT where it occurs. If otherwise 0 < p < 2, the modulus of ellipticity becomes innity whenever |Du| = 0 and the equation is said to be singular at those points (x, t) ∈ UT where it occurs.

25

1.4.1 Physical interpretation and well posed problems A second-order elliptic PDE generalizes Laplace's and Poisson's equations. As in the history of Laplace's equation, the solution u typically represents the density of some quantity, as a chemical concentration for example. Pn The term i,j=1 ai,j uxi xj represents the diusion of u within U set, the coefcients ai,j describing P the anisotropic, heterogeneous nature of the medium. In particular L1 := − ni=1 ai,j uxi is the diusive ux density, and the ellipticity condition implies L1 ·Du ≤ 0, which means that the ow is from Pnregions of higher to lower concentration. The rst-order term L2 · Du = i=1 bi uxi represents the transport within U set, and the zeroth-order term cu describes the local increase or decrease of the chemical. On the other hand, general second-order parabolic PDEs describe, within U region, in physical applications the time-evolution of the density of some quantity u, say as before a chemical concentration. This is to describe a physical situation, with usual boundary conditions, and so we need also to dene what is a well posed problem. We say that a given problem for a partial dierential equation is well-posed, if

• The problem has a solution; • The solution is unique; • The solution depends continuously on the data given in the problem (for example, boundary data) Last condition is very important for physical applications, as it is preferable that our solution changes by a small rate when the conditions dening the problem vary a little. We will call a solution u of a given partial dierential equation classical when owing partial derivatives up to the request of derivation of the partial dierential operator involved in the equation, while a integral notion of weak solution will be developed properly to each kind of equation in order to make the problem we face a well-posed one.

References and further steps Our main reference for physical interpretation and construction of problems with PDEs is [12], while a detailed description of the classic equation of Mathematical Physics can be found in [6]. Partial dierential equations have a vast literature and the study of their model equation has a classic one. We refer for a brief introduction to [16], [30], [9] and for the theory of linear PDEs we refer to [59], [47]. For degenerate and singular PDEs we owe much to [15],[19], [22]. 26

1.5 The variational approach: nonlinear analysis We consider a formal equation of the form

L[u] = 0

(1.21)

for L[·] a given partial dierential operator, possibly nonlinear and for u the unknown. The Calculus of Variations concerns variational problems, which are problems such as (1.21), but where the possibly nonlinear operator L[·] is a derivative of an appropriate energy functional I[·] such that L[·] = I 0 [·]. So now the beginning equation is

I 0 [u] = 0

(1.22)

Example 7. Given a surface, nd the shortest curve between two given points. If the surface is given by a parametric representation x = x(u, v),y = y(u, v),z = z(u, v) and if we employ the customary notation for the rst fundamental form E = x2u + yu2 + zu2 , F = xu xv + yu yv + zu zv , G = x2v + yv2 + zv2 then the length L of a curve on the surface, dened by the equation v = v(u) between two points u1 , u2 is given by the integral Z

u1

L=

p E + 2F v 0 + Gv 0 2 du

u0

The problem consists in nding functions v(u) which yield extrema of this integral, the popular geodesic curves. Let us suppose now U ⊂ Rn is a bounded open set with a smooth boundary ∂U ∈ C 1 , and let us admit we are given a smooth function L : U¯ × R × Rn → R,

L = L(x1 , ..., xn , y, z1 , ..., zn ) = L(x, y, z) for z ∈ Rn , y ∈ R, x ∈ U that we will call as it is usual in mathematical physics the Lagrangian. Let I[·] have the explicit form Z I[w] = L(x, w(x), Dw(x))dx U

¯ → R satisfying some boundary conditions as for smooth functions w : U w|∂U = g . If we suppose that a particular smooth function u belonging to the set of smooth functions satisfying the boundary condition happens to be a minimizer of I[·] , we see that u is a solution of a certain nonlinear partial dierential equation, said the Euler- Lagrange equation. 27

Proposition 1.19. (Euler-Lagrange equation) If among the set of smooth functions satisfying the boundary condition w|∂U = g there is one which minimizes the functional I[·] then it satises inside U the nonlinear PDE −

n X


∂xi Lzi (x, u, Du) + Ly (x, u, Du) = 0

(1.22)

i=1

Proof. Firstly choose any smooth function v ∈ Co∞ and consider the function of a real variable i(τ ) := I[u + τ v] (τ ∈ R). Since u is a minimizer of I[·] and u + τ v = u = g on ∂U we observe that i(·) has a minimum in τ = 0, so that i0 (0) = 0. This derivative is called rst variation and if explicitly computed gives 0

i (τ ) =

Z X n

Lzi (x, u + τ v, Du + τ Dv)vxi + Ly (x, u + τ v, Du + τ Dv)v dx

U i=1

meaning with Lzi the derivative of L on the i-th coordinate of the variable z ∈ Rn , for Ly = Dy L and with Dx L := (Lx1 , Lx2 , ...., Lxn ). Lettig τ = 0 and using the nullity of the rst derivative we get 0

0 = i (0) =

Z X n

Lzi (x, u, Du)vxi + Ly (x, u, Du)vdx

U i=1

and then as v has compact support inside U by integrating by parts we get the vanishing of the boundary integral and what remains is

Z 0= U



−

n X

 (Lzi (x, u, Du))xi + Lz (x, u, Du) v dx

i=1

so that using the fundamental Lemma of Calculus of Variations we deduce from this equation that as it holds for every test function v the function u solves the announced PDE. In the same spirit we could continue our calculations by computing the second derivative of I[·] at the function u (called second variation ), and we will nd that we would must have i00 (0) ≥ 0. Computing this derivative and letting τ be 0 we get another integral inequality holding for all test functions. Then using the right function, as [30] does in its section 8.1.3 or as [39] presents in its rst chapter, we will deduce that operator L must have those necessary conditions (called by Giacquinta Legendre-Hadamard 28

conditions ) that will be replaced and guaranteed by stronger basic convexity assumptions on the Lagrangian in order to obtain an existence theory: n X

Lzi zj (x, u, Du)bi bj ≥ 0, ∀b ∈ Rn , x ∈ U

(1.23)

i,j=1

Remark 10. Observe that the Euler-Lagrange equation associated with the energy functional I[·] is a quasilinear, second-order PDE in divergence form. There is a vast literature concerning these equations, but we refer in this thesis to [42] , part II . 1

Example 8. Let the Lagrangian be L(x, y, z) = (1 + |z|2 ) 2 : then the energy is Z 1 (1 + |Dv|2 ) 2 dx

I[v] = U

and note that this is the area of the graph of the function v : U → R. Its associated Euler-Lagrange equation is the notorious Minimal Surface equation: n  X i=1

vxi



(1 + |Dv|2 )

1/2

= 0, ∀x ∈ U xi

As the left side of the equation is n times the mean curvature of the graph of the function, this equation obliges a minimal surface to have zero mean curvature. Following classical theory, existence of minimizers for the functional I[·] is given by condition (1.5) that implies for sure the coercivity of the Lagrangian, and so it is called coercivity condition. With a simple look to Poincaré's inequality, and recalling that ∂U is smooth by our hypothesis, it easily follows that we can ask just the Lagrangian to satisfy L(x, y, z) ≥ α|z|p −β for α > 0 and β ≥ 0 for all z ∈ Rn ,1 < q < ∞, y ∈ R and x ∈ U to have both an inferior bound for L and coercivity. Let us now dene the class of admissible functions w for the functional I[w]: as the wider is this class the more are the candidates to be minimizers, we will enlarge our space of smooth functions to n o A := w ∈ W 1,p (U )| w∂U = g, in the sense of traces

Remark 11. Note that the set of admissible functions for our problem is a closed convex set in W 1,p (U ) since if U is bounded and has a regular boundary 29

Trace Theorem applies ( [30] chap. 5, or [8], chap. IX.2) and there exists a continuous linear operator T : W 1,p (U ) → Lp (∂U ) such that T (u) = u∂U ∈ Lp (∂U ) by which A = T −1 ({g}): a closed set since Lp (∂U ) is a metric space. About convexity its sucient to observe that we are on a subset of a vector space whose dening condition is stable under convex linear combinations as αg + (1 − α)g = g when we are on ∂U . Let us recall some theorems seen in the paragraph of convexity: we know that if the Lagrangian L satises its coercivity conditions, if the mapping p → L(x, y, z) is convex for each y ∈ R, x ∈ U , then I[·] is lower semicontinuous on a weakly closed topological subspace V ⊆ W 1,p (U ) and moreover a minimizer is guaranteed by Theorem 1.4. On a weakly closed subspace V ⊆ W 1,p (U ) there exists under these conditions at least a function u ∈ V solving I[u] = minw∈V I[w]. So if we suppose A to be non-empty, as it is weakly closed in W 1,p (U ) we can set it to be V and win an existence deal for a minimizer in the admissible class. We just demonstrated the following theorem:

Theorem 1.20. Assume that the Lagrangian L satises the coercivity condition and that it is convex in the variable z. If the set A is non-empty then there exist a minimum in A for the energy I[·]. In general there may be more than one minimizer, but if we require further assumptions we will ensure that the existing minimizer is unique.

Theorem 1.21. Suppose the Lagrangian L does not depend on the function itself L = L(x, z) and that the mapping z → L(x, z) is uniformly convex for each x ∈ E i.e. there exists a number γ > 0 such that n X

Lzi zj (x, z)bi bj ≥ γ|b|2 , ∀z, b ∈ Rn , x ∈ U

i,j=1

Then then a minimizer u ∈ A of I[·] is unique. Proof. Let I[u1 ] = inf A I = I[u2 ] so that u1 , u2 are both admissible minimiz2 =: u˜ ∈ A. By uniform convexity ers for I . Then as A is convex we have u1 +u 2 assumption we have for every x ∈ U, z1 , z2 ∈ Rn : α |z1 − z2 |2 2 2 Let us use a rst time z1 = Du1 and z2 = Du1 +Du and a second time 2 Du1 +Du2 z1 = Du1 and z2 = obtaining two inequalities, and integrating over 2 U we obtain Z Z Du1 + Du2 Du1 − Du2 α I[u1 ] ≥ I[˜ u]+ Dz L(x, )·( )dx+ |Du1 − Du2 |2 dx 2 2 8 U U L(x, z1 ) ≥ L(x, z2 ) + Dz L(x, z2 ) · (z1 − z2 ) +

30

Z Z α Du1 + Du2 Du2 − Du1 )·( )dx+ I[u2 ] ≥ I[˜ u]+ Dz L(x, |Du1 − Du2 |2 dx 2 2 8 U U Then we make the half-addition of I[u1 ] and I[u2 ] and we obtain the second inequality Z inf A I + inf A I I[u1 ] + I[u2 ] α = = I[u1 ] |Du1 − Du2 |2 dx ≤ I[u1 ] ≤ I[˜ u]+ 8 U 2 2

u] by the minimizer u1 . This means that the integral is zero Ras I[u1 ] ≤ I[˜ |Du1 − Du2 |2 dx = 0 and so Du1 = Du2 in W 1,p (U ). Next, consider U |u1 − u2 | ∈ W01,p (U ) as they both belong to A: in such a space Poincaré's inequality holds and so ||(u1 − u2 )||1,p = 0. We derived Euler-Lagrange equation using the fact that our minimizer u was smooth. Now we are interested in proving that if we enlarge our class of functions to the set A and we undertake some suitable growth conditions on the Lagrangian L then it is still true that a minimizer for the functional I[·] in this wider class of admissible functions satises the boundary-value problem for the Euler-Lagrange equation ( P − ni=1 (Lzi (x, u, Du))xi + Ly (x, u, Du) = 0 x ∈ U (1.24) u = g, x ∈ ∂U

Denition 1.18.

An admissible function u ∈ A is a weak solution for the boundary-value problem for the Euler-Lagrange equation provided

Z X n

(Lzi (x, u, Du))vxi + Ly (x, u, Du)v dx = 0

(1.25)

U i=1

∀v ∈ Wo1,p (U ) This denition is motivated by the following argument. Suppose these growth conditions to hold for a constant K (that can be chosen as the greatest between the three eventually given) and all z ∈ Rn , p > 1, y ∈ R, x ∈ U

|L(x, y, z)| ≤ K(|z|p + |y|p + 1) ( |Dz L(x, u, z)| ≤ K(|z|p−1 + |y|p−1 + 1) |Dy L(x, u, z)| ≤ K(|z|p−1 + |y|p−1 + 1).

(1.26) (1.27)

Remark 12. We will see in a second moment that these conditions are called isotropic, and that there exist under non isotropic conditions (anisotropic ones) a close theory which has not yet been unied. The presentation of this theory is the aim of our last chapter. 31

Let us suppose u smooth for a while and let us make a step back: we multiply Euler-Lagrange equation (1.22) by a test function φ ∈ Co∞ and we integrate by parts obtaining the equality Z X n (Lzi (x, u, Du))φxi + Ly (x, u, Du)φ dx = 0 (1.28) U i=1

Now assume u ∈ W 1,p (U ) and observe that by (1.27) we have 0

|Dz L(x, y, z)| ≤ K(|z|p−1 + |y|p−1 + 1) ∈ Lp (U ) p for p0 = p−1 the conjugate exponent to p as p10 + p1 = 1 and equivalently observe 0 that the y - derivative of the Lagrangian lies in the same space Lp (U ) using the second growth condition. Consequently we can use the Dominated Convergence theorem and infer that equation (1.28) holds ∀v ∈ Wo1,p (U ). Next we state that minimizers of I[·] living in the class A are actually solutions of the boundary-value problem associated to Euler-Lagrange equation (1.22), this time being understood as an equation valid for almost every x ∈ U , as a solution u ∈ Wo1,p (U ) is not exactly a function.

Theorem 1.22. If the Lagrangian L satises growth conditions (1.26) and (1.27) and if u ∈ A is a minimizer for the energy functional I[u] = min I[w] A

then u is a weak solution of the Euler-Lagrange equation. Proof. The demonstration is similar to the one previously given to get EulerLagrange equation, but it uses nite-dierence method, as we are not any

more supposing u to be smooth and dierentiation under integrals has to be taken care of with approximation arguments. For any v ∈ Wo1,p (U ) and τ ∈ R we can set i(τ ) := I[u + τ v], as growth condition (1.26) guarantees us that the integral is bounded. This real valued function represents as already seen a rst variation of the energy I : for τ 6= 0 we write the dierence quotient Z L(x, u + τ v, Du + τ Dv) − L(x, u, Du) i(τ ) − i(0) = dx (1.29) τ τ U and for almost every x ∈ U n

L(x, u + τ v, Du + τ Dv) − L(x, u, Du) X lim = Lzi (x, u, Du)vxi +Ly (x, u, Du)v τ →0 τ i=1 32

almost everywhere, as L is smooth and v ∈ Wo1,p (U ), and more on

L(x, u + τ v, Du + τ Dv) − L(x, u, Du) 1 = τ τ 1 = τ

Z 0

1

n X

Z 0

1

d L(x, u + sv, Du + sDv)ds ds

Lzi (x, u + sv, Du + sDv)vxi + Ly (x, u + sv, Du + sDv)vds

i=1

p

p0

Now, for conjugate exponents p, p0 yields Young inequality ab ≤ ap + bp0 and by this and growth conditions (1.26) follows in a few calculations the bound

L(x, u + τ v, Du + τ Dv) − L(x, u, Du) p p p p ≤ C(|Du| + |u| + |Dv| + |v| + 1) τ stays in L1 (U ) as U has a nite Lebesgue measure. Thus by Dominated Convergence theorem i0 (0) exists and since by our assumption function i(·) has a minimum for τ = 0, we will have two estimates of it Z L(x, u + τ v, Du + τ Dv) − L(x, u, Du) u(τ ) − u(0) 0 = lim dx 0 = i (0) = lim τ →0 U τ →0 τ τ

=

Z X n

Lzi (x, u, Du)vxi + Ly (x, u, Du)vdx

(1.30)

U i=1

And this concludes the proof as it shows u to be a weak solution. We saw in rst section that under coercivity hypothesis on the Energy functional and convex one on the function to be integrated, we are sure to get a minimum for the functional in a closed set. Next we introduced the notion of weak solution and we showed that each minimizer of the functional is a weak solution, thus now we know that under suitable growth conditions the minimum found in rst section is also a weak solution. A natural question that may arise is that the Euler−Lagrange equation will have other solutions which are not minima of the functional I[·]. But there is a special case for which each weak solution is necessarily a minimizer of I[·], and it is given as usual by a convexity hypothesis.

Proposition 1.23. If the map (y, z) → L(x, y, z) is convex for every x ∈ U (as in Denition 1.2), then each weak solution to the Euler − Lagrange equation associated to it is a minimizer of the energy functional I[·] 33

Proof. Suppose u ∈ W 1,p (U ) solves −

Pn

i=1 (Lzi (x, u, Du))xi + Ly (x, u, Du) = 0 in weak sense and select another w ∈ W 1,p (U ). By convexity of (y, z) → L(x, y, z) we get for L = L(x, y, z) and each x ∈ U

L(x, u, Du)+Dz L(x, u, Du)·(Dw−Du)+Dy L(x, u, Du)·(w−u) ≤ L(x, w, Dw) Now by integrating it over U and testing with (w −u) ∈ W 1,p (U ) the identity Z X n 0= Lzi (x, u, Du)vxi + Ly (x, u, Du)v dx U i=1

we get I[u] ≤ I[w], as u is a weak solution which satises equation, we proved our proposition.

Euler-Lagrange

Example 9. (Hamilton-Jacobi equations) Consider the initial-value problem for the Hamilton-Jacobi equation for u : Rn × [0, ∞) → R the unknown u = u(x, t) , H : Rn → R the given Hamiltonian and g : Rn → R the initial data ( ut + H(Du) = 0, in Rn × (0, ∞) u = g, on Rn × {t = 0}

Hamilton-Jacobi equation does not in general have a smooth solution u lasting for all the times t > 0 as the method of characteristics may fail. We remember that two of the characteristic equations associated with the Hamilton-Jacobi PDE are

x˙ = D H(p, x) p˙ = −D H(p, x)

(

p

x

It's a classic result that these ODE can be derived in a completely dierent way from a variational principle. Essentially if we assume that function x(·) is of class C 2 and that it is a critical point of the energy functional

v

Z

v v

t

L( ˙ (s), (s)) ds

I[ (·)] := 0

being L = L(q, x)the Lagrangian function, for any two xed points x, y ∈ Rn x(·) belonging to the admissible class of functions

v v v then the function x(·) will satisfy for 0 ≤ s ≤ t Euler-Lagrange equation d − (D L(x˙ (s), x(s))) + D L(x˙ (s), x(s)) = 0 ds A := { (·) ∈ C 2 ([0, t]; Rn )s.t (0) = y, (t) = x}

p

x

34

This vector equation consisting of n coupled second-order ODE can be converted into Hamilton's equations: a system of 2n rst-order ODE. We may rst dene the Generalized Momentum for 0 ≤ s ≤ t, corresponding to the position x(·) and velocity x(·) ˙ :

q(s) := D L(x˙ (s), x(s)), p

then we may suppose that for q as a smooth function of (p, x) the equation q = Dp L(p, x) can be uniquely solved for q = (p, x). Then we dene the Hamiltonian H associated to the Lagrangian L for p, x ∈ Rn as

q

q

q

H(p, x) := p · − L( (p, x), x)

being function q(p, x) implicitly dened by our assumptions. Just by using Euler-Lagrange equation there is a classical argument (see for instance [30]) showing that functions x(·), p(·) satisfy Hamilton- Jacobi equations

x˙ (s) = D H(p(s), x(s)) p˙ (s) = −D H(p(s), x(s))

(

p

x

The most famous case is Newton's law for the motion of a particle of mass m > 0 , moving in a force eld f generated by a potential φ. In this context we dene Lagrangian function as the dierence between the kinetic and potential energies. Assuming to be in a simply connected domain, force eld generation by the potential writes f := −Dφ, and taking as Lagrangian L(p, x) = 21 m|p|2 − φ(x) its corresponding Euler-Lagrange equation is

x

fx

(1.31)

m¨(s) = ( (s))

The corresponding Hamiltonian is the total energy H(p, x) = i.e. the sum of potential and kinetic energy.

1 |p|2 2m

+ φ(x),

We discussed until this moment the existence and uniqueness of minimizers for the energy I[·], dening the class of weak solutions of Euler-Lagrange equation and their relation with minimizers of the energy functional. The classic method continues in proving the regularity of weak solutions so that from weak solutions we will be able to return to the desired strong ones. This will be the main focus of this work: we will analyze how regularity theory develops under always more restrictive hypothesis.

35

In 1900 D. Hilbert posed the following two problems in his well-known lecture delivered before the International Congress of Mathematicians at Paris:

• (20th problem) "Has not every regular variation problem a solution,

provided certain assumptions regarding the given boundary conditions are satised, and provided also if need be that the notion of a solution shall be suitably extended?"

• (19th problem) "Are the solutions of regular problems in the Calculus

of Variations always necessarily analytic?"

These two problems have originated such a great deal of work that it would be very dicult even to quote the dierent contributions. An interesting survey on these problems until 1997 has been made by Marcellini in[67], and a more recent one can be found in [70]. We discuss in this last part of Preliminaries the smoothness of minimizers to our energy functional by making a number of heavily simplifying assumptions, the most notably being that our Lagrangian L(x, y, z) depends only on z , L = L(z). This kind of functionals are referred as autonomous functionals. To do so, we suppose our functional to have for a f ∈ L2 (U ) the form Z I[v] := {L(Dv) − vf } dx U

We will consider p = 2, such that necessary growth conditions (1.27) to admissible weak solutions to Euler Lagrange equation will become |Dz L(z)| ≤ C(|z| + 1) and the equation will be, for such a u ∈ A and for all v ∈ H01 (U )

Z X n

Z f v dx

Lzi (Du)vxi dx =

U i=1

(1.32)

U

With appropriate hypothesis we will now show that if u ∈ H 1 (U ) is a weak solution of non linear partial dierential equation (1.32) above, then 2 (U ). Our appropriate hypothesis for this purpose are actually u ∈ Hloc

• L is autonomous • the boundedness of second derivative |D2 L(z)| ≤ C

∀z ∈ Rn

(1.33)

• uniform convexity of L , as specied in (1.3) n X

Lzi ,zj (x)bi bj ≥ α|b|2 ,

i,j=1

36

∀b ∈ Rn

(1.34)

This second hypothesis is a non-linear analogue of the classical uniform ellipticity condition for linear PDE, and the analogy is not casual as the demonstrating process that follows will be similar to the classic one for local H 2 regularity for solutions of linear second order elliptic PDE. We will give here this proof because its technique is extremely useful in the theory that will follow, as we will see in analysis of [63], [64] and in various other contexts.

Theorem 1.24. Let u ∈ H 1 (U ) be a weak solution of the nonlinear partial dierential equation in U −

n X

(Lzi (Du))xi = f

(1.35)

i=1

and let L ∈ C (R ) satisfy previous two conditions of boundedness of the 2 (U ). second derivative and uniform convexity. Then u ∈ Hloc 2 If in addition u has zero trace on boundary ∂U ∈ C , then u ∈ H 2 (U ) with continuity of the data ||u||H 2 (U ) ≤ C||f ||L2 (U ) ∞

n

Proof. Let us consider xed an open set V ⊂ U compactly contained and

let W ⊂ U another open set compactly contained in U which compactly contains V so that V ⊂⊂ W ⊂⊂ U . Now take a smooth cut-o function ζ such that it equals 1 constantly in V , whose support is contained in W and 0 ≤ ζ ≤ 1 in W .Let |h| > 0 be small enough and for k ∈ {1, 2..., n} dene by dierence quotients the function

w := −Dk−h (ζ 2 Dkh u)

(1.36)

An interesting technical point to understand is the power to assign to ζ : it will be useful in following calculations to dene such a w with ζ of power two. Now, as w ∈ H01 (U ), let us test it into

Z X n

Lzi (Du)(−Dk−h (ζ 2 Dkh u))xi

Euler-Lagrange PDE (1.32) to get

Z dx =

U i=1

f (x)(−Dk−h (ζ 2 Dkh u)) dx (1.37)

U

By this, using the identity (1.20) of dierence quotients we have

A :=

Z X n U i=1

Dkh Lzi (Du)(ζ 2 Dkh u)xi

Z dx = −

f (x)(Dk−h (ζ 2 Dkh u)) dx =: B

U

(1.38) Now by smoothness of L it is possible to get a linear integral equation that will serve us in a silver plate a bound for the dierence quotient of the weak 37

gradient of the solution, and with this bound and the second property of dierence quotients in Proposition 1.18 we get the existence of D(Du) and its membership on L2 (U ). So we will have u ∈ H 2 (V ) for each V ⊂⊂ U and 2 (U ). Now let us compute the dierence quotient of the membership u ∈ Hloc Lzi by using regularity of L and Fundamental Theorem of Calculus:

Dkh (Lzi (Du(x))) =

1 h

Z 0

1

n X

1

Z

1 h

0

Lzi (Du(x + hek )) − Lzi (Du(x)) = h

d Lz (sDu(x + hek ) + (1 − s)Du(x))ds = ds i

Lzi zj (sDu(x + hek ) + (1 − s)Du(x))(uxj (x + hek ) − uxj (x))ds =

j=1 n X

ahij (x)Dkh (uxj (x))

i, j ∈ {1, 2, .., n}

j=1

R1 if we set ahij (x) := 0 Lzi zj (sDu(x + hek ) + (1 − s)Du(x))ds Substituting the linear expression found for Dkh (Lzi (Du(x))) in left-hand side of equation (1.38) we nd n Z n Z X X 2 h h h ζ ai,j Dk uxj Dk uxi dx+ A1 +A2 = ahi,j Dkh uxj Dkh (u)2ζζxi dx = A = B i,j=1

U

i,j=1

U

(1.39) Uniform convexity assumed by hypothesis means i,j=1 Lzi zj (z)bi bj ≥ θ|b|2 for each z, b ∈ Rn so that by Fubini-Tonelli's theorem and since the integral in the denition of ahi,j is taken in a set whose measure is unitary, it implies uniform ellipticity of ahi,j ,

Pn

A1 =

n Z X i,j=1

ζ 2 ahi,j Dkh uxj Dkh uxi dx

U

Z ≥θ

2

ζ 2 |Dkh Du| dx

U

Next, using the bound C for the second derivatives of the Lagrangian L and 0 Young's inequality ab ≤ ap + C()bp we get by p = p0 = 2 Z Z Z C 2 2 h h 2 h |Dkh u| dx |A2 | ≤ C ζ|Dk (Du)| |Dk (u)|dx ≤  ζ |Dk (Du)| dx +  U U U which gives us a lower bound for A2 so that by taking  = 2θ our inequalities for A1 , A2 imply Z Z θ 2 2 h A≥ ζ |Dk (Du)| dx − C |Du|2 dx 2 U U 38

Now with the help of rst property of dierence quotients of Proposition 1.18, as ||Dh u||Lp (V ) ≤ C||Du ||Lp (U ) , we estimate this integral with the use of Young inequality on each singular addendum of the multiplication and then clustering Z Z Z C 2 2 |B| ≤ C |f ||v|dx ≤ |f | dx +  |Dk−h (ζ 2 Dkh u)| dx  U U U Z Z C 2 ≤ |f |2 dx +  |D(ζ 2 Dkh u)| dx  U U Z Z Z C 2 2 2 h ≤ |f | dx + { |Dk u)| dx + ζ 2 |Dkh Du| dx}  U W W Z Z Z C 2 ≤ |f |2 dx + { |Du)|2 dx + ζ 2 |Dkh Du| dx}  U U U Z Z θ 2 ζ 2 |Dkh Du| dx + C (|f |2 + |Du|2 )dx ≤ 4 U U choosing  = 4θ and a suitable C big enough. Finally using inequalities for A1 , A2 , B we have Z Z 2 2 h ζ |Dk Du| dx ≤ C (|f |2 + |Du|2 )dx (1.40) U

U

Now, since ζ = 1 on V we have Z Z 2 h |Dk Du| dx ≤ C (|f |2 + |Du|2 )dx U

(1.41)

U

for k ∈ {1, 2, .., n} and a sucienlty small h. Thus by the second property of dierence quotients we have that as Dkh Du ∈ L2 (V ), it implies Du ∈ H 2 (V ) for each V ⊂⊂ U , proving the rst part of the theorem.

39

1.5.1 The bootstrap argument As a last preliminary, here we present the so called " bootstrap" argument, adopted to demonstrate that if L is smooth then under previously adopted suitable conditions u is innitely dierentiable too. This has an easier proof in the case of linear second-order elliptic PDE, as the equation maintains its linearity under dierentiation formulas and an iteration procedure can be started and iterated on an equation of the same type. Nevertheless this method can not be applied to nonlinear partial dierential equations such as those coming from Calculus of Variations, because the non linear structure implies this method of iteration to fail as we do not obtain an equation similar to the precedent under linear and dierential operations. Much deeper ideas are necessary, as we present here in the standard "boothstrap" argument . Firstly we show that the weak partial derivative of our solution solves a linear second-order elliptic PDE. Let us consider for clarity Euler-Lagrange equation (1.28) with f ≡ 0. Select a function v ∈ Co∞ (U ) and for a chosen k ∈ {1, 2...n} set w := −vxk in Euler2 Lagrange equation . As we know by precedent theorem that u ∈ Hloc (U ), we can integrate by parts and get

Z X n

Lzi zj (Du)uxk xj vxi dx = 0

(1.42)

U i,j=1

If we set u ˜ := uxk and for i, j ∈ {1, 2...n} ai,j (x) := Lzi zj (Du(x)), then12 for each V ⊂⊂ U we can pass to limit in previous equality to get for each v ∈ H01 (V ) Z X n ai,j (x)˜ uxj vxi dx = 0 (1.43) U i,j=1

This means by Fundamental Lemma of Calculus of Variations, that u ¯ ∈ 1 H (U ) is a weak solution in V of the linear second order PDE,

−

n X

(ai,j (x)˜ uxj )xi = 0

(1.44)

i,j=1

which is elliptic as L has been assumed to be convex. Now comes to light the importance of next section: we cannot apply to this one (1.44) the regularity theory for linear elliptic PDE of second order (see [42]), because hypothesis are not enough as we know just that L is smooth 12 Note

that ai,j = aj,i thanks to Schwartz Theorem.

40

and its second derivative is bounded (1.29) but no continuity or classic differentiability is assumed on ai,j (x). We can just deduce directly from our hypothesis (1.33) that ai,j ∈ L∞ (V ). There is an old theorem of Schauder 0,α 0,α ([42], sec. 6.4) saying that if ai,j ∈ Cloc (U ), then we have u˜ ∈ Cloc (U ) which 1,α means u ∈ Cloc (U ). As ai,j = Lzi zj (Du) with L smooth we would have 1,α 3,α ai,j ∈ Cloc (U ), and another Schauder's estimate would imply u ∈ Cloc (U ), and so on. The "bootstrap" argument can be continued on and so we can dek,γ (U ) for every positive integer k , and so that u ∈ C ∞ (U ). duce from it that Cloc 0,α Schauder's fundamental hypothesis was ai,j ∈ Cloc (U ), but until this point we have just the boundedness of these coecients , ai,j ∈ L∞ (V ). Thus Hilbert's problems remained open, at least until Schauder's theorem hypothesis could not be fullled. Finally, a theorem given in 1957 by E.De Giorgi and an equivalent theorem given in 1958 of J.Nash found in two completely dierent approaches answered this last question.

References and further steps We mainly follow [30] chap.8, [39] on its rst chapter for what concerns fundamentals of Calculus of Variations and chap. VII for regularity results, [12] chap.4 and [44] in its entire work. We follow [8] in its last three chapters for enlightening functional approach to Calculus of Variations. Hölder continuity of solutions given by De Giorgi-Nash-Moser theorem will be investigated deeply in next chapter and we refer to its references.

41

2

Regularity on isotropic growth conditions

2.1 De Giorgi Theorem The brilliant and pioneering work of Ennio De Giorgi covers the hypothesis missing by Schauder's theorem, in order to answer to Hilbert's problems. His work gives not only the Holder-continuity of solutions of linear secondorder elliptic PDE in divergence form n X

(ai,j (x)uxj )xi = 0

(2.1)

i,j=1

in an open set U ⊂ Rn but also it gives the denition of a wider set of functions that satisfy a certain integral inequality, called Caccioppoli's inequality. For this set of functions the conclusion of the theorem is valid, and weak solutions to our problem are spontaneously members of this set: so we can obtain Holder continuity by their membership to this set. We give in this rst exposition a simple presentation of this ideas in the case of homogeneous natural growth conditions on ai,j (x), as a continuation of previous chapter. We suppose thus that ai,j (x) := Lzi zj (D(u(x)), and conditions (1.33),(1.34) on L give classical ellipticity to ai,j matrix: 2

λ|η| ≤

n X

ai,j (x)ηi ηj ,

|ai,j (x)| ≤ Λ

i,j=1

where λ, Λ > 0 are the lower and the upper modulus of ellipticity of ai,j . 2 Supposing (1.33),(1.34) we know by the theory developed that u ∈ Hloc (U ), the solution owes bounded weak second derivatives, and its rst derivatives are solution of the equation just mentioned (2.1) or its alter-ego (1.43). Subsequently we dene DGO(U), the homogeneous De Giorgi Class of functions on U. We will denote by {u > k} the set of points x ∈ U for which holds u(x) > k . De Giorgi's method is based on the geometric idea that boundedness (and in a second step even continuity) of a measurable function u, dened over U , can be investigated via the analysis of the decay of its level sets.

Denition 2.1.

Let U ⊂ Rn an open set, x0 ∈ U and choose r ∈ R+ , k ∈ R, and set A(k, x0 , r) := Br (x0 ) ∩ {u > k} to be the super-level set of u . A 1,2 function u ∈ Wloc (U ) belongs to the homogeneous De Giorgi set DGO+ (U ) if there exists a C > 0 for it satises for ∀k ∈ R, almost every x0 ∈ U and every 0 < r < R < d(x0 , ∂U ), the integral inequality Z Z C 2 |u(x) − k|2 dx (2.2) |Du| dx ≤ 2 (R − r) A(k,x0 ,R) A(k,x0 ,r) 42

In a parallel way, dening the sub-level set of u as

B(k, x0 , r) := Br (x0 ) ∩ {u < k}, 1,2 a function u ∈ Wloc (U ) belongs to homogeneous De Giorgi set DGO− (U ) if it satises for ∀k ∈ R, almost every x0 ∈ U and every 0 < r < R < d(x0 , ∂U ) the integral inequality Z Z C 2 |Du| dx ≤ |u(x) − k|2 dx (2.3) 2 (R − r) B(k,x0 ,r) B(k,x0 ,R)

In conclusion we dene homogeneous De Giorgi's class as

DGO(U ) := DGO+ (U ) ∩ DGO− (U )

Remark 13. Previous integral condition (2.2) is equivalent to the following Z

C dx ≤ (R − r)2

+ 2

|D(u(x) − k) | Br (x0 )

Z

2

|u(x) − k)+ |

dx

(2.4)

BR (x0 )

on same hypothesis on x0 , k, r, R, since the weak gradient of the positive part of (u(x) − k) by property (1.11) of W 1,p (U ) is the weak gradient of u in points of the set {u > k}. Analogous consideration can be made for integral equation (2.3) by using (u(x) − k))+ . So the integrals of (2.4) are made upon A(k, x0 , r) := Br (x0 ) ∩ {u > k}. Finally, we want to remark that |A(k, x0 , R)| + |B(k, x0 , R)| = |BR (x0 )| Next we introduce subsolutions and supersolutions of our equation of 1,2 interest: a function u ∈ Wloc (U ) is a weak solution of (2.1) if

Z X n

ai,j (x)uxj vxi dx = 0

(2.5)

U i,j=1

is valid for each v ∈ Wo1,2 (U ).

Denition 2.2.

A function u ∈ W 1,2 (U ) is said to be a weak supersolution (resp. subsolution) of the previous equation if for each v ∈ Wo1,2 (U ) it satises

Z X n

ai,j uxj vxi dx ≥ 0(resp. ≤)0

(2.6)

U i,j=1

It is evident that u is supersolution

i −u is a subsolution and vice-versa. 43

As previously announced, we can now show that solutions to equation (2.1) are members of DGO(U ).

Theorem 2.1. If u is a weak subsolution (resp. a weak supersolution) of (2.1) with natural homogeneous elliptic growth conditions on ai,j (x) given by (1.33),(1.34), then u ∈ DGO(U )+ (resp. u ∈ DGO(U )− ). Consequently a weak solution of (1.21) lies in De Giorgi class. Proof. Let u be a weak subsolution, x0 ∈ U, 0 < r < R < d(x0 , ∂U ) and let η ∈ Co∞ (U ) be a cut-o function between Br (x0 ) and BR (x0 ) such that

C . Let us test our equation (2.6) with ψ := vη 2 and with |D(η)| ≤ R−r v = (u − k)+ η 2 (computation will be possible thanks to fourth property of Sobolev functions) obtaining, as u is a subsolution Z X n ai,j uxj {(u − k)+ η 2 }xi dx U i,j=1

Z =

η

2

U

n X

n X

Z ai,j uxj vxi dx + 2

ηv{ U

i,j=1

ai,j uxj ηxi } dx ≤ 0

i,j=1

and considering boundedness of ai,j (x) 13 we get Z Z n n X X 2 η ai,j uxj ηxi } dx ≤ ai,j uxj vxi dx ≤ −2 ηv{ U

U

i,j=1

Z |ηv||

2 U

n X

i,j=1

Z ai,j uxj ηxi | dx ≤ 2C

ηv|Du||Dη| dx

(2.7)

U

i,j=1

so that weRcan bound on Rthe left the square integral gradient of the soR 2 of the 2 2 2 2 lution, as U |Dv| dx = Br (x0 ) η |Dv| dx ≤ U η |Dv| dx, and considering that integrals are taken in U ∩ {u > k} with Z Z Z n X 2 2 2 C1 η |Dv| dx ≤ η ai,j uxj vxi dx ≤ 2C ηv|Du||Dη| dx U

U

U

i,j=1

≤ 2C||ηDv||L2 (U ) ||vDη||L2

(2.8)

just by a shot of Holder's inequality at the third step. Dividing for ||ηDv||L2 (U ) and making the square power of all we obtain by (2.8) Z Z 2 |Dv| dx = η 2 |Dv|2 dx ≤ C||vDη||2L2 (U ) = Br

U

13 This

condition is actually too strong: it would be sucient to have ai,j (x) ∈ L∞ (U ) i.e. essential boundedness.

44

= C||vDη||L2 (BR (x0 )) ≤

C 2 2 ||v||L2 (BR (x0 ) (R − r)

(2.9)

all thanks to the right cut-o function and our main equation (2.5), and so getting the desired integral inequality. The desired inequality for supersolutions is given by the fact that u is a subsolution i -u is a supersolution so that if u is a supersolution we have Caccioppoli's inequality for -u in DGO+ (U ). Now, as the super-level set of the opposite function −u is exactly the sub-level set of u, just by using as a test function φ := (u − k)− η 2 and making the same steps with integrals in U ∩ {u < k} we have that supersolutions lie in DGO(U )− . Finally, a weak solution is automatically both a subsolution and a supersolution, so it lies in intersection of their functional spaces which actually is DGO(U )

Remark 14. We remark here that constant C of the theorem is mainly the ratio of lower and upper ellipticity coecients of ai,j (x); precisely C1 and C of equation (2.8). On a successive step, we prove local essential boundedness of De Giorgi's set elements. This is a crucial step, as it will be derived from this result the boundedness of essential oscillation of its elements.

Denition 2.3.

Let p be a number 0 < p < ∞, (X, M, µ) a measure space ¯ and g : X → R a measurable function whose t-super-level set is L+ (g, t)) := g −1 ((t, ∞]). The essential supremum of g is dened as

M = ess supx∈X g(x) := inf {t ∈ R|µ(L+ (g, t)) = 0}

(2.10)

Analogously with the t-sub-level set of g L− (g, t)) := g −1 ([−∞], t)) can be dened its essential inmum as

m = ess inf x∈X g(x) := sup{t ∈ R|µ(L− (g, t)) = 0}

(2.11)

The essential oscillation of g in X is dened as

w = ess oscx∈X g(x) := ess supx∈X g(x) − ess inf x∈X g(x) = M − m (2.12) When considering a measurable function f ∈ Lp (U ), we will denote usually the essential extremum or inmum with supU f or inf U f as in this case no confusion may arise because such an element f is not exactly a function, but an equivalence class of functions coinciding almost everywhere. On a rst step we prove local essential boundedness of the solution: this will be fundamental when we will bound essential oscillation from above, and yet unveils future tools. The following integral estimate will be crucial to start the recursive argument. 45

Proposition 2.2. If u ∈ DGO(U )+ then for each k ∈ R, x0 ∈ U , r, R such that 0 < r < R < dist(x0 , U ) the there exists a constant C=C(n) such that the following integral estimate occurs Z

2 Cn n |(u(x) − k) | dx ≤ |A(k, R)| (R − r)2 Br

+ 2

Z

2

|(u(x) − k)+ | dx

(2.13)

BR

Proof. Set v := (u − k)+ for simplicity, and let 0 < r < r¯ < R < dist(x, ∂U ) so that Br ⊂ Br¯ ⊂ BR . Consider a cut-o function η between Br¯ and BR

C1 , η ≡ 1 in Br¯ and η ≡ 0 out of BR . Considering that for which |Dη| ≤ (R−r) 2n 1,2 ηv ∈ Wo (U ) we can apply Sobolev's inequality with 2∗ = n−2 to get the second inequality of the following ones, while whose rst passage is due to Hölder inequality. Z  2∗2 Z 2 2∗ 2 |v| dx |A(k, r)|1− 2∗ = |v| dx ≤ Br

Br

Z

2∗

 2∗2

2 n

Z

2∗

 2∗2

2

|ηv| dx |A(k, r)| n ≤ Br Br Z  Z  2 2 2 n |D(ηv)|2 dx |A(k, r)| n |D(ηv)| dx |A(k, r)| ≤ Cn Cn |v| dx

|A(k, r)| =

Br¯

Br 2

Now, thanks to the fact that |D(vη)| ≤ (|vDη| + |ηDv|)2 ≤ 2(|vDη|2 + |ηDv|2 ) we have, by using properties of cut-o function and Caccioppoli's inequality in the second passage14 , Z  2 |D(ηv)|2 dx |A(k, r)| n ≤ Cn Br¯ n Z  Z o 2 2 Cn |vD(η)| dx + |ηD(v)|2 dx |A(k, r)| n ≤ Br¯ Br¯ Z o n C  Z 2 1 2 2 n ≤ |D(v)| dx |A(k, r)| Cn |v| dx + (R − r)2 Br¯ Br¯ Z Z n C  o 2 2C2  1 2 2 Cn |v| dx + |v| dx |A(k, r)| n = 2 2 (R − r) BR (R − r) BR Z  2 Cn max{C1 , 2C2 }  2 n |v| dx |A(k, r)| (R − r)2 BR So we obtained Z Z  2 Cn  2 2 |v| dx |A(k, r)| n |v| dx ≤ 2 (R − r) BR Br 14 Note

that (R − r¯) = (R − R+r 2 )=

R−r 2

.

46

Subsequently we treat integral averages as functions of radius and magnitude 15 k on which are dened the level sets A(k, r). We set the following notation for each h ∈ R, Br ⊂⊂ U

Z

|u(x) − k|2

u(k, r) :=

(2.14)

A(k,r)

This function represents the averaged oscillation of the function around the index of magnitude k : the idea is to show that for every k , r < R chosen there exists a number z ∈ R such that u(k − z, r) is zero, in order to get the nullity of the set (that we call of magnitude ) where the solution is bigger than k − z and so to obtain a bound for the essential supremum in the ball considerated for the set of magnitude. Next proposition and theorems will clarify these assertions. Now, thanks to last proposition we already have some peculiar properties to show for functions u(k, r).

Proposition 2.3. For every h > k,x ∈ U xed, 0 < r < R < dist(x, ∂U ) we have |A(h, r)| ≤ u(h, r) ≤

1 u(k, R) (h − k)2

(2.15)

2 Cn n 2 |A(k, R)| u(k, R) (R − r)

(2.16)

Proof. The proof is a simple application of the fact that if h > k, then (u(x) − h)+ ≤ (u(x) − k)+ and A(h, r) ⊂ A(k, r) for the rst inequality and an easy use of Proposition 2.2.

Now we may consider a, b ∈ R+ , and multiplying the a-power of (2.15) to b-power of (2.16) we get our main inequality for iteration:

|A(h, r)|a u(h, r)b ≤

Cnb

2b

2a

2b

(h − k) (R − r)

|A(k, R)| n u(k, R)a+b

(2.17)

We dene two functions which will give us the possibility to compute our algebraic iteration: they still represent the measure of points for which we have a certain magnitude, weighted on the integral average around the pick of magnitude. For every h ∈ R, x ∈ U , 0 < r such that Br (x) ⊂⊂ U , set   P (h, r) := |A(h, r)|a u(h, r)b /rn (2.18) call magnitude the property of being grater than a predened constant, as it happens in {u > k} 15 We

47

  a P¯ (h, r) := |{u ≥ h} ∩ B¯r | u(h, r)b /rn

(2.19)

It is possible to rewrite the second member of inequality (2.17) as a unique = aα and a + b = bα and solving power of P (k, R): imposing α > 0 s.t. 2b n the second order q equation coming from this easy system we nd a positive solution α =

1 2

+

n+8 4n

and so

P (h, r) ≤

Cnb 2a

2b

(h − k) (R − r)

P (k, R)α

(2.20)

Remark 15. (technicalities) • Note that as its set of denition is bigger, we have P (h, r) ≤ P¯ (h, r); • For rn > 0 let (kn )n∈N , (rn )n∈N be an increasing sequence and a decreasing one s.t. kn → k0 and rn → r0 , then P (kn , rn ) −→n→∞

P¯ (k0 , r0 )

(2.21)

• observe that the choice of such an α determines a, b uniquely.

Lemma 2.4. Let β > 0 and let {xi }i∈N be a sequence of real positive numbers such that xi+1 ≤ CB i x1+β i

for a C > 1 and B > 1. 1 1 If x0 ≤ C − β B − β2 , we have i

xi ≤ B − β x0 ,

thus

lim xi = 0

i→∞

Proof. We proceed by induction. The assert is obviously true if i = 0, and if we assume it to hold for i then 1

1+β

xi+1 ≤ CB i x1+β ≤ CB i (B − β x0 ) i

= CB i(1−

1+β ) β

1

x1+β = (CB β xβ0 )B − 0

i+1 β

x0

so that our inequality is valid also for (i + 1) To maintain loyalty to De Giorgi technique, we present the alter-ego of this general Lemma, considered in our context.

48

Lemma 2.5. ( Vanishing of super-level set, bounding magnitude) Fix a suitable16 R0 > 0 . Then for any k0 ∈ R chosen and for any σ ∈ (0, 1) there exists a t = t(k0 , σ) ∈ R such that (2.22)

P (k0 + t, R0 − σR0 ) = 0

with t2b =

α (2a+2b) α−1 ca

2

σ 2a R02a

P (k0 , R0 )α−1

Proof. The idea of the proof is to consider the special sequences • the increasing kn = k0 + t −

t 2n

→

• the decreasing rn = R0 − σR0 +

σR0 2n

k0 + t and →

R0 − σR0

λ=

2a + 2b α−1

and to demonstrate by induction that

P (kn , rn ) ≤ C

P (k0 , R0 ) , 2λn

(2.23)

where the initial step of induction is given by equation (2.20) with initial values and the inductive pass is given by the same equation. Thanks to second point of previous remark we have that

lim P (kn , rn ) = P¯ (k0 + t, R0 − σR0 ) ≤ 0

n→∞

and by rst point of the same remark we get the desired formula. This can be demonstrated analogously with an application of precedent Lemma, whose hypothesis are fullled by (2.23). Quantitative expression for t is then determined to accomplish the second hypothesis of Lemma (2.4). 17 Finally we are able to bound the essential supremum of the solution in the half ball thanks to the weighted L2 norm of the oscillation of the function around a chosen magnitude point k0 .

Theorem 2.6. (Bounding the solution in the Half Ball) Let u ∈ DGO(U )+ and x a suitable radius r > 0. Then, for every k0 ∈ R we have the following estimate for the essential supremum in the half ball 1/2  |A(k , r)|  α−1 1 Z 2 0 2 |u − k0 | sup u ≤ k0 + c n r A(k0 ,r) rn Br/2 16 A

(2.24)

suitable radius will be always referred as a radius far enough from the boundary

r < dist(x, ∂U ) 17 To

be more precise they are the same required for hypothesis in preceding Lemma 2.4, with choice of C,B that we are up to show in Theorem 2.6

49

Proof. (Step 1)

Let us put β = α − 1 > 1 and let us dene the sequences

• ki := k0 (1 − • ri := 2r (1 +

1 ) 2i

1 ) 2i

increasing s.t. k0 = limi→∞ ki , (ki+1 − ki ) =

k0 ; 2i+1

decreasing s.t. r0 = r/2 = limi→∞ ri , |ri+1 − ri | =

r 2i+2

• P (ki , ri )i∈N decreasing, because the measure of the set of its denition is less and less as radius decreases and magnitude increases. Then by (2.20) we have that

P (ki , ri ) ≤ Cnb

P (ki−1 , ri−1 )β+1 2a r 0 ( 2ki+1 ) ( 2i+2 )2b

= CB i P (ki−1 , ri−1 )β+1

b 22a+2b Cn ,B k02a r2b

= 2(2a+2b) as desired for Lemma 2.4 with xi = P (ki , ri ). b R Now we need to nd ko such that r1n |A(k0 , r)|a A(k0 ,r/2) |u − k0 |2 dx = with C =

1

−

1

P (k0 , r0 ) ≤ C − β B β2 in order to have sucient condition of Lemma 2.4. This means we need b Z 1 1 − 1 a |u − k0 |2 dx = P (k0 , r0 ) ≤ C − β B β2 P (k0 , r/2) = n |A(k0 , r)| r A(k0 ,r/2) which brings us to

18 β

|A(k0 , r)| 2  1 k0 ≥ H( ) rn rn |A(k0 , r)| H( ) rn

α−1 2

Z

2

|u − k0 |

β 2ab

=

A(k0 ,r/2)

1 Z α−1 2ab 2 |u − k0 | rn A(k0 ,r/2)

1 and Now remember by the denition of α that a + b = b(α), so ab = (α−1) b 1 ¯ exponent (α − 1) 2a = 2 . Taking H = H + 1 we have a right amount for k0 :

¯ |A(k0 , r)| ) k0 = H( rn

α−1 2

 12 1 Z 2 |u − k0 | rn A(k0 ,r/2)

(2.25)

Now we can nally apply Lemma 2.4 to get that 18 with

α(2a+2b) a

(2a+2b) a b (1/2) a

b

and B, C as previously dened. Notice the important point that k0 , r appear only in C constant, from where the are expressed. H=

2

2

Cn2a

50

1 Z 1/2  |A(k , r)|  α−1 2 0 2 0 = lim P (ki , ri ) = P (k0 , r/2) = n |u − k0 | n i→∞ r A(k0 ,r/2) r which by the fact that (u − k0 ) > 0 in A(k0 , r/2) means that necessarily |A(k0 , r/2)| = 0 which means that

¯ |A(k0 , r/2)| ) sup u(x) ≤ k0 = H( rn Br/2

α−1 2

1 Z  21 2 |u − k0 | rn A(k0 ,r/2)

(Step 2) The problem in our rst step is that we do not seem to be free to chose a k0 ∈ R with this strategy. However if from the very beginning of the proof ˜0 , we set ki := k˜0 (1 − 21i ) increasing s.t. k0 = k˜0 = limi→∞ ki , (ki+1 − ki ) = 2ki+1 ˜ with k0 = k0 + d for a d ∈ R to be chosen later, we will obtain condition

|A(k0 , r)| ) d ≥ −k0 + H( rn

α−1 2

1 Z α−1 2ab 2 |u − k0 | rn A(k0 ,r/2)

that will be satised for

¯ |A(k0 , r)| ) d = H( rn

α−1 2

1 Z  12 2 |u − k0 | rn A(k0 ,r/2)

(2.26)

and so

1 Z 1/2  |A(k + d, r)|  α−1 2 0 2 0 = lim P (ki , ri ) = P (k˜0 , r/2) = n |u − k0 − d| i→∞ r A(k0 +d,r/2) rn and

Z  |A(k , r/2)|  α−1  21 2  1 0 2 ˜ ¯ supBr/2 u(x) ≤ k0 = k0 +d = k0 +H |u − k0 | rn rn A(k0 ,r/2)

Corollary 2.7. If u ∈ DGO(U ) then u ∈ L∞ loc (U ). More precisely ||u||L∞ (Br/2 )

n 1 Z o 21 2 ≤C |u| dx . |Br | Br 51

(2.27)

Proof. By preceding theorem choosing k0 = 0 and calling the volume of unitary ball wn rn we get

1 Z  21 n |A(0, r)| o α−1 2 2 sup u ≤ C n ≤ |u| dx n r A(0,r) r Br/2  1 Z 1 Z  12 α−1  12 2 2 2 C n |u| dx wn ≤ C |u| dx r A(0,r) wn rn A(0,r) As also −u ∈ DGO(U ), we have

 1 Z  21 2 sup(−u) ≤ C | − u| dx wn rn A(0,r) Br/2 thus

 21  1 Z 2 |u| dx inf u ≥ −C Br/2 wn rn A(0,r)

so that the L∞ norm of u in the half ball is bounded and so the essential oscillation w(r/2). Now we pass to show the Holder-continuity of elements in DGO(U ). Let us suppose that k0 (u) := M (2r)+m(2r) and that |A(k0 , r)| ≥ 21 |Br |. Last 2 inequality will be useful to dene a special function with compact support v in following argument, for whom is valid Poincaré's inequality (1.16). It can be replaced surely for whatever 0 < γ < 1 with|A(k0 , r)| ≥ (1 − γ)|Br | and proof is easily adaptable to this more general case.

Remark 16. (More than a half ball) If we suppose u ∈ DGO(U ) we can always assume |A(k0 (u), r)| ≥ 21 |Br |. Indeed, 1 1 k0 (u) = (sup u + inf u) = − (inf (−u) + sup(−u)) = −k0 (−u) Br 2 Br 2 Br Br

Thus {u ≤ k0 (u)} = {−u ≥ k0 (−u)} and if for absurd assumption we would have |A(k0 (u), r)| = |{Br ∩ {u ≥ k0 (u)}| < 21 |Br | then |Br ∩ {(−u) ≥ k0 (−u)}| ≥ 12 |Br | and so our hypothesis is veried for (−u) function . Replacing u by (−u) there is no changing in our assumption u ∈ DGO(U ). Let h > k > k0 and let us set

  h − k v(x) := min{u(x), h} − min{u(x), k} = u(x) − k   0

52

x ∈ {u ≥ h} x ∈ {k < u < h} x ∈ {u ≤ k} (2.28)

¯ r) = Br − A(k, r) whose measure is not Note that v vanishes in a set B(k, zero as assumption |A(k0 , r)| ≥ 21 |Br | implies |suppBr v| = |Br − A(k, r)| ≥ |Br ∩ {u < k0 } ≥ 12 |Br | 19 . As we have just seen that v has compact support in Br , so we can apply Poincaré inequality (1.16) to v with p = 1 and we observe that Dv = Du in {k < u < h} Z

|v|

n n−1

dx

 n−1 n

Z

Z

≤ Cn

|Dv|dx = Cn

(2.29)

|Du|dx

Br

Br

A(k,r)−A(h,r)

Now calculating and using the notation M (2r) for the essential supremum in the ball of radius 2r,

(h − k)|A(h, r)|

n−1 n

=

Z

(h − k)

n n−1

 n−1 Z n dx =

A(h,r)

dx

 n−1 n

≤

A(h,r)

Z

1/2

|Du|dx ≤ Cn |A(k, r) − A(h, r)|

Cn

v

n n−1

Z

A(k,r)−A(h,r)

2

|Dv| dx

A(k,r)−A(h,r) 1/2

Cn |A(k, r) − A(h, r)|

Z

2

|Du| dx

 21

≤

A(k,r) 1/2

Cn |A(k, r) − A(h, r)|

C Z r2

1/2

Cn |A(k, r) − A(h, r)|

A(k,r)

 21 |M (2r) − k| |A(k, 2r)| ≤ 2

C r

 21 (u(x) − k) dx ≤ 21 2

19 Being

2

h > k > k0 we have that A(h) ⊂ A(k) ⊂ A(k0 ) are reverse in order. Holder inequality. 21 As u ∈ DGO(U ). 20 By

53

 21

≤ 20

Cn (|A(k, r)| − |A(h, r)|)

1/2

C r2

We take its square to obtain

(h − k)2 |A(h, r)|

2n−2 n

2

|M (2r) − k| wn r

n

 21

≤ Cn rn−2 |M (2r) − k|2 (|A(k, r)| − |A(h, r)|)

(2.30)

Now we dene the most important sequences of magnitudes for our purpose. Consider M (2r) − k0 ki = M (2r) − −→i→∞ M (2r) (2.31) 2i

Remark 17. Simple calculations show 0 ; • ki − ki−1 = M (2r)−k 2i • M (2r) − ki−1 = • ki = M (2r) +

M (2r)−k0 2i−1

m(2r) 2i+1

;

, k0 =

M (2r)+m(2r) 2

and k∞ := limi→∞ ki = M (2r);

• A(k∞ , r) = A(M (2r), r) ⊂ A(ki , r) ⊂ A(ki−1 )

∀i ∈ N (next gure)

By using the rst property remarked above in (2.30) and second one recursively in next passages we get 2n−2 2n−2 |M (2r) − k0 |2 2 n n = (k − k ) |A(k , r)| ≤ |A(k , r)| i i−1 i i 22i Cn rn−2 |M (2r) − ki |2 (|A(ki−1 , r)| − |A(ki , r)|) =

|M (2r) − Cn rn−2 22i−2

k0 |2

(|A(ki−1 , r)| − |A(ki , r)|)

(2.32)

and simplifying both left and right members of previous inequality we have

|A(ki , r)|

2n−2 n

≤ Cn rn−2 (|A(ki−1 , r)| − |A(ki , r)|)

(2.33)

This means that the measure of super-level sets is controlled with the measure of the dierence of that set and the previous one.

54

By its denition 0 = |A(k∞ , r)| = |A(M (2r), r)| = limi→∞ |A(ki , r)| as {A(ki , r)} is a decreasing chain of subsets and Lebesgue measure's additivity gives the limit. This means that given a positive number ( 1j ) for j ∈ N we are able to nd an m ¯ ∈ N such that for each m > m ¯ |A(km , r)| ≤ ( 1j ). Let us suppose j and so m ¯ xed and let m > m ¯. If i ≤ m then |A(km , r)| ≤ |A(ki , r)|, thus summing from 1 to m we obtain 2n−2 2n−2 2n−2 Pm Pm n n ≥ = m|A(km , r)| n , and so sumi=1 |A(ki , r)| i=1 |A(km , r)| ming eq (2.33) from 1 to m we have

m|A(km , r)|

2n−2 n

≤

m X

|A(ki , r)|

2n−2 n

≤ Cn r

n−2

m X

(|A(ki−1 , r)| − |A(ki , r)|) =

i=1

i=1

(2.34)

Cn rn−2 (|A(k0 , r)| − |A(km , r)|) being a telescopic sum and reindexing ki+1 = ki . Finally, n

|A(km , r)| ≤ (Cn r

n−2

(|A(k0 , r)| − |A(km , r)|) n

n 2n−2

1
2n−2 ≤ m n 1
2n−2 ≤ m

n 1
2n−2 (Cn r (|Br |)) ≤ (Cn rn−2 (wn rn )) 2n−2 m n
2n−2 n 1 Cn r (2.35) m We state this result as a Lemma as it is fundamental tool for next theorem.

n−2

n 2n−2

Lemma 2.8. (De Giorgi Lemma) and ki as dened in (2.31). Let x ∈ U , r > 0 suitable. Set k0 = M (2r)+m(2r) 2 1 If |A(k0 , r)| ≥ 2 |Br | then there exists m ¯ ∈ N s.t. ∀m > m ¯ we have n

|A(km )| ≤ Cn r

n

1
n−2 m

(2.36)

As a consequence comes the following

Theorem 2.9. If u ∈ DGO(U ) then for each suitable radius r > 0 there exist a constant k < 1 independent from r for which holds r w( ) ≤ kw(2r) 2

(2.37)

i )| Proof. Consider preceding Lemma 2.8, it says that C |A(k ≤ Cn rn

n

1 n−2 −→0 i |A(kn 1 ¯ )| C rn < 2 . By




when i goes to innity, so there exists n ¯ ∈ N such that Theorem 2.6 with k0 = ki we have 1 Z 1/2  |A(k , r)|  12 r i 2 M ( ) = sup u ≤ ki + C n |u − ki | ≤ n 2 r r Br/2 A(ki ,r) 55

1 1/2  |A(k , r)|  21 i 2 ki + C n |A(ki , r)|(M (2r) − ki ) ≤ n r r |A(ki , r)| ki + C (M (2r) − ki ) < rn 1 1 M (2r) − m(2r) M (2r) − kn¯ = kn¯ + M (2r) = M (2r) − kn¯ + (2.38) 2 2 2 2n¯ +2 where we used at the ending passage the third property of Remark 17 and previously the existence of n ¯ number. Subtracting from both members m( 2r ) we get an inequality of the type r r M (2r) − m(2r) r r w( ) = M ( ) − m( ) ≤ M (2r) − − m( ) ≤ n ¯ 2 2 2 2 2 (M (2r) − m(2r))(1 − For 0 < k = kn¯ = (1 −

1 ¯ +2 ) 2n

1 2n¯ +2

) = kw(2r)

(2.39)

< 1 depends only on the number n ¯.

Lemma 2.10. Let h : [0, ∞) → [0, ∞) an increasing function such that for a chosen R > 0 there exist two positive constants C1 , C2 < 1 such that h(C1 r) ≤ C2 h(r)

(2.40)

for each r ≤ R. Then there exists α > 0 s.t. h(r) ≤

1  r α h(R) C2 R

(2.41)

for each r ≤ R. Proof. For whatever innitesimal decreasing sequence of numbers {xn }n∈N →

0 given a 0 < t < x1 there exists 1 < k ∈ N s.t. xk < t < xk−1 . This comes just by noticing that as {xn } is innitesimal and 0 < t then there exists necessarily m ¯ ∈ N s.t. for each n > m ¯ we have xn ≤ t, and taking now k := min{n > m|x ¯ n < t} it is easy recognizable by the denition of minimum that xk−1 > t and that's it. Now, as 1 < C sequence cn = C1n−1 is strictly decreasing and innitesimal. Chose r ≤ R and as c1 = 1 > Rr we know by precedent discussion that there exists k ∈ N : ck+1 < Rr < ck . By monotony of h iterating k times our hypothesis inequality we have h(r) ≤ h(ck R) ≤ C2 h(ck−1 R) ≤ ........ ≤ C2k h(R) 56

(2.42)

Now as C1k+1
−1 + logC1 ( Rr ) and as C2 < 1 we get r

1 logC1 [C2logC1 ( R ) ] h(r) ≤ ≤ C h(R) = = C2 1 1 logC1 ( Rr )·logC1 [C2 ] 1  r logC1 C2 = C1 h(R) = h(R) (2.43) C2 C2 R Set α = logC1 C2 and reading left and right side of this inequality yields the proof. C2k h(R)

log ( r ) C2−1 C2 C1 R h(R)

Remark 18. Note that if C1 < C2 then α ∈ (0, 1), as logarithm in this case is a decreasing function. Theorem 2.11. ( De Giorgi Theorem) If u ∈ DGO(U ) then there exists α ∈ (0, 1) such that locally u ∈ C 0,α (U ). Proof. First note that w(r) := M (r) − m(r) is an increasing function , next

x a suitable R > 0 and note that Theorem 2.8 provide inequality

(2.44)

w(C1 r) ≤ C2 w(r) for each r ≤ R and with C1 = 41 , 0 < C2 = (1 − precedent Lemma to get

w(r) ≤

1  r α w(R), C2 R

1 ¯ +2 ) 2n

< 1. So we can apply

∀r ≤ R

(2.45)

¯ big enough to have C2 > C1 so that by As C2 = (1 − 2n¯1+2 ) we can choose n previous remark we know that α ∈ (0, 1). Now x x, y ∈ BR : |x − y| = r < R. Then as by precedent discussion w(R) has been shown to be bounded and by (2.45)  w(R)  1  r α w(R) = |x − y|α = k|x − y|α α C2 R C2 R (2.46) 0,α So u ∈ C (BR ) for any choice of suitable R > 0. |u(x) − u(y)| ≤ w(r) ≤

Remark 19. We observe that the constant α depends only on the constant C of Denition 2.1 of De Giorgi Classes (see equations (2.2), (2.3)) which on itself depends only on the ellipticity ratio of ai,j as shown by demonstration of Theorem 2.1. Indeed by Theorem 2.9 is guaranteed the existence of n¯ ∈ N s.t. C |A(krnn¯ ,r)| < 12 . In De Giorgi Theorem we choose α = logC1 C2 with

57

C1 = 14 , 0 < C2 = (1 − 2n¯1+2 ) < 1. Thus if C → ∞ then n ¯ → ∞ and C2 → 1, α → 0. Precisely, if boundedness and ellipticity are given by coecients n X

|ai,j (x)| ≤ C

ai,j (x)bi bj ≥ C1 |b|2

∀b ∈ Rn

i,j=1

then we have lim α = 0

C →∞ C1

Thus the Holder continuity proved in this theorem is critical with respect to this ellipticity ratio CC1 when tending to ∞.

Corollary 2.12. Let u ∈ W 1,2 (U ) be a weak solutions of Euler-Lagrange equation Z n X

U

 (Lzi (x, u, Du))φxi dx = 0

i=1

for L = L(x, y, z) as in previous chapter, associated to the functional Z L(Du(x))dx

I[u] = U

where the Lagrangian L ∈ C ∞ (U ) is supposed to satisfy to conditions (1.33),(1.34) of uniform convexity and upper bound of second derivatives. Then u ∈ C ∞ (U ) is a smooth function. Proof. As conditions (1.33),(1.34) are satised we know by Theorem 1.24

that solution u lies in Hilbert space H 2 (U ) = W 2,2 (U ). Next we proceed as in "bootstrap" argument of Preliminaries chapter, to obtain by iterative application of Schauder's estimates our thesis.

Remark 20. A not homogeneous result can be found easily by observing that Schauder's Theorem has a not homogeneous version. The only hypothesis more to require is the regularity of the term which now would stay on the right of the equality. References and Further steps De Giorgi insights opened the way to the non linear theory, and they are the cornerstones of what is nowadays called Non Linear Potential Theory (see [86]). De Giorgi's proof rested on a new method, based on the idea of proving regularity properties of solutions via the analysis of the decay and 58

density properties of their level sets; a method that became pervasive in the whole regularity theory. We refer to [13], [3] for the basic outline of this demonstration, and to seventh chapter of [44] for some improvement that we adapted to the homogeneous case. We adapted results of [44],[40] to obtain a simpler and concise demonstration of Theorem 2.6. The version we presented here of De Giorgi Theorem is a simplied one relatively to [13], as in his paper De Giorgi uses several Lemmas using approximation of semilinear functions, while we derived Caccioppoli's inequality by testing the equation with cut o functions. It must then be said that this result has been proved by De Giorgi in the case of linear equations, but surprisingly as his method was enough abstract and there was no use of linearity, Olga A. Ladyzhenskaya and Nina N. Uraltseva extended this result also for quasilinear equations in [55]. This technique has been improved to demonstrate reguliarity of minima of calculus of variations, see for instance [40] and [44]. This means that the same regularity found for solutions of linear elliptic equations in divergence form can be proved also for each quasi-minima of functionals of the calculus of variations satisfying suitable growth assumptions, without using the Euler-Lagrange equation which may possibly not exist if the Lagrangian is not smooth enough. This opened the way to a new method: to demonstrate that a wide class (as quasi minima of functionals surely is) of solutions has somewhat property, it would have been be of common use to demonstrate it for functions in De Giorgi class. Next section is entirely devoted to this approach.

2.2 De Giorgi classes We point the beginning of this section announcing some ne properties that DGO(U ) functions have in common with harmonic functions.

Theorem 2.13. (Harnack's Inequality) Let U ⊂ Rn a bounded open connected set and let V ⊂⊂ U a compactly contained open set. If u ∈ DGO(U ), there exist a constant depending only on V, U such that sup u ≤ C(V, U ) inf u V

V

(2.47)

Furthermore, classical consequences of Harnack's inequality come also for DGO(U ) functions.

Theorem 2.14. (Strong Maximum Principle) Let U be a connected set and u ∈ DGO(U )− . If u has an interior minimun point, then it is constant. 59

Corollary 2.15. (Liouville Theorem) Let u ∈ DGO(Rn ) be bounded below.Then, u is constant We will derive these results with more generality, for a wider class of functions to whom can be proved also that minima of regular functionals belong. Precisely this results hold for functions belonging to the homogeneous p-De Giorgi class of functions, that we present in this section. Following [41] there is a unied regularity theory for both elliptic partial dierential equations and minima of regular functionals of calculus of variations. Let us consider on an open set U ∈ Rn the functional Z F (u, U ) = F (x, u, Du)dx U

in which F (x, u, z) is a Caratheodory function satisfying the growth conditions

|z|p − b(x)|u|µ − a(x) ≤ F (x, u, z) ≤ M |z|p + b(x)|u|µ + a(x) ∗

where for α > np and β > p∗p−µ , a(x) ∈ Lα (U ), b(x) ∈ Lβ (U ) are two nonpn is well dened for p < n, negative functions, M > 0, 1 < p ≤ µ < p∗ = n−p as for p ≥ n and results of immersion of the Preliminaries chapter every such a function has an Hölder continuous version. Let us assume also that 1 = np − , β1 = 1 − pµ∗ − . α

Denition 2.4.

1,p A function u ∈ Wloc (U ) is a sub-quasi-minimum of the functional F above, with constant Q ≥ 1, if for every compactly supported function W 1,p (U ) 3 φ ≤ 0 we have

F (u, K) ≤ QF (u + φ, K) 1,p Conversely a super-quasi-minimum of F is a function u ∈ Wloc (U ) s.t. for every 0 ≤ ψ ∈ W 1,p (U ) with compact support in U s.t.

F (u, K) ≤ QF (u + φ, K) 1,p Finally, a quasi-minimum of F (x, u, z) is a function u ∈ Wloc (U ) which has both previous characteristics.

Next theorem says that each sub-quasi-minimum of F is in fact a member of a general Caccioppoli 's inequality (whose proof can be found in [44]).

60

Theorem 2.16. (Caccioppoli inequality for sub-quasi-minima) Let u ∈ W 1,p (U ) be a sub-quasi-minimum of the functional F (x, u, z) dened above, and let growth conditions announced to be satised. Then there exists R0 > 0 depending on the p∗ norm of u and the β norm of b(x) such that for every x0 ∈ U ,every 0 < r < R < min(R0 , dist(x0 , ∂U )) and every k ≥ 0 we have Z

c |Du| dx ≤ (R − r)p A(k,r) p

Z

p

(u−k)p dx+c(||a||α +k p R−n )|A(k, R)|1− n +

A(k,R)

Remark 21. The theorem yet stated remain valid if u is a sub-quasi-minimum with Dirichlet conditions on ∂U . This happens when for every function ψ ≤ u with ψ − u ∈ Wo1,p (U ) and compact K = supp(ψ − u)we have F (u, K) ≤ QF (ψ, K)

and the trace of u on ∂U is a bounded function. This induce to dene the non-homogeneous p-De Giorgi classes.

Denition 2.5.

1,p Let u ∈ Wloc (U ). We say that u belongs to the De Giorgi (U, c, M, , R , k ) if for every couple of concentric balls Br ⊂ class DG+ 0 0 p BR ⊂⊂ U , with R < R0 and for every k ≥ k0 ≥ 0 we have Z Z p c p (u − k)p dx + c(M p + k p R−n )|A(k, R)|1− n + |Du| dx ≤ p (R − r) A(k,R) A(k,r)

We can dene similarly the De Giorgi class DG− p to be the class of functions 1,p + u such that −u ∈ DGp . These are elements of the linear space Wloc (U ) such that for every 0 < r < R ≤ R0 and k ≤ −k0 we have Z Z p c p |Du| dx ≤ (u − k)p dx + c(M p + k p R−n )|B(k, R)|1− n + p (R − r) B(k,R) B(k,r) for sets A, B dened as in previous section on De Giorgi Theorem. Finally we − indicate with DGp = DG+ p ∩ DGp the p-De Giorgi non-homogeneous class.

Remark 22. We can simplify considerably the inequalities above by a homothety to reduce to the case R = 1 and by some substitutions to get for n n p p y = Rx, w(x) = v(y), v = u + M R , h = k + M R . A surprising characteristic of De Giorgi classes is that Caccioppoli's inequalities dening them contain practically all the information deriving from the minimum properties of the function u for what concerns Holder continuity and boundedness. As the demonstration of such assertions are very similar to the ones introduced in previous and next section, we refer to [44] for proofs for the sake of readability. 61

Theorem 2.17. Let u be function in DG+p (U ). Then, u is locally bounded from above in U, and for every x0 ∈ U , 0 < r < R ≤ min(R0 , dist(x0 , ∂U )) and for every q > 0 there exists a constant c(q) such that n sup u(x) ≤ c(q) Br

1 (R − r)n

Z

uq+ dx

 1q

1

n

+ k0 + ||a|| p R p

o

BR

for quantities R0 , k0 introduced in the denition above. A similar inequality holds for DG−p (U ) functions.

Corollary 2.18. Let u ∈ W 1,p (U ) be a sub-quasi-minimum of functional F (x, u, z), satisfying growth conditions as above. Then, u is locally bounded from above in U . Similarly, every super-quasi-minimum of F is locally bounded from below and so every quasi-minimum of F is locally bounded in U. Always with similar techniques to previous section it can be proved continuity of De Giorgi class functions.

Theorem 2.19. Let u ∈ DGp (U, c, M, , R0 , k0 ), then u is locally Holder continuous in U and for each x0 ∈ U and for suitable 0 < r < R ≤ min(R0 , dist(x0 , ∂U )) we have the existence of a γ ∈ (0, 1) such that the estimate o n r γ oscBr u =: w(r) ≤ c

R

holds.

w(u, R) + M rγ

In their paper [23], Di Benedetto and Trudinger proved Harnack's inequality for functions belonging to p-De Giorgi classes, and hence for quasi-minima of integral functionals. Here we present a similar approach by [44].

Theorem 2.20. (Harnack's inequality for quasi-minima) Let u be a positive function belonging to DGp (U, c, M, , R0 , k0 ) with k0 = 0 and let r < R20 , such that the ball of radius 3r is contained in U , then sup ≤ c(inf u + M rα ) Br

Br

As quasi-minima of the integral functional F are spontaneously members of the p-De Giorgi class, previous inequality holds in particular for them. To demonstrate this Theorem we need a Lemma that can be found in [44] (chap. 7), which is a technical generalization of the technique used in Lemma 2.8. We will demonstrate our assert for cubes Q(l), where l is the side of the cube. 62

Lemma 2.21. Let u ∈ DG−p (U, c, M, , R0 , k0 ) be a positive function. Then, for every γ ∈ (0, 1) and for every T > 21 there exists a positive constant c(γ, T ) such that22 if |B((θ, 1)| ≤ γ|Q(2T )| for some θ > 0 then inf u ≥ c(γ, T )θ Qt

1−γ ). with c(γ, T ) = λ(1 − (2T )n

Proof. (of Theorem 2.20)

We begin by showing that

u(x0 ) ≤ c

inf u(x)

Q(x0 ,R)

which can be equivalently be expressed with

v(x) :=

u(x) ≥c>0 u(x0 )

in Q(x0 , R). As remarked in Remark 21 we can assume without loss of generality that M = 0, R = 1. The general case will follow by writing u + M Rs instead of u. Observe that v function satises Caccioppoli's inequality and then Theorem of Holder continuity 2.19 applies to v , giving us the estimate  r γ  r γ ≤ c||v||∞,Q(x,R) oscQ(x,r) v ≤ c · oscQ(x,R) v R R for each x ∈ U and every 0 < r < R < 12 dist(x, ∂U ). Now let Z(τ ) = (1 − τ )−δ , where δ > 0 will be chosen later, and let τ0 be the largest value of τ for which ||v||∞,Q(x0 ,τ ) ≥ Z(τ ). Since left-hand side of the preceding relation is bounded, and the right-hand side diverges as τ → 1, we have 0 ≤ τ0 < 1. Let x¯ ∈ Q(x¯0 , τ0 ) be such that v(¯ x))||v||∞,Q(x0 ,τ0 ) ≥ (1 − τ0 )−δ . We have

||v||∞,Q(¯x, 1−τ0 ) ≤ ||v||∞,Q(¯x, 1+τ0 ) < Z( 2

2

1 + τ0 ) = 2δ (1 − τ0 )−δ . 2

On the other hand, using the estimate of Holder continuity n r γ o osc v ≤ c w(v, R) + M rγ R 22 Remember

that sets B(k, r) were introduced in last section as the intersection between ball of radius r and the lower-level set of u − k function. Here we make a soft adaptation in considering these sets as the intersection of the lower-lever set of u − k and the cube of side r.

63

on cubes Q(x, R) with M = 0,R =

1−τ0 2

and r = R for  < 1 we get

oscQ(¯x, 1−τ0 ) v =: w(r) ≤ c||v||∞,Q(¯x, 1−τ0 ) β ≤ c2δ (1 − τ0 )−δ β 2

2

and hence

v(x) ≥ v(¯ x) − oscQ(¯x, 1−τ0 ) v ≥ (1 − τ0 )−δ (1 − c2δ β ) 2

0 for every x ∈ Q(¯ x, 1−τ ). Choosing  = 2

1 c2δ+1

we obtain

1 v(x) ≥ (1 − τ0 )−δ 2 0 0 ). We can now apply Lemma 2.21 with R = 1−τ ,T = 2 and in Q(¯ x, 1−τ 2 2 1 −δ θ = 2 (1 − τ0 ) . We have in this way γ = 0 and c = c(0, 2) so

v(x) ≥

c(0, 2) (1 − τ0 )−δ 2

Iterating this argument we get for every ν

v(x) ≥

cν (1 − τ0 )−δ 2

in Q(¯ x, 2ν−1 (1 − τ0 )). Let ν be such that 2 ≤ 2ν−1 (1 − τ0 ) < 4. We have log2 c  8 cν ≥ (1 − τ0 ) and so

1  8 log2 c (1 − τ0 )−δ−log2 c 2  in Q(¯ x, 2) ⊃ Q(x0 , 1). We now choose δ = −log2 c so that we have  = 2tc and therefore 1  16t log2 c v(x) ≥ 2 c in Q(x0 , 1), which gives us the estimate v(x) ≥ K desired at the beginning. v(x) ≥

Finally, let Qr = Q(x1 , r) be a cube contained in U , for a suitable r, and let x0 ∈ Q¯r be such that u(x0 ) = supQr u(x). Taking R = 3r we obtain by precedent inequality that

sup u(x) ≤ c Qr

inf Q(x0 ,R)

and so we obtained our thesis. 64

≤ c inf u(x) Qr

In the homogeneous case formulas become more elegant, and classical properties of harmonic functions can be derived.

Denition 2.6.

1,p We say that the function u ∈ Wloc (U ) belong to the homogeneous p-De Giorgi class when for a positive constant M > 0, it satises for each k ∈ R the inequalities Z Z M p |Du| dx ≤ (u − k)p dx p (R − r) A(k,r) A(k,R) Z Z M |Du|p dx ≤ (u − k)p dx p (R − r) B(k,r) B(k,R)

for every couple of concentric balls of suitable radius. In particular this happens when u is a quasi-minimum of a functional of the kind Z F (u, U ) = F (x, u(x), Du(x))dx U

with growth conditions

|z|p ≤ F (x, u, z) ≤ M |z|p .

Theorem 2.22. (Harnack's inequality for DGOp functions) Let U be a bounded and connected open set of Rn and let V ⊂⊂ U . Let u ∈ DGOp (U ),p > 1, be a positive function Then there exists a constant c = c(V, U ) such that sup u ≤ c inf u V

V

Proof. Let Q1 , .., Qn be a nite family of cubes such that any two consecutive

cubes Qi ∩ Qi+1 6= ∅ and that

sup u = sup u, V

inf u = inf u V

Q1

Qn

We can assume that each of these cubes has a side of length r < R0 and that the cubes of sides 6r are contained in U . For each cube Qi we can write preceding Harnack inequality for DGp (U )function with M = 0:

sup u ≤ c · inf u. Qi

Qi

On the other hand, since any two consecutive cubes intersect, we have

inf ≤ sup u Qi

Qi+1

and so the chain goes on for n steps and conclusion follows in classical way. 65

Two classical consequences of Harnack's inequality are particularly interesting.

Theorem 2.23. (Strong Maximum Principle for DGOp functions) Let U be a connected set, and u(x) ∈ DGOp (U ). If u has an interior minimum point, then u is constant in U . Proof. Homogeneous Caccioppoli's inequalities do not change if substituting

u + α at the place of u, so we can suppose minU u = 0. If we let E be the set of points of V in which u assumes its minimum value 0 by our hypothesis E 6= ∅ and by demonstrated continuity of u it is also a closed set, because it is the preimage of a point in a Hausdor space. If Q is a cube with center in a point of E and appropriate side, we have by Harnack's inequality for every η > 0 being still u + η > 0 a positive function η = inf (u + η) ≥ c sup(u + η) ≥ c sup u Q

Q

Q

that letting η → 0 we must have u = 0 in Q. So E is also an open set and E = U by connection gives the vanishing of u.

Theorem 2.24. (Liouville Theorem for DGOp functions) Let u ∈ DGOp (Rn ) be a function bounded from below. Then u is constant. Proof. Let α = inf u > −∞. By writing u − α instead of u we can assume

α = 0 and therefore by Strong Maximum Principle that u > 0 in Rn . From Harnack's inequality we have for every R > 0 that sup ≤ c inf u QR

Qr

If we let R → ∞ the right hand side tends to zero and therefore u is every equal to zero.

Reference and further steps The development of this small section is an adjustment of the theory presented in [44] and [18].

66

2.3 A Geometric proof by expansion of positivity Previous section about De Giorgi's Theorem for weak solutions intended as elements of Hilbert Space W 1,2 (U ) and growth conditions were weighted upon this intention. Now we enlarge our choice of a weak solution to more general Banach spaces. If we x a p > 1 and we dene a weak solution as

Denition 2.7.

A function u ∈ W 1,p (U ) is said to be a weak supersolution (resp. subsolution) of the previous equation if for each v ∈ Wo1,p (U ) it satises

Z X n

ai,j uxj vxi dx ≥ 0(resp. ≤ 0)

(2.48)

U i,j=1

With this assumption natural homogeneous growth conditions to be looked for are (P p n i,j=1 ai,j (x, y, z)zi zj ≥ m|z| (2.49) |ai,j (x, y, z)| ≤ M |z|p−1 for m, M > 0 positive constants. We also deal with more generality on the equation morphology: we abandon linearity of the equation and we suppose to work with dierential equation div(a(x, u, Du)) = 0 (2.50) for a(x, y, z) : U × R × Rn → Rn a Caratheodory vector eld 23 satisfying the homogeneous structure conditions (P p n i=1 ai (x, u, Du)uxi ≥ m|Du| |a(x, u, Du)| ≤ M |Du|p−1 a(x, u, Du) = (a1 (x, u, Du), ....., an (x, u, Du)) (2.51) In this section we present a geometrical proof of De Giorgi's Theorem for a general p > 1 based on the concept of expanding positivity24 , combining De Giorgi's and Moser's techniques with a mono-dimensional Poincaré's inequality. Supposing weak solutions of elliptic equations (2.50) with growth conditions (2.51) are locally bounded25 , we prove their Holder continuity. Firstly, to give substance to our theory, we briey introduce a popular example of equation in divergence form which satisfy growth conditions (2.51). 23 Which

is Denition 1.5 in the case Ω = U × R for each component ai of the vector eld. 24 Positivity and magnitude are synonyms: they mean the property of our solution to be greater than a certain a priori established constant 25 For the boundedness we refer to Theorem 3.24 of next chapter,which, proved in greater generality, can be easily adapted to our case by considering isotropic growth conditions.

67

Example 10. (p-Laplace operator) Dene for a function u ∈ W 1,p (U ) where U ⊂ Rn is an open bounded set the p − Laplacian as ∆p (u) := div(|Du|p−2 Du)

(2.52)

which can be rewritten with scalar convention in n  X n X [ (uxi )2 ] k=1

p−2 2

n X j=1

i=1

u xj



(2.53) xk

A weak solution u of p-Laplacian equation must satisfy for each φ ∈ Wo1,p (U ) Z

|Du|p−2 Du · Dφ dx = 0

(2.54)

U

This p-Laplace equation is the Euler-Lagrange equation associated to the request of minimizing the energy integral Z

|Dv|p dx

U

over all v ∈ W 1,p (U ) satisfying the boundary condition v − h ∈ Wo1,p (U ) for a given function h : Rn → R. This quasilinear partial dierential elliptic operator of second order has been studied deeply in literature, but our main reference for it is [57], which provides a complete overview, and [31], in which C 1,α (U ) regularity is proved by combining De Giorgi and Moser techniques. Holder continuity of its solutions is proved in what follows, just by the assumption of local boundedness of u. This operator is very useful in a number of continuum mechanics and stochastic applications. As a leading example,an historically posed problem is the non-Newtonian ltration, whose equation is ut = ∆p (u)

This can be seen as a particular case of the turbolent ltration in porus media equation  

∂ρ = cα λdiv |Dρn |p−2 Dρn ∂t where θ > 0 and constants n > 0, p > 1 must satisfy np > 1. Scaling out the constants in this equation gives ut = ∆p (un ) which for n = 1 gives back the θ

non Newtonian ltration equation. In the paper [69] numerical schemes for its solution are provided for the application in the engineering practice.

68

Secondly we state main theorem of this section.

Theorem 2.25. Let u be a locally bounded weak solution of equation n X

(ai (x, u, Du))xi = 0

i=1

where ai = ai (x, y, z) is suposed to be a Caratheodory function26 dened in U × R × Rn satisfying the homogeneous structure conditions (2.51). Then u is locally Holder continuous in U and there exist constants α, c determined just by initial data 27 with 0 < α < 1 and c > 0 such that for every open set V compactly contained in U and for every x1 , x2 ∈ V we have |u(x1 ) − u(x2 )| ≤ c||u||L∞ (V )

 |x − x | α 1 2 dist(V, ∂U )

(2.55)

The idea The main idea of the proof is the following. For a suitable choice of radius R in our domain V, take a big ball of radius 4R around a chosen point x0 . We rst nd an x1 ∈ BR (x0 ) and constant c > 0 for which positivity can be clustered in a ball centered in x1 and radius cR : in BcR (x1 ) we will prove that 1 u > µ− + w 8 for w essential oscillation of u in B4R (x0 ) and µ− its essential inmum there

in.

26 Which 27 Initial

is Denition 1.5 in the case Ω = U × R data will be always referred as U, n, p, ai,j , m, M .

69

Subsequently, once positivity has been concentrated in that small ball, we demonstrate by using a Measure Theoretical Lemma that the set of negativity points of u solution along a long thin cylinder can be reduced to have zero measure; so we demonstrate that positivity holds for this long cylinder whose length is enough to get o from BR (x0 ): more precisely the length of cylinder centered in x0 is 2R.

Finally we observe that making an orthogonal change of variables mapping our direction vector of the cylinder to the x- axis, we obtain a solution of an equation of the same type of (2.50), possibly with a dierent dierential operator but still respecting same hypothesis of growth (2.51). So we are able to establish our expansion of positivity along every cylinder centered in x1 and long 2R whose direction vector follows a rational coordinate versor: we can cover the ball BR (x0 ) with an open cover of cylinders in which holds our positivity request u > µ− + 16e1 s w. By compactness we can extract a nite number of such cylinders for which is true that u > µ− + 16e1 s w and so we have for a special s that inf{u(x) : x ∈ BR (x0 )} ≥ µ− +

70

1 w 16es

(2.56)

Setting w(r) := w(4R) = supB4R (x0 ) u(x)−inf B4R (x0 ) u(x) =: µ+ −µ− , a = by previous inequality we get w(ar) = 28 w(R) ≤ µ+ − µ− −

1 1 w = (1 − )w = bw(r) s 16e 16es

1 4

(2.57)

Thus hypothesis of underwritten Criterion of Holder Continuity are fullled, and our solution u is locally Holder continuous. Remark 23. A last word has to be spent in honor of V domain. Since now we proved Holder continuity on Balls but we may deal with a dierent domain. Let V ⊂ U be an open connected subset whose closure V¯ ⊂ U is compact, as U is bounded, and let x, y ∈ V . As V has compact closure in a complete metric space it is totally bounded i.e. for each choice of a radius  there exists a nite number n of balls of such a radius covering it. So if we take  = 21 dist(V, ∂U ), there exist n balls of radius  covering V ; as we  suppose V to be connected, we have the existence of n points {xi }ni=1 such ¯ that xi ∈ V ∩ Bi ∩ Bi+1 . Eventually reordering the indices we can suppose x ∈ B1 ∩ V¯ , y ∈ Bn ∩ V¯ . Now let [u]α,i be the Holder-seminorm of u in Bi ∩ V¯ , we have |u(x) − u(y)| ≤ |u(x) − u(x1 )| +

nX  −2

|u(xi ) − u(xi+1 )| + |u(xn −1 ) − u(y)| ≤

i=2 n X i=2

[u]α,i α ≤ max [u]α n α ≤ K|x − y|α i=1...n

(2.58)

where last inequality is due to naming constant terms (maxi=1..n [u]α n ) = K and supposing x, y do not lien in the osame ball (otherwise local continuity applies). Once the partition V¯ ∩ Bi has been chosen for a suitable i=1...n choice of radius , the constant K does not depend on the choice of the points x, y and this allows us to take the supremum on V¯ domain. Lemma 2.26. (Criterion for Holder Continuity) Let R be a suitable radius so that B¯R (x0 ) ⊂ U and for r < R let w(r) be the essential oscillation of a bounded u in Br (x0 ). If there exist a, b ∈ (0, 1) such that ∀ r < R holds w(ar) ≤ bw(r)

(2.59)

Then for α = log( ab ) we have for each r < R w(R)  r α w(r) ≤ b R 28 w(R)

= sup{u(x) : x ∈ BR (x0 )} − inf{u(x) : x ∈ BR (x0 )}

71

(2.60)

and hence u is Holder continuous in BR (x0 ) Proof. Once observed that w(r) essential oscillation is an increasing function,

statement (2.60) follows immediately from Lemma 2.10 as our hypothesis fulll its hypothesis. Then thanks to boundedness of u we have w(R)  r α ||u||L∞ (Br (x0 ))  r α w(r) ≤ ≤ (2.61) b R b R and this is (2.55) once r = |x1 − x2 | is established and R < dist(BR (x0 ), ∂U ) must be chosen in a suitable way.

The proof Let u be a locally bounded weak solution of equation n X

(2.62)

(ai (x, u, Du))xi = 0

i=1

with structure condition (2.49) and with ai = ai (x, s, z) supposed to be a Caratheodory function. Let 0 < r < R < dist(V, ∂U ) be suitable radius such ¯R (x0 ) is contained in U . Let us denote as usual by w(R), µ− , µ+ that closure B essential oscillation, inmum and supremum of u in BR (x0 ). Given constants 0 < a, H < 1 to be xed later, we dene the function i h 1 1 − (2.63) G(u) := (u − µ− + aw(R)H)p−1 (w(R)H)p−1 + which is zero when quantity in squared parenthesis is negative. Thus this function vanishes for u > µ− + (1 − a)w(R)H while it is positive a.e. in the set A := {x ∈ BR (x0 ) : u(x) ≤ µ− + (1 − a)w(R)H} (2.64) The denition of this function is properly adopted to demonstrate following Lemma.

Lemma 2.27. (Logarithmic Lemma) Let u be a weak solution of the equation in divergence form (2.50) with growth conditions (2.51). Then there exists a constant c1 depending only upon the data such that Z Br (x0 )

  D log+

 p Hw(R) dx ≤ c1 u − µ− + aw(R)H

Z

|Dφ|p dx

(2.65)

BR (x0 )

where log+ = (log(·))+ is the positive part of log function, 0 < r < R < dist(V, ∂U ) as usual and φ ∈ C 1 (U ) a nonnegative cut o function such that φ ≡ 1 in Br (x0 ) vanishing out of BR (x0 ). 72

Proof. We test our main equation (2.50) in weak solution form

with ψ := G(u)φp , obtaining as G is positive only on A set29 Z 0= ai (x, u, Du)(G(u)φp )xi =

R P U

aj (x, u, Du)ψxj dx

U

D(u)φp + (u − µ− + awH)p A  i 1 1 +pD(φ)φp−1 − dx (u − µ− + awH)p−1 (wH)p−1 Z

h ai (x, u, Du) (1 − p)

and now by using growth conditions (2.51) we obtain Z |D(u)φ|p (p − 1)m dx ≤ p A (u − µ− + awH) Z   1 dx (2.66) pM |D(u)φ|p−1 |D(φ)| (u − µ− + awH)p−1 A R R R Now we use Young's inequality γ p−1 zdx ≤ 1 γ p dx+c(1 ) z p dx to obtain from the right side member of precedent inequality

|D(u)φ|p−1 |D(φ)| dx ≤ p−1 A (u − µ− + awH) Z Z |D(u)φ|p pM {1 |Dφ|p }dx} p dx + c(1 ) A (u − µ− + awH) A Z

pM

so that as m < M our previous inequality becomes Z  p |D(u)φ| (p − 1 − 1 )m dx ≤ p−1 A (u − µ− + awH) Z Z p pM c(1 ) |Dφ| dx ≤ C |Dφ|p dx A

(2.67)

(2.68)

BR (x0 )

Now we choose 1 > 0 such that (p − 1 − 1 ) > 0 and by properties of our cut o function φ we have Z Z  p |D(u)| dx ≤ c1 |D(φ)|p dx (2.69) u − µ + awH − A∩Br (x0 ) BR (x0 29 Einstein's

summation convention is adopted.

73

and calculating  h D log

i Hw D(u) = u − µ− + awH u − µ− + awH

(2.70)

and respecting positive parts we obtain the thesis as Z p Hw(R) ) dx = D(log+ ( u − µ− + aw(R)H Br (x0 ) Z Z p  |D(u)| dx ≤ c1 |D(φ)|p dx u − µ + awH − A∩Br (x0 ) BR (x0

(2.71)

Remark 24. On the same hypothesis of previous Lemma, dening superlevel sets A(k,r) and sub-level sets B(k,r) as the intersection of the positivity set of u with cylinders Cr,l (x0 ), CR,L (x0 ) instead of balls Br (x0 ), BR (x0 ), and by using a cut-o function modied so that vanishing conditions are respected on cylinders, we obtain a similar inequality for solutions of equation (2.50) with growth conditions (2.51) Z Cr,L (x0 )

D(log+ (

Z p Hw(R) ) dx ≤ c1 |Dφ|p dx u − µ− + aw(R)H CR,L (x0 )

(2.72)

for suitable 0 < l < L, 0 < r < R. Next Lemma provide a Caccioppoli's type inequality for weak solutions of (2.50) with growth conditions (2.51).

Lemma 2.28. (Caccioppoli's type inequality) Let u be a locally bounded weak solution of (2.50) whose dierential operator satises hypothesis (2.51). Then there exists a constant c2 depending only on the data and such that Z

p

Z

p

φ |D(u − k)− | dx ≤ c2 U

|Dφ|p |(u − k)− |p dx

(2.73)

U

Proof. Let us consider the weak form of our equation (2.50):

R

a(x, u, Du) · D(φ)dx = 0 and let us test it with the function ψ = φ (u − k)− for whatever choice of a positive φ ∈ Co1 (U ). Now, since D(ψ) = (u − k)− pφp−1 D(φ) + φp D(u), we have Z Z p−1 p(u − k)− φ a(x, u, Du) · D(φ)dx + φp a(x, u, Du) · D((u − k)− )dx = 0 p

U

U

74

U

(2.74)

But then we can wisely use our growth conditions (2.51) and Young inequality30 to get Z Z p p φp−1 |D(φ)| |(u − k)− ||(u − k)− |p−1 dx ≤ m φ |D(u − k)− | dx ≤ pM U

U

Z

Z

p

p

|D(φ)|p |(u − k)− |p dx

φ |D(u − k)− | dx + C()pM

pM U

(2.75)

u

from which choosing 
0 we obtain our thesis.

Now let's recall our algebraic Lemma (2.4) from previous section that will be extremely useful in the technique presented here, which is similar to the one already seen.

Lemma 2.29. Let β > 0 and let {xi } be a sequence of real positive numbers such that xi+1 ≤ CB i x1+β i

for a C > 1 and B > 1. 1 1 If x0 ≤ C − β B − β2 , we have i

xi ≤ B − β x0 ,

thus

lim xi = 0

i→∞

Next is a De Giorgi Type Lemma, as we use the same iterative technique on sub-level sets as we did previously for super-level sets, generalizing it to our p-ellipticity context (2.51).

Lemma 2.30. (De Giorgi type) Chosen x0 ∈ U , R0 < dist(V, U ), k0 arbitrarily, there exists a number ν > 0 depending only upon the data, such that if |{x ∈ BR0 (x0 ) : u(x) ≤ ko + µ− }| < ν then |{x ∈ B R0 (x0 ) : u(x) ≤ 2

ko 2

+ µ− }| = 0 −n

2n2

Number ν can be determined quantitatively as ν = C7 p 2 p R0n wn with wn as usual the volume of the unitary ball and C7 the constant that will be shown in proof. Proof. Let C(k, R) = BR (x0 ) − A(k, R) where A(k, R) is the super level set

of De Giorgi class DGO+ i.e. C(k, R) := {x ∈ BR (x0 ) : u(x) ≤ k}. Now we introduce a sequence of radius and magnitudes 30 See

its version in precedent Lemma.

75

• Rj :=

R0 2

+

R0 2j+1

kj :=

k0 2

k0 2j+1

+

+ µ−

kj+1 = kj −

• Cj := {x ∈ BRj (x0 ) : u(x) ≤ kj }with C∞ = ∩j∈N Cj = {x ∈ BR∞ (x0 ) = BR/2 : u(x) ≤ • Yj := Y0 =

|Cj | |BRj (x0 )|

k0 2

k0 ; 2j+2

+ µ = k∞ };

with

|{x∈BR0 (x0 ):u(x)≤ko +µ− }| (R0 )n wn

and Y∞ =

|{x∈BR0 /2 (x0 ):u(x)≤ k2o +µ− }| (R0 /2)n wn

The strategy is to apply Lemma 2.20, but to do this we need to demonstrate that it is satised the relation xi+1 ≤ CB i x1+β for C > 1 and B > 1 and i β > 0. k0 and as kj+1 ≤ kj in Cj+1 Observe that in Cj+1 holds u(x) ≤ kj+1 = kj − 2j+2 we have |u − kj | > |kj+1 − kj | so that Z |u − kj |p dx ≥ |kj+1 − kj |p |Cj+1 | (2.76) Cj+1

As BRj+1 ⊂ BRj follows Cj+1 ⊂ Cj as also kj+1 < kj for each j ∈ N. Thus if we have φj ≡ 1 on Cj+1 and if we set γ = | minBR0 (x0 ) φj | we have

1 γ|Ck+1 | ≤ |kj+1 − kj |p

Z

1 |u − kj | dx ≤ |kj+1 − kj |p Cj+1 p

Z

φpj |u − kj |p dx

Cj

(2.77) We take φj ∈ CU1 a cut o function such that φj ≡ 1 in BRj+1 (x0 ) and φj ≡ 0 out of BRj (x0 ) and such that its derivative can be bounded with 4c3 2j c3 = a power of j : |D(φj )| ≤ Rj −R for a positive constant c3 . These R0 j+1 will be referred as growth estimates for φj in what follows. Now we consider np to get the embedding of Gagliardofor p < n Sobolev exponent p∗ = n−p ∗ 1,p Nirimberg-Sobolev (Teorem 1.8) for Wo (U ) ⊂ Lp (U ). To apply this, rst use Holder inequality on (2.77)

n 1 γ|Cj+1 | ≤ (kj − kj+1 )p

Z

p∗

(φj |u − kj |) dx

o pp∗

p

|Cj |1− p∗ ≤

Cj

p Z cn |Cj |1− p∗ |D(φj (u − kj ))|p dx ≤ p (kj − kj+1 ) Cj p Z Z o cn |Cj |1− p∗ n p p p p |Dφ | |u − k | dx + φ |D(u − k )| dx ≤ 31 j j j j (kj − kj+1 )p Cj Cj

31 Using

Cacioppoli's inequality Lemma.

76

p

cn |Cj |1− p∗ (kj − kj+1 )p p

Z

|Dφj |p |u − kj |p dx ≤ 32

Cj

cn |Cj |1− p∗ 2jp (kj − kj+1 )p R0p

Z

(kj − u)p dx ≤ 33

Cj p

cn 2p(j+2) |Cj |1− p∗ 2jp p k0 |Cj | k0 p R0p

(2.78)

so that taking left side of this inequality and right side we deduce p

cn |Cj |2− p∗ 22p(j+1) |Cj+1 | ≤ R0p Finally, as R20 ≤ Rj+1 ≤ R0 holds 2 − pp∗ = 1 + np we have

wn R0n 2n

≤ |BRj+1 (x0 )| ≤ wn R0n and34 as

p

Yj+1

(2.79)

p

|Cj+1 | c6 |Cj |1+ n 22p(j+1) 2n c7 |Cj |1+ n (22p )j = ≤ ≤ = p |BRj+1 (x0 )| R0p R0n (Rjn )1+ n p

= c7 (22p )j (Yj )1+ n

(2.80)

so that choosing β = np for p < n, because otherwise with p > n by Morrey's embedding we would already have Holder Continuity, we can apply algebraic Lemma 2.20 to obtain the desired thesis. On a next step, we show a Measure theoretical Lemma, which roughly speaking asserts that if the set where u is bounded away from zero occupies a sizable portion of Br (x0 ), then there exists at least one point x1 and a neighborhood Bηr (x0 ) such that u remains large in a large portion of Bηr (x0 ). Thus the set where u is positive clusters about at least one point of Br (x0 ).

Lemma 2.31. (Measure Theoretical Lemma)35 Let Br (x0 ) be a ball in Rn centered in x0 , let u ∈ W 1,1 (Br (x0 ) be a nonnegative function that for some positive γ, χ, and 0 < α < 1 satises ||u||W 1,1 (Br ) ≤ γχrn−1

|{x ∈ Br (x0 ) : u(x) ≥ χ}| ≥ α|Br |

32 Using

(2.81)

growth estimates for φj . k0 k0 ≤ k0 and (kj − kk+1 )) 2j+2 . that (kj − u) ≤ (kj − µ− ) = k20 + 2j+1 34 Where w is the measure of the unitary n-ball. n 35 For the sake of readability we give its proof at the end of the paragraph in "References and further steps". 33 Consider

77

Then for every 0 < δ, β < 1 there exists an x1 ∈ Br (x0 ) and a η = η(α, δ, γ, β, n) ∈ (0, 1) satisfying (2.82)

|{x ∈ Brη (x1 ) : u(x) ≥ βχ}| ≥ (1 − δ)|Brη |

Let us x x0 ∈ U and R > 0 such that the closure of the ball B4R (x0 ) is contained in U . Let us consider the essential oscillation in B4R (x0 )

w := w(4R) := supB4R (x0 ) u − inf B4R (x0 ) u =: µ+ − µ− . We proceed with the aim to demonstrate that the oscillation w is reduced by a xed quantity in the ball BR (x0 ). To obtain this we need the use of precedent Measure Theoretical Lemma 2.21 to give fulllment to De Giorgy Lemma hypothesis on the reduction of the rst inductive step and obtain nally the vanishing of the measure of negativity point for the limit of the sequence of sets. This will lead us to cluster the positivity of u in a small ball. We observe preliminarily that there are two possibilities:

1 1 |{x ∈ BR (x0 ) : u(x) ≥ µ− + w}| ≥ |BR | or 2 2 1 1 |{x ∈ BR (x0 ) : u(x) < µ− + w}| ≥ |BR | 2 2

(2.83)

Let us assume that the rst alternative is valid, the second one can be studied analogously. We know that u ≥ µ− : our rst goal is to nd a ball inside BR (x0 ) where u ≥ µ− + 81 w a.e.

Lemma 2.32. (Clustering positivity) If |{x ∈ BR (x0 ) : u(x) ≥ µ− + 12 w}| ≥ 21 |BR | holds, then there exists a point x1 ∈ BR (x0 ) and a constant 0 < 0 < 1 such that 1 u(x) > µ− + w 8

(2.84)

a.e. in B0 R (x1 )

Proof. We need an estimate of W 1,1 (BR (x0 )) norm of u to apply the Measure

Theoretical Lemma and to this aim we use Caccioppoli's inequality (2.73) to second step of next inequality, with appropriate φ, k . Let k = µ− and φ ∈ C01 (B2R (x0 ) a cut o function taking value 1 in BR (x0 ), vanishing outside B2R (x0 ) and such that |Dφ| ≤ R2 . We get by a wise use of Holder inequality ( p1 + p−1 = 1) p

Z |Du|dx ≤ BR (x0 )

Z

p

|Du| dx

BR (x0 )

78

 p1

|BR (x0 )|

p−1 p

≤

C

Z

p

p

|Dφ| |(u − µ− | dx

 p1

|BR (x0 )|

p−1 p

(2.85)

B2R (x0 )

Thus, since u(x) − µ− ≤ w we obtain

Z |Du|dx ≤ Cw

Z

BR (x0 )

p

|Dφ| dx

 p1

|BR (x0 )|

p−1 p

≤

B2R (x0 )

Cw |BR (x0 )| = CwRn−1 (2.86) R still by growth conditions of our cut-o function and by using the volume of n-ball. We now are able to apply Measure Theoretical Lemma 2.22 to the function (u − µ− ) with β = 12 , χ = 21 w and 0 < δ < 1 to be chosen properly to give life to De Giorgi type Lemma starting conditions. So we get the existence of a x1 ∈ BR (x0 ) and 0 < η < 1 s.t. 1 |{x ∈ BηR (x1 ) : u(x) > µ− + w}| ≥ (1 − δ)|BηR | 4

(2.87)

1 |{x ∈ BηR (x1 ) : u(x) ≤ µ− + w}| < δ|BηR | 4

(2.88)

which means

Here we can apply De Giorgi type Lemma 2.21 with R0 = ηR, and dening as in referring Lemma

• k0 = µ− + 41 w, • Rj :=

R0 2

+

R0 = ηR;

R0 2j+1

kj :=

k0 2

+

k0 2j+1

+ µ−

kj+1 = kj −

• Cj := {x ∈ BRj (x0 ) : u(x) ≤ kj }with C∞ = ∩j∈N Cj = {x ∈ BR∞ (x0 ) = BR/2 : u(x) ≤ • Yj := Y0 =

k0 2

k0 ; 2j+2

+ µ = k∞ };

|Cj | with |BRj (x0 )| |{x∈BR0 (x0 ):u(x)≤ko +µ− }| (R0 )n wn

and Y∞ =

|{x∈BR0 /2 (x0 ):u(x)≤ k2o +µ− }| (R0 /2)n wn

So, if we choose properly δ = ν , where ν is the constant needed36 by referred Lemma 2.21, we obtain that

Y0 = 36 Note

|{x ∈ BηR (x1 ) : u(x) ≤ µ− + 14 w}| |C0 | = µ− + 81 w almost everywhere in B η2 R (x1 ) which corresponds to the assertion we needed with 0 = η2 Now let us pass to the expansion on a direction. Let x1 be the central point of the positivity ball B0 R , and let us pass to a two-dimensional notation for the point: x1 = (¯ x, y¯) with x¯ ∈ R and y¯ ∈ Rn−1 . We denote by Bn−1,R (¯ y) n−1 the ball in R with center y¯ and radius R > 0 and by CR,L (x1 ) the cylinder of base Bn−1,R (¯ y ) and length L in x-direction. To be more precise:

• B(n−1,R) (¯ y ) := {y ∈ Rn−1 : |y − y¯| < R}; • CR,L (x1 ) := {(x, y) ∈ R × Rn−1 : |x − x¯| < L, y ∈ Bn−1 (¯ y )}

Remark 25. It is noteworthy that the cylinder of radius 0 R, length 3R and center x1 is still contained inside our originally xed ball B4R (x0 ).

We dene a key-set which will allow us to expand positivity of u. By using half of the radius of the positivity ball B0 R (x1 ) we dene for a parameter s to be xed later the set Dn−1,s (x1 ) ⊂ Rn−1 of y ∈ B(n−1,0 R/2) (¯ y ) for whom there exist at least an x in interval [¯ x − 5R/2, x¯ + 5R/2] giving a negativity bound. In other words Dn−1,s (x1 ) is the set of point of the base of the cylinder

80

CR, 5 R (x1 ) for which positivity is contradicted by a constant to be chosen in 2 relation to s number. n 1 o Dn−1,s (x1 ) := y ∈ Bn−1,0 R/2 (¯ y ) : ∃x ∈ [¯ x−5R/2, x¯+5R/2], u(x, y) ≤ µ− + s 8e = n 5R 1 o 5R B(n−1,0 R/2) (¯ y )− y ∈ B(n−1,0 R/2) (¯ y ) : ∀x ∈ [¯ x− , x¯+ ], u(x, y) > µ− + s 2 2 8e Now we show that we can reduce negativity in cylinder C0 R/2, 5 R (x1 ) as much 2 as we want just by choosing s big enough.

Lemma 2.33. (Reducing Negativity) For every positive constant 0 < ν0 < 1 there exist a positive real sν such that (2.91)

|Dn−1,s (x1 )| ≤ ν0 |Bn−1,0 R/2 (¯ x)|

Proof. Let y ∈ Dn−1,s (x1 ), then by denition of Dn−1,s (x1 ) there exists an

x ∈ [¯ x −5R/2, x¯ +5R/2] such that u(x, y) ≤ µ− + 8e1s or simply u(x, y)−µ− ≤ 1 . By our previous positivity clustering we know that for almost each 8es y ∈ Bn−1,0 R/2 (¯ y ) holds u(¯ x, y) − µ− > 18 w. So, as u(x, y) − µ− + 8e1s w ≤ 8e2s w, we have 1 w + 8e1s w u(¯ x, y) − µ− + 8e1s w es + 1 8 ≥ = (2.92) 2 u(x, y) − µ− + 8e1s w 2 8e1s w Subsequently as es−1 = ee < e2 + 12 = e 2+1 using the fact that logarithm37 is increasing and previous inequality we have  u(¯ x, y) − µ− + 8e1s w  38 es + 1 s − 1 ≤ log+ ( ) ≤ log+ = 2 u(x, y) − µ− + 8e1s w s

s

x ¯

x

1 w 8



− log+



1 w 8



= u(¯ x, y) − µ− + 8e1s w 1 1   Z x¯+ 25 R d  w w d 8 8 log+ dt ≤ log + 1 dt dt u(t, y) − µ− + 8es u(t, y) − µ− + x ¯− 52 R log+

Z



s

u(x, y) − µ− +

1 8es

1 8es

 dt

by Fundamental Theorem of Calculus. Now we integrate preceding inequality of left term and right one over the set Dn−1,s (x1 ), and using the inclusion Dn−1,s (x1 ) ⊂ Bn−1,0 R/2 (¯ y ) we obtain

(s − 1)|Dn−1,s (x1 )| ≤ 37 Logarithm

here is meant to be in basis e, observe also that if s > 3 then log( e 2+1 ) = e +1 log+ ( 2 ) is a positive function. 38 The number 1 w has just a stylistic meaning, as we need to recall Hw term in Loga8 rithmic Lemma. s

s

81

Z

x ¯+ 52 R

nZ

B(n−1,0 R/2) (¯ y)

x ¯− 52 R

nZ

Z

B(n−1,0 R/2) (¯ y)

 D log+

x ¯+ 52 R

x ¯− 52 R

1 w 8

u(t, y) − µ− +

1 8es

 o dt dy ≤ 39 1

 D log+

1 w 8

u(t, y) − µ− +

1 8es

 p op dtdy ·

n o p−1 p · 5R |B(n−1,0 R/2) | =

nZ (x1 ) 5 0 R/2, 2 R)

C(

 D log+

1 w 8

u(t, y) − µ− +

1 8es

 p o p1 n o p−1 p 5R |B(n−1,0 R/2) | dx

Now we make a step back to consider Logarithmic Lemma 2.17, with H = 1 , a = e−s and in particular Remark 17 as we are dealing with cylinders. 8 Imposing for the test function that |Dφ| ≤ 02R , we obtain

 Z (s − 1)|Dn−1,s (x1 )| ≤ C

p

|Dφ| dx

 p1 

 p−1 p 5R |B(n−1,0 R/2) | ≤

C0 R,3R (x1 )

 p1   p−1 2  p C|C0 R,3R (x1 )| 5R |B(n−1,0 R/2) | ≤ 0 R/2  p−1  p1  2  C p 5R |B(n−1,0 R/2) | C|B(n−1,0 R) R| ≤ |B(n−1,0 R/2) | 0 R/2 0

(2.93)

= 1. Finally the thesis comes at hand when taking s big still as p1 + p−1 p C enough to give (s−1) < ν0 . 0 Comparing Lemma of Reduction of Negativity to De Giorgy type Lemma, it is noteworthy that for each 0 < ν0 < 1 as small as requested, we can nd an s > 0 big enough to get

|Dn−1,s (x1 )| ≤ ν0 |B(n−1,0 R/2) (¯ y )|

(2.94)

This Lemma says that on C0 R/2,5R (x1 ) holds u(x) > µ− + 8e1s w for a set whose measure can managed to be as close to the one of the cylinder it is contained in as we want, letting s increase. Now we use this fact to apply De Giorgy type Lemma adapted to cylinders, to have positivity in a cylinder C0 R/4,4R (x1 ) being the limit of a sequence of cylinders, and thus to enlarge positivity in cylinder length dimension, that is enough to pass over BR (x0 ). 39 Use

for this passage Holder inequality with p1 + p−1 p = 1 with respect to both variables.

82

Remark 26. It is to be emphasized that previous Lemma on its own does not give us positivity almost everywhere in C0 R/2,5R (x1 ) because by letting s → ∞ we get the obvious inequality u(x) > µ− which can not be used directly for our purposes. Lemma 2.34. (Positivity a.e. along a Cylinder) If u is a locally bounded weak solution of (2.62) and for it holds |{x ∈ BR (x0 ) : u(x) ≥ µ− + 12 w}| ≥ 12 |BR | then in C0 R/2,4 (x1 ) we have almost everywhere u(x) ≥ µ− +

1 w 16es

(2.95)

for s chosen as in the proof that follows. Proof. Take a z ∈ [¯x−2R, x¯+2R] and consider around it the ball B0 R/2 (z, y¯) ⊂

Rn . As 0 < 0 < 1, the ball considered lies in the cylinder B0 R/2 (z, y¯) ⊂ C0 R/2,5R (x1 ) in which is valid by precedent Lemma 2.23 inequality u(x) ≤ |Dn−1,s (x1 )| ≤ ν0 . We demonstrate µ− + 8e1s w for a set of small measure |B(n−1, (¯ y )| 0 R/2) that |{(x, y) ∈ B0 R/2 (z, y¯) : u(x, y) ≤ µ− +

1 w}| ≤ ν|B0 R/2 | 8es

(2.96)

for number ν of De Giorgi type Lemma40 , used with R0 = 0 R/2 and k0 = 1 1 w in order to obtain u(x, y) > µ− + 16e ¯). s w for every (x, y) ∈ B0 R/4 (z, y 8es As the choice of z ∈ [¯ x − 2R, x¯ + 2R] was arbitrary, we will deduce from it 1 that u(x, y) > µ− + 16es w for every (x, y) ∈ C0 R/2,4 (x1 ).

To proof inequality (2.96), we observe that 40 Actually

the idea is to use De Giorgi type Lemma for each ball B0 R/2 (z, y¯) ⊂

C0 R/2,5R (x1 ) for a choice of z ∈ [¯ x − 2R, x ¯ + 2R]

83

B0 R/2 (z, y¯) ⊂ [¯ x−

0 R , x¯ 2

+

0 R ] 2

× B(n−1,0 R/2) (¯ y)

from which we get the obvious inequality between these two sets

|{(x, y) ∈ B0 R/2 (z, y¯) : u(x, y) ≤ µ− +

1 w}| ≤ 8es

0 R 0 R 1 ,z + ] × B(n−1,0 R/2) (¯ y ) : u(x, y) ≤ µ− + s w}| ≤ 41 2 2 8e 0 R 0 R 1 |{y ∈ B(n−1,0 R/2) (¯ y ) : ∃x ∈ [z − ,z + ], u(x, y) ≤ µ− + s w}|0 R ≤ 2 2 8e |Dn−1,s (x1 )|0 R ≤ ν0 |B(n−1,0 R/2) (¯ y )|0 R ≤ |{(x, y) ∈ [z −

ν

|Bn,1 | 0 R |Bn−1,0 R/2 | ≤ ν|B0 R/2 (z, y¯)| |Bn−1,1 | 2 |B

|

n,1 C just choosing an s big enough to get (s−1) ν. < ν0 ≤ 2|Bn−1,1 | 0 This starts the iteration of De Giorgi type Lemma and as previously said gives us our thesis.

Conclusion of the proof of Theorem 2.16 The result follows by fulllment of Criterion for Holder Continuity42 : we need to show that for some 0 < a, b < 1 holds w(ar) ≤ bw(r) ∀r < 4R. Let us show how found positivity carries this hypothesis. 1 By an application of Lemma 2.24 we show that u(x) > µ− + 16e s w a.e. in whatever cylinder centered in x1 ∈ BR (x0 ) point around whom has been clustered positivity (Lemma 2.23), with radius 04R and having direction 41 Left-hand 42 Lemma

set is smaller than Dn−1,s × [z − 0 R/2, z + 0 R/2]. 2.17.

84

given by every chosen versor ζ pq of Rn owing rational coordinates. Indeed if ϕis an orthogonal change of variables that maps versor ζ pq to x-axis, the functionv = u(ϕ(x)) is a locally bounded weak solution of an equation of the same type of (2.50) with growth conditions (2.51) for the new operator and dened in B4r (x0 ): orthogonality leaves unvaried the kind of growth 1 conditions. Therefore by Lemma 2.25 we have v ≥ µ− + 16e in s w a.e. 1 C0 R/4,2 (ϕ(x1 )), that means that we have u ≥ µ− + 16e w a.e. in a cylinder s centered in x1 , with length 43 2R, radius 0 R/4 and having direction ζ pq . As each cylinder is long enough to get over BR (x0 ), we have a countable open ¯R (x0 ). Being this a compact set, we can extract covering of the closed ball B from our covering a nite number of cylinders subcovering the ball BR (x0 ) and we have 1 inf u(x) ≥ µ− + w (2.97) BR (x0 ) 16es This is enough to fulll hypothesis of Holder Continuity Criterion: choose a = 41 and estimate for r = 4R,

w := w(r) = sup u − inf u =: µ+ − µ− B4R (x0 )

B4R (x0 )

 1 1  w = 1 − w(r) 16es 16es which provides inequality w(ar) ≤ bw(r) ∀r < 4R with b = (1 − the Theorem is proved. w(ar) = w(R) ≤ µ+ − µ− −

(2.98) 1 ), 16es

and

2.4 The vectorial case: a counterexample We showed in this chapter the theory concerning partial dierential elliptic isotropic equations in scalar form, and we showed with simplicity on a rst step and with more generality on a second step how regularity and boundedness can be demonstrated. The vectorial case is dierent, as no hope can be exercised in nding a general boundedness result. Indeed De Giorgi himself in [?] with a singular matrix ai,j , Giusti and Miranda [45] with even an analytic matrix ai,j , Necas [74] and many others found important counterexamples. We explain briey the simple counterexample made by De Giorgi, showing that above theorems 2.11 and 2.16 cannot be extended to case N > 1. Let us consider an energy integral whose integrand function is a positive denite quadratic form on rst derivatives of a whatever vector 43 Length

are preserved by orthogonal change of variables.

85

function u : Rn → Rn whose i-component will be denoted with a top index ui . Precisely we take

I(u) :=

Z h X n U

bh,k (x)uhxk

2

+

h,k=1

n  X

ukxh

2 i dx

h,k=1

Let us consider a measurable, real vector valued function u : Rn → Rn 1,2 belonging to the Sobolev space Wloc (U, Rn ). This function is a candidate to be a minimum of I(u) if for every vector valued test function φ ∈ Co∞ (U, Rn ) it results Z h X n n n  X  X i I(u, g) = bh,k (x)uhxk bh,k (x)gxhk + uhxk gxhk dx = 0 U

h,k=1

h,k=1

h,k=1

The example proposed is a particular case of integral I(u), i.e.

J(u) =

Z h X n U

(uhxk )2 h,k=1





+ (n − 2)

n X h=1

uhxh

n X xh xk h  2 i u dx = 0 +n 2 xk |x| h,k=1

whose Euler-Lagrange equation44 is, for h ∈ {1, .., n} and ∆2 the usual Laplacian, n n h X X xi xj i i i u + EL(u) = (n − 2) (n − 2) uxi + n 2 xj x |x| h i=1 i,j

+n

n h X xh x k  k=1

|x|2

(n − 2)

n X i=1

uixi

n X xi xj i i +n u + ∆2 uh = 0 2 xj |x| xk hi,j=1

Subsequently in order to extend the vector valued form of Proposition 1.23, De Giorgi states the following Lemma 1,2 Lemma 2.35. Let u ∈ Wloc (U, Rn ) be a vector valued function, with continuous derivatives in every point x ∈ Rn , |x| = 6 0 dierent from zero. If u satises Euler-Lagrange equation EL(u) written above, then it is a minimum of the energy integral J(u).

Proof. The proof given by De Giorgi is a simple consequence of approxima-

tion. Take g ∈ Co∞ (Rn , Rn ), α ∈ Co∞ (Rn , R) such that α = 1 in a neighborhood of the origin. Then we dene ∀t ∈ R

gtα (x) := g(x)(1 − α(tx)) 44 In

its vector-valued form, see for example [44]

86

1,2 (U, Rn ) as t → ∞ and as Then we have that gtα converges to g in Wloc α ∞ n n gt ∈ Co (R , R ) then by Euler-Lagrange equation it follows

I(u, gtα ) = 0 and our thesis comes by approximation of the energy integral. Another approach to this proof is due to the theory developed in Preliminaries, and the pacic realization that a such kind of function u stands in W 1,2 (U, Rn ) and it is a minimum of integral J(u) i it satises the EulerLagrange equation, thanks to the fact that J(u) functional is convex. The counter example found by De Giorgi is exactly the function

u(x) :=

x |x|α

which is a member of W 1,2 (Rn , Rn ) and minimizes J(u) for every α verifying the equation  n 2 (2n − 2)2 α + + (αn + α2 ) = 0 2 as it satises its Euler-Lagrange equation also if it is not continuous in x = 0. Indeed we have

• ∆2 (xh |x|α ) = (αn + α2 )xh |x|α−2 ;  Pn  α = (n + α)|x|α ; x |x| • i i=1 xi

xi xj i,j=1 |x|α

•

Pn

•



|x|α

 xh

•

Pn

k=1





xi |x|α

 xj

= (α + 1)|x|α ;

= αxh |x|α−2 ;

xh xk |x|α−1

So if we chose α :=

 xk n 2



= (n + α − 1)xh |x|α−2

1− √

1 (2n−2)2 +1



the dened function u minimizes

the integral J(u) but it is discontinuous in x = 0 for every n ≥ 5.

87

References and further steps The geometric proof presented here is entirely based on a geometrical revision of [37], whose passages are here presented with illustrations and wide explanation of details. Particular attention has been given to the proof of Measure theoretical Lemma which can be found in [21], [17], and its extension to BV functions [83]. The important dierence with De Giorgi's proof is due to this lemma and the logarithmic one, which imply clustering of positivity and the other properties which allow us to reduce negativity . This is the key to Apply De Giorgi Type Lemma to every appropriate ball in the cilinder and get positivity everywhere in it. Space expansion technique is extremely useful for demonstrating Holder continuity of solutions once we know solutions are bounded: we showed this simple fact in (2.98), which can be found also in [36], whose strategy consists just on demonstrating the space expansion of positivity. This goes hand in hand with the enormous importance that Harnack's inequalities have in demonstrating Holder continuity: Moser's method indeed consists in nding previously a Harnack's inequality for solutions and then to apply it in order to get continuity. A further step would be to apply this method to anisotropic equations,in order to obtain holder regularity. In [36] is presented an application of this method in the particular anisotropic case where anisotropy is of the more studied type, with rst (n − 1) variables of the Lagrangian obeying to a p-condition and last one to a q -condition.

Appendix We give here for completeness sake the proof of Measure Theoretical Lemma, in the more general context of bounded variation functions, then by membership of W 1,1 (U ) functions to BV (U ) the proof will be concluded. Simple properties of BV (U ) functions can be found in [59] and a wider explanation is conducted in [88]. For ρ > 0, denote by Kρ (y) ⊂ RN a cube of edge ρ centered in y.

Let u ∈ BV (Kρ ) satisfy kukBV (Kρ ) ≤ γρN −1

and

|[u > 1]| ≥ α|Kρ |

(2.99)

for some γ > 0 and α ∈ (0, 1). Then, for every δ ∈ (0, 1) and 0 < λ < 1 there exist xo ∈ Kρ and η = η(α, δ, γ, λ, N ) ∈ (0, 1), such that |[u > λ] ∩ Kηρ (xo )| > (1 − δ)|Kηρ (xo )|.

88

(2.100)

Partition of the cube It suces to establish the Lemma for u continuous and ρ = 1. For n ∈ N partition K1 into nN cubes, with pairwise disjoint interior and each of edge 1/n. Divide these cubes into two nite subcollections Q+ and Q− by

Qj ∈ Q+

⇐⇒

Qi ∈ Q−

⇐⇒

α |Qj | 2 α |[u > 1] ∩ Qi | ≤ |Qi | 2 |[u > 1] ∩ Qj | >

and denote by #(Q+ ) the number of cubes in Q+ . By the assumption X X |[u > 1] ∩ Qj | + |[u > 1] ∩ Qi | > α|K1 | = αnN |Q| Qj ∈Q+

Qi ∈Q−

where |Q| is the common measure of the Ql .

From the denitions of the classes Q± ,

αnN
1] ∩ Qj | X |[u > 1] ∩ Qi | α + < #(Q+ )+ (nN −#(Q+ )). |Qj | |Qi | 2 − +

Qj ∈Q

Therefore

Qi ∈Q

#(Q+ ) >

α nN . 2−α

¯ + of Q+ . A cube Qj belongs to Q ¯ + if Consider now a subcollection Q 2α Qj ∈ Q+ and kukBV (Qj ) ≤ kukBV (K1 ) . (2 − α)nN Clearly α ¯ +) > #(Q nN . (2.101) 2(2 − α) Fix δ, λ ∈ (0, 1). The idea of the proof is that an alternative occurs. ¯ + such that there is a subcube Q ˜ ⊂ Qj where Either there is a cube Qj ∈ Q

˜ ≥ (1 − δ)|Q| ˜ |[u > λ] ∩ Q|

(2.102)

¯ + there exists a constant c = c(α, δ, γ, η, N ) such that or for any cube Qj ∈ Q kukBV (Qj ) ≥ c(α, δ, γ, λ, N ) 89

1 nN −1

.

(2.103)

¯ + , we can add (2.103) Hence if (2.102) does not hold for any cube Qj ∈ Q over all such Qj . Therefore taking into account (2.101), we have α c(α, δ, γ, N )n ≤ kukBV (K1 ) ≤ γ. 2−α and for n large enough this fact leads to an evident absurdum.

Proof of the Measure Theoretical Lemma when N = 2 ¯ + . Without loss of generality we may assume (xo , yo ) = Let K 1 (xo , yo ) ∈ Q n (0, 0). Assume that α (2.104) [u > 1] ∩ K n1 > |K n1 | 2 2α kukBV (K 1 ) ≤ kukBV (K1 ) . (2.105) n (2 − α)n2 1 1 , ) let Y(x) 2n 2n the cross section of the set [u > 1] ∩ K 1 with lines parallel to y -axis, through n the abscissa x, i.e. Denote by (x, y) the coordinates of R2 and, for x ∈ (−

Y(x) ≡ {y ∈ (−

1 1 , ) such that u(x, y) > 1}. 2n 2n

Therefore

Z |[u > 1] ∩ K 1 | ≡ n

1 n 2

|Y(x)|dx.

1 − 2n

α Since, by (2.104), |[u > 1] ∩ K 1 | > |K 1 |, n 2 n 1 1 there exists some x ˜ ∈ (− , ) such that 2n 2n |Y(˜ x)| ≥

α . 4n

(2.106)

Dene

Ax˜ ≡ {y ∈ Y(˜ x) such that ∃x ∈ (−

1 1 (1 + λ) , ) such that u(x, y) ≤ }. 2n 2n 2

Note that for any y ∈ Ax˜ the variation along the x direction is at least (1 − λ) α . If |Ax˜ | ≥ , we have that the BV norm of u in K 1 is at least n 2 8n α(1 − λ) and therefore (2.103) holds. 16n 90

α , we have that there exists at least a y˜ ∈ Y(˜ x) such that 8n (1 + λ) 1 1 u(x,y˜) ≥ for any x ∈ (− , ). 2 2n 2n Dene If |Ax˜ | ≤

Ay˜ ≡ {x ∈ (−

1 1 1 1 , ) such that ∃y ∈ (− , ) such that u(x, y) ≤ λ}. 2n 2n 2n 2n

Note that for any x ∈ Ay˜ the variation along the y direction is at least (1 − λ) . 2 δ(1 − λ) δ we have that the BV norm of u in K 1 is at least If |Ay˜| ≥ n n 2n and therefore (2.103) holds. δ we have that |[u > λ] ∩ K 1 | ≥ (1 − δ)|K 1 | and therefore If |Ay˜| ≤ n n n (2.102) holds. Summarising either (2.102) or (2.103) hold. Therefore the alternative occurs and the case N = 2 is proved.

Proof of the Measure Theoretical Lemma when N > 2 Assume that Lemma 1.1 is proved in the case N = m and let us prove it when N = m + 1. Let z a point of Rm+1 . To make to notation easier, write z = (x, y) where x ∈ R and y ∈ Rm . ¯ + . Without loss of generality we may assume z = (0, 0). Let K 1 (z) ∈ Q n Assume that α (2.107) [u > 1] ∩ K n1 > |K n1 | 2 2α kukBV (K 1 ) ≤ kukBV (K1 ) . (2.108) n (2 − α)nm+1

1 1 , ) consider the m -dimensional cube centered in 2n 2n (x, 0), orthogonal to the x−axis and with edge n1 and denote this cube with ¯ 1 (x). Dene A¯ as the set of the x ∈ (− 1 , 1 ) such that K n 2n 2n α ¯ 1 (x) > |K ¯ 1 (x)| [u > 1] ∩ K n 4 n For any x ∈ (−

and

kukBV (K¯ 1 (x)) ≤ n

16 kukBV (K1 ) . (2 − α)nm 91

It is possible to prove that

α . 8n ¯ 1 (¯ ¯ 1 (¯ Let x ¯ ∈ A¯ and apply Lemma 1.1 to K x) (we can do so because K x) n n is a m-dimensional set). ¯ 1 (¯ So we get the existence of a constant η0 > 0 and a point yo ∈ K x) such n that if we dene the set ¯ ≥ |A|

(1 + λ) } 2 η0 ¯ η0 (¯ where K , centered in x , y 0 ) denotes the m−dimensional cube of edge n n (¯ x, y0 ) and orthogonal to the x−axis, we have ¯ η0 (¯ A ≡ {(¯ x, y) ∈ K x, y0 ) such that u(¯ x, y) ≥ n

δ η0 |A| ≥ (1 − )( )m . 2 n

(2.109)

Dene

B ≡ {y ∈ A such that ∃x ∈ (−

1 1 , ) such that u(x, y) ≤ λ}. 2n 2n

(1 − λ) Note that for any y ∈ B the variation along the x direction is at least . 2 δ η0 If |B| ≥ ( )m , we have that the BV norm of u in K 1 is at least n 2 n δ(1 − λ) η0 m ( ) and therefore (2.103) holds. 4 n δ η0 If |B| ≥ ( )m , taking in account (2.109) we have that in the cylinder 2 n 1 1 ¯ η0 (0, y0 ) the measure of the set where u(x, y) ≥ λ is greater (− , ) × K n 2n 2n ηom than (1 − δ) m+1 . Therefore (2.102) holds in a suitable subcube of K 1 . n n Summarising either (2.102) or (2.103) hold. Therefore the alternative occurs and the case N > 2 is proved.

Such kind of result has natural application in regularity theory for solutions to PDE's, a good reference for these applications is [19]. As already said before for the Moser approach developed, this purely theoretical method is useful in the case of anisotropic operators where it would be necessary a development of a new approach tailored on the structure of the operator, as has already been done in this geometric proof of holder continuity of solutions. Next chapter is devoted to this rapid developing subject. 92

3

Regularity on anisotropic growth conditions

3.1 Introduction and counterexamples In previous chapter we dealt with energy integrals of the form45 Z I[w] = L(x, w(x), Dw(x))dx U

¯ × R × Rn → R the being w : U ⊂ Rn → R a function, L(x, y, z) as L : U n Lagrangian, for z ∈ R , y ∈ R, x ∈ U . Supposing L coercive L(x, y, z) ≥ ν|z|p − M

ν, M > 0, 1 < p < n;

(3.1)

with growth conditions

|L(x, y, z)| ≤ K(|y|p + |z|p + 1)

(3.2)

( |Dy L(x, y, z)| ≤ K(|y|p−1 + |z|p−1 + 1) |Dz L(x, y, z)| ≤ K(|y|p−1 + |z|p−1 + 1). whose Euler-Lagrange equation was n X

(Lzi (x, w, Dw))xi + Ly (x, w, Dw) = 0

i=1

Via this equation the variational problems can be transformed into partial dierential equations whose weak solutions u ∈ W 1,p (U ) must satisfy Z X n (Lzi (x, w, Dw))φxi + Ly (x, w, Dw)φ dx = 0 ∀φ ∈ Wo1,p (U ) U i=1

or

Z (X n U

) ai (x, w, Dw)φxi + b(x, w, Dw)φ dx = 0,

∀φ ∈ Wo1,p (U )

(3.3)

i=1

when calling Lzi (x, w, Dw) =: ai (x, w, Dw),Ly (x, w, Dw) =: b(x, w, Dw).This has brought us to consider PDEs in divergence form as

div(a(x, w, Dw)) = b(x, w, Dw) We have seen that exists a satisfying theory for such operators L, called by virtue of growth conditions (3.1),(3.2) isotropic. 45 Without

risk of misunderstanding in next denition p will stand for the minimum growth, but in this context is the common elliptic type growth.

93

Nevertheless now observe that none of the integrands below

L1 (v, U ) :=

Z X n

ai (x)|Di v|pi dx

U i=1

1 ≤ ai (x) ≤ M, Z L2 (v, U ) :=

(3.4)

1 < p := p1 ≤ p2 ≤ ... ≤ pn =: q

|Dv|p + a(x)|Dv|q dx,

0 ≤ a(x) ≤ M

(3.5)

1 < p ≤ p(x) ≤ q

(3.6)

U

Z L3 (v, U ) ≡ Dp(x) (v) :=

|Dv|p(x) dx,

U

satises at the same time coercitivity and growth conditions (3.1),(3.2) above. All these integrands, which are of relevance in mathematical physics46 , satisfy a more general property.

Denition 3.1. We say that the operator L(x, z) : U × Rn → R satises (p,q)-growth conditions if for chosen 1 < p ≤ q and suitable positive ν, M we have

ν|z|p − M ≤ L(x, z) ≤ M (1 + |z|q ),

1 1 every minimizer of the integral Z

I(u) :=

L(Du(x))dx

(3.8)

U

is locally Holder continuous in U . A priori, there would not be anything preventing us by inferring that regularity and boundedness results proved in last chapter for the isotropic case occur also in anisotropic growth conditions. Unfortunately, in two celebrated papers Mariano Giaquinta in [38] and Paolo Marcellini in [62] answered armatively to the question whether isotropic growth conditions would have been necessary for the Holder continuity and the boundedness of the solutions.

Counterexample 2. (Marcellini) Marcellini was the rst to initiate a systematic study of such anisotropic problems in a series of seminal papers [62],[63],[64],[67]. n−1 ) > 2 and consider the functional Let n > 3, q > 2( n−3 Z n X n−1 o 1 1 I(u) = (uxi )2 + |uxn |q dx 2 i=1 q U

(3.9)

to simplify notations, we consider an open bounded set U ⊂ Rn , included in the half-space {x ∈ Rn : xn > 0}. The local minimum found by Marcellini is given by 1

(xn )q  q−2 u(x) = c Pn−1 2 i=1 (xi ) 

(3.10)

2 with c = ( n−1 − q−2 )( q−2 )q−1 > 0 thanks to our choice of q . q−1 q We observe that the function u is unbounded in a neighborhood of the line Pn−1 1 2 {xi = 0, i = 1, ..., n − 1}, that the Lagrangian L(z) = 2 i=1 (xi ) + 1q |xn |q as shown in previous counterexample is convex, and that it is of class48 C [q] (Rn ), being a polynomial Pn−1 2 if q is even. By calculations below we see that in the set n {x ∈ R : i=1 xi 6= 0, xn > 0} our function u(x) is a classic solution of Euler-Lagrange equation, n−1 X

(uxi )xi + (|uxn |q−2 uxn )xn = 0

i=1

Indeed, 48 [q]

denotes the integer part of q .

95

(3.11)

•

Pn−1 i=1

(uxi )xi = q−1

n−1 o X − q−2 2(c)1/(q−2) n q − 1 q/(q−2) 2 = 2 − (n − 1) (xn ) xi q−2 q−2 i=1 q−1

1/(q−2)

2(c) q−2

n−1 X − q−2 2 o q/(q−2) 2 (q − 1) − − (xn ) xi q−1 q−2 i=1

n

n − 1

h i • (|xn |q−2 uxn )xn = Dxn (uxn )q−1 = q−1

n−1 X − q−2 2(c) (q − 1)  q q−1 q/(q−2) 2 (xn ) xi q−2 q−2 i=1 q−1 q−2

and they are opposite quantities as



q−1 q−2



1 = 1 + ( q−2 ), and ( n−1 − q−1

2 ) q−2

=

q q−1 ) c( q−2 1 h 2(c) q−2

q−1

n − 1

2  2(c) q−2  q q−1 i − − + =0 q−2 q−1 q−2 q−2 q−2

n−1 Condition q > 2( n−3 ) ensures us that u ∈ H 1,2 (U ) and that uxn ∈ Lq (U ), as 2 there exist a constant C limiting xn 's increase such that |u(x)| ≤ C(|x|n−1 )− q−2 , q 2 |uxi (x)| ≤ C(|x|n−1 )− q−2 , |uxn (x)| ≤ C(|x|n−1 )− q−2 .

To prove that u(x) is a weak solution of Euler-Lagrange equation, Marcellini adapts a method originally set by De Giorgi (see Lemma 2.35). For a given parameter t > 0, take a cut-o function gt : R → [0, 1] being zero out of the ball of radius 2t centered in zero and taking value 1 inside half of this ball, i.e. in [−t, t]; we also set the request 49 |g0 t (s)| ≤ 2t , ∀s ∈ R. Now, to see that u(x) is a weak solution of Euler Lagrange equation associated to our Lagrangian L in whole set U , let φ ∈ Co1 (U ). We can associate to φ the function φt (x) := φ(x)(1 − g(|x|n−1 )),

n−1 X |x|n−1 =: ( |xi |2 )1/2

(3.12)

i=1 49 We

saw in Proposition 1.5 how to construct the cut-o function we are invokating.

96

and observe that φt ∈ Co1 (U ) and that φt = 0 in a neighborhood of the line x1 = x2 = ... = xn = 0 while it perfectly approximates φ: limt→0 ||φt − φ||Ho1,2 (U ) = 0,

limt→0 ||(φt )xn − φxn ||Lr (U ) = 0,

∀r ∈ [1, ∞)

Since φt ∈ Co1 (U ) and ad u is a classic solution of Euler-Lagrange equation for each x ∈ V := U − {xi = 0, i = 1, .., n}, we have Z X n−1

Z

|uxn |(q−2) uxn (φt )xn dx = 0

uxi (φt )xi dx +

V i=1

(3.13)

V

which can be extended to U as φt has compact support in V . Now if we pass to the limit for t → 0 we obtain by approximation the same integral equation with φ function, as desired. Finally, as L is a convex function then our weak solution u by Proposition 1.23 also a minimizer for the energy I(u).

Remark 28. (Degeneracy) It is an interesting point that this example is concerned with degenerate inapproaches zero, the Euler-Lagrange equation tegrals: for instance, when uxnP n−1 (uxi )xi + (|uxn |q−2 uxn )xn = 0 becomes deassociate to the LagrangianL , i=1 generate elliptic, losing ellipticity in the xn -direction. This context will be taken under analysis in next section 3.3 (Additional Structure) where we will study anisotropy intended as dierent growths in each direction. Counterexample 3. (Giaquinta) The example of Giaquinta is given by the functional I(u) =

Z X n−1 U

whose integrand L(Du) = sian

i=1

 1 |uxi |2 + |uxn |4 dx 2

Pn−1

|uxi |2 +  2 0 0 2  HL (Du) =  0 0 0 ... 0 ... i=1

1 |u |4 2 xn

... 0 2 0 0

(3.14)

is strictly convex as its Hes-

 0 0 .. 0   .. 0   2 0  0 6u2xn

is clearly positive denite where the equation is not degenerate and satisfying the (3.7)-growth conditions c0 (|z|2 − 1) ≤ L(z) ≤ c2 (|z|4 + 1) 97

as actually one can see by n−1

X 2 1 1 c0 (|z| − 1) ≤ |z| − (zn ) + |zn |4 = |zi | + |zn |4 ≤ c2 (|z|4 + 1) . 2 2 i=1 2

2

2

The rst inequality being given by negativity of the polynomial −P21 zn4 +zn2 −1 < 2 0, and the second inequality is easily recognizable setting x = n−1 i=1 |zi | and using both inequalities z 2 − z + 1 > 0 and z 2 > 21 |zn |4 . Moreover one easily sees that for n ≥ 6 the function √ u(x) :=

n−4 x2n 24 (Pn−1 u2 )1/2 i=1 xi

(3.15)

makes the above functional (integral) nite50 and it is a solution of EulerLagrange equation associated to this functional. We show this just by showing the minimizing eect of u: thanks to Proposition 1.19 we will know that such a minimizer is obliged to solve Euler-Lagrange equation. For every φ ∈ W 1,2 (U ) such that supp(φ) ⊂⊂ U , Z hX n−1  U

uxi

2

i=1

+

1  4 i dx ≤ 51 u xn 2

√ Z hX 2 n−1  √ 1  n − 4 2xn 4 i n − 4 x2n xi  + dx = 24 |x|2n−1 2 24 |x|n−1 U i=1 Z X n−1 U

≤

xi

2

i=1

Z  U

x4n i (n − 4)2 x4n + dx ≤ 122 · 4 |x|2n−1 124 · 2 |x|2n−1

h n − 4

uxn

n−4i + 2 dx ≤ 4 12 · 2

4 h 1

Z hX n−1 U

i=1

Z U

(n − 4)x4n h 1 n − 4 i + dx 122 · |x|2n−1 4 122 · 2

(uxi + φxi ) )2 +

4 i 1 uxn + φxn dx 2

This solution u(x)is clearly unbounded in each xi -hyperplane passing by 0 by the rst (n − 1) coordinates. As counterexamples demonstrate, we cannot expect energy functionals I(u) with general (p, q)-growth conditions to have solutions sharing same 50 Just

by calculating derivatives and seeing that lim|x|→0 L(Du) = k gives a constant, and remembering we are integrating on a bounded domain U . Pn−1 2 1/2 51 We denote by |z| . n−1 =: ( i=1 |zi | )

98

properties of regularity and boundedness as isotropic energy functionals do. This eld of research is still largely open, and a lot of work has been done in supposing certain specic properties the gap of the ratio q/p > 1, or working directly with the model problem of some Lagrangian, as we will see shortly. Our attempt here is to draw a coherent picture of the state-of-art of this rapidly developing eld. As is usual in pure mathematics but very unusual in research papers of this subject, we shall prefer to present the subject with the more generality available and then to show gradually increasing the hypothesis main results and some applications. The literature concerning this domain is very vast and there is no hope to cluster all the progresses made in every direction in these few pages: for this and other motivations we apologize for all that ne material which will not nd its room here, together with missed quotations of important contributions. The study of regularity of minima of functionals with non standard growth conditions of (p,q) -type was initiated by Marcellini [64], [67], who rst identied a condition that, under enough regularity of the Lagrangian L, ensures the regularity of minima. Under suitable smoothness and convexity of the Lagrangian L we are able to shift by the problem of regularity of minimizers of calculus of variations to regularity of solutions of elliptic equations. Indeed this possibility of shifting from variational approach to PDEs, presented52 in Theorem 1.22 and Proposition 1.23, need not to worry about the left term of growth conditions (3.6) as they use convexity of (y, z) → L(x, y, z) for each x, and upper bounds on derivatives of the Lagrangian (1.6), (1.7). We will see in section 3.3 that there is a proper setting for (weak ) solutions to Euler-Lagrange equation also when anisotropy is spread in each direction.

3.2 Dierent growths from above and below In this section we make the following assumptions on the Lagrangian, being HL the Hessian matrix of L(x, y, z):

C 2 (Rn ) regularity ;

• (z) → L(z) ∈ C 2 (Rn ) • ν|z|p ≤ L(z) ≤ M (1 + |z|q ), • ν(1 + |z|2 )

(p−2) 2

1 2 and for p > 2 assume that n q < p n−2

(3.17)

n Then the gradient of u is locally bounded, i.e. Du ∈ L∞ loc (U, R ).

This is a very important point for regularity of non degenerate functionals with (p,q) -growth conditions. This because once we have got the local boundedness of the gradient, we may rewrite locally the (p, q)-growth condition ν|z|p ≤ L(z) ≤ M (1 + |z|q ) as an isotropic (p, p)-growth condition ν|z|p ≤ L(z) ≤ M (1 + |k|q ) = C p and apply the theory previously developed. 1,q (U ), while theory of direct It has to be noted also that it is required u ∈ Wloc methods developed in Preliminaries56 ensure us only to have an a priori min1,p imizer in Wloc (U ). To get rid of this integrability gap, if we require a more strict dependence of the ratio pq , we have the desired membership of u to the ideal Sobolev Space to work with. 55 Although

we enunciate here them in stronger hypothesis than in the available versions, in the following discussion we will give the more general version. 56 We refer to Theorem 1.4.

100

1,p (U ) be a local minimizer of the functional I(v), Theorem 3.2. Let u ∈ Wloc under the previous assumptions of smoothness of L, (p, q)-growth and convexity. Assume moreover that for p > 2

q (n + 2) < p n

(3.18)

1,q (U ). Then u ∈ Wloc

Next pages are devoted to the proof of these results and their improvements, the motivation of this being the presentation of the techniques adopted, that we believe are of a certain technical thickness.

Proof of theorem 3.1 Here we give a more general proof, supposing ai = ai (x, Du) has dependence also in x, allowing it to be just a locally Lipschitz continuous function in the whole domain U × Rn and having the following assumptions replacing the continuity of derivatives

|aizj (x, z) − ajzi (x, z)| ≤ M (1 + |z|2 ) |aixk (x, z)| ≤ M (1 + |z|2 )

p+q−4 4

p+q−2 4

∀i, j ∈ {1, 2, ...n}

(3.19) (3.20)

∀i, k ∈ {1, 2, ...n}

The rst condition's aim is to replace with more generality Schwartz theorem, and with it the variational condition aizj = ajzi and allowing us to take a non continuously dierentiable ai (x, Du). The second hypothesis replaces our supposed non-dependence of ai by x-variable, allowing us to write ai (x, Du) but with a bound to the growth of the x-derivative depending on the growth of Du. Precisely we proof the following more general theorem. 1,q Theorem 3.3. Let u ∈ Wloc (U ) be a weak solution of equation n  X

 ai (x, Du) =b

(3.21)

xi

i=1

and let us suppose b ∈ L∞ loc (U ), (p, q)-growth conditions 2

ν(1 + |z| )

(p−2) 2

2

|λ| ≤

n X

(ai (x, z))zj λi λj ≤ M (1 + |z|2 )

(q−2) 2

|λ|2

(3.22)

i,j=1

together with conditions (3.19), (3.20) of growth on x-derivative and on uniformity of symmetric derivatives of ai (x, Du). 101

If 2 < p ≤ q < and for θ =

2q np−(n−2)q

n p n−2

1,∞ (U ), then u ∈ Wloc

there exist c, β > 0 independent of u such that 2 1/2

sup (1 + |Du| )

 ≤C

x∈Br

1 ||(1 + |Du|2 ||Lq (BR ) β (R − r)

θ

(3.23)

for every suitable r, R such that 0 < r < R ≤ r + 1.

Next Lemma allows us to estimate growth conditions (3.19) and (3.20) with quadratic z -growth conditions.

Lemma 3.4. Under assumptions (3.19),(3.20) and for constants ν, M 2

ν(1 + |z| )

(p−2) 2

2

|λ| ≤

n X

(ai (x, z))zj λi λj ≤ M (1 + |z|2 )

(q−2) 2

|λ|2

(3.24)

i,j=1

we have, for every λ, η ∈ Rn , P 1/2 P i (x, z)λ λ (1 + |z|2 )(q−2)/4 |η|, a • i,j aizj (x, z)λj ηi ≤ C i j i,j zj √  for C = M + 2√Mm ; 1/2 P P i pn i (1+|z|2 )q/4 , with C = M m • i,j azj (x, z)λi λj i axk (x, z)λi ≤ C

Proof. By using right inequality of hypothesis (3.24) it is easily shown that |aizj (x, z)| ≤ M (1 + |z|)(q−2)/2

(3.25)

Now dene bi,j (x) = 12 (aizj + ajzi ), ci,j (x) = 21 (aizj − ajzi ) and observe that aizj = bi,j inequality and then (3.24) to P+ ci,j . Applying rstly Cauchy-Schwartz P | i,j bi,j λj ηi | and applying (3.20) to | i,j ci,j λi ηj | and then dominating with left hand side of hypothesis (3.24) we obtain easily the desired estimate. Next, following the same method of Theorem 1.24 of Preliminaries, we demonstrate that u ∈ H 2 (V ) for each V ⊂⊂ U by showing that the dierence57 quotients Dh (D(u)) are bounded in L2 (V ) norm. But this strategy will give us much more: by this calculations we will be able to extract some precious inequalities bounding (1 + |Du|2 ) that will be of extreme relevance 57 Notation

is referred to Denition 1.13 of Preliminaries.

102

for its L∞ estimate. The rst steps are very similar: let V ⊂⊂ U ,h > 0 suciently small, set for a predenite direction ei for simplicity58

Dh u(x) := Dsh u(x) =

u(x + hes ) − u(x) h

(3.26)

and test the weak form of equation (3.21) i. e. equation (3.3) with (3.27)

φ = D−h (η 2 gα,k (Dh u))

where η ∈ Co1 (V ) and gα,k ∈ C 1 (R) a properly constructed function of a real variable, dened for |t| ≤ k by (3.28)

gα,k (t) := t(1 + t2 )(α−2)/2

and for t > k extended linearly to be a continuously dierentiable function. It has been built in [64] to have the following properties.

Proposition 3.5. (Properties of gα,k function) Let gα,k : R → R the function previously dened, and let Gα,k (t) := Then, Gα,k has the following properties ∀α ≥ 2, k > 0 • |gα,k (t)| ≤ (1 + t2 )(α−1)/2 ;

2 (t) gα,k 0

gα,k (t)

.

• gα,k (t) ≤ (α − 1)(1 + t2 )(α−2)/2 ; 0

• limk→∞ gα,k (t) ≥ (1 + t2 )(α−2)/2 ; 0

• there exists a constant cα,k s.t. Gα,k (t) ≤ cα,k (1 + t2 ) , ∀t ∈ R;  2 (α−2)/2 2 α/2 • Gα,k (t) ≤ (1 + t ) · 2 1+k k2 By using φ as a test function in ) Z (X n ai (x, u, Du)φxi + b(x, uDu)φ dx = 0, U

∀φ ∈ Wo1,b (U )

(3.29)

i=1

we obtain, by property (1.20) of dierence quotients59 Z X Z n  0  h i 2 h D a (x, Du) η gα,k D uxi + 2ηηxi gα,k dx = b(x)D−h (η 2 gα,k )dx U i=1

U

(3.30)

58 While

dierently in Preliminaries Dh u meant the vector of components Dih u, we prefer here for easiness of notation to omit the i component. 0 59 Denoting with g h h α,k (D u) = gα,k and gα,k (D u) = gα,k .

103

As in Theorem 1.24 , using equality

Du + thDh (Du) = (1 − t)Du(x) + tDu(x + hes )

(3.31)

we compute

1 D a (x, Du) = h h i

1 h

1

Z 0

1

Z

d i a (x + thes , (1 − t)Du(x) + tDu(x + hes ))dt (3.32) 0 dt ! Z 1 n X d i a (x + thes , Du + thDh (Du))dt = aixs + aizj Dh uxj dt dt 0 j=1 (3.33)

So we obtain that 1

Z Z U

0

η 2 gα,k

A := 0

1 2

− 1

2ηgα,k U

0

η gα,k U

−

aizj Dh (uxi )Dh (uxj )dtdx =

i,j

Z Z

Z Z

X

0

n X

0

i

n X

aixs

+

+

n X

! aizj Dh (uxj )

ηxi dtdx

j=1

i=1

Z

aixs Dh (uxi )dtdx

b(x)D−h (η 2 gα,k )dx := B + D + E

(3.34)

U

We estimate separately each term A, B, D, E in previous equation. Let us start from A, that we estimate using ellipticity assumption (3.22), precisely the left-hand inequality: Z Z 1 X 0 A := η 2 gα,k aizj Dh (uxi )Dh (uxj )dtdx ≥ U

Z Z m U

1

0

i,j

0

η 2 gα,k (1 + |Du + thDh (u)|2 )(p−2)/2 |Dh (Du)|2 dtdx

(3.35)

0

Next we evaluate B by use of second inequality of Lemma 3.4 and Young's 1 2 inequality |ab| ≤ a2 + 4 b,

Z Z |B| = U

0

1

0

η 2 gα,k

n X i

104

aixs Dh (uxi )dtdx ≤

1

Z Z

n X

0

2

η gα,k

C 0

U

!1/2 aizj Dh (uxi )Dh (uxi )

(1 + |Du + thDh (Du)|2 )q/4 dtdx

i,j 1

n Z Z ≤ C U

Z Z

C 4

+

U

n X

0

η 2 gα,k

0 1

! aizj Dh (uxi )Dh (uxj ) dtdx

i,j

o 0 η 2 gα,k (1 + |Du + thDh (Du)|2 )q/2 dtdx

(3.36)

0

Next we evaluate n X

1

Z Z

2ηgα,k

D= 0

U

aixs +

i=1

n X

! aizj Dh (uxj ) ηxi dtdx = D1 + D2

j=1

where for evaluating D1 we use assumption (3.20) on the growth of ai (x, z) x-derivative60 and the fact that (p−1)+(q−1) ≤ (q−1)+(q−1) = (q−1) and that 4 4 2 0 1/2 gα,k = (Gα,k gα,k )

Z Z n 1 X |D1 | = 2ηgα,k aixs ηxi dtdx ≤ U 0 i=1

1

Z Z

2η|Dη||gα,k |(1 + |Du + thDh (Du)|2 )(p−q+2)/4 dtdx ≤

nM U

0

Z Z

1

2η|Dη||gα,k |(1 + |Du + thDh (Du)|2 )(q−1)/2 dtdx

nM U

(3.37)

0

and in evaluating D2 we use the rst inequality of Lemma 3.4 and denition of Gα,k (t) in Proposition 3.5 to get Z Z n 1 X i h |D2 | = 2ηgα,k azj D (uxj )ηxi dtdx ≤ U 0 i,j=1

Z Z

1

2ηgα,k U

0

! 12

n X

aizj Dh (uxi )Dh (uxj )

i,j=1

Z Z ≤C U 60 Remember

(1+|Du+thDh (Du)|)

0

1

n

0

η 2 gα,k

n X

! 21 aizj Dh (uxi )Dh (uxj )

i,j

by (3.33) that the argument of ai is Du + thDh (Du).

105

·

(q−2) 4

|Dη|dtdx

  21 o (q−2) · Gα,k (1 + |Du + thDh (Du)|2 ) 2 |Dη|2 dtdx ≤ n n Z Z 1 X 0 aizj Dh (uxi )Dh (uxj )+ C η 2 gα,k U

Z Z

0

i,j

1

o (q−2) C h 2 2 2 Gα,k (1 + |Du + thD (Du)| ) |Dη| dtdx (3.38) 4 u 0 We use nally to evaluate E integral, the bound for b(x) the property that the dierence quotient has smaller Lp norm than the usual derivative along ei direction: Z −h 2 |E| = b(x)D (η gα,k )dx ≤ U Z ∂ 2 ||b||L∞ (V ) ∂xs (η gα,k dx ≤ U Z 0 |Dh uxs |dx ≤ ||b||L∞ (V ) (2η|ηxs ||gα,k | + η 2 gα,k U nZ (2η|Dη||gα,k |dx+ ||b||L∞ (V ) U Z Z o 1 2 0 h 2 0  η gα,k |D uxs | dx + η 2 gα,k dx (3.39) 4 U U Now by taking  suciently small, we deduce that there exist a positive constant C such that Z Z 1 0 η 2 gα,k (1 + |Du + thDh (u)|2 )(p−2)/2 |Dh (Du)|2 dtdx ≤ A ≤ C U

0

|B| + |D| + |E| ≤ 1

Z Z Z Z

U 1

0 η 2 gα,k (1 + |Du + thDh (Du)|2 )q/2 dtdx

0

2η|Dη||gα,k |(1 + |Du + thDh (Du)|2 )(q−1)/2 dtdx

+ 0

U

Z Z + u

1

Gα,k (1 + |Du + thDh (Du)|2 )

(q−2) 2

|Dη|2 dtdx

(3.40)

0

We give same exponent in last inequality in this way. By using Young inp q equality ab ≤ ap + bq with p1 = 1q , 1q = (q−1) we have q

|gα,k |(1 + |Du + thDh (Du)|2 )

(q−1) 2

≤

q |gα,k |q q − 1 + (1 + |Du + thDh (Du)|2 ) 2 q q (3.41)

106

and by using Young inequality with

1 p

= 2q , 1q =

(q−2) q

we have

q/2

h

2

Gα,k (1 + |Du + thD (Du)| )

(q−2) 2

so we obtain

Z Z

1

0

η 2 gα,k (1 + |Du + thDh (u)|2 )(p−2)/2 |Dh (Du)|2 dtdx ≤

C U

2Gα,k q − 2 q ≤ + (1 + |Du + thDh (Du)|2 ) 2 q q (3.42)

0

A ≤ |B| + |D| + |E| ≤ Z Z U

1 0 η 2 gα,k (1 + |Du + thDh (Du)|2 )q/2 dtdx

0

1

 |gα,k |q q − 1 h 2 2q + 2η|Dη| + (1 + |Du + thD (Du)| ) dtdx q q U 0 " q/2 # Z Z 1 2G q q − 2 α,k |Dη|2 + + (1 + |Du + thDh (Du)|2 ) 2 dtdx = q q U 0   Z Z 1 q−1 h 2 2q 2 0 2 (q − 2) (1+|Du+thD (Du)| ) η gα,k + 2η|Dη| + |Dη| dtdx q q U 0 # Z Z 1" q/2 G |gα,k |q α,k + 2|Dη|2 + η|Dη dtdx (3.43) q q U 0 Z Z



If we consider the case α = 2, for every k > 0, t ∈ R we have

g2,k (t) = t,

0 g2,k (t) = 1,

G2,k = t2

and if we take η as a cut-o function such that in V ⊂⊂ U , η ≡ 1, then previous inequality simplies considerably in the second inequality here below Z Z 1 C |Dh (Du)|2 dtdx ≤ V 1

Z Z

(1 + |Du + thDh (u)|2 )

C V

0 (p−2) 2

|Dh (Du)|2 dtdx ≤

0

Z Z V

1

q

(1 + |Du + thDh (Du)|2 ) 2 dtdx ≤

0

107

1

Z Z

q

(1 + |(1 − t)Du(x) + tDu(x + hes )|2 ) 2 dtdx ≤ 61

0

V

1

Z Z V

Z Z V

q

(1 + (|Du(x)| + |Du(x + hes )|)2 ) 2 dtdx =

0

1

q

(1 + Du(x)2 + 2|Du(x)Du(x + hes )| + Du(x + hes )2 ) 2 dtdx ≤

0

(3.44)

|V | + 4||Du||Lq (V )

considering that (1 + |Du + thDh (u)|2 ) > 1 and using Minkovski inequality in last passage. So we have obtained ||Dh Du||L2 (V ) < K and by quotient dierence properties we can infer that u ∈ H 2 (V ).

Lemma 3.6. Under assumptions of Theorem 3.3, there is a constant C such that for every α for which the right hand side is nite holds Z η

n X

2

U

(1 + |uxs |2 )

(α+p−4) 2

|D(uxs )|2 dx ≤

s=1

Z C(α − 1) U

(η 2 + |Dη|2 )

n X (α+q−2) (1 + |uxs |2 ) 2 dx

(3.45)

s=1

Proof. Let us go to the limit for h → 0 in (3.43) ,then Dominated convergence applies and

• Du + thDh (Du) = (1 − t)Du(x) + tDu(x + hes ) → Du in Lq (V ) , by continuity in Lq (V ) of the translation; • gα,k (Dh u) → gα,k (uxs ) in Lq (V ) as gα,k is a linear function; • Gα,k (Dh u) → Gα,k (uxs ) in Lq/2 (V ) as gα,k is quadratic; • limit can be done to left-hand side of inequality, because of its lower semicontinuity62 ; using relations of Proposition 3.5 and using Fatou's Lemma, we obtain Z (p−2) (α−2) C η 2 (1 + |uxs |2 ) 2 (1 + |Du|2 ) 2 |D(uxs )|2 dx ≤ U

Z (α − 1)

η 2 (1 + |uxs |2 )

(α−2) 2

U 61 As

62 See

t ∈ (0, 1).

remark 20.

108

q

(1 + |Du|2 ) 2 dx+

Z

η|Dη|(1 + |uxs |2 )

+

(α−1) 2

(1 + |Du|2 )

(q−1) 2

dx ≤

U

Z

α

|Dη|2 (1 + |uxs |2 ) 2 (1 + |Du|2 )

+2

(q−2) 2

dx ≤

U

(3.46) Next we sum up with respect to s = 1, 2....n and we observe that there exists constants C, K such that for l = 2q , (q−1) , (q−2) 2 2 n n X X 2 l 2 l K (1 + |uxs | ) ≤ ((1 + |Du| ) ≤ C (1 + |uxs |2 )l s=1

(3.47)

s=1

So we obtain Z n n X (α−2) X (p−2) η2 K (1 + |uxs |2 ) 2 (1 + |uxs |2 ) 2 |D(uxs )|2 dx ≤ U

s=1

s=1

Z n n n X (α−2) X q (1 + |uxs |2 ) 2 dx+ C (α − 1) η 2 (1 + |uxs |2 ) 2 U

Z +

η|Dη| U

(1 + |uxs |2 )

(α−1) 2

s=1

Z

2

|Dη|

+2

s=1

s=1

n X

U

n X

n X

(1 + |uxs |2 )

(q−1) 2

dx

s=1 n o X (q−2) (1 + |uxs |2 ) 2 dx ≤ (1 + |uxs | ) 2

α 2

s=1

s=1

(3.48) Finally we recover to the formula n X

Ysa

s=1

n X s=1

Ysb

63

 n  n(n − 1) X a+b Ys ≤ 1+ 2 s=1

(3.49)

valid for every Ys ≥ 0 and a, b > 0, and we apply it with Ys = 1 + |uxs |2 . So that our inequality becomes64

Z U

η2

n X

(1 + |uxs |2 )

(α+p−4) 2

|D(uxs )|2 dx ≤

s=1

63 Whose 64 Note

proof can be found in ([64], Lemma 2.9). that (α+q−2) = (α−2) + 2q = (α−1) + (q−1) = 2 2 2 2

109

α 2

+

(q−2) 2

.

n X (α+q−2) (1 + |uxs |2 ) 2 dx+ η

Z

n

2

C (α − 1) U

s=1

Z +

η|Dη| U

(1 + |uxs |2 )

(α+q−2) 2

dx

s=1

Z

n X

2

|Dη|

+2

n X

U

2

(1 + |uxs | )

(α+q−2) 2

o

dx ≤

s=1

Z

2

C(α − 1)

2

(η + |Dη| ) U

n X

(1 + |uxs |2 )

(α+p−2 2

dx

(3.50)

s=1

using Holder inequality for the integral with mixed term η|Dη| and choosing an appropriate constant C.

Lemma 3.7. Let 0 < r < R ≤ r + 1 suitable for having Br ⊂ BR ⊂⊂ U balls of their radius being compactly contained in U . Then there exists a constant C such that for each of such radius and for every α ≥ 2 we have Z

n X

(1 + |uxs |2 )

2∗ (α+p−2) 4

! 22∗ ≤

dx

Br s=1

Cα3 (R − r)2

where 2∗ =

2n (n−2)

Z

n X

(1 + |uxs |2 )

(α+q−2) 2

(3.51)

dx

BR s=1

is the Sobolev conjugate exponent.

Proof. We calculate the gradient of (1 + |uxs |2 ) |D[η(1 + |uxs |2 )

(α+p−2) 4

(α+p−2) 4

,

]|2 ≤

(α+p−4) (α+p−2) (α + p − 2)2 2 η (1 + |uxs |2 ) 4 |Duxs |2 + |Dη|2 (1 + |uxs |2 ) 2 (3.52) 4 Since p ≤ q , by precedent Lemma we have that there exists C such that

Z X n

|D[η(1 + |uxs |2 )

(α+p−2) 4

]|2 dx ≤

U s=1

Cα

3

Z

2

2

(η + |Dη| ) U

n X s=1

110

(1 + |uxs |2 )

(α+q−2 2

dx

(3.53)

If we apply Sobolev's inequality for each s = 1, ...n,

Z

[η(1 + |uxs |2 )

(α+p−2) 4

 22∗

2∗

≤

] dx

U

Z

|D[η(1 + |uxs |2 )

C

(α+p−2) 4

(3.54)

]|2 dx

U

Now by these two inequalities, P by Minkovsky P inequality applied to them and∗ applying also the inequality ns=1 Ysa ≤ ( ns=1 Ys )a with our exponent a = 22 we get ! 22∗ Z n X 2∗ (α+p−2) ∗ 4 (1 + |uxs |2 ) η2 dx ≤ U

s=1

! 22∗ Z h i 22∗ (α+p−2) η 2 (1 + |uxs |2 ) 4 dx ≤ U n Z X

Cα

[η(1 + |uxs | )

(α+p−2) 4

2∗

 22∗ dx ≤

] dx

U

s=1 3

2

Z

2

2

(η + |Dη| ) U

n X

(1 + |uxs |2 )

(α+q−2) 2

dx

(3.55)

s=1

and if we take η as a cut-o function between Br and BR , this meaning η ≡ 1 2 on Br , η ≥ 0 in BR and vanishing out of BR , with as usual |Dη| ≤ (R−r) we obtain

Z

n X

(1 + |uxs |2 )

2∗ (α+p−2) 4

! 22∗ dx

≤

Br s=1

Cα

3

Z BR

(1 +

n (α+q−2) 2 2 X (1 + |uxs |2 ) 2 dx ) (R − r) s=1

(3.56)

which is our thesis if we consider that R − r ≤ 1 as 0 < r < R ≤ r + 1. This argument has been set to get started the same technique that we used in chapter 2 for De Giorgi Theorem65 . Indeed, for 0 < r0 < R0 ≤ r0 + 1 suitable radius, let us dene for every k ∈ N

α1 = 2 65 See

αk+1 := (αk + p − 2)

2∗ + (2 − q) ∀k ≥ 1 2

for instance the proof of Theorem 2.1 and denition of De Giorgi classes.

111

(3.57)

R0 − r0 2k and, not casually,let's dene for those Rk

(3.58)

Rk := r0 +

n X

Z Ak :=

(1 + |uxs |2 )

(αk +q−2) 2

! (α

1 k +q−2)

(3.59)

dx

BRk s=1

Under these notations, previous Lemma 3.7 can be rewitten as

 Ak+1 =

Cαk3 22(k+1) (R0 − r0 )2

α

1 k +p−2

(αk +q−2) (αk +p−2)

· Ak

(3.60)

Now, we will have to iterate previous equation on Ak+1 so that we would need to understand if the resulting exponent is nite,

θ :=

∞ Y (αk + q − 2) k=1

(3.61)

(αk + p − 2)

Lemma 3.8. Let αk be the sequence yet dened, then we have the following two equivalent representation formulas for αk X k−2  ∗ i 2 2∗ αk = 2 + p − q 2 2 i=0 

(3.62)

for each k ≥ 2  αk = 2 +

2∗ 2

!

k−1

−1

p(2∗ /2) − q (2∗ /2) − 1



(3.63)

for each k ≥ 1. Furthermore, θ is nite and is given by q (2∗ /2) − 2 θ= p (2∗ /2) − (q/p)

(3.64)

Proof. The proof of formulas (3.62) and (3.63) is a verication that they are

equivalent to each other and we can so demonstrate (3.62) for induction using just the denition of αk . Once we have got these formulas at hand, by using the denition of αk we have ∞ Y (αk + q − 2)

(αk + p − 2) k=1

 =q

112

2∗ 2

k−1

1 αk + p − 2

(3.65)

and so by (3.63) we deduce that ∞ Y (αk + q − 2)

=

(αk + p − 2) k=1

p+

q(2∗ /2)k−1 h ∗ i
p(2 /2)−q 2∗ k−1 − 1 2 (2∗ /2)−1



(3.66)

Finally, since 2∗ /2 > 1 , when we pass to the limit for k → ∞ we obtain our thesis. Finally to give end to our demonstration we demonstrate that there are positive constants β ,C such that for each k ≥ 1  θ 1 Ak+1 ≤ C A1 (3.67) (R0 − r0 )β We remember that 1  (α +q−2) R (αk +q−2) Pn k 2 2 dx ; • Ak := BR s=1 (1 + |uxs | ) k

• R∞ = r0 and limk→∞ αk = ∞; R  1q Pn 2 2q | ) (1 + |u • A1 = BR dx xs s=1 0

• limk→∞ ||v||Lk (Br ) = ||v||L∞ (Br0 ) = ess supBr0 v 0

This will mean, passing to the limit k → ∞ and as (αk + q − 2) → ∞, exactly that it converges to the essential supremum,

sup(1 + |Du|2 )1/2 = lim C Br0

k→∞

Z

n X

(1 + |uxs |2 )

(αk +q−2) 2

! (α dx

1 k +q−2)

≤

Br0 s=1

Z

!θ 1q

n X

1 C (1 + |uxs |2 ) 2 q dx βθ (R0 − r0 ) BR0 s=1  θ 1 2 C ||(1 + |Du| ||Lq (BR ) (R − r)β

≤ (3.68)

Thus, if we show (3.67) we obtain our desired bound for the essential supremum of the gradient. To this aim we assume without loss of generality that A1 ≥ 1 and that constant C ≥ 217 . We iterate

 Ak+1 =

Cαk3 22(k+1) (R0 − r0 )2

α

113

1 k +p−2

(αk +q−2) (αk +p−2)

· Ak

(3.69)

to get

θ

Ak+1 ≤

Aθ1

 k  Y Cα3 22(i+1) αi +p−2 i

i=1 P

−2θ

Aθ1 C(R0 − r0 )

(R0 − r0 )2 1 i αi +p−2

≤ (3.70)

= Aθ1 C(R0 − r0 )−θβ

setting

β=

∞ X i=1

∞

X 2 ≤2 ai + p − 2 i=1



2∗ i−1 2



p(2∗ /2) − q (2∗ /2) − 1

−1 =

where inequality is due to (3.63) and

C = exp θ

∞ X log(Cα3 22(2i+1) )

2∗ p(2∗ /2) − q (3.71)

! (3.72)

i

αi + p − 2

i=1

With this, Theorem 3.1 is proved. Now we give the idea of the strategy66 to prove that the minimality of u 1,p 1,q boosts its integrability from Wloc (U ) to Wloc (U ), provided

(n + 2) q < p n

(3.73)

We start by Euler-Lagrange equation for the functional I(u) = that we write compactly as Z D(L(Du))D(ψ(x))dx

R U

L(D(u))dx (3.74)

U 1,p ∀ψ ∈ Wloc . Now we consider for a suitable radius R a family of perturbed functionals Z Z Z q Iσ (u, BR ) = L(Du)dx + σ |Du| dx = Lσ (Du)dx (3.75) BR

for

BR

Lσ (Du) = L(Du) + σ|Du|q

U

(3.76)

Now let us consider u ∈ C , a mollication of u and let v,σ ∈ W (BR ) the solution of the minimum problem Z inf Lσ (Dv)dx (3.77) 1,q ∞

u +Wo (BR

66 Details

1,q

BR

of the proof can be found in the paper of Mingione,Leonetti and Esposito [29].

114

so that for a suitable sequence of choices of mollicators u we have a sequence of minimizers v,σ for our σ - problem lying in -closed convex set u + Wo1,q (BR ). As v,σ are minimizers of (p, q)-variational integrals and lie in W 1,q (U )o , we can apply Theorem 3.1 to infer that they are individually essentially bounded above, and using the important inequality67 (Z )θ Z Z

|Dv,σ |q dx ≤ C

Br

|Du |q dx + 1

L(Du)dx + σ BR0

(3.78)

BR

it is easily veriable that sequence v,σ is equibounded. Compactness arguments apply when computing the limit σ → 0, or limit  → 0, giving us Du as weak limit of Dv,σ in Lp (BR ). Then we pass to the limit in (3.78), thus getting by semicontinuity our desired thesis θ Z Z q L(Du)dx + 1 (3.79) |Du| dx ≤ C BR

Br

Example 11. The previous regularity result of Theorem 3.3 can be applied, for example, to the Euler-Lagrange equation of the functional68 Z F (u) :=

(1 + |Du|2 )α(x) dx

(3.80)

U

This situation looks quite more general but if α(x) is a (Hölder) continuous function in U , then in every ball included in our domain BR for a suitable radius, growth conditions (3.22) are satised because of the Theorem of Weierstrass. It is just necessary to set p := inf BR α(x) and q := supBR α(x).

Example 12. Another application of Theorem 3.3 consist in elliptic equations of the form n X

(a(x)uxi )xi + |uxn |q−2 uxn


xn

= b(x)

(3.81)

i=1

with a locally bounded b(x)) and with a(x) a positive locally Lipschitz continuous function . When q is close enough to 2 Theorem 3.3 applies and we have the desired regularity. 67 We

will not give the proof of this important tool here for brevity and simplicity of exposition: this precious inequality is obtained with a similar technique already showed here by a careful use of dierence quotients in the setting of fractional Sobolev spaces, and a detailed proof can be found in ([29], Claim 2) and improved with the result of Theorem 3.1, or more generally in [67] in a more general framework. 68 This kind of functionals have been deeply studied by V.V.Zhikov in [89], with the methods of averaging.

115

Remark 29. (The Lavrentiev Phenomenon (LP) The term Laurentiev phenomenon refers to a surprising result rst demonstrated in 1926 by M. Lavrentiev [56], in the context of non autonomous functionals. There it was shown that it is possible for the variational integral of a two-point Lagrange problem, which is sequentially weakly lower semicontinuous on the admissible class of absolutely continuous functions, to possess an inmum on the dense subclass of C 1 admissible functions that is strictly greater than its minimum value on the full admissible class. Let us make an example in dimension one. Suppose we have a standard functional in the calculus of variations Z F (y) =

b

f (t, y(t), y 0 (t))dt

a

with a Lagrangian function f : [a, b] × R × R → R. If we consider the minimization of F on the set of functions such that y(a) = ya and y(b) = yb , belonging to dierent classes as AC([a, b]) (absolutely continuous functions or equivalently functions in W 1,1 ([a, b]), Lip([a, b]) (Lipschitz functions) or C 1 ([a, b]), the so called Lavrentiev phenomenon can happen, that is for instance inf

{y∈AC([a,b])|y(a)=ya ,y(b)=yb }

F (y)
0.

Lavrentiev Phenomenon typically exhibits when under (p,q)-growth conditions. Roughly speaking, LP occurs for a map in W 1,p (U, Rn ) when it is 1,q not possible to nd a sequence of more regular maps {vn }n∈N ∈ Wloc (U, Rn ) 1,p such that vn → v in Wloc (U, Rn ). When the Lavrentiev Phenomenon occurs for a local minimizer u then it follows that it is not possible to realize locally minimizing sequences {un } for the energy integral I with more regular maps 1,q {un } ∈ Wloc (U, Rn ). LP is a clear obstruction to minimality, since if u is 116

1,q (U, Rn ), then by the very denition there is a minimizer such that u ∈ Wloc not LP for u. It is interesting and signicant that LP never occurs when p = q and when L(x, z) = L(z), so that LP results from the coupling of the (p,q)-growth with the dependence on x of the integrand. For an exhaustive list of examples we cite [60] and for an explanation on how the Lavrentiev Phenomenon can be dened as a relaxation of the functional L see [11].

References and further steps Our main reference to prove boundedness results for (p, q)-growth conditions is the pioneering work of Marcellini [64], while for the higher integrability of minimizers we refer to [29]. Lavrentiev phenomenon is studied in [61], [34] and further improvements on how to exclude it can be found in [89].

3.3 Additional Structure Anisotropy of growth conditions can be expressed in more than a way: a generalization of (p, q)-growth conditions may be given by variable growth exponents as ν|z|p(x) ≤ L(x, z) ≤ M (1 + |z|q(x) ) (3.82) with continuous p(x), q(x) and 1 < a ≤ p(x) ≤ q(x) ≤ b < ∞ for a, b ∈ R, as for example the functional L3 in (3.6). Applications of precedent theorems in local case has already been given when exponent functions p(x), q(x) are continuous and equals in Example 8. The energy functional Z |Du|p(x) dx (3.83) F (u) := U

comes to light when considering a number of models from Mathematical Physics, as homogeneization of strongly anisotropic materials69 , electrorheological uids 70 , temperature dependent viscosity uids and image processing models. Its model equation is referred as the p(x)-Laplacian equation, that is formally
div |Du|p(x)−2 Du = f (3.84) with p : U → (1, ∞) bounded and continuous. 69 [89]

V.V.Zihkov's pioneering work shows that thanks to the particular structure of (3.5) there exists a condition called the log-continuity assumption which provokes the absence of Lavrentiev Phenomenon fot the functional (3.6). 70 A detailed and comprehensive presentation of the mathematical modeling of electrorheological uids can be found in [81].

117

Example 13. (Electrorheological uids [81]) Electrorheological uids are special viscous liquids characterized by their ability to undergo signicant changes in their mechanical properties when an electron eld is applied. This property they owe can be exploited in many technological applications, as clutches and shock absorbers just to name a few. As the behaviour of electrorheological uids is strongly inuenced by non-homogeneous electric elds, in [77] M. Ruºi£ka and K.R. Rajagopal develop a model that takes into account the interaction of the electro-magnetic elds and the moving liquid and thus the electric eld is treated as a variable which has to be determined. This interaction between the electro-magnetic eld and the moving material isdecribed on the basis of the "dipole currentloop" model developed in [76] by Y.H. Pao in 1978 . They start with the general balance laws for mass, linear momentum, angular momentum, energy the second law of thermodynamics in the form of the Clausius-Duhem inequality and Maxwell's equations in their Minkovskian formulation. Then they choose dependent and independet variables, reecting the nature of the process under analysis, and impose the requirement of Galilean invariance of both the constitutive relations and balance laws. Then they simplify the system by incorporating the physical properties of an electrorheological uid: • it can be considered approximatively as a non-conducting dielectric; • the mechanical response does not change if a magnetic eld is applied.

On a last step they carry out a dimensional analysis and subsequent approximations, restricting the validity of the resulting system to a certain but typical situations. If the uid is assumed to be incompressible, it turns out that the relevant equations are71 described by div(E + P) = 0 curl(E) = 0 ρ0

∂v − div(S) + ρ0 [∇v]v + ∇φ = ρ0 f + [∇E]P ∂t div(v) = 0

(3.85)

(3.86)

where E is the electric eld, P the polarization, ρ0 the density, v the velocity, D its symmetric gradient, S the extra stress, φ the pressure and f the 71 We

abandon the letter "D" for the derivative, which in continuum mechanics is used to be employed for denoting the material derivative and we use the common nabla operator.

118

mechanical force. In the last section of the book [81] it is proposed a model capable of explaining many of the observed phenomena, having the form S = α21 ((1 + |D|2 )

p−1 2

− 1)E ⊗ E + (α31 + α33 |E|2 )(1 + |D|2 )

+α51 (1 + |D|2 )

p−2 2

(DE ⊗ E + E ⊗ DE)

p−2 2

D+ (3.87)

where αi,j are material constants and where the material function p depends on the strength of the electric eld E2 and satises (3.88)

1 < p∞ ≤ p(|E|2 ) ≤ p0 < ∞

Since the material function p which essentially determines S, depends on the magnitude of the electric eld |E|2 , we have to deal with an elliptic (in case of steady ows for example) or parabolic system of partial dierential equations with the over called (p,q)-growth conditions, i.e. the elliptic operator S satises at maximal range S(D, E) · D ≥ c0 (1 + |E|2 )(1 + |D|2 ) |S(D, E)| ≤ c1 (1 + |D|2 )

p0 −1 2

p∞ −2 2

|D|2 ,

|E|2

(3.89)

Since the solution E of the Maxwell's equations is in general not constant it si clear from the form of the extra stress S that the canonical functional setting are the spaces Lp(x) (U ) and W 1,p(x) (U ), the so-called generalized Lebesgue and generalized Sobolev spaces, endowed with a suitable norm (see [81], chap.2). We point out that the generalized Lebesgue and Sobolev spaces have a lot in common with the classic ones, but however there are also many open fundamental questions, which are well understood in the classic theory. For example, it is not known, even for very "nice" functions p(x), whether smooth functions are dense in the space W 1,p(x) (U ). For as more generalized is this new condition, it bounds every component of z vector at the same way. Here we present another growth condition which remedies this lack of directional anisotropy, and that has determined the development of an amazing expansion of Sobolev Spaces theory. These growth conditions are

ai (x, u, Du)uxi ≥ C1,i |uxi |pi

(3.90)

|ai (x, u, Du)| ≤ C2,i |uxi |(pi −1)

(3.91)

for Lpi = ai (Du, u, x), pi > 1 and C1,i , C2,i > 0 ∀i = 1, 2...n.

119

The prototype for this kind of anisotropic elliptic equations is the pi - Laplacian equation n X
|uxi |(pi −2) uxi x = 0 ∀x ∈ U (3.92) i

i=1

In many applications it is natural to work with the concept of anisotropic Sobolev spaces, where the various weak partial derivatives of u are integrable with dierent exponents. This theory is consistent, as there exists non negative non trivial solutions to the over written pi laplacian operator, as is demonstrated in [28] for U = Rn an extremal case. Let us assume U ⊂ Rn to be an open bounded domain, and without loss of generality assume pmax = max{1,2..,n} {pi } = pn , pmin = min{1,2..,n} {pi } = p1 , p1 ≤ p2 ≤ ... ≤ pn , p = (p1 , ..., pn ). As usual we denote by p¯ the harmonic average of pi s as n

1X 1 p¯ = n i=1 pi

Denition 3.2.

We dene the

(3.93)

anisotropic Sobolev spaces

W p (U ) := {v ∈ Lp1 (U ) : vxi ∈ Lpi (U )} p

W (U ) := {v ∈ Lpn (U ) : vxi ∈ Lpi (U )}  W p (U ) := v ∈ L1 (U ) : vxi ∈ Lpi (U )

(3.94) (3.95) (3.96)

with the respective norms,

||v||W ,p (U ) :=

n X

||vxi ||Lpi (U ) + ||v||Lp1 (U )

(3.97)

||vxi ||Lpi (U ) + ||v||Lpn (U )

(3.98)

i=1

||v||W p (U ) :=

n X i=1

We dene as usual the closure of Co∞ (U ) in these spaces, with respect their norms, W po (U ) := W p (U ) ∩ Wo1,1 (U ) p

p

W o (U ) := W (U ) ∩ Wo1,1 (U )

(3.99)

Remark 30. We remark the obvious inclusion p

W (U ) ⊂ W p (U ) It is suggestive to study the possibilities of extending the classical GagliardoNirenberg-Sobolev and Morrey inequalities for the anisotropic Sbobolev spaces p W p (U ) and W (U ). This constitutes an anisotropic generalization of the embedding theorems, such as presented in our rst chapter. We now introduce here below some useful general properties of functions belonging to anisotropic Sobolev spaces, as developed by [84]. 120

Proposition 3.9. (Properties of W 1,p (Rn ) functions) • (Generalised Holder inequality [48]) Let 0 < pj ≤ ∞, 0 < qj ≤ ∞ for 1 ≤ j ≤ n, p = (p1 , p2 , ..., pn ), q = (q1 , q2 , ..., qn ). If u ∈ Lp and v ∈ Lq then uv ∈ Lr , where ( 1 = p1j + q1j 1≤j≤n rj (3.100) r = (r1 , r2 , ..., rn )

and also

(3.101)

||uv||r ≤ ||u||p · ||v||q

• (Homogenization of the product [48])P Let p1 , p2 , ....pn ∈ (0, ∞] such72 that ni=1 p1i = 1r . Then if wi ∈ Lpi , n n Y Y ||wi ||Lpi (3.102) w i ≤ r i=1

i=1

L

Thus, if v ∈ W 1,p the product of its derivatives lies in Lr , Qn

i=1

vxi ∈ Lr

• (Generalized Interpolation) Let p1 , p2 , ....pn ∈ (0, ∞] and let θ1 , θ2 , ..., θn ∈ (0, 1) denote Pn θiweights with unitary sum θ1 + θ2 + ... + θn = 1. Let us dene p˜ = i=1 pi .Given w1 , w2 , .., wn measurable functions on U, inequality ||

n Y

θi

|wi | |Lp˜(U ) ≤

i=1

n Y

|||wi |θi ||

θi

L pi (U )

i=1

(3.103)

This means that taking w1 = w2 = ... = wn we would have ||w||p˜ ≤

n Y

||w||θpii

(3.104)

i=1

• (Sobolev Type Troisi [84]) Pn inequality, n¯ p 1 1 Let p¯ = n i=1 pi the harmonic mean of p0i s, let p¯∗ = n−¯ . With notap tion introduced adopted, if p¯ < n, there exists a constant C depending on n, p such that ∀v ∈ Co1 (Rn ), ||v||Lp¯∗ (Rn ) ≤ C

n Y

||vxi ||Lpi (Rn )

(3.105)

i=1 72 For those p

i = ∞ we will stipulate the convention for the functional sets A, A(Rn ) = A.

121

1 ∞

= 0. We also denote for simplicity

Best constants and extremal functions for the Sobolev embedding in Rn are studied in [84], [85],[28]. In the case of a bounded domain U , which we are interested in, its geometry plays a crucial role. The direct approach leads to embedding theorems that hold only for functions dened on a rectangular domain. The reason is that the extension Theorem (Proposition 1.6, last property), which is used for the classical case, "mixes" the derivatives and the extended function no more belongs to the given anisotropic Sobolev space. We announce here the two key theorems on Sobolev and Morrey embeddings, for whose reference we redirect the reader to [78], [49].

Theorem 3.10. (Anisotropic Rectangular Sobolev Embedding) Let Ω ⊂ Rn be a rectangular domain, and p = (p1 , .., pn ) with

Let p¯∗ be dened as p¯∗ =

n X 1 ≥ 1. p i=1 i

(3.106)

n¯ p n − p¯

(3.107)

if p¯ < n

or chosen arbitrarily in [1, ∞) otherwise. p Then the space W (Ω) is continously embedded into Lp¯∗ (Ω). Theorem 3.11. (Anisotropic Rectangular Morrey Embedding) Let Ω ⊂ Rn be a rectangular domain, and p = (p1 , .., pn ) with n X 1 < 1. p i i=1

(3.108)

Then the space W p (Ω) is continuously embedded into C 0,β (Ω) with α ≤ 1, 0 1 and that when r is an even integer ∂Ωr is analytic near the origin, where our problem will arise. A simple calculation shows that for 0 < γ < 1r the function u(x1 , x2 ) = x−γ 1 satises 0 = ux2 ∈ Lp2 (Ωr ), ux1 ∈ Lp1 (Ωr ) as for example 2 −

124

2 = ||ux1 ||L1 (Ωr ) (1 − γr) (3.115)

with 1 ≤ p1
0 we have P 1 + hqi + nj=1 |zj |qj L(zi + h) − L(zi ) ≤C (3.121) L zi ≤ h h 1  Pn qj qi we calcuand reverse inequalities for −h. Now, for h = 1 + j=1 |zj | late ! !− q1 n n i X X |Lzi | ≤ 2C 1 + |zj |qj · 1 + |zj |qj = j=1

= 2C

1+

n X

j=1

1− q1 i

! |zj |qj

= 2C

j=1

1+

n X

!1/qi0 |zj |qj

(3.122)

j=1 0

This means that Lzi (Du) ∈ Lqi (U ) since uxj ∈ Lqj (U ). Now by the same 0 argument of Theorem 1.22, replacing W 1,q (U ) with W 1,q (U ) and W 1,q (U ) 0 with W 1,q (U ), we can use nite-dierences method to infer our thesis, Z L(Du + τ Dv) − L(Du) i(τ ) − i(0) = dx (3.123) τ τ U The convergence of the limits is guaranteed by the condition Lzi (Du) ∈ 0 Lqi (U ) and Young inequality, which is still valid for each direction ab ≤ aqj qj

+

73 We

q0

b j qj0

now it has to be applied for each index of the sum. Thus by

will refer in future to Lq as the anisotropic conjugate space of Lq . 0

126

Dominated Convergence theorem i0 (0) exists and since by our assumption function i(·) has a minimum for τ = 0, we have the two estimates of Z L(Du + τ Dv) − L(Du) u(τ ) − u(0) 0 0 = i (0) = lim = lim dx = τ →0 τ →0 U τ τ Z X n = Lpi (Du)vxi + Lz (Du)vdx (3.124) U i=1

And this concludes the proof as it shows u to be a weak solution in the sense of 3.120.

Denition 3.4. A function u ∈ W 1,q (U ) is a weak solution for the boundaryvalue problem for the anisotropic Euler-Lagrange equation provided Z X n

(Lpi (Du))vxi

dx = 0

(3.125)

U i=1

∀v ∈ Wo1,q (U ).

Example 14. (Anisotropic pi -Laplacian) The prototype of pi Laplacian equation 3.92 has been studied exhaustively by [35],[24] and many others. Its equation can be expressed as n X

|uxi |(pi −2) uxi


xi

=0

∀x ∈ U

i=1

The p-Laplacian operator weights partial derivatives with dierent powers, so that the underlying functional framework involves the anisotropic Sobolev spaces presented above. Critical exponents for embeddings of these spaces into Lq have two distinct expressions according to whether the anisotropy is concentrated next to some pi or spread out. Existence results in the subcritical case are strongly inuenced by this phenomenon of the embeddings. A function u ∈ W 1,q (U ) is thus a weak solution to the anisotropic EulerLagrange equation if for each φ ∈ Wo1,q (U ) we have Z X n


|uxi |(pi −2) uxi φxi dx = 0

(3.126)

U i=1

Under suitable but quite strong hypothesis on the Lagrangian function we are able to nd conditions to get Lipschitz continuity. This work is due to Marcellini [63] in 1988, who showed with a similar approach to the one we gave in previous section about (p,q) -growth conditions the following theorem. 127

Theorem 3.16. Consider an autonomous integral of the Calculus of Variations of the form Z I(u) =

(3.127)

L(Du(x))dx U

where L ∈ C 2 (Rn ), U ⊂ Rn is a bounded open set and u : U → R the unknown solution, a scalar function of the Sobolev space W 1,p (U ) for some p > 1. Let L be such that m

n X



pi

|zi | ≤ L(z) ≤ M 1 +

n X

i=1

2

m|λ| ≤

n X

|zi |pi



(3.128)

i=1



Lzi zj (z)λi λj ≤ M 1 +

i,j=1

n X

 |zi |pi −2 |λ|2

(3.129)

i=1

for every z, λ ∈ Rn , where m, M are positive constants and 2 ≤ pi
0. An analogous result holds if we deal with (u − k)+ . Proof. Let us consider the equation in weak form for u ∈ W 1,[p,q] (U ): Z ap (z, u, Du)φx dz + U 1,[p,q]

∀φ ∈ Wo

Z X n−1

aq,i (z, u, Du)φyi dz = 0

(3.151)

U i=1

(U ), and let us test it with the test function φ(z) = θ(z)q (u − k)− ,

f or θ ∈ Co1 (U )

(3.152)

Now we make some calculations to nd

Dx φ = qθq−1 (u − k)− Dx θ + θq Dx u Dyi φ = qθq−1 (u − k)− Dyi θ + θq Dyi u

(3.153)

and so we nd Z Z q−1 0= ap (z, u, Du)qθ (u − k)− (Dx θ)dz + ap (z, u, Du)θq (Dx u)dz+ U

Z X n−1

aq,i (z, u, Du)qθ

U

q−1

(u − k)− (Dyi θ)dz +

U i=1

Z X n−1 U i=1

132

aq,i (z, u, Du)θq (Dyi u)dz (3.154)

We now consider repeatedly structure conditions (3.140)-(3.143) to obtain n−1 X

|aq,i (z, u, Du)| ≤ C1 (n − 1)|Dy u|q−1 + C2

(3.155)

i=1

and so we have

Z C0

  θq |Dx (u − k)− |p + |Dy (u − k)− |q dz ≤

U

C1

Z 

 qθq−1 |(u − k)− ||Dx θ||Dx (u − k)− |p−1 dz+

U

C1

Z 

 q(n − 1)θq−1 |(u − k)− ||Dy θ||Dy (u − k)− |q−1 dz+

U

C2

Z 

qθ

q−1

 |(u − k)− ||Dx θ| dz+

U

C2

Z 

q(n − 1)θ

q−1

Z  |(u − k)− ||Dy θ| dz + 2C2 θq dz

U

(3.156)

U p−1

Now we write θq−1 = θq p θq−p p and we apply Young's inequality |ab| ≤ p = p˜ and p = tildeq to get ap˜ + C()bq˜with p−1

Z C0

  θq |Dx (u − k)− |p + |Dy (u − k)− |q dz ≤

U

Z

q

Z

p

θq−p |Dx θ|p |(u − k)− |p dz+

θ |Dx (u − k)− | dz + C(1 )

1 U

U

Z

q

Z

q

|Dy θ|q |(u − k)− |q dx+

θ |Dy (u − k)− | dz + C(2 )

+2 U

U

Z +3

θ

q−p

p

Z

p

|Dx θ| |(u − k)− | dz + C(3 )

U

Z

q

U

Z

q

|Dy θ| |(u − k)− | dz + C(4 )

+4

θq dz+

U

q

Z

θ dz + 2C2 U

θq dz

(3.157)

U

the result follows then by choosing M = maxi=1,2,3 {i , C(i )} and taking C = CM0 .

133

Remark 33. (Homogeneous Caccioppoli's inequality) In a parallel way to the denition of DGOp we observe by formula (3.156) here that if the operator a(z, u, Du) were homogeneous i.e. if C2 = 0 then Caccioppoli's generalized inequality has the easier form Z

  θq |Dx (u − k)− |p + |Dy (u − k)− |q dz ≤

U

C

Z n

o θq−p |Dx θ|p |(u − k)− |p + |Dy θ|q |(u − k)− |q dz

(3.158)

U

for any constant k > 0 To give an anisotropic version of the De Giorgi Lemma (2.8) and (2.30), it is necessary to introduce the intrinsic geometry related to the anisotropic elliptic equation. It is a geometry induced by the operator itself. q

q−p

Let R0 , k0 > 0 be given numbers and Rx = R0p k0 p , and then let us dene

Qj := {z = (x, y) : |x| < rj , |y| < Rj },

rj =

Rx R0 R0 Rx + j+1 , Rj = + j+1 2 2 2 2

and observe that q

q−p

|Qj | = cj R0p k0 p R0n−1 →j→∞ c˜∞ > 0 as cj are equibounded positive constants converging to a strictly positive constant c∞ > 0. Another thing to observe is that n R0 o Rx , |y| < Q∞ = z = (x, y) : |x| < 2 2 is a cylinder of half of the original radius and a particular length weighted on the radius itself and in the constant k0 , with anisotropic powers. Dene

Ds,j := {z = (x, y) ∈ Qs : u(z) ≤ kj },

kj :=

k0 k0 + j+1 + µ− 2 2

being µ− as usual the essential inmum of u in U , and dene

Zj :=

|D2j,j | | |Q2j

Our question to be answered concerns the behaviour of Zj for large j . Let us denote meanwhile n o k0 D∞ := ∩j∈N D2j,j = z ∈ Q∞ : u(z) ≤ + µ0 2 134

If we demonstrate that for large j Zj goes to zero, we would get the vanishing of the set D∞ as Q∞ has always positive measure. This would give us

u(z) >

k0 + µ0 , 2

in Q∞

Lemma 3.21. (De Giorgi type Lemma) There exists a number ν > 0 depending only on the data but u, R0 , k0 such that either (η =) k0 < R0 (= C3 r)

(3.159)

lim Zj = 0

(3.160)

or, if Z0 < ν then j→∞

Remark 34. (Homogeneous De Giorgi type Lemma) If the operator is homogeneous the alternative does not occur. In this case the situation is very similar to De Giorgi type Lemma (2.30): it is possible to prove that there is an absolute number ν > 0 depending only upon the data but in R0 , k0 , u such that if Z0 < ν then Zj converges to zero as j becomes big. Conditions to verify De Giorgy type Lemma i.e Z0 < ν are this time a little harder to be found than we did in Lemma 2.33 (Reducing Negativity) because of the persistence of the x variable in the left side of inequality dening our intrinsic anisotropic version of

n 1 o Dn−1,s (x1 ) := y ∈ Bn−1,0 R/2 (¯ y ) : ∃x ∈ [¯ x−5R/2, x¯+5R/2], u(x, y) ≤ µ− + s 8e In this case the intrinsic geometry we used is working against us. So we give denition of a new function weighted on the precedent which solves an equation of the same kind with same kind of growth conditions as we had until now. This function will be dened so that the x dependence in the left hand side of our set of denition vanishes and thus an expansion of positivity for it is aviable via De Giorgi type Lemma: this will imply an expansion of positivity for the function we began with, and the achievement of the proof for expansion of positivity.

135

3.3.3 Boundedness of weak solutions When we have no hypothesis on the regularity of the Lagrangian, we are naturally brought to take under consideration just the anisotropic EulerLagrange equation without referring to the integral to be minimized. Thus we consider the situation by the only point of view of elliptic equations in divergence form as (3.161)

div(A(x, u, Du)) = B(x, u, Du)

with A(x, u, z) = (A1 (x, u, z), ..., A2 (x, u, z)), B(x, u, z) measurable for every function u ∈ W 1,1 (U ), on an open bounded U ⊂ Rn . In this formulation a weak solution of the preceding equation 3.161 will be referred as a function u ∈ W p (U ) for p = (p1 , ..., pn ) such that for each φ ∈ Wop (U ) we have the following integral identity

Z nX n U

o Ai (x, u, Du)φxi (x) + B(x, u, Du)φ(x) dx

(3.162)

i=1

Next we give some integral properties of functions in W p (U ) whose "nonautonomous" product integral is nite.

Proposition 3.22. (Some integral inequalities) • Let us consider p = (p1 , .., pn ), α = (α1 , .., αn ) two n-vectors of natural numbers with αi ≥ 0, pi ≥ 1 ∀i = 1, ..n , and u ∈ Wo1,1 (U ) such that n Z X U

i=1

then, if have

Pn

1 i=1 pi

(3.163)

|u|αi |uxi |pi dx < ∞

> 1, then u(x) ∈ L (U ) for q = q

 Pn  αi i=1 1+ pi P   n 1 −1 i=1 p

, and we

i

||u||Lq ≤ C

n Z Y

n Z X i=1

pi

1/pi α n 1+ p i i=1 i

P

|u| |uxi | dx

(

)

≤

U

i=1

C

αi

!P |u|αi |uxi |pi dx

Pn 1 i=1 pi α n 1+ p i i=1 i

(

)

(3.164)

U

with constant C = C(αi , pi , n) while for ni=1 p1i = 1, then u ∈ Lq (U ) for every q > 0 we have previous inequality with constant C = C(αi , pi , n, q, |U |); P

136

• Let ∂U be a Lipschitz continuous regular boundary and u ∈ W 1,1 (U )

such that

n Z X

If

1 i=1 pi

pi

|u| |uxi | dx +

U

i=1

Pn

αi

n Z X U

i=1

> 1 and maxi=1,..n (αi + pi ) ≤ q for q =

Pn 

 α 1+ p i i  1 −1 i=1 p

i=1

P n

i

then u(x) ∈ Lq (U ) and we have ||u||Lq ≤ C

n Z Y i=1

C

n Z X i=1

U

αi

(3.165)

|u|αi +pi dx < ∞

pi

αi +pi

|u| |uxi | dx + |u|

 P 1/pi α n (1+ i ) dx i=1 pi ≤

U

|u|αi |uxi |pi dx +

n Z X i=1

!P |u|αi +pi dx

Pn 1 i=1 pi α n 1+ p i i=1 i

(

)

(3.166)

U

with constant P C = C(αi , pi , n, |U |). If instead ni=1 p1i = 1, then u ∈ Lq (U ) and for every q > 0 we have the previous inequality with constant C = C(αi , pi , n, q, |U |). Proof. In the case u(x) is a bounded function, the proof is completely anal-

ogous to the one of Theorem 1 of ([25], chap.2). If u(x) is an unbounded function, then the conclusion is obtained by applying the inequality found for the bounded case to the function ( u(x) |u(x)| ≤ N w(x) = (3.167) N |u(x)| > N

and passing to the limit as N → P ∞ together with the use of Fatou's Lemma. The rst assertion in case of ni=1 p1i = 1 can be easily deduced from item 1 because in this case q → ∞. For the second point the essential idea is to apply the method of the partition of unity for each point x0 ∈ ∂U for which there exists by hypothesis a ball whose intersection with ∂U is representable in the form xi = Γ(x1 , ..., xi−1 , xi+1 , ..xn ) for some 1 ≤ i ≤ n, where the function Γ satises a Lipschitz condition in the corresponding domain of its arguments. We are now able to state the following theorem of boundedness of weak solutions of the divergence equation, under several anisotropic growth conditions given in big generality that we are up to describe . 137

Suppose u(x) is a weak solutions of equation in divergence form 3.161 in the ball B2R ⊂ U and make the following assumptions:

A(x, u, z) · z ≥

n X

|zi |pi − d(x)|u|δ − g(x)

(3.168)

|zi |pi + b0 (x)|u|µ0 + e0 (x)

(3.169)

i=1 n X

|Ai (x, u, z)|

pi pi −1

≤ a1

n X i=1

i=1

|B(x, u, z)| ≤

n X

ci (x)|zi |pi −1 + w(x)|u|γ + f (x)

(3.170)

i=1

|A(x, u, z)| ≤ a

n X

|zi |pi −1 + b(x)|u|µ + e(x)

(3.171)

i=1

where pi ≥ 1,δ, µ0 , µ, γ ≥ 0, a1 , a positive constants. Let us set 75   Pn 1 n  k = P i=1 pi  p = max pi p¯ =  P n n i=1..n 1 1 i=1 pi − 1 i=1 pi − 1

(3.172)

For 1 ≤ s ≤ min(k, pr¯ ) with r = max(µ0 , p, δ, (γ + 1), (µ + 1)) let us suppose s that {b(x), e(x), e0 (x), ci (x), [ci (x)]pi , w(x), f (x), d(x), g(x)} ∈ L s−1 (U ) are nonnegative functions.

Theorem 3.23. Under previous assumptions on anisotropic non-homogeneous growth of A(x, u, z), B(x, u, z) and if n X 1 > 1 p¯ > r p i=1 i

then

(Z ess sup |u(x)| ≤ C BR

|u(x)|sr dx

(3.173)

 sθ1

) +1

(3.174)

B2R

−sr where θ = p¯k−s and C(pi , n, µ0 , µ, δ, γ, R, a) depends on these arguments and s s−1 on the L (U )- norms of the functions [ci (x)]pi , w(x), f (x), d(x), g(x), b(x), e(x). 75 We abandon in this section the notation for p ¯ as the harmonic average and pn for the maximum of the pi s, to give space to these new formulations given in the unique enviroment of elliptic equations in divergence form.

138

Proof. First we show how to homogenize the growth conditions with some substitutions. Let

1

1

1

M = ||e(x)||Lµs¯(U ) + ||f (x)||Lγ s¯(U ) + ||g(x)||Lδ s¯(U ) + 1

(3.175)

s with s¯ = s−1 . Let us set u ¯(x) = |u(x)| + M . Then, our growth conditions76 written i terms of u ¯ are n X A(x, u, z) · z ≥ |zi |pi − g¯(x)|¯ u|δ (3.176)

|A(x, u, z)| ≤ a

i=1 n X

|zi |pi −1 + ¯b(x)|¯ u|µ

(3.177)

i=1

|B(x, u, z)| ≤

n X

u|γ ci (x)|zi |pi −1 + f¯(x)|¯

(3.178)

i=1

Now, for xed numbers q ≥ 1 and l ≥ M we dene the functions ( u¯q M ≤ u¯ ≤ l Fl,q (¯ u) := q−1 q ql u¯ − (q − 1)l l ≤ u¯
Gl,q (u) := F (¯ u)F 0 (¯ u)p−1 − q p−1 M qp−p+1 sign(u)

(3.179) (3.180)

for −∞ < u < +∞ . Denote for brevity Fl,q = F and Gl,q = G. Let η = η(|x|) be a smooth nonnegative function, nite in the sphere B2R whose properties will be better dened a posteriori, and let us set (3.181)

φ(x) = η p G(u)

On the set {x ∈ B2R : |u| = 6 l−M } almost everywhere we have by Proposition p−1 1.6 that φxi = (pη ηxi G(u) + η p G0 (u)uxi ). Next, observe that p

p

[F 0 (¯ u)] ≤ G0 (u) ≤ [F 0 (¯ u)] where

( G0 (u) :=

p−1

F (¯ u) [F 0 (¯ u)]

(pq−p+1) 0 F (¯ u)p q 0 p

F (¯ u)

≤ |G(u)|

|u| < l − M |u| > l − M

(3.182) (3.183)

Thus by homogenized growth conditions and these last inequalities, writing the weak solution equation with φ we get n X

φxi (x)Ai (x, u, Du) + φ(x)B(x, u, Du) ≥

i=1 76 Where

¯b(x) = b(x) + M −µ e(x),f¯(x) = w(x) + M −γ f (x),g¯(x) = d(x) + M −δ g(x) and their norms are bounded in Ls¯(U ) thanks to Minkovski inequality.

139

n X


pη p−1 ηxi G(u) + η p G0 (u)uxi Ai (x, u, Du) + η p G(u)B(x, u, Du) ≥

i=1 0

p

p

η F (¯ u)

n X

pi

0

p

p

δ

|¯ uxi | −pη F (¯ u) |¯ u| g¯(x)−apη

p−1

0

p−1

|Dη|F (¯ u)F (¯ u)

n X

|¯ uxi |pi −1

i=1

i=1

−pη p−1 |Dη|F (¯ u)F 0 (¯ u)p−1 |¯ u|µ¯b(x) − η p F (¯ u)F 0 (¯ u)p−1

n X

ci (x)|¯ uxi |pi −1

i=1

−η p F (¯ u)F 0 (¯ u)p−1 |¯ u|γ f¯(x)

(3.184)

while almost everywhere on the set {x ∈ B2R : |u| = l − M } we have the relations φxi = pη p−1 ηxi G and in particular uxi = u ¯xi = 0 so that the precedent inequalities are still valid almost everywhere in this set, with the suitable simplications. Therefore we integrate the previous inequality over B2R and we take into account the denition of weak derivative. This gives

Z

p

0

p

η F (¯ u) B2R

Z

p

0

p

n X i=1

Z

δ

η F (¯ u) |¯ u| g¯(x)dx + ap

p

+

η

p−1

0

p−1

|Dη|F (¯ u)F (¯ u)

B2R

B2R

Z

|¯ uxi |pi dx ≤

η p F (¯ u)F 0 (¯ u)p−1

B2R

n X

η p F (¯ u)F 0 (¯ u)p−1 |¯ u|γ f¯(x)dx+

B2R

η p−1 |Dη|F (¯ u)F 0 (¯ u)p−1 |¯ u|µ¯b(x)dx

p

|¯ uxi |pi −1

i=1

Z pi −1 dx+ ci (x)|¯ uxi |

i=1

Z

n X

(3.185)

B2R

To estimate the integrals on the right side of precedent inequality we apply Holder and Young's inequalities by a standard argument with

p¯ 1 ab ≤ ap¯ + 0 p¯ p¯

 p¯0 1 0 bp¯ , 

∀ a, b ≥ 0,

1 1 + 0 =1 p¯ p¯

(3.186)

where  > 0 depends as a constant by the same quantities of the constant C announced in the enunciate of the theorem. Let us choose now as usual   0 ≤ |x| ≤ r 1 η = η(|x|) = 0 (3.187) |x| ≥ r + σ   linear r ≤ |x| ≤ r + σ 140

Finally we take into account that F (¯ u) ≤ u¯q and F 0 (¯ u) ≤ q¯ uq−1 , u¯ ≥ 1 and we get

Z

0

p

F (¯ u)

n X

Br

pi

−p

p

Z

sr+sp(q−1)

|¯ uxi | dx ≤ Cq (1 + σ )

|¯ u|

 1s dx

(3.188)

Br+σ

i=1

Assuming that the right hand side of previous inequality if nite, we let l → ∞ in left hand side of it, and applying Fatou's Lemma we obtain77 n Z X

p(q−1)

|¯ u|

Z

−p

pi

 1s dx

(3.189)

τ = p¯ − kθ = sr − sθ

(3.190)

|¯ uxi | dx ≤ C(1 + σ )

|¯ u|

Br

i=1

sr+sp(q−1)

Br+σ

Now let

h = sp(q − 1) + sθ,

m=

k s

and substituting we have

Z

τ +mh

|¯ u|

1  mh

dx

−p

≤ C(1 + σ )

s h

Z

τ +h

|¯ u|

Br

 h1

(3.191)

dx

Br+σ

Now let us set

h = sθmν ,

ν ∈ N,

r + σ = rν = (1 + 2−ν )R, Z

τ +sθmν

|¯ u|

Θν =

r = rν+1 , σ = R2−(ν+1)

1  sθm ν dx

(3.192)

Brν

Then we have

ν+1

(3.193)

Θν+1 ≤ C mν Θν inequality that we iterate to get

Θν+1

P∞

≤ C(

ν+1 ν=0 mν

)Θ = C 0

Z

τ +sθ

|¯ u|

 sθ1 dx

Z =C

B2R

 sθ1 |¯ u| dx sr

B2R

by convergence of the serie, for a chosed m ≥ 2. Therefore , as ν → ∞ we ν get by convergence of the Lsθm norm to L∞ one that

Z ess sup |¯ u| ≤ C BR

sr

 sθ1

|¯ u| dx B2R

Finally, going back to |u(x)| = u ¯(x) − M we obtain our thesis. 77 Recall

the denition of Fl,q (¯u).

141

(3.194)

References and further steps Our main reference for the boundedness of anisotropic elliptic equations are [50],[82], but there are several dierent equivalent versions as [51],[52] in the context of Sobole-Orcliz spaces. It is noteworthy that last result presented for boundedness can fulll the hypothesis of Corollary 3.19. However conditions (3.172)-(3.173) ask a request on pi 's distance 78 that we would prefer to avoid. To this goal, we formulate an hypothesis on a possible development, that is to apply Sobolev's embedding theorem for semi-rectangular domains and the generalized Caccioppoli's inequality, to get an estimate of the type (2.13) on an anisotropic version, and then to apply the iterative De Giorgi's argument on a previously well dened geometry. This approach has already been performed by [20] in the case of intrinsic cylinders with Troisi's inequality on a open bounded domain that now, by Theorem 3.10, we know working perfectly on intrinsic cylinders as they are semi-rectangular domains. Using this method to obtain boundedness without restrictions on pi 's, the expansion of positivity method used for this special class of anisotropic equations is worthy of interest for an extension to more general growth conditions.

References [1] R. A. Adams, Anisotropic Sobolev inequalities, ƒasopis pro p¥stování matematiky, vol. 113(1988) No. 3, 267-279. [2] R. A. Adams, J. J. F. Fournier [3] L. Ambrosio,

Sobolev spaces, Academic Press (2003).

Lecture Notes on Elliptic Partial Dierential Equations.

[4] J. M. Ball, Convexity conditions and Existence theorems in nonlinear elasticity, Archive for Rational Mechanics and Analysis, vol. 63 n.4, (1977) p.337-403 . [5] L. Beck, Elliptic Regularity Theory: a First Unione Matematica Italiana, Springer. [6] Bitsadze,

Course,Lecture Notes of the

Equations of Mathematical Physics, MIR Publishers Moskow.

[7] L. Boccardo, P. Marcellini, L∞ -regularity for variational problems with sharp non standard growth conditions, Bollettino della Unione Matematica Italiana, 4-A,219-225, (1990). 78 Namely

that p¯ > r.

142

[8] H. Brezis, Analyse fonctionnelle, Collection Mathematiques appliquees pour la maitrise, deuxième tirage, Masson 1983. [9] H. Brezis, Functional Analysis, Sobolev Equations, Universitext, Springer.

Spaces and Partial Dierential

[10] H. Brezis, F. Browder, Partial Dierential tury, Adv. in Math. 135, 76-144(1998).

Equations in the 20th Cen-

[11] G. Buttazzo, V.J. Mizel, Interpretation of the Lavrentiev phenomenon by relaxation, Journal of Functional Analysis, vol. 110, issue 2, 434-460 (December 1992). [12] R. Courant, D.Hilbert, Methods of Mathematical Interscience Publishers, INC., New York.

Physics,Volumes I,II

[13] E. De Giorgi, Sull'analiticità e la dierenziabilità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino Ser. III 3, 2543 (1957). [14] E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Unione Mat. Ital. IV. 1, 135137 (1968). [15] E. DiBenedetto,

Degenerate Parabolic Equations, Springer-Verlag.

[16] E. DiBenedetto, Birkhauser.

Partial Dierential Equations, second edition,

[17] E. DiBenedetto, V.Vespri, On the singular Arch.Rat. Mech. Anal. 132(1995), 247-309.

equation β(u)t = ∆u,

[18] E. DiBenedetto, U. Gianazza, Some properties of De Giorgi Rend. Istit. Mat. Univ. Trieste, Volume 48 (2016), 245-260.

Classes,

[19] E. DiBenedetto, U. Gianazza, V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics, Springer.

Remarks on local boundedness and local holder continuity of local weak solutions to anisotropic p-laplacian type equations, JEPE vol. 2,2016,p.157-169.

[20] E. DiBenedetto, U. Gianazza, V. Vespri

[21] E. DiBenedetto, U. Gianazza, V. Vespri, Local clustering of the non-zero set of functions in W 1,1 (E), Rend.Lincei Mat.Appl 17(2006) 223-225.

143

[22] E. DiBenedetto, J.M. Urbano, V.Vespri, Current Issues on Singular and Degenerate Evolution Equations, Handbook of Dierential Equations: Evolutionary Equations,pages 169-286, Verlag, North-Holland. [23] E. DiBenedetto, N. S.Trudinger, Harnack inequalities for Quasi-minima of Variational integrals, Ann.Ist.Henri Poincaré,Analyse non -Linéaire (4),295-308, (1984).

Local Holder Continuity of weak solutions for an anisotropic elliptic equation, Nonlinear Dier. Equ. Appl. 20(2013) 463-

[24] A. Di Castro, 486.

Some integral inequalities and the solvability of degenerate quasi-linear elliptic systems of dierential equations Matem. Sb.,64,

[25] Yu. A. Dubinskii,

no.3, 458-480,1964.

[26] J.F. Eastham and J.S. Peterson, The Finite Element Method in Anisotropic Sobolev Spaces, Elsevier, Computers ans Mathematics with Applications 47(2004), 1775-1786. [27] M. Eleuteri, P. Marcellini, E. Mascolo, Regularity for scalar without structure conditions, Adv. Calc. Var. 2018, preprint.

integrals

[28] A. El Hamidi, J.M.Rakotoson, Extremal functions for the anisotropic Sobolev inequalities, Ann. I.H.Poincaré- AN 24 (2007) 741-756.

Higher Integrability for Minimizers of Integral Functionals with (p, q) Growth, Journal of Dierential

[29] L. Esposito, F. Leonetti, G. Mingione, Equations 157, 414-438 (1999).

[30] L. Evans, Partial Dierential matics, vol 19, AMS.

Equations, Graduate Studies in Mathe-

A new proof of local C 1,α regularity for solutions of certain degenerate elliptic p.d.e., Journal of Dierential Equations Volume 45,

[31] L. Evans,

Issue 3, September 1982, pages 356-373. [32] L. Evans,

An introduction to stochastic dierential equations,AMS.

[33] S. Fornaro, F. Paronetto,V. Vespri, Disuguaglianza di Harnack per equazioni paraboliche, quaderno 2/2008, Università del Salento, Dipartimento di Matematica "Ennio De Giorgi".

144

Examples of the Lavrentiev Phenomenon with Continuous Sobolev Exponent Dependence, Journal of Convex Analysis, Volume 10

[34] M. Foss,

(2003), No. 2, 445-464.

Existence and non existence results for anisotropic quasilinear equations, Ann.Ist.H.Poincaré Anala.

[35] I. Fragalà, F. Gazzola, B. Kawohl, Non Linéaire 21(2004), 715-734.

[36] F. D. Gamze, P. Marcellini, V. Vespri,Space

expansion for a solution of an anisotropic p-Laplacian equation by using a parabolic approach, Riv.Math.Univ. Parma,vol.5 (2014), 93-111.

An alternative approach to Holder-continuity of solutions to some elliptic equations, Nonlinear Anal-

[37] F. D. Gamze, P. Marcellini, V. Vespri, ysis 94 (2014) 133-141, Elsevier.

[38] M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Mathematica 59,245-248 (1987), Springer-Verlag.

Multiple Integrals in the Calculus of Variations and nonlinear elliptic systems, Annals of Mathematics studies, Princeton Univer-

[39] M. Giaquinta,

sity Press, 1983.

[40] M. Giaquinta and E. Giusti, Quasi-Minima, Ann.Ist.H.Poincaré, Analyse Non Lineaire, 1, (1984),79-107. [41] M. Giaquinta and E. Giusti, On the regularity integrals, Acta Math. 148 (1982), 31-46.

of minima of variational

[42] D.Gilbarg, N.S. Trudinger, Elliptic Partial Dierential Equations of Second Order, second edition, Springer-Verlag. [43] E. Giusti, Equazioni ellittiche dell'Unione Matematica Italiana.

del second'ordine,Pitagora, Quad.

[44] E. Giusti, Direct Methods in the Calculus of Variations, World Scientic. [45] E. Giusti, M. Miranda, Un esempio di soluzioni discontinue per un prob-

lema di minimo relativo ad un integrale regolare del calcolo delle variazioni, Boll.Unione Mat. Ital. IV Ser 1(1968), 219-226.

[46] M. E. Gurtin, An introduction to continuum Science and Engeneering, vol. 158.

145

mechanics, Mathematics in

[47] Q. Han, F. Lin, Elliptic Partial Dierential Equations, Courant Institute of Mathematical Sciences (1997), AMS. [48] G. H. Hardy, J. E. Littlewood, G. Pólya, Press (1934).

Inequalities, Cambridge Univ.

A note on the anisotropic generalisations of the Sobolev and Morrey embedding theorems, Monatshefte für Mathe-

[49] J. Haskovec and C. Schmeiser,

matik, September 2009, Volume 158, Issue 1, pp. 7179.

The boundedness of generalized solutions of elliptic dierential equations, Vestnik Moskovskogo Universiteta, Vol.25, No.5, pp.44-

[50] I. M. Kolodii, 52, 1970.

On the boundedness of generalized solutions of elliptic dierential equations, Russ. Math. Surv. 38, 186 (1983).

[51] A.G. Korolev,

Boundedness of the generalized solutions of elliptic equations with nonpower nonlinearities, Moskow Wood Technology Institute.

[52] A.G. Korolev,

Translated from Matematicheskie Zametki, Vol. 42,No.2, pp. 244-255, August (1987).

[53] S.N. Kruzhkov and I. M. Kolodii, On anisotropic Sobolev spaces, 38, 188 (1983).

the theory of embedding of

Boundary value problems for nonlinear partial dierential equations in anisotropic sobolev spaces (English), ƒasopis pro

[54] A. Kufner, J.Rakosnik,

p¥stování matematiky, vol. 106 (1981), issue 2, pp. 170-185 .

[55] O. A. Ladyzhenskaya, N. N. Ural'tseva,Linear ans Quasilinear Elliptic Equations, Academic Press (translated by Scripta Technica, inc.). [56] M. Lavrentiev, Sur Quelques Problèmes du Calcul di Mat. (Ann.Mat. Pura Appl.), 4 (1926), 7-28. [57] P. Lindqvist, ematics 2016.

des Variations, Ann.

Notes on the p-Laplace Equation, Springer Briefs in Math-

[58] V. Liskevich, I. Skrypnik, Holder regularity of solutions to an anisotropic elliptic equation, Nonlinear Analysis 71 (2009),1699-1708, Elsevier.

Dispense del corso di Analisi III and Dispense del corso di Istituzioni di Analisi Superiore.

[59] R. Magnanini,

146

[60] B. Mania, Sopra (1934), 147-153.

un esempio di Lavrentie, Boll. Un. Mat. Ital. 13

[61] C. Mantegazza, Some Univ. Napoli, preprint.

elementary questions in calculus of variations,

Un exemple de solution discontinue d'un probleme variationnel dans le cas scalaire,(1987-1988), Università degli Studi di Firenze.

[62] P. Marcellini,

Regularity of Minimizers of Integrals of the Calculus of Variations with Non Standard Growth Conditions, communicated by E.

[63] P. Marcellini,

Giusti,Università degli Studi di Firenze.

Regularity and Existence of Solutions of Elliptic Equations with p,q-Growth Conditions, Journal of Dierential Equations 90,1-

[64] P. Marcellini, 30 (1991).

[65] P. Marcellini, Regularity for some scalar variational problems under general growth conditions,Journal of optimization theory and applications: vol 90 No.1 pp. 161-181, July 1996.

On the denition and the lower semicontinuity of certain quasiconvex integrals,Annales de l'I.H.P. ,section C, tome 3, n 5(1986), p

[66] P. Marcellini, 391-409.

[67] P. Marcellini, Alcuni recenti sviluppi nei problemi Hilbert,Boll.Un.Mat.Ital. 11-A(1997), 323-352. [68] E. Mascolo,

19esimo e 20esimo di

Appunti di Analisi Funzionale.

[69] N. E. Mastorakis, H. Fathabadi, On the Solution of p-Laplacian for nonNewtonian uid ow, WSEAS trans. on Math., Issue 6,Volume 8, June 2009.

Regularity of minima: an invitation to the dark side of the calculus of variations, Applications of Mathematics, 51(2006), No.4,

[70] G. Mingione, 355-425.

[71] C.B. Morrey, Second order elliptic systems of partial dierential tions, Annals of Math. Studies,n. 33, Princeton Un. Press (1954).

equa-

[72] J. Moser, A New Proof od De Giorgi's Theorem Concerning the Regularity Problem for Elliptic Dierential Equations Communications on Pure and Applied Mathematics ,vol XIII,457-468 (1960). 147

[73] J. Nash, Continuity May 1958.

of solutions of parabolic and elliptic equations, 26

Example of an irregular solution to a nonlinear elliptic system with analytic coecients and conditions for regularity, Theor. Nonlin.

[74] J. Necas,

Oper. Contr. Aspects. Proc. 4th Int. Summer School. Akademie-Verlag, Berlin 1975 , pp.197-206.

Imbedding theorems for functions with partial derivatives considered in various metrics, Amer.Math.Soc.Transl. (2) Vol.90,

[75] S. M. Nikol'skii, 1970.

[76] Y.H. Pao,Electromagnetic Forces in Deformable Continua, Mechanics Today (s.Nemat-Nasser,ed) ,vol.4, Pergamon Press, 1978, pp. 209-306. [77] K.R. Rajagopal and M.Ruºi£ka, On the Modeling of Electrorheological Materials, Continuum Mechanics and Thermodynamics (2000). [78] J.Rakosnik ,Some 4(46), pp. 471-497. [79] F. Rosso,

remarks to anisotropic Sobolev spaces II, Mat. Sb

Dispense del corso di Istituzioni di Fisica Matematica.

[80] W. Rudin,

Principles of Mathematical Analysis, Mc Graw-Hill.

[81] M. Ruºi£ka, Electrorheological ory, Springer.

Fluids: Modeling and Mathematical The-

Global Boundedness of Solutions of Anisotropic Variational Problems, Bollettino U.M.I. (7) 5-A (1991), 345-352.

[82] B. Stroolini,

[83] A. Telcs, V. Vespri, A quantitative Lusin theorem for functions Springer INdAM series 13, Geometric methods in PDE's. [84] M. Troisi, Teoremi di inclusione Ricerche di Mat. 18, 3-24, (1969).

in BV,

per spazi di Sobolev non isotropi,

[85] M. Troisi, Ulteriori contributi alla teoria degli isotropi, Ricerche Mat. 20, pp. 90-117 (1971).

spazi di Sobolev non

On the weak continuity of weak operators and applications to potential theory, Amer. J.Math. 124 (2002), 369-410.

[86] N. S. Trudinger, X. J. Wang, [87] J.M. Urbano, The ematics, Springer.

Method of Intrinsic Scaling, Lecture Notes in Math148

[88] W. P. Ziemer, Weakly Dierentiable functions- Sobolev Spaces and Functions of Bounded Variation, Graduate texts in Mathematics 120.

Averaging of functionals of the Calculus of Variation and elasticity theory, Mtah,USSR Izvestiya, Vol 29 (1987) 29-33.

[89] V. V. Zikhov,

Acknowledgments Finally the always pleasant task of acknowledgments, by me and my advisor professor Vincenzo Vespri, to whom I express my greatest gratitude, here deserves its due space. Throughout these months various colleagues, students and professors, and many other people have helped us with constructive criticisms, suggestions, advice and encouragement to the production of this work. Special thanks are due to professor Paolo Marcellini, who sided us from the very beginning, for his suggestions during the whole work and for showing us that the special case is more often harder than the general one, as expressed by Polya's anecdote. Another fundamental sustain has to be acknowledged to professor Elvira Mascolo, who joined us and contributed substantially on the presentation of anisotropic Sobolev spaces, as well as on encouragement given by her powerful enthusiasm on this research domain. We thank professor Fabio Rosso for his advice on the application of operators with anisotropic growth conditions to a special class of uids: the consistence of a mathematical theory improves when a down-on-earth use of it can be given. We thank engineer Lucia Cataldo for her help with illustrations and gures, to which is to be attributed a more uent exposition of very technical methods. Finally we thank students and colleagues met in the conference "Metric Analysis and Regularity" held in Catania, for their brilliant wit shared with us, and students and colleagues of our Ulisse Dini department, with a special acknowledgment to phd student Tommaso Di Marco who gave sustain to the student, for answering to "those questions that a student does not have the courage to ask to his teacher".

To all, our sincerest thanks.

149