Elliptic and Parabolic Equations with Discontinuous

Elliptic and Parabolic Equations with Discontinuous Coefficients. A. Maugeri, D. K. Palagachev, L. G. Softova Copyright © 2002 Wiley-VCH Verlag GmbH & Co. KGaA ISBNs: 3-527-40135-0 (Hardback); 3-527-60086-8 (Electronic)

Elliptic and Parabolic Equations with Discontinuous Coefficients Antonino Maugeri Dian K. Palagachev Lubomira G. Softova

MATHEMATICAL RESEARCH Volume 109

Elliptic and Parabolic Equations with Discontinuous Coefficients Antonino Maugeri Dian K. Palagachev Lubomira G. Softova

)WILEY-VCH Berlin • Weinheim • New York • Chichester Brisbane • Singapore • Toronto

Authors: Prof. Dr. Antonino Maugeri, Dipartimento di Matematica, Universita di Catania, Catania, Italia Prof. Dr. Dian K. Palagachev, Dipartimento di Matematica, Politecnico di Bari, Ban, Italia Dr. Lubomira G. Softova, Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria 1st edition Library of Congress Card No.: applied for

Die Deutsche Bibliothek - CIP Cataloguing-in-Publication-Data A catalogue record for this publication is available from Die Deutsche Bibliothek

ISBN 3-527-40135-0 ISSN 0138-3019

This book was carefully produced. Nevertheless, authors, editors, and publishers do not warrant the information contained therein to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details, or other items may inadvertently be inaccurate. All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form - by photoprinting, microfilm, or any other means - nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.

© WILEY-VCH Verlag Berlin GmbH, Berlin (Federal Republic of Germany), 2000 Printed on non-acid paper. The paper used corresponds to both the U.S. standard ANSI Z.39.48 - 1984 and the European standard ISO TC 46. Printing: GAM Media GmbH, Berlin Bookbinding: Druckhaus ,,Thomas Miintzer", Bad Langensalza Printed in the Federal Republic of Germany WILEY-VCH Verlag Berlin GmbH Buhringstr. 10 D-13086 Berlin Federal Republic of Germany

To our children and our families

Preface The book presents various results concerning second order elliptic and parabolic equations with discontinuous coefficients. Special emphasis is placed on the solvability and regularity theory of the principal boundary value problems both for linear and nonlinear operators. It is intended for all the specialists working in the area of partial differential equations, for post-graduate students and university students specializing in nonlinear analysis, PDE, harmonic analysis, functional analysis. We would be glad if this book could become a token to our friend Filippo Chiarenza, who passed on in 1996 at a very young age. Filippo was very involved in the topics dealt in this book, and we have always been influenced by him as a human being and as a scientist. We are particularly grateful to Mrs. Gesine Reiher from Wiley-VCH, Berlin, for the cooperation and detailed revision of the manuscript. The book was printed directly from a l^T^X output prepared by ourselves, and we therefore would like to apologize to the readers for any prospective misprints. July 2000

The authors

Contents Introduction

11

Chapter 1. Boundary Value Problems for Linear Operators with Discontinuous Coefficients 15 1.1. Examples of Nonsolvable BVPs for Linear Operators with Discontinuous Coefficients 15 1.2. The Cordes Condition 19 1.2.1. Solvability in W 2 ' P (H), p 2 22 1.2.2. Regularity in Morrey Spaces 23 1.2.3. Higher Summability of the First Derivatives 28 1.3. Operators with Sobolev Coefficients 34 1.4. Miranda-Talenti Estimate 39 1.5. Elliptic Equations in the Plane 46 1.5.1. Dirichlet Problem 46 1.5.2. Oblique Derivative Problem 48 1.6. Cauchy-Dirichlet Problem for Parabolic Operators 57 1.6.1. The Parabolic Cordes Condition . 57 1.6.2. A Local Regularity Result in Morrey Spaces 63 1.6.3. Higher Summability of the Derivatives 70 1.6.4. Parabolic Operators with Sobolev Coefficients 77 1.7. Oblique Derivative Problem for Parabolic Operators with Measurable Coefficients 83 1.7.1. The Case of Two Spatial Variables 83 1.7.2. Multidimensional Problems 89 1.8. Elliptic Oblique Derivative Problems 92 1.8.1. Operators with Measurable Coefficients 92 1.8.2. Operators with Sobolev Coefficients 100 Chapter 2. Linear and Quasilinear Operators with VMO Coefficients 103 2.1. Preliminaries and Auxiliary Results 105 2.1.1. The Elliptic Case 105 2.1.2. The Parabolic Case 113 2.2. Regular Oblique Derivative Problem — a Special Case 116 2.2.1. An Extension Lemma 118

10

Contents

2.3.

2.4.

2.5.

2.6.

2.2.2. Boundary Representation Formula 2.2.3. Boundary A Priori Estimate 2.2.4. Global Sobolev Regularity and Solvability Regular Oblique Derivative Problem — the General Case 2.3.1. Global A Priori Estimate 2.3.2. Uniqueness Criterions 2.3.3. Strong Solvability Singular Oblique Derivative Problem 2.4.1. Higher Sobolev Regularity 2.4.2. Proof of Theorem 2.4.1 Parabolic Oblique Derivative Problem 2.5.1. Small Time Representation Formula for the Solution 2.5.2. W^(QT) A Priori Estimate 2.5.3. Unique Solvability Quasilinear Operators with VMO Coefficients 2.6.1. A Priori Estimates 2.6.2. Solvability of the Quasilinear Problem

122 125 132 135 136 139 140 142 144 149 153 156 160 164 165 167 173

Chapter 3. Nonlinear Operators with Discontinuous Coefficients

177 3.1. Nonlinear Cordes Condition 177 3.1.1. Campanato's (A2) Condition 177 3.1.2. The Theory of Near Mappings 183 3.1.3. Near Operators in Hilbert Spaces 186 3.2. Nonlinear Elliptic Systems 188 3.2.1. Nonlinear Operators and Near Mappings 188 3.2.2. A Theorem on Local Solvability 192 3.2.3. Dirichlet Problem for Nonlinear Elliptic Equations in the Plane 193 3.2.4. Other Boundary Conditions 205 3.2.5. Regularity Theory. Higher Integrability and Holder Continuity 206 3.3. Parabolic Systems 217 3.3.1. Strong Solvability Results 217 3.3.2. Regularity Theory. Higher Integrability of D2u and Dtu . . . 220 3.3.3. Basic Parabolic Operators. Holder Continuity 223

Appendix

A.I. Functional and Real Analysis Tools A.2. Maximum Principles

235

235 237

Bibliography

241

Index

253

Functional Spaces and Their Respective Norms

255


Index (A) condition

177

2

(A ) condition

Cordes condition 12f. elliptic

178, 183

vectorial

182

2

(A *) condition (Ap) condition

12, 20, 68

for systems

190 191

69

nonlinear

181

parabolic

58

Aleksandrov-Bakel'man-Pucci maximum principle 238f.

differential operator elliptic

basic operators

parabolic

11 57, 154

elliptic 210

strictly elliptic

parabolic

uniformly elliptic

BMO

223

24, 105, 113

11 11

Dini continuity 108

Caccioppoli type estimates

25, 63,

207, 223f.

elliptic regularization

149

elliptic regularizing property

104, 117

Calderon-Zygmund inequality 39 Calderon-Zygmund kernel 108f. parabolic 114

144

finite type vector field 144

variable 114 Campanato space

finite order of contact

24, 107, 221

Caratheodory condition

13, 177

fundamental estimates

212ff.

fundamental solution of elliptic operator

122

Cauchy-Dirichlet problem 57, 83, 115

of parabolic operator

113

Cauchy-Neumann problem 83, 153 commutator

109ff., 114

conormal derivative problem 36, 82 controlled growth 210

7/(r)

106

Gagliardo-Nirenberg inequality 236

Index

254

Gehring-Giaquinta-Modica Lemma

operator monotone

209 generalized reflection

110, 115

infinite type vector field 144

Kn property

183ff.

strictly monotone 186

Gronwall inequality 236

Jensen inequality

near

186

parabolic boundary

115, 238

Poincare inequality

236

pointwise Hormander condition

236

Riesz potential

23, 221

112

236

Riesz-Schauder theory

100, 141

Robin problem 135 Leray-Schauder fixed point theorem

235 Shapiro-Lopatinskii compatibility condition Marcinkiewicz space

39, 106

method of continuity

235

50, 104

solution classical

Miranda-Talenti estimate (inequality)

21, 39

generalized (variational) strong

Morrey space

24, 67, 107, 221

12

weak

11, 15

12, 16, 116, 154 11, 15

strong monotonicity condition 179 natural structure conditions 167 near operators

183ff.

Nemyckii operator

tangency set

237

142

tangential gradient

41

neutral type vector field 143

VMO oblique derivative problem regular

VMO-modulus of continuity

104, 117, 136, 155

singular (degenerate)

24, 106, 114

104, 142

volume potential

113

106


Introduction The general goal of the present book is the comprehensive development of the solvability and regularity theory of second order elliptic and parabolic equations with discontinuous coefficients and nondivergence principal part. To fix the ideas, consider the linear equation Cu = aij(x)DXiXju

=f(x)

(1.1)

where the coefficients a^(x) and the right-hand side f(x) are defined on a bounded domain ft C M n , n > 2. The ellipticity of the operator C will be expressed, as usual, by the requirement of positive definiteness of the coefficient matrix A(x) — {û(x)}^-_1. C is strictly elliptic if X(x) > const > 0 in H,

(1.2)

where X(x) is the minimum eigenvalue of A(x), while uniform ellipticity means boundedness of the spectrum of A(x), < const

in

ft

(1.3)

with A(x) being the maximum eigenvalue of A(x). Suppose for a moment that C has a divergence structure jC'u = DXi(a^(x)DXiu)=f(x).

(1.4)

If a u 's are sufficiently smooth, then C1 reduces to C modulo a term containing lower order derivatives of u. However, if the coefficients are only bounded and measurable functions in ft, aij(x) EL 0 0 (ft),

(1.5)

and / is integrable in ft, then one defines a weak (generalized or variational) solution to (1.4) as a function u belonging to L 2 (ft) along with its weak derivatives DXiu, i = 1,... , n, (u p (n) for suitable p G (1, oo), satisfying the equation Cu = / almost everywhere (a.e.) in 0 and that concept makes sense also for / G Lp($l) and a u satisfying (1.5). Unfortunately, the solely assumptions (1.2) and (1.5) are not enough to ensure strong solvability of (I.I) as shown by C. Pucci (see Section 1.1). (The planar case, n = 2, propounds a remarkable exception of this feature, when the boundary value problems for (I.I) are solvable without any additional requirements.) A precise analysis of Pucci's example suggests that a reasonable strong solvability theory of (I.I) cannot be built without suitable additional hypotheses on the principal coefficients a u of £. These should be either regularity assumptions on a u 's or conditions on the spectrum of the matrix A(x) stronger than (1.2) and (1.3). For instance, if a1-7' are continuous functions in 17, a^(z)GC0(ft),

(1.6)

p

a satisfactory theory (known as "L -theory") exists, which provides solvability and regularity for (I.I) in Sobolev spaces W2'p(£l) for any p > 1. To put the final touches to these brief notes, we forward the reader to the classical monographs of Ladyzhenskaya and Ural'tseva [LU 1], Ladyzhenskaya, Solonnikov and Ural'tseva [LSU], Miranda [MR 5] and Gilbarg and Trudinger [GT] for all respective results and details. Our principal objective here is to develop an analogue of the Lp-theory of elliptic and parabolic equations, weakening the continuity assumption (1.6) on the principal coefficients of the operators under consideration. The motivation to deal with that topic is linked to the fact that mathematical modeling of numerous physical and engineering phenomena leads to boundary value problems for discontinuous elliptic or parabolic operators which require strong solutions. Moreover, the study of nonlinear problems in Sobolev classes bring along problems with L°° data. Chapter 1 proposes a relatively self-contained survey on the results concerning the principal boundary value problems for linear elliptic and parabolic equations with discontinuous coefficients. Special attention is paid to the so-called Cordes condition, introduced by H. O. Cordes in the study of Holder continuity of solutions to (I.I). It

Introduction

13

replaces (1.6) with the requirement that the eigenvalues \i(x) of the matrix A(x) do not scatter too much. The Cordes condition enabled G. Talenti to derive strong solvability in PV 2 ' 2 (H) of the Dirichlet problem for the operator C and his work was followed by various studies dealing with other boundary value problems for elliptic and parabolic operators. Another class of discontinuous coefficients considered in Chapter 1 is that introduced by C. Miranda and formed by functions belonging to the Sobolev space Wl>n(Sl) (Dxaij G L n + 2 , Dtaij G L< n + 2 )/ 2 in the case of parabolic operators as studied by F. Guglielmino). We have collected the central results regarding solvability and regularity theory both in Sobolev and Morrey functional spaces of linear operators with either Cordes or Wl'n principal coefficients. Although these two types of discontinuity are substantially different, the approaches in studying boundary value problems are unified on the base of the elegant Miranda-Talenti inequality which permits an exact calculus of the constants appearing in the L2-a priori bounds. The rest of the tools employed in Chapter 1 consists of suitable algebraic estimates, imbedding results of special type, fixed point theorems and approximation. We should emphasize on the fact that the analytic implements exploited restrict the considerations in the frameworks of VF2'2 or W2>p with p greater than, but sufficiently close to 2. Chapter 2 deals with Lp-theory of elliptic and parabolic operators with principal coefficients au belonging to the Sarason class VMO of functions whose integral oscillations over balls shrinking to a point converge uniformly to zero. In two pioneer articles of early '90s, F. Chiarenza, M. Frasca and P. Longo succeeded to modify the classical methods for obtaining Lp estimates of solutions to (I.I) which allowed them to move from (1.6) into a^ G VMO. Roughly speaking, the approach goes back to A. Calderon and A. Zygmund and makes use of an explicit representation formula for the second derivatives D2u of any solution to (I.I) in terms of a singular integral acting on /, and a commutator acting on the same derivatives D2u but having as density a1^ (x) — ali(y). Thus, if the coefficients ali have a "small integral oscillation" (that is, au G VMO), then the Lp-norm of D2u is bounded in terms of Lp-norm of / and this holds for any p G (l,oo). Taking into account the fact that VMO contains as proper subsets both C°(fi) and W 1 ) n (n), the Lp-theory of operators with VMO principal coefficients really generalizes what was known before 1990. After a brief survey on the results concerning Dirichlet problem for linear elliptic and parabolic operators, we develop the Lp-theory, for all p G (l,oo), of the regular oblique derivative problem for operators with VMO coefficients. Indeed, the representation formula contains now additional terms due to the first order boundary operator and this requires machinery different from that applied to Dirichlet problem. Although the respective results are not surprising, they are new and applicable also to singular oblique derivative problems. We complete Chapter 2 with a section dealing with Dirichlet problem for quasilinear elliptic operators with principal coefficients belonging to VMO with respect to the independent variables. Chapter 3 is devoted to nonlinear elliptic and parabolic systems prescribed by Caratheodory's functions (that is, functions which are measurable with respect to the independent variable and continuous in dependent ones). The ellipticity of the nonlinear operators should be understood in the sense of a condition introduced by S. Campanato which implies the usual uniform ellipticity and ensures, roughly speak-

14

Introduction

ing, that the operator under consideration behaves like the Laplacian, both considered as mappings from PF2'2 into L 2 . In other words, Campanato's ellipticity condition is a nonlinear version of the Cordes condition. We sketch the main notions and results of the theory of near mappings and apply it to the study of boundary value problems for nonlinear systems. By the aid of local energy estimates of Caccioppoli type, various regularity results in Sobolev, Morrey and Holder spaces are derived for the solutions of the systems considered. The book closes with Appendix which collects real- and functional-analysis results, maximum principles and useful inequalities used throughout the exposition. Basic Notations N = set of positive integers; Z — set of all integers; E = set of real numbers; E+ = set of positive real numbers; [q] = integer part of q G E. En = Euclidean n-dimensional space, n > 2, with points x — (zi, . . . ,x n ); \x\ — 1 /9

\\x\\n = (ElLi X1) J (x I y)n = the scalar product^ E"j=i ^2/t(jjn = volume of the unit ball in En . For two n x n matrices A = {aij}lj=l , B = {bij}%j=1 set: (A\B)n = E"j=i

aij

*>ij]

For a point set 5 define: dS = boundary of 5, 5 = closure of S = S U 5J denote: Ker^t = the kernel (the null-space) of .4 = {u e It: A(u) - 0}, Im^4 = the range of A = ^l(il), Coker^l = the cokernel of A = 93/im>f Throughout the book the standard summation convention on repeated indices is adopted.


Chapter 1

Boundary Value Problems for Linear Operators with Discontinuous Coefficients The general goal of the present chapter is to provide a relatively self-contained survey on the Lp-theory of boundary value problems for linear elliptic and parabolic operators with discontinuous coefficients. Special emphasis is placed on operators with principal coefficients satisfying the Cordes condition or belonging to appropriate Sobolev spaces. By the aid of various analytic techniques we show how these requirements imply strong solvability and regularity for the problems under consideration.

1.1.

Examples of Nonsolvable BVPs for Linear Operators with Discontinuous Coefficients

The variational approach for the partial differential equations deals with divergence form equations such as Di (aij(x)DjU + 6*(z)u) + ci(x)Diu + d(x)u = Difi(x)+f(x),

(1.1)

where x E H, fi C E n . A weak (generalized, variational) solution of (1.1) is any function u € Wl^($l] satisfying / ^(aij(x)Dju + bi(x)u)Di(p-

(ci(x)Diu-d(x)u}^dx

Then the weak solution to (1.1) needs only to be once weakly different iable, but one has the advantage that the solutions of boundary value problems for (1.1) exist under

16

Chapter 1. BVPs for Linear Operators with Discontinuous Coefficients

very general and minimal assumptions on the data, such as aij A|£| 2

l

a.a. x € 0, V f G E n ,

(1.2)

p

and 6%c*,d, / ,/° belonging to suitable L (0) spaces in such a way that the bilinear form associated to (1.1) is coercive. Very different is the situation when one looks for strong solutions, for example, of the linear Dirichlet problem

a^(x)DijU = g(x) a.e. 0,1 u=0

l

on L 2 (0) for which the Dirichlet problem (1.3) does not admit a solution. Let n > 2 and set 0 = {x G En : \x\ < I } . Consider the operator ,

n n

( -

with Kronecker's 5^ and set A for the coefficient matrix

The operator C is strictly elliptic since for each £ G En and almost all x G 0 we have = (1 - An) /

i

2

- , l=l

Considering the eigenvalues \i(x) of the matrix A it results \i(x) > A a.e. in 0 and, taking into account the invariance of the trace of A, Tr A — ]CILi î(x} — lî=i ail(x}, we get

Further, having in mind the invariance of Tr.A 2 , i.e., YJi=i tf(x) = we obtain

Y%j=i(aij(x))2

i=l

/

\

n

i=l 2 , \22n

(1 - An) + A n + 2A(1 - An)

A 2 n(n- 1) - 2A(n - 1) + 1.

^2

1.1.

Examples of Nonsolvable BVPs

17

Then it is easy to verify that

due to the choice of A (0 < A < Ê^y). Setting W*?(ti) = W2>2(ty n W^2(fl) we note that the solvability of (1.3) is equivalent of finding u G W7"^2 such that Cu = g for almost all x G fi. To show the nonsolvability of the Dirichlet problem (1.3), we are going to construct a function g G L 2 (f£) for which (1.3) is not solvable. More precisely, we shall show that the linear and continuous operator

is not an isomorphism. Let us choose A' = A(n — !)/(! — A(n — 1)). It results A' < n/2 — 1 and then we take a number // such that A7 4-1 < IJL < n/2. Set

9k\ ) — gk(\ \) — S - ._ I

„

|X|

II

i

I .

11 f\j \ \JU\ \ J.,

and I I

0) with bounded coefficients. Define further the function ui(x) — x\x - 1. Then DijUi = A(A - 2)xiXj\x\x~4 since 2 — n/2 < A. Moreover, CUl = \x x~2 [A2 4- (n - 2)A + b\(X - 1)] = 0 a.e. fl. Therefore, the Dirichlet problem £u = 0 a.e. 0,

u = 0 on dft

1.2. The Cordes Condition

19

admits two solutions u\(x) and u^(x) — 0 belonging to W^2(fl). Let us note that in the examples presented above the dimension n of the space is always strictly greater then 2. The two-dimensional case presents an extraordinary exception which will be analyzed later (see Section 1.5). In fact, it has been proved by Bernstein in [B] and by Talenti in [TL 1] that the solely condition (1.2) (i.e., uniform ellipticity of the operator and essential boundedness of its principal coefficients) is enough to ensure isomorphic properties of C.

1.2.

The Cordes Condition

Some additional considerations (see [TL 3]) on the operator £ = fef (1 - An) + Â Z? tf ,

0 < A < -,

can help us to understand how the Cordes condition arises and what its meaning is. We have already shown that if n > 2 and 0 Anx): > Xdx) — 1, Q \^ z V ; 4^ v ; f

\~^ X 2 /

\

V^ A

/

N

-1

Ai(rc) > A a.e. \l. n ;_ A

/

N

A

^

i = l.....n

(1.6) In fact, the last supremum coincides with n

max^ /

n

Xj i / x>i —— J.,

- (n-l)A2

^ — ±,... ,i

/

n-l

( ^-^

a/j ^ /\,

2

/ -i

V^

\

V^

î

A

>. A

1.2.


19

admits two solutions u\(x) and u^(x) — 0 belonging to W^($l}. Let us note that in the examples presented above the dimension n of the space is always strictly greater then 2. The two-dimensional case presents an extraordinary exception which will be analyzed later (see Section 1.5). In fact, it has been proved by Bernstein in [B] and by Talenti in [TL 1] that the solely condition (1.2) (i.e., uniform ellipticity of the operator and essential boundedness of its principal coefficients) is enough to ensure isomorphic properties of C.

1.2.


Some additional considerations (see [TL 3]) on the operator £ = fef (1 - An) + Â Z? tf ,

0 < A < -,

can help us to understand how the Cordes condition arises and what its meaning is. We have already shown that if n > 2 and 0 Xdx) — 1, Q \^ 4^ v ;

A

/

N

A

^

Ai(rc) > A a.e. \l. n ;_

i = l.....n

(1.6) In fact, the last supremum coincides with

* = = lj x *- A j ^ = i 3 - - - 3

max

- (n-l)A2


20


Now, (1.6) yields 71

-,

V" A?(z) p(ty fl W 0 1>p (n) equipped with the norm

/ =

/ (E

\JQ

i,j=l

It is well known (cf. [GT]) that for each / G I/ p (ft), p > 1, the Dirichlet problem for the Poisson equation AM = /,

u G W2'"(n)

admits a unique solution satisfying the estimate IHIjy 7 2 .p(n) < C ( p ) \ \ f \ \ L P ( Q ) with a constant C(p) depending also on n and fi. Moreover, if 0 is a convex domain, (7(2) — 1 in virtue of Lemma 1.2.2 (the Miranda-Talenti estimate). Then the Laplacian A: W^j p (fl) —> I/ p (f£) is an isomorphism for each p > 1. Let us denote by A-^p) the inverse operator A"1: Lp(ti) -> W^f(n). It follows from Lemma 1.2.2 that || A"1 (2) || < 1 and fixing a number r G (2, -f oo), by means of the convexity of the norms, we obtain

with l/p = a/2 -f (1 - a)/r, a G (0, 1). Indeed, the inequality (1.14) is valid also for r G (1,2). Then the following result holds (see [CM 5]). Theorem 1.2.3. Under the assumptions of Theorem 1.2.1 there exist two real numbers po and pi, 1 < PQ 1 as p —> 2. Therefore, there exists pi £ [2,r] such that 1

for each p £ [2,pi]. Analogously, there is a number p0 6 (1,2) such that for each p £ (p 0 ,2] the above estimate holds. Now taking / £ Z/ p (ft), p £ (po,Pi), and consider the mapping T: W%*(Sl) -» W^fl) denned as before by setting U = Tw, w £ with [/ being the unique solution of the problem AE7 = of + (A - a£)w,

C7 £ ^"(17).

Proceeding similarly as in the proof of Theorem 1.2.1, we shall prove that T is a contraction. Thus, taking into account (1.13) we obtain for each couple w 1,^2 €

* / Jv, ^ 0 P (Q)'

Since ||A x(p)11\/l — e < 1, the assertion follows and the problem (1.15) admits a unique solution. The estimate (1.16) follows by observing that Aw = af + (A - aC}u and

D

1.2.2.

Regularity in Morrey Spaces

The next regularity assertion involves the Morrey and Campanato spaces L p ' A (fJ) and £ P ' A (0), respectively, whose definitions and main properties are sketched below. Let Q be a bounded domain of W1 and let u be a real- valued function defined on n. Set £(zV) = { z £ R n : \ X - X Q \ x(£l) for 0 < A < n. If n < A < n + p, then with a - ^—-

and where sup

that L°°(n) C £ p ' n (ft) C L p (ft) for all p>l. We should mention that £ p ' n (ft) = £ 1 ' n (f]) = BMO and the space BMO (functions with bounded mean oscillation) will play very important role in what follows in Chapter 2 together with its subset VMO (functions with vanishing mean oscillation). The Morrey regularity result is the next one (see [GI]) Theorem 1.2.7. Under the assumptions of Theorem 1.2.1, let q be the positive number q=

sup

log ((1 -I- \/l - e)x + \/l -e) ^ — L.

(i.i 8 )

/or eac/i /(x) G L 2 ' A (Q) ; A < nq, the solution of the Dirichlet problem (1.9) fulfills the bound n

( n).

(1-19)

1.2. The Cordes Condition

25

Proof. The result is achieved by the aid of some preliminary results (Caccioppoli type estimates) having interest in themselves. Lemma 1.2.8. Let v(x) be a harmonic function on the ball B(R) = B(0,R). Then for each p and a, 0 < p < a < R, it results \D fdx l ljV Let us point out the crucial importance of the fact that in the last estimate the constant on the right-hand side equals 1 and the exponent of the power (p/cr) n is exactly the dimension of the space En . Lemma 1.2.9. Let v(x) be a harmonic {unction on the semisphere B+(R) = B(R) fl [xn > 0} and such that v(x) - 0 on TR - dB+(R) n [xn - 0}. Then for any 0 < p < a < R, it results

Lemma 1.2.10. (cf. [KN]) Let (p(t) be a nonnegative, nondecreasing function defined on the interval [0, S] such that for each couple of numbers p and R, 0 < p < R < S it results (p(p) < ((I + >/l -e)(p/R)x + ^/l - e) 0 and 0 < e < 1. Let q be the number given by (1.18). Then there exists a positive constant C(e) such that for each couple p, R, 0 < p < R < 8, one has

Lemma 1.2.11. (cf. [CM 4]) Let (p(i) be a nonnegative, nondecreasing function defined on the interval [0, S] such that for each p and R, 0 < p < R < 5, it results

with A,B,a and (3 positive constants, a — /? > 0. Then there exists a constant C depending on A and a — /3 such that for each p and H, 0 < p < R < 5, the following estimate holds

The proof of Theorem 1.2.7 is achieved, following the steps below. (Recall that C is an elliptic operator satisfying the Cordes condition.)

26


Step 1: Let u(x) G W2^(B(R}) be a solution to the equation Cu = 0. Then for each 0 < p < R it results

n 2

\DijU\2dx

.

(1.20)

Proof. Split the function u into the sum u — v -f ^, where v solves the Dirichlet problem Av = 0 a.e. £(#), i; - w G and w is a solution of = (A-a/> with a(x) - ^=1

flii

a.e.B(R),

(^) Ei!j=i(flij' W) 2 -

Tt follows from

Lemma 1.2.8 that

whereas, using the Miranda-Talenti estimate (Lemma 1.2.2) and (1.13), we have

r

/

JB(R]

n

r

Y] \DijW\2dx < / {J=1

J B(R]

|(A - aC)u\2dx 0 and (1.28) yields /V E (DljU)2dx < (l + ^) (l + -)a 2 /" + [(l + H V i + ^ ) ( i _ e ) +

ff J

I g> Jo

Thus, choosing cr = i/(l + ?y)(l — e) we get

»J = 1

+ [ 0 small enough, (1.29) reads f 0* ]T (DijU)2dx n £(ty for each e > 0. Recall that a^'s are coefficients of an elliptic operator £ such that unicity breaks down for the associated Dirichlet problem. Thus, a u G W 1 > n ~ e (ft) does not imply that £ is an isomorphism. On the other hand, if one considers £ = a^(x)Dij with coefficients a^ G Wl'n+£(ti), then a** G C°>a(tt) and £ is isomorphism between W^(ti) = W 2 ' 2 (ft) r}W^2(ty and L 2 (ft). The limit case aij €

wl>n(Sl)

(1.35)

has been studied by Miranda in [MR 2]. Existence and regularity results have been proved for solutions of the Dirichlet and conormal derivative problems associated to the operator £ = alj(x)Dij under the assumptions (1.35) and the usual uniform ellipticity condition n

A | £ | 2 < £^(:r)^ 2. Moreover, if q < n the estimate (1.40) holds with a constant C depending also on the Lipschitz constants of ali 's in a neighborhood of dfl. (Recall that yc — (xi, . . . , xn) is the vector of unit outward normal to It is worth noting that, the Neumann problem in general does not admit solutions if the condition c(x] < CQ < 0 fails. However, if a solution does exist, the estimate (1.40) holds with right-hand side ||/||L9(n) + I Proof. Only a sketch of the proofs of Theorems 1.3.1 and 1.3.2 will be proposed. Further, we will prove the results assuming aij G C°(ty, u G C2(H) n C 3 (fi). Indeed, in the case of continuous coefficients the Dirichlet and conormal derivative problems admit unique strong solutions (cf. [LU 1], [GT]). Thus, our assertions will follow by the aid of usual density arguments. As consequence of (1.36) Miranda (see [MR 1]) shows the validity of the following estimate n

n

A2 ]T (DijU)2 < f2 + J] (a*V - aârs)Dk(DiUDrsu).

(1.42)

1.3.

Operators with Sobolev Coefficients

Setting Ok = (a u DiuDju) A2

37

with a real positive &, we derive from (1.42) „

n

„

n

< \ 8kf2dx + I 9k

/ ek Y^ (Dijufdx

"

r

]T DiuDrsuDj6k(aijars

+ /

J£l i

•r

P

+ I e °

-airajs)dx

s=±

n k

J

JT DiuDrsUXjâi* - aijars)da

Y;

DiuDrsuDj(aijars

-airais)dx.

ij,r,s=l

(1.43)

Let Pk = /n (E"=i(Diu)2)k E"j=i(Diju)2dx and denote by Ji, J2, J3, J4 respectively, the four integrals at the right-hand side of (1.43). It results

Jip (^) (the closure of (70° (H) with respect to the norm of W 2 ' p ((7)), the Calder on- Zygmund inequality (see [CZ 1], [ST 1], [GT])

1.4. Miranda-Talenti Estimate

39

Wl>n C VMO (see [CFL 1], [CFL 2]), these results will really improve what was derived by Miranda. Another question to arise is whether the above results remain true under the more general assumption n/e 0 < 0 < 1. aij € w°' (Sl) (The symbol We'n/e stands for the Besov (or fractional Sobolev) space, see [AD]). Apart from some particular cases ([CAZ, n = 3, 9 = 3/4], [CA, Wl-*/™(tt), p G (6 - e,6), n > 3], see also [Z 1], [Z 2]), this question first posed by Miranda, has now an affirmative answer. In fact, as it will be seen in Chapter 2, also the space W e > n / 0 (fi), 0 < 6 < 1, is contained in VMO and both the Dirichlet and oblique derivative problem are always solvable in W2'p(ft) for each p G (1, oo), assuming a u G VMO(fi). It is worth noting that the Dirichlet problem for the operator C = a^(x)Dij has been studied in VF 2 ' 2 (n) n W^2^) under less restrictive assumptions than a u G Wl>n(ty. For instance, the paper [CT] deals with solvability and Ty2'2-estimates of solutions to Dirichlet problem in the special situation when a u (x) = alj(xn). Employing divergence-form-equations techniques of [LU 1], Franciosi and Fusco derived existence and regularity results both in W 2 ' 2 (ft)fWo' 2 (ft) ([FF 1]) and w£'cp(n) ([FF 2]) if a" G L°°(n) and Dk(a^) G Ln(ti) for 1 < k < n - 1. Finally, making use of rearrangement and symmetrization techniques, Alvino, Buonocore and Trombetti were able to improve Miranda's result assuming a u G L°°(n) and Dk(a^) G ICakW for 1 < /c < n (cf. [AT], [ABT]). Recall, that the Marcinkiewicz space L^ eak (fi) = Lp'°°(n) = weak-Lp((7) is defined according to the norm

- sup

f \u\dx.

JE

(1.50)

Remark 1.3.4. The assumption a u G Wl'n($l] is a sufficient condition ensuring that the solution to the equation aijDijU = / belongs to W 3 ' 2 (fi) if / G Wl^(^l) (cf. [V]).

1.4.

Miranda-Talenti Estimate

As already stated in Lemma 1.2.2, the bound Y" (Dijufdx -J=l

61)

42


with

Making use of (1.61), we can evaluate the second tangential derivatives urs = (ur)s = (Dru — UQ>cr}s = Drsu -- —-— xs - v,Q8xr - w 0 ( = DrsU - U0rXs - 6>Cr>C

Moreover Drsu = urs + uor>cs

_ t=i

(1.62)

The terms (x r ) s can be evaluated by means of the coefficients Bî, z , f c = l , . . . , r a — 1, of the second fundamental form on d£l. Namely,

dx

r

/, «o\ (1-63)

-

r

Further, it follows from |x| = 1 that X>r^ = 0.

(1.64)

l

r=l

Thus (1.63) and (1.64) yield

Moreover, in virtue of (1.60) and (1.65) one gets

dxs dxr

~*mrIt is worth noting that the vectors dx/dti are tangential to (9H, that is,

E

ixr

OXf

f)+.

— Q

TOT*

7 — 1

'' '*'

r=l

and therefore

_

dx

ki — ~" Oj.

dx_ _ d ' ~oT~ — "oT"

dx

77

"I

( L66 )

1.4. Miranda-Talenti Estimate

43

whence

Now the product DiuDrsu that appears at the integrand (airajs - aijars)DiUDrs

(1.67)

of the surface integral (1.55) becomes DiuDrsu =

(1.68)

Setting \i — Vâ lu|| La(n) for all x',x" G K, \x' - x"\ < dist (K, 9fi)/2. It is reasonable that the result above holds true also up to the boundary.

1.5.2.

Oblique Derivative Problem

Although the linear planar Dirichlet problem can be considered exhaustively studied, different and more complicate is the situation regarding the oblique derivative problem. Our goal now is to show that when dealing with planar oblique derivative problems, as in the Dirichlet case, the only assumptions (1.77) and (1.78) are sufficient to ensure strong solvability provided suitable compatibility conditions hold (compare with the requirements imposed in Theorem 1.3.2 in multidimensional domains). Let il be a C2-smooth planar domain. Then the boundary d£l is a closed curve with parameterization

xi=xi(ip),

x2=x2(p),

c2) for the unit outward normal to d£} and let x(^) be the curvature of 0 on 917 implies

6(L) - 6(0) J0 2?r 2?r Remembering that h is an integer, it follows =

In other words, d6/d(p - x > 0 on 0) when the parameter (p increases. We will show that dO/dp — x > 0 °n d^t is an essential condition in order to have the estimate (1.94). For, consider the vector field i — (cos u;, sin a; ) with a fixed angle uj. Then (1.83) gives du dO . r.

^ = ^-* = °

V

^M-

The function u(x\, #2) — #1 sin a; — x% cos a; is a classical solution of the problem Aw = 0

in H,

However, |n| - / \Du\2dx > f (kuYdx = 0. JQ JQ That is, the inequality (1.94) does not hold true. Remark 1.5.10. Consider an elliptic operator C = a^(x)D{j with coefficients satisfying (1.77). It follows from Remarks 1.2.17 and 1.2.19 that for each A > (A 2 + A 2 )/2AA there exists a constant B > A such that (DijU) Then the results of Lemmas 1.5.6 and 1.5.8 hold also for the operator C. Precisely, if dO/dp - x > 0 on 0 on 9ft. Then the difference of any two solutions of the problem (1.99) is affine function. That is, u\ — u 0, d6/d(p - x £ 0 on 9ft. T/ien tte V7 2 ' 2 (ft)solution of the problem (1.99) is unique modulo an additive constant. The above considerations allow to derive an existence result (cf. [TL 2]) for the problem (1.99). Proposition 1.5.13. Assume (1.77), (1.78), 9ft G C2, i G C x (9ft), d0/d(p - x > 0 on 9ft and h — 0. TTien /or eac/i / G I/2 (ft) the problem (1.99) admits a solution u G Jy 2)2 (ft) which is unique modulo an additive constant. Proof. Set V(0) for the closure of \u G C 2 (ft): -^ = const

on 9ft 1

with respect to the norm in VF 2>2 (ft). Further, define an equivalence relation "=" on V(0) as follows: u = v if and only if u — v = const, and denote by V the quotient space V(0)/^. Setting

it is a simple matter to verify that || • \\y is a norm in V and V is Banach space with respect to it. In fact, let {wk} be a Cauchy sequence in V. By means of the Poincare inequality (A.2), we get \\w\\v < \\w - wn\\w**(n) < C\\w\\v.

(1.100)

56


Therefore, {w'k} = {wk — (wk)n} is a Cauchy sequence in l/F 2)2 (ft). Hence, there exists aw G W2>2(fy such that \\w'k - w\\W2,2(ty —>> 0

as

k -^ oo.

Later, w'k G V(0) and V(0) is a closed space. Therefore u> G F(0) and (1.100) implies w\\v < \\w'k - w\\w2>2(ty -> 0 as

k -> oo.

This proves the completeness of V. The solvability of the problem (1.99) will be proved by the aid of the continuity method. For this goal, we will prove first solvability in ]/F 2 ' 2 (ft) of a similar problem for the Poisson equation

Aw = f(x) du

— = const Let {fk} be a sequence of Co°(ft) functions with L 2 (ft)-limit f ( x ) . Denote by C 2 (ft) Pi C l ( f t ) a solution of the classical problem (1.102)

— =

As it was mentioned above for the problem (1.88), the index of (1.102) is zero (h = 0!) and an appropriate choice of the constants Ck ensures existence of a unique (modulo an additive constant) classical solution Uk G C 2 (f]) nC 1 (fi) of (1.102). Lemmas 1.5.6 and 1.5.8 imply \\Um ~ Uk\\V < C\\fm -

fk\\L*(Sl)

with a constant C depending only on H. The completeness of V yields existence of u G V(0) such that \\Uk - u\\v -> 0 as k -> +00. Moreover, IIAtz - /|| L 2 (n) < ||/ - M| L 2 (n) + V2\\u - uk\\v, whence Aw = / almost everywhere in ft. Further, the weak derivatives D\u and Dû have traces on 9ft, which are L p ( 1. This means that {duk/dt} tends in L p ( *)> ~*r*the mapping T:

Wl'1(Q)-^Wl'1(Q)

defined by where U(x,t) is the unique solution of the problem AC/ - DtU = af + Aw - D t w - a£*w,

U G W^Q).

In view of (1.107) and Lemma 1.6.1, T results a contraction mapping. In fact, \\Twi — Tw 2, is a solution of the equation Aw - Dtu — 0 in Q(cr), then

for any r G (0, 1).

66


Proof. We have u G C°°(Q(ea)) for any e G (0,1). Therefore, the estimate (1.122) holds in Q(eo) for the function u and any its derivative. In particular, »*,'.'«(«,)) ^"Nl^cH.,))

Vr6(0,l)

and we get (1.123) passing to the limit as e -> 0. D The Cordes condition permits to extend the last result to the case of parabolic operators with variable coefficients. Lemma 1.6.8. Suppose (1.104), (1.105) and (1.106) to be fulfilled in the cylinder Q* = B(r) x (— T, 0) and let u G W2>l(Q*), 2 < p < p 0) be a solution of the equation aij(x,t)DijU

- Dtu = 0 a.e. Q*

with PQ defined in Theorem 1.6.3. Then, for each r G (0, 1) and for each a G (0, CTO], cr0 — min{r, x/Tj, it results \\U\\ o < , l~q- T n * /P |M| o " V^Qfr,)) - ^fl^C(p) " "w*S(Q(a))

where q =

\og((I + ^T^C(P))x + Vl^C(p)) ^ ( ^ . , c J sup -^---—^- and C(p) is

m Theorem 1.6.3. Proof. Consider the restriction of w on (2(, where iu is the solution of Cauchy-Dirichlet's problem - Dtw = (Atz - aa^Diju) - (1 - a)J9 t w in while i> verifies A u - J D t i ; = 0 inQ(t7),

v e W*

By means of (1.116) and (1.107) we get

a

K 2 u|| P ,Q we have « « »$,(„>< C2r""/" « „ »^(

u(x,0) = 0on

and equip W^l(Q) with the norm

(

EllAÎI ij^l

dxdt

where || • ||AT stands for the Euclidean norm in RN . In these settings the MirandaTalenti estimate becomes \\DijU(x,t)\\2n,Ndx < i

\\&u(x,t)\\2Ndx

J

Vt G (0,T)

n

and the results presented above remain valid also for systems defined by the operator £*.

1.6.3.

Higher Summability of the Derivatives

Also in the case of parabolic operators we are able to prove regularity results concerning higher summability of the derivatives of solutions to the Cauchy-Dirichlet problem. For the sake of brevity we will denote as before by || • || g>gi the norm in the space and \\Du\\q,qi = ]£ \\DiU\\g,gi, 1=1

The following result holds true.

\\D2u\\qtqi =

1.6.

Cauchy-Dirichlet Problem for Parabolic Operators

71

Theorem 1.6.12. Let ft be a bounded and convex domain ofW1 with boundary dtl £ C2 and let the coefficients a^(x, t) satisfy the assumptions (1.104), (1.105) and (1.106). Suppose f ( x , i ) to be a function belonging to L 9l (0,T, L g (fJ)) with real numbers q and qi such that 2 < g, q\ < oo, 0 < nq\ + 2q — qq\ < nq.

0, that is, if and only if p > 0 and \6\ < 2^/p/n. The unity is an eigenvalue of multiplicity n2 - 1 of the matrix associated to the form (1.130) while • • i the remaining two eigenvalues are obtain xi_

- I)2 = (Au - Dtu)2 +

,

and

• Thus, we • p(Dtu)2

>yu) 2 - DauDjju) + (2 + 0 and ff (x) < 0 in view of the convexity of Q, we obtain n

V (Dj(DiuDiju) - Di(DiuDjju)) O^dxdt < 2/z / J iJ=i ® Analogously, one gets

n r = -2p, \ 6»~l V DiuDijuDjuDtudxdt--2

JQ

I

f / \Du(x,T)\2dx (M +1) Jn

1.6.


73

and therefore,

l - V ^ - D 2 * ^ f (\D*ur + 2 JQ

(Dtu}*)o»dx

< I (Aw - Dtu)29»dxdt + 2/x / Awi^"1 V JQ JQ ^ ,

n

- 2(1 + p + /(! + /?)(! — e) and that value equals (1.132) Moreover, the infimum of (1.132) with respect to 77 > 0 is

We are going now to determine J and p in such a way that

If we want (1.133) to be verified for e = 1 and any //, then we have to choose 8 — —p. After that, giving to p the value 2/(n + 1) for which the denominator of (1.133) takes its maximum, (1.133) would be satisfied if

(n 4- l)e - n — (n — !)>/! - e On the other hand, if we require (1.133) with p, = 0 to be verified for each e G (0, 1), we should choose 5 = 0 and p = 1. For such values of 6 and /?, (1.133) holds if (L134)

Further on, (n + !)£" — n — (n — 1)\/1 —

1.6.


75

if 0 < £ < £ * , where 3n - 3 + \/9n2 -f 6n - 3 2(3n - 1) '

Hence, taking IJL < (p(e), 0 < e < e* and setting a — r = >/(! -f 77) (1 - e) in (1.131), we have to choose 77 in such a way that

2/yi -f 7?\/l - £ + (1 -f r?)(l - e) + /x 1 2 "^ 2' That choice of 77 is possible in view of (1.134) and therefore (1.131) implies / (\D2u\2 + (Dtu)'2} e^dxdt + f \Du(x,T)\2^+^dx JQ JQ n.

Moreover, if q^ > n in addition, then

\u(X',f) - u(x",t")\ < c\\f\\q,gi (\x' - x'r + \f - n for each couple (x',tr), (x" ,t") G Q, where a = 2 - n/q - 2/qi. We refer the reader to the article [M 1] for the entire proof of Theorem 1.6.13.

1.6.


1.6.4.

77

Parabolic Operators with Sobolev Coefficients

The boundary value problems for linear parabolic operators with Sobolev coefficients were first studied by Guglielmino ([GL 1], [GL 2]). To begin with, consider the parabolic equation aij(x,t)DijU

- Dtu = f ( x , t )

a.e. Q

(1.137)

and suppose the following assumptions are fulfilled for n > 3: ai) ali G L°°(Q) and there exist two positive constants A and A such that

for almost all (x,i) G Q and all £ G E n . 0,2) Dkdli G L°°(0,T, L n (£7)) for any z,^ = 1,... ,n and /c = 1,... ,n, Dta u G a 3 ) Setting

there exists a nondecreasing function £Q(CT) defined for a > 0, lim(7_>0+ e^cr) = 0 and such that for almost all t G (0,T) it results \A(x,t)\ndx ?

for any £? C Q with |£| < cr. a 4 ) 17 is a bounded domain of En with boundary 3fi G C2 . Then the following result holds. Theorem 1.6.14. Under the above assumptions, let f G Lqi (0,T, L 9 (fi)) witft 2 < g, gi < oo; 0 < nî H- 2q — qq\ < nq. Then the Cauchy-Dirichlet problem aijDijU- Dtu = f

a.e. Q,

u£W%l(Q)

admits a unique solution. Moreover, there exists a constant C nondecreasing with respect to T and depending on (7, A, A, £0(0) and q such that \\Du\\23*

q2

-f \\Du\\q2^ -f |p2u||2,2 (1.138)

78


Proof. As already did, we start with considering the Cauchy-Dirichlet problem with regular coefficients ali . Then, there exists a unique strong solution as says [LSU]. Our goal will be to derive (1.138) for this solution after that the assertion will follow by density arguments. Consider the linear orthogonal transformation

which transforms the quadratic form a^^j into its canonic form p^rjj2. orthogonal mapping Pir =

Then the

cijcrsqi

transforms the quadratic form aikarspirpks -- Dtuaikpik + -(Dtu)2 n n into pi' pip'qj, -

Since

we get

with m = min {A2/2, l/2n}. Hence, setting py- = DijU and choosing appropriately qtj's above, we obtain aikarsDiruDksu - -utaikDiku + -(Dtu)2 > m(\D\\2 + (Dtu)2}. V ' (1.139) Now, in view of (1.137) we get aikarsDikuDrsu - 2DtuaikDiku + (Dtu)2 = f2 in Q. Thus (1.139) can be rewritten as

m (\D2u\2 + (Dtu)2} < f2 + aikars(DiruDksu - DikuDrsu) 2 - -} DtuaikDiku + ( - - l) (Dtu)2

= f2 + aikars(Dr(DiuDssu) DtualkDiku + 1 1

n'

Dtuf.

1.6.


79

Moreover, m(\D2u\2 + (Dtu)2) < f2 + (airaks - aikars^Dk(DiuDrsu) + aikDk(DiUDtu) - aikDiuDkiu + 11 - n -\ Dtuf V )

(1.140)

and setting 6^ — (alkDiuDku)^ with a nonnegative exponent ^ to be specified later, we reach from (1.140) e» + 0»(airaks -aikars)Dk(DiuDrsu) QV

Dtu) + —Dt(aik}DiuDku

Integrating over Q and taking into account the boundary conditions, we get m / 6

JQ

< { f29»dxdt + j dt I JQ Jo

0»(airaks -aikars}DiuDrsu>ckd(7

- f DiuDrsuDkv(e»(airaks-aikars)}dxdt[ Di J JQ JQ - h i / e»Dt(aik}DiuDkudxdt + f 1 -nn-) / * JQ JQ JQ \\ /)JQ

f26»dxdt.

This way, D

i

II T^n.(

T M I ^ M + l)

x" /"*l I

rp + ||L/W(-, i J|i£,2( M +i)(n) —

\

7 l

I

where Ci is a constant depending on A, A, // and P ^ = / |L>u| 2/x (|jD 2

j1= y ^ / 0 dxA, T . lr jj. ir /DM / ir J — / at / 6r (a a /*

2

Jo

2

J d$i

— a ik aT

fcs

27

I

37

i

47

I 57

/

\


80

J3 = n I 0»-l(aikars -aâ JQ (aikars - air aks}dxdt Q

-- [ 6»DtuDk(aik)dxdt, JQ

Q J4 - 2/i / JQ J5 = -

0»-lfaikajhDiuDhuDjk udxdt

Q

Q

The Miranda estimate (1.46) yields

while assumption as) produces J3 < C3

for each positive number L. As it concerns the integrals J^ and s, we obtain J^

2 and the estimate (1.138) takes on the form ||Z?«||,.,,. + ||I>«||,./2,oo + l|0 2 «l|2,2 + ||Au||2,2 < K\\f\\q,q

with 2 < # < 4 , 2 < g * < 4g/(4 — q). Further, the solution u satisfies Nkoo < CII/112,2,

esssup \u\ < C||/||g,9, Q

2 < q < 4,

\u(x'J) - u(x", t")\ < C7||/||M (\x> - x"\a + \t' - *'r/2(a+1)) , 0 < a < 2 - 4/g, 2 < q < 4. Condition a^) can be further generalized (see [AR 2], [AR 3]) to Dkaij e L^(0,T,L p (n)),

Dt G

with n < p < oo. For p = n we obtain 02), while if p = n + 2 one gets ^' G L n + 2 (Q),

Aa^' G L^(Q).

82


Remark 1.6.17. Results similar to the above stated hold also for initial-boundary value problems with conormal derivative boundary condition n

aijDiUXj

=0

on 90

supposing in addition that the coefficients a u of the parabolic operator are Lipschitz continuous in a neighborhood of dft x (0,T). Remark 1.6.18. Indeed, the results obtained remain valid (see [GL 2], [AR 3]) for equations with lower order terms

+ c(z, t)u - Dtu = /(z, t) imposing the next assumptions on bl and c when n > 3 y(z, t) G L (0, T, L n (ft)) Sl c(z, t) G L (0, T, L s (ft))

f o r n < r < oo, for 2 < 5, si < oo.

Theorems 1.6.14 and 1.6.15 continue to hold if n2qql nq(qi - 2) and one of the next four conditions is verified: a)

nqi+2q-2qqi

> 0, s > g,8l = ^,

2nq(qi-2)

6) ngi + 2g - 2