LINEAR ELLIPTIC EQUATIONS WITH LOWER ... - Project Euclid

Differential and Integral Equations

Volume 26, Numbers 5-6 (2013) , 623–638

LINEAR ELLIPTIC EQUATIONS WITH LOWER-ORDER TERMS Flavia Giannetti, Luigi Greco, and Gioconda Moscariello Dipartimento di Matematica e Applicazioni “R. Caccioppoli” Universit` a degli Studi di Napoli “Federico II” Via Cintia – 80126 Napoli , Italy (Submitted by: Haim Brezis) Abstract. We consider second-order linear elliptic equations with lowerorder terms whose coefficients lay in Marcinkiewicz spaces. We prove existence, uniqueness and regularity of a solution to the Dirichlet problem.

1. Introduction The aim of this paper is to study some questions arising for a class of elliptic equations whose model appears in the stationary diffusion-convection problems. We consider the second-order linear elliptic operator L u = − div(A∇u + ue) + b · ∇u + cu

(1.1)

in a bounded domain Ω of RN , N ≥ 3. We suppose that A : Ω → RN ×N is a matrix-valued function verifying A ∈ L∞ (Ω; RN ×N )

(1.2)

which is elliptic, in the sense that for almost every x ∈ Ω and all ξ ∈ RN A(x)ξ · ξ ≥ |ξ|2 .

(1.3)

Concerning the vector fields e and b and the function c, in view of the Sobolev embedding theorem in Lorentz spaces (see Section 2), natural assumptions are N (1.4) e, b ∈ LN,∞ (Ω; RN ), c ∈ L 2 ,∞ (Ω). We shall consider the Dirichlet problem for the operator L. Under assumptions (1.2) and (1.4), for NN−1 < p < N we can think of L as follows: 1,p −1,p (Ω), L : W0 (Ω) → W

Accepted for publication: December 2012. AMS Subject Classifications: 35J15, 35J25. 623

(1.5)

624

Flavia Giannetti, Luigi Greco, and Gioconda Moscariello 0

where for all u ∈ W01,p (Ω) and v ∈ W01,p (Ω) Z hL u, vi = [(A∇u + ue) · ∇v + vb · ∇u + cuv] dx. For a given F ∈

(1.6)

Ω p L (Ω; RN ),

a solution to the problem ( L u = − div F, in Ω u = 0, on ∂Ω

(1.7)

is a function u ∈ W01,p (Ω) satisfying the equation in the sense of distributions, that is, such that Z hL u, vi =

F · ∇v dx,

(1.8)

Ω 0

for all v ∈ D(Ω). Clearly, (1.8) extends to all v ∈ W01,p (Ω). We point out that both diffusion and convection terms may appear in the equation of problem (1.7). Our interest is on the effect of lower-order terms − div(u e) + b · ∇u + c u

(1.9)

on the properties of the principal part operator L0 u = − div(A∇u). We first consider invertibility of L. In spite of the apparently tiny generalization, a number of new difficulties arise under our assumptions (1.4) compared with the classical case, treated by Stampacchia [13] (see also [4, 7]), N b, e ∈ LN (Ω, RN ), c ∈ L 2 (Ω). The operator L0 is coercive on W01,2 (Ω): hL0 u, ui ≥ k∇uk22 . Lower-order terms can cause lack of coercivity. One way to overcome this difficulty is to consider L u + λ u for λ > 0 large enough. In the classical case, λ can be chosen so that L u + λ u is coercive. In general, this is no N longer the case if b, e ∈ LN,∞ or c ∈ L 2 ,∞ . The problem is that the norm in Marcinkiewicz spaces is not absolutely continuous: functions can have large norm even if restricted to subsets of arbitrarily small Lebesgue measure. In the classical case, lower-order terms (1.9) define a compact operator W01,p (Ω) → W −1,p (Ω). To prove this, first we assume that the coefficients N are bounded. Then, if b, e ∈ LN and c ∈ L 2 , we approximate them by bounded functions in the corresponding spaces. The corresponding operators will converge to (1.9) in the norm of the space of bounded linear operators of

Linear elliptic equations with lower-order terms

625

W01,p (Ω) into W −1,p (Ω). Bounded functions are not dense in Marcinkiewicz N spaces. For general b, e ∈ LN,∞ and c ∈ L 2 ,∞ , the operator (1.9) is not compact. In [2] problem (1.7) has been solved in W01,2 (Ω) assuming the coefficients of lower-order terms in Marcinkiewicz spaces with a control on the norms. In the present paper, under assumptions (1.2)–(1.4), we show that the operator L has the same invertibility properties as the principal part L0 , provided the coefficients e, b, and c of lower-order terms have distance to L∞ sufficiently small. Under the same assumption, we also study regularity of solutions of the Dirichlet problem. More precisely, we show that the solution u ∈ W01,2 (Ω) to problem (1.7) with F ∈ Lp (Ω; RN ), 2 < p < N , ∗ belongs to Lp (Ω). We stress that, for all 1 < r < ∞ and 1 ≤ q < ∞, the Lorentz space r,q L (Ω) is contained in the closure of L∞ (Ω) in Lr,∞ (Ω). Therefore, our results extend those of [13, 7, 4, 11]. 2. Preliminary results Let Ω be a bounded domain in RN . For a measurable E ⊂ Ω, we denote by |E| its Lebesgue measure. For a measurable function f : Ω → R we denote by µf (λ) = |{x ∈ Ω : |f (x)| > λ}| its distribution function and by f ∗ (t) = inf{λ : µf (λ) ≤ t} its decreasing rearrangement; see [3]. Clearly, f ∗ (t) = 0 if t > |Ω|. For 1 < p < ∞ and 1 ≤ q < ∞, we consider the quantity nZ ∞ dt o1/q kf kp,q = [t1/p f ∗ (t)]q t 0 and for q = ∞ the obvious modification kf kp,∞ = sup {t1/p f ∗ (t)}. Lp,q

0 0. We note the equality

for every measurable E ⊂ Ω. We remark that for any p ∈ (1, ∞), L∞ is not dense in Lp,∞ . We define the distance of a given f ∈ Lp,∞ to L∞ as distLp,∞ (f, L∞ ) = inf∞ kf − gkp,∞ . g∈L

(Actually, since k kp,∞ is not a norm, distLp,∞ is only equivalent to a metric.) To find a formula for the distance, we consider the truncation operator. For k > 0, we set y min{|y|, k}. (2.5) Tk (y) = |y| Then distLp,∞ (f, L∞ ) = lim kf − Tk f kp,∞ . k→∞

Indeed, ∀g ∈ L∞ , ∀k ≥ kgk∞ , we have for almost every x ∈ Ω, |f (x) − g(x)| ≥ |f (x) − Tk f (x)|. For further comments on the distance to L∞ and some applications, we refer to [5]. Example 2.1. Let Ω be the unit ball of RN and p ∈ (1, ∞). The function f (x) = |x|−N/p belongs to Lp,∞ . Setting ωN = |Ω|, for k > 0 and λ > 0 we compute µf −Tk f (λ) = ωN (λ + k)−p and

( (t/ωN )−1/p − k, (f − Tk f )∗ (t) = 0,

0 < t < ωN k −p t ≥ ωN k −p .

Hence 1/p

kf − Tk f kp,∞ = ωN does not depend on k.

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Flavia Giannetti, Luigi Greco, and Gioconda Moscariello

On the contrary, for all 1 ≤ q < ∞, starting with the definition of k kp,q , a simple application of Lebesgue dominated convergence theorem shows that L∞ is dense in Lp,q . Hence, for 1 ≤ q < ∞, Lp,q , and in particular the Lebesgue space Lp , is contained in the closure of L∞ in Lp,∞ . The closure of L∞ coincides with the closure of C0∞ . The elements of the closure can be characterized by the condition of having absolutely continuous norm; see [3, Section 1.3]. We note an equivalent definition of kf kp,q when q < ∞: Z ∞ kf kqp,q = p λq−1 µf (λ)q/p dλ. 0

As a consequence, for fixed k > 0, we have Z k 1 λq−1 µf (λ)q/p dλ = kTk f kqp,q . p 0

(2.6)

Indeed, ( µf (λ), µTk f (λ) = 0,

0 0 such that, if distLN,∞ (e, L∞ ) < ε, then

distLN,∞ (b, L∞ ) < ε,

dist

L

N ,∞ 2

(c, L∞ ) < ε,

1,p −1,p (Ω) L : W0 (Ω) → W

is invertible. Corollary 3.3. Assume e and b in the closure of L∞ in LN,∞ , c in the N closure of L∞ in L 2 ,∞ , and that (3.4) or (3.5) holds. Then 1,p −1,p (Ω) L : W0 (Ω) → W

is invertible for all p ∈ (N 0 , N ) ∩ (q 0 , q). Proof of Theorem 3.2. As we have the Fredholm alternative, it suffices to show injectivity. We first assume p ≥ 2 and condition (3.4). In this case, we do not need any restriction on distances (3.3). We follow [4] and use as test functions in equation L u = 0 truncations v = Tk u, where Tk is defined by (2.5). As u v ≥ 0, by (3.4) and (3.6) Z [A∇u + u (e − b)] · ∇v dx ≤ 0. Ω

Therefore, using the ellipticity condition (1.3), Z Z 2 |∇Tk u| dx ≤ k |e − b| |∇Tk u| dx, Ω

0 N 2−2 , the adjoint operator x ∗ L v = −∆v − div γ v 2 |x| is not surjective on W01,2 (Ω). Condition (3.4) holds in this case.

632


4. Regularity Now we discuss the regularity of the solution u ∈ W01,2 (Ω) to the equation L u = g − div F Lp (Ω, RN )

with F ∈ and g ∈ Lp∗ (Ω), 2 < p. As usual, 1/p∗ = 1/p + 1/N . We no longer assume that the principal part 1,p −1,p (Ω) L0 : W0 (Ω) → W

is an isomorphism: no regularity for A and p is far from 2, thus in general we cannot conclude u ∈ W01,p (Ω). In [13, Section 4] Stampacchia considered an operator L defined by (1.1) N with coefficients b, e ∈ LN (Ω, RN ) and c ∈ L 2 (Ω), satisfying c − div e ≥ const. > −∞

(4.1)

in the sense of distributions, and proved that ⇒

2 N.

(4.2)

See also [4]. Our next result generalizes Stampacchia’s in the case 2 < p < N . In the following theorem we do not assume sign condition (3.4) or (3.5). Theorem 4.1. For every fixed p ∈ (2, N ) there exists ε = ε(p) > 0 such that, if distLN,∞ (e, L∞ ) < ε,

distLN,∞ (b, L∞ ) < ε,

dist

L

N ,∞ 2

(c, L∞ ) < ε

and u ∈ W01,2 (Ω) solves the equation L u = g − div F with F ∈ Lp (Ω, RN ) and g ∈ Lp∗ , then ∗

u ∈ Lp . Moreover, we have the estimate kukp∗ ≤ C (kF kp + kgkp∗ + kuk2 )

(4.3)

for some constant C = C(p, N ) > 0. ∗

Remark 4.2. The conclusion u ∈ Lp would follow by Sobolev embedding if u ∈ W01,p (Ω), but in general this latter property fails, for 2 < p < N .


633

Corollary 4.3. Assume e and b in the closure of L∞ in LN,∞ and c in the N closure of L∞ in L 2 ,∞ . Then, for all p ∈ (2, N ), L u = g − div F ∗

with F ∈ Lp (Ω, RN ) and g ∈ Lp∗ , implies u ∈ Lp . Proof of Theorem 4.1. As can be expected, taking g in the larger space Lp∗ ,p will actually suffice. Initially, we assume that kekN,∞ , kbkN,∞ , and kck N ,∞ are sufficiently 2 small. We use v = u − Tk u

(4.4)

as a test function in the equation, where k > 0. In this way, we have Z Z (A∇u · ∇v + u e · ∇v + v b · ∇u + c u v) dx = (g v + F · ∇v) dx. (4.5) Ω

Ω

By the definition of v, we can rewrite the left-hand side as Z Z 2 [A∇v · ∇v + v (e + b) · ∇v + c v ] dx + k sgn u (e · ∇v + c v) dx, (4.6) Ω

Ω

By H¨ older’s inequality and Sobolev’s inequality, Z |v (e + b) · ∇v + c v 2 | dx ≤ (S2 ke + bkN,∞ + S22 kck N ,∞ ) k∇vk22 . 2

Ω

Hence, in view of ellipticity condition (1.3), if S2 ke + bkN,∞ + S22 kck N ,∞ < 1/3, 2

then Z Ω

[A∇v · ∇v + v (e + b) · ∇v + c v 2 ] dx ≥

2 k∇vk22 . 3

By (4.5), (4.6), and (4.7), we get Z Z 2 2 k∇vk2 ≤ k (|e| |∇v| + |c| |v|) dx + (|g| |v| + |F | |∇v|) dx. 3 Ω Ω

(4.7)

(4.8)

Now we use some ideas from [9]. Let ϕ(k) = k 2γ , where γ = p∗ /2∗ − 1 > 0. We multiply both sides of (4.8) by ϕ0 (k) and integrate with respect to k on the interval [0, K], where K > 0. As |∇v| = v = 0 when |u| ≤ k, ∇v = ∇u

634


when |u| > k, and |v| ≤ |u|, by Fubini’s theorem we get Z K Z K Z 0 2 2γ−1 ϕ (k) k∇vk2 dk = 2γ k dk |∇u|2 dx 0

|u|>k

0

Z =

|∇u|2 |TK u|2γ dx

Ω

and similarly Z K Z Z ϕ0 (k) k dk (|e| |∇v| + |c| |v|) dx ≤ (|e| |∇u| + |c| |u|) |TK u|2γ+1 dx 0

Ω

Z

K

ϕ0 (k) dk

0

Ω

Z

Z (|g| |v| + |F | |∇v|) dx ≤

Ω

(|g| |u| + |F | |∇u|) |TK u|2γ dx.

Ω

Therefore, from (4.8) we get 2 k|TK u|γ ∇uk22 ≤ 3

Z

(|e| |∇u| + |c| |u|) |TK u|2γ+1 dx

Ω

(4.9) Z

2γ

(|g| |u| + |F | |∇u|) |TK u|

+

dx.

Ω

We estimate each term in the right-hand side. By Hölder’s inequality, Z |e| |∇u| |TK u|2γ+1 dx ≤ k|TK u|γ ∇uk2 kekN,∞ k|TK u|γ+1 k2∗ ,2 Ω

Z Ω

|c| |u| |TK u|2γ+1 dx ≤ ku |TK u|γ k2∗ ,2 kck N ,∞ k|TK u|γ+1 k2∗ ,2 2

Z

|g| |u| |TK u|2γ dx ≤ ku |TK u|γ k2∗ ,2 kgkp∗ ,p k|TK u|γ k

Ω

Z

|F | |∇u| |TK u|2γ dx ≤ k|TK u|γ ∇uk2 kF kp k|TK u|γ k

Ω

Moreover, as ∇[u |TK u|γ ] ≤ (1 + γ) |∇u| |TK u|γ , using Sobolev’s inequality we get ku |TK u|γ k2∗ ,2 ≤ (1 + γ) S2 k|TK u|γ ∇uk2 .

2p p−2

2p p−2

.


635

From (4.9) we easily find 2 k|TK u|γ ∇uk2 ≤ [kekN,∞ + (1 + γ) S2 kck N ,∞ ] k|TK u|γ+1 k2∗ ,2 2 3 + [kF kp + (1 + γ) S2 kgkp∗ ,p ] k|TK u|γ k

2p p−2

.

In the left-hand side we note that |∇u| ≥ |∇TK u| and use Sobolev’s inequality to get 1 k|TK u|γ+1 k2∗ ,2 ≤ k|TK u|γ ∇TK uk2 ≤ k|TK u|γ ∇uk2 . (1 + γ) S2 Hence, assuming (1 + γ) S2 [kekN,∞ + (1 + γ) S2 kck N ,∞ ] < 1/3 2

we have 1 k|TK u|γ+1 k2∗ ,2 ≤ (C kF kp + C 2 kgkp∗ ,p ) k|TK u|γ k 2p p−2 3 with C = (1 + γ) S2 . The above estimate can be rewritten as 1 γ 2 kTK ukγ+1 ∗ (γ+1),2(γ+1) ≤ (C kF kp + C kgkp∗ ,p ) kTK uk 2p . 2 γ p−2 3 Noticing that the choice of γ yields 2p 2∗ (γ + 1) = γ = p∗ p−2 and 2(γ + 1) < p∗ , we get 1 kTK ukp∗ ≤ C kF kp + C 2 kgkp∗ ,p , (4.10) 3 and we conclude letting K → ∞. Now, we replace smallness of norms of e, b, and c with smallness of distances to L∞ . To this end, we rewrite equation L u = g − div F as − div[A∇u + u (e − e0 )] + (b − b0 ) · ∇u + (c − c0 ) u = (4.11) [g − (c0 − div b0 ) u] − div[F − u (e0 − b0 )] for suitable vector fields e0 and b0 , and a function c0 of class C 1 . Using Sobolev’s embedding theorem, we see that F − u (e0 − b0 ) ∈ Lr and g − (c0 − div b0 ) u ∈ Lr∗ , where r = min{p, 2∗ }. By the above argument, if ke − e0 kN,∞ ,

kb − b0 kN,∞ ,

kc − c0 k N ,∞ , 2

636

Flavia Giannetti, Luigi Greco, and Gioconda Moscariello ∗

are small enough, then u ∈ Lr . In case r = p we are done. In case r = 2∗ < p, now we notice that F − u (e0 − b0 ) ∈ Lr1 and g − (c0 + div b0 ) u ∈ L(r1 )∗ , where r1 = min{p, 2∗∗ }, and repeat the argument. It is clear that we get ∗ u ∈ Lp in a finite number of times. ∗

∗

Remark 4.4. Actually the proof yields |u|p /2 ∈ W01,2 (Ω). Moreover, if e, b, and c have small norms, then in the right-hand side of (4.3) the term kuk2 can be dropped. Example 4.5. For N − 2 > γ > 0 and Ω the unit ball,  x  −∆u − div γ u 2 = N (2 + γ) in Ω |x|  u=0 on ∂Ω has the distributional solution u = |x|−γ − |x|2 . ∗

The right-hand side in the equation is smooth, while u ∈ Lp (Ω), 2 < p < N , if and only if γ < Np − 1. We have e = γ |x|x2 and distLN,∞ (e, L∞ ) is proportional to γ. This shows that ε in Theorem 4.1 really depends on p. We notice that assuming sign condition (3.4) or (3.5), the proof of Theorem 4.1 can be simplified and less restrictive conditions on the distances to L∞ of the coefficients are required. Proposition 4.6. Let L be defined by (1.1), with coefficients satisfying (1.2), (1.3), and (1.4). Moreover, assume sign condition (3.4) or (3.5). Then, for every fixed p ∈ (2, N ) there exists ε = ε(p) > 0 such that, if distLN,∞ (e − b, L∞ ) < ε and u ∈ W01,2 (Ω) solves the equation L u = g − div F with F ∈

Lp (Ω, RN )

∗

and g ∈ Lp∗ , 2 < p < N , then u ∈ Lp .

Indeed, by (3.6), under assumption (3.4) we get Z Z [A∇u · ∇v + u (e − b) · ∇v] dx ≤ (g v + F · ∇v) dx, Ω

Ω

and we can argue as in the proof of Theorem 4.1. Under assumption (3.5) we get Z Z [A∇u · ∇v + v (b − e) · ∇u] dx ≤ (g v + F · ∇v) dx, (4.12) Ω

Ω


637

and even [13, Lemma 4.1] suffices to prove the result. We only remark that the equation L u = g − div F can be rewritten as in (4.11) for e0 of class C 1 , with b0 = 0 and c0 = div e0 , so that (c − c0 ) − div(e − e0 ) = c − div e ≥ 0. Remark 4.7. Theorem 4.1 allows us to consider the duality solution of Stampacchia, in order to prove existence of a solution to the problem L u = g, for g ∈ Lp∗ (Ω), with

N N −1

u ∈ W01,p (Ω)

< p < 2. For details we refer to [13].

Boundedness of solution in case p > N cannot be inferred merely assuming that the coefficients have sufficiently small distances to L∞ , as the following example shows. Example 4.8. Let Ω be the unit ball and consider the operator L u = −∆u − div(u e) with e=

x . |x|2 (1 − log |x|)

Note that e ∈ LN . By a direct calculation it is easily seen that the unbounded function u(x) = 1 − log |x| − |x|2 satisfies u ≡ 0 on ∂Ω, and L u ∈ C 0 (Ω). Here we prove the following: Theorem 4.9. Let L be defined by (1.1), with coefficients satisfying (1.2), (1.3), (1.4), and (3.5) or (4.2). Under these assumptions, there exists ε > 0 such that, if distLN,∞ (e − b, L∞ ) < ε and u ∈ W01,2 (Ω) solves the equation L u = g − div F with F ∈ Lp (Ω, RN ) and g ∈ Lp∗ , p > N , then u ∈ L∞ . Proof. If (3.5) holds, then we have (4.12), and the argument of [13] can be extended to prove boundedness. In particular, the result holds for the operator u 7→ − div A∇u + b · ∇u,

638


which satisfies (3.5). Under assumption (4.2), by Theorem 4.1 we have u e ∈ Ls and c u ∈ Ls∗ , for some s > N . We write the equation L u = g − div F as − div A∇u + b · ∇u = (g − c u) − div(F − u e) and use the previous case.

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