CUBO A Mathematical Journal Vol.12, N¯o 01, (133–148). March 2010
Well-Posedness Results for Anisotropic Nonlinear Elliptic Equations with Variable Exponent and L1-Data Stanislas OUARO Laboratoire d’Analyse Math´ematique des Equations (LAME), UFR. Sciences Exactes et Appliqu´ees, Universit´e de Ouagadougou 03 BP 7021 Ouaga 03 Ouagadougou, Burkina Faso email :
[email protected]
ABSTRACT We study the anisotropic boundary value problem −
N X ∂ ∂ ai x, u = f in Ω, u = 0 on ∂xi ∂xi i=1
∂Ω, where Ω is a smooth open bounded domain in RN (N ≥ 3) and f ∈ L1 (Ω). We prove the existence and uniqueness of an entropy solution for this problem.
RESUMEN Estudiamos el problema de valores en la frontera anisotropico −
N X ∂ ∂ ai x, u = f ∂xi ∂xi i=1
en Ω, u = 0 sobre ∂Ω, donde Ω es un dominio abierto suave do RN (N ≥ 3) y f ∈ L1 (Ω). Proveamos la existencia y unicidad de una soluci´ on de entrop´ıa para este problema.
Key words and phrases: Anisotropic; variable exponent; entropy solution; electrorheological fluids. Math. Subj. Class.: 35J20, 35J25, 35D30, 35B38, 35J60.
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Introduction
Let Ω be an open bounded domain of RN (N ≥ 3) with smooth boundary. Our aim is to prove the existence and uniqueness of an entropy solution to the anisotropic nonlinear elliptic problem N X ∂ ∂ − a x, u = f in Ω i ∂xi ∂xi i=1 (1.1) u = 0 on ∂Ω,
where the right-hand side f ∈ L1 (Ω). We assume that for i = 1, ..., N the function ai : Ω × R → R is Carath´eodory and satisfies the following conditions: ai (x, ξ) is the continuous derivative with respect to ξ of the mapping ∂ Ai (x, ξ) such that Ai : Ω × R → R, Ai = Ai (x, ξ), i.e. ai (x, ξ) = ∂ξ the following equality and inequalities holds Ai (x, 0) = 0,
(1.2)
for almost every x ∈ Ω. There exists a positive constant C1 such that pi (x)−1
|ai (x, ξ)| ≤ C1 (ji (x) + |ξ|
)
(1.3) ′
for almost every x ∈ Ω and for every ξ ∈ R, where ji is a nonnegative function in Lpi (.) (Ω), with 1/pi (x) + 1/p′i (x) = 1. There exists a positive constant C2 such that ( p (x) C2 |ξ − η| i if |ξ − η| ≥ 1 (ai (x, ξ) − ai (x, η)) . (ξ − η) ≥ (1.4) p− i C2 |ξ − η| if |ξ − η| < 1, for almost every x ∈ Ω and for every ξ, η ∈ R, with ξ 6= η, and p (x) |ξ| i ≤ ai (x, ξ).ξ ≤ pi (x)Ai (x, ξ)
(1.5)
for almost every x ∈ Ω and for every ξ ∈ R. We also assume that the variable exponent pi (.) : Ω → [2, N ) are continuous functions for all i = 1, ..., N such that: p¯(N − 1) p¯(N − 1) < p− , i < N (¯ p − 1) N − p¯
N − X 1 p+ p¯ − N i − pi − 1 < , − > 1 and − p ¯ (N − 1) p p i i=1 i
N 1 1 X 1 + = , p− i := ess inf pi (x) and pi := ess sup pi (x). x∈Ω p¯ N i=1 p− x∈Ω i We introduce the numbers N (¯ p − 1) ∗ Nq N (¯ p − 1) q= , q = = . N −1 N −q N − p¯
(1.6)
where
(1.7)
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A prototype example that is covered by our assumptions is the following anisotropic (p1 (.), ..., pN (.))harmonic equation: p (x) p (x)−2 Set Ai (x, ξ) = (1/pi (x)) |ξ| i , ai (x, ξ) = |ξ| i ξ where pi (x) ≥ 2. Then we get the following equation: N X ∂ − ∂xi i=1
! ∂ pi (x)−2 ∂ u =f ∂xi u ∂xi
which, in the particular case when pi = p for any i ∈ {1, ..., N } is the anisotropic p(.)-Laplace equation. For the proof of existence of entropy solutions of (1.1), we follow [2] and derive a priori estimates q∗ n for the approximated solutions un and the partial derivatives ∂u ∂xi in the Marcinkiewicz spaces M −
and Mpi q/p¯ respectively (see section 2 or [2] for definition of Marcinkiewicz spaces). The study of nonlinear elliptic equations involving the p−Laplace operator is based on the theory of standard Sobolev spaces W m,p (Ω) in order to find weak solutions. For the nonhomogeneous p(.)-Laplace operators, the natural setting for this approach is the use of the variable exponent Lebesgue and Sobolev spaces Lp(.) (Ω) and W m,p(.) (Ω). Variable exponent Lebesgue spaces appeared in the literature for the first time in a 1931 article by Orlicz[21]. After [21], Orlicz abandoned the study of variable exponent spaces to concentrate on the theory of function spaces that now bears his name (Orlicz spaces). After Orlicz’s work (cf. [21]), H. Hudzik [14], and J. Musielak [20] investigated the variable exponent Sobolev spaces. Variable exponent Lebesgue spaces on the real line have been independently developed by Russian researchers, notably Sharapudinov[26] and Tsenov [27]. The next major step in the investigation of variable exponent Lebesgue and Sobolev spaces was the comprehensive paper by O. Kovacik and J. Rakosnik in the early 90’s [16]. This paper established many of basic properties of Lebesgue and Sobolev spaces with variables exponent. Variable Sobolev spaces have been used in the last decades to model various phenomena. In [5], Chen, Levine and Rao proposed a framework for image restoration based on a Laplacian variable exponent. Another application which uses nonhomogeneous Laplace operators is related to the modelling of electrorheological fluids. The first major discovery in electrorheological fluids was due to Willis Winslow in 1949 (cf. [28]). These fluids have the interesting property that their viscosity depends on the electric field in the fluid. They can raise the viscosity by as much as five orders of magnitude. This phenomenon is known as the Winslow effect. For some technical applications, consult Pfeiffer et al [22]. Electrorheological fluids have been used in robotics and space technology. The experimental research has been done mainly in the USA, for instance in NASA laboratories. For more information on properties, modelling and the application of variable exponent spaces to these fluids, we refer to Diening [6], Rajagopal and Ruzicka [22], and Ruzicka [24]. In this paper, the operator involved in (1.1) is more general than the p(.)−Laplace operator. Thus, the variable exponent Sobolev space W 1,p(.) (Ω) is not adequate to study nonlinear problems of this type. This leads us to seek entropy solutions for problems (1.1) in a more general variable exponent Sobolev space which was introduced for the first time by Miha¨ılescu et al [18].
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As the right-hand side in (1.1) is in L1 (Ω), the suitable notion of solution is a notion of entropy solution (cf. [3]). The remaining part of this paper is organized as follows: Section 2 is devoted to mathematical preliminaries including, among other things, a brief discussion of variable exponent Lebesgue, Sobolev, anisotropic spaces and Marcinkiewicz spaces. Existence of weak energy solution for (1.1) where f ∈ L∞ (Ω) was proved in [14]; We will also briefly recall the results of [14] in section 2. The main existence and uniqueness result is stated and proved in section 3.
2
Mathematical Preliminaries
In this section, we define Lebesgue, Sobolev and anisotropic spaces with variable exponent and give some of their properties. Roughly speaking, anistropic Lebesgue and Sobolev spaces are functional spaces of Lebesgue’s and Sobolev’s type in which different space directions have different roles. Given a measurable function p(.) : Ω → [1, ∞). We define the Lebesgue space with variable exponent Lp(.) (Ω) as the set of all measurable function u : Ω → R for which the convex modular Z p(x) ρp(.) (u) := |u| dx Ω
is finite. If the exponent is bounded, i.e., if p+ < ∞, then the expression |u|p(.) := inf λ > 0 : ρp(.) (u/λ) ≤ 1
defines a norm in Lp(.) (Ω), called the Luxembourg norm. The space (Lp(.) (Ω), |.|p(.) ) is a separable Banach space. Moreover, if p− > 1, then Lp(.) (Ω) is uniformly convex, hence reflexive, and its ′ 1 dual space is isomorphic to Lp (.) (Ω), where p(x) + p′ 1(x) = 1. Finally, we have the H¨older type inequality: Z uvdx ≤ 1 + 1 |u| |v| ′ , (2.1) p(.) p (.) p− p′− Ω ′
for all u ∈ Lp(.) (Ω) and v ∈ Lp (.) (Ω). Now, let n o W 1,p(.) (Ω) := u ∈ Lp(.) (Ω) : |∇u| ∈ Lp(.) (Ω) , which is a Banach space equipped with the norm
kuk1,p(.) := |u|p(.) + |∇u|p(.) . An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the modular ρp(.) of the space Lp(.) (Ω). 1,p(.) Next, we define W0 (Ω) as the closure of C0∞ (Ω) in W 1,p(.) (Ω) under the norm kuk1,p(.) . Set C+ (Ω) = p(.) ∈ C(Ω) : min p(x) > 1 . x∈Ω
1,p(.)
Furthermore, if p(.) ∈ C+ (Ω) is logarithmic H¨older continuous, then C0∞ (Ω) is dense in W0
(Ω),
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1,p(.)
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1,p(.)
that is H0 (Ω) = W0 (Ω) (cf. [13]). Since Ω is an open bounded set and p(.) ∈ C+ (Ω) is logarithmic H¨older, the p(.)−Poincar´e inequality |u|p ≤ C |∇u|p(.) 1,p(.)
holds for all u ∈ W0
(Ω), where C depends on p, |Ω|, diam(Ω) and N (see [13]), and so kuk := |∇u|p(.) , 1,p(.)
is an equivalent norm in W0
(Ω). Of course also the norm kukp(.)
N X ∂ := ∂xi u
p(.)
i=1
1,p(.)
1,p(.)
is an equivalent norm in W0 (Ω). Hence the space W0 (Ω) is a separable and reflexive Banach space. 1,p(.) Let us present a natural generalization of the variable exponent Sobolev space W0 (Ω) (cf. [18]) that will enable us to study the problem (1.1) with sufficient accuracy. → → First of all, we denote by − p (.) : Ω → RN the vectorial function − p = (p1 (.), ..., pN (.)). The → − 1, p (.) anisotropic variable exponent Sobolev space W0 (Ω) is defined as the closure of C0∞ (Ω) with respect to the norm
kuk→ − p (.)
N X ∂ := ∂xi u i=1
.
pi (.)
− 1,→ p (.) The space W0 (Ω), kuk→ − p (.) is a reflexive Banach space (cf. [18]). Let us introduce the following notations: − → → + − − − P + = (p+ 1 , ..., pN ), P − = (p1 , ..., pN ),
and
+ − − − + − − P++ = max p+ 1 , ..., pN , P− = max p1 , ..., pN , P− = min p1 , ..., pN , P−∗ =
N
N X i=1
1 −1 p− i
, P−,∞ = max P−+ , P−∗ .
We have the following result (cf. [18]): Theorem 2.1. Assume Ω ⊂ RN (N ≥ 3) is a bounded domain with smooth boundary. Assume relation (1.6) is fulfilled. For any q ∈ C(Ω) verifying 1 < q(x) < P−,∞ for all x ∈ Ω,
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then the embedding
− 1,→ p (.)
W0
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(Ω) ֒→ Lq(.) (Ω)
is continuous and compact. We Also recall the result of the study of problem (1.1) for the right-hand side f ∈ L∞ (Ω) (cf. [15]). We first recall the definition of weak energy solution of (1.1) for the right-hand side more regular i.e f ∈ L∞ (Ω). − 1,→ p (.) Definition 2.2. Let f ∈ L∞ (Ω); a weak energy solution of (1.1) is a function u ∈ W0 (Ω) such that Z X Z N − ∂ ∂ 1,→ p (.) ai (x, u). ϕdx = f (x)ϕdx, for all ϕ ∈ W0 (Ω). (2.2) ∂x ∂x i i Ω i=1 Ω
We have proved among other results in [15] the following Theorem Theorem 2.3. Assume (1.2)-(1.6) and f ∈ L∞ (Ω). Then there exists a unique weak energy solution of (1.1). Finally, in this paper, we will use the Marcinkiewicz spaces Mq (Ω)(1 < q < ∞) with constant exponent. Note that the Marcinkiewicz spaces Mq(.) (Ω) in the variable exponent setting was introduced for the first time by Sanchon and Urbano (see [25]). Marcinkiewicz spaces Mq (Ω)(1 < q < ∞) contain the measurable functions g : Ω → R for which the distribution function λg (γ) = |{x ∈ Ω : |g(x)| > γ}| , γ ≥ 0, satisfies an estimate of the form λg (γ) ≤ Cγ −q , for some finite constant C > 0. The space Mq (Ω) is a Banach space under the norm Z t 1 ∗ 1/q ∗ kgkMq (Ω) = sup t g (s)ds , t 0 t>0 where g ∗ denotes the nonincreasing rearrangement of f : g ∗ (t) = inf {γ > 0 : λg (γ) ≤ t} . We will use the following pseudo norm kgkMq (Ω) = inf C : λg (γ) ≤ Cγ −q , ∀γ > 0 , ∗
which is equivalent to the norm kgkMq (Ω) (see [2]). We have the following Lemma (for the proof, see [2, proof of Lemma 2.2]). → − 1, P
Lemma 2.4. Let g be a nonnegative function in W0 − (Ω). Assume p¯ < N , and that there exists a constant c such that − N Z X ∂g pi (2.3) ∂xi dx ≤ c(γ + 1), ∀γ > 0. i=1 {|g|≤γ}
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Then there exists a constant C, depending on c, such that kgk
M
3
N (p−1) ¯ N −p ¯
(Ω)
≤ C.
Existence and Uniqueness of Entropy Solution
In this section, we study the problem (1.1) for a right-hand side f ∈ L1 (Ω). In the L1 setting, the suitable notion of solution for the study of (1.1) is the notion of entropy solution (cf. [3]). We first define the troncation function Tt by Tt (s) := max {−t, min {t, s}}. Definition 3.1. A measurable function u is an entropy solution to problem (1.1) if, for every − 1,→ p (.) t > 0, Tt (u) ∈ W0 (Ω) and Z X N
ai (x,
Ω i=1
− 1,→ p (.)
∂ ∂ u). Tt (u − ϕ)dx ≤ ∂xi ∂xi
Z
f (x)Tt (u − ϕ)dx,
(3.1)
Ω
for all ϕ ∈ W0 (Ω) ∩ L∞ (Ω). − 1,→ p (.) Remark 3.2. A function u such that Tt (u) ∈ W0 (Ω) for all t > 0 does not necessarily belong in W01,1 (Ω). However, it is possible to define its weak gradient, still denoted by ∇u. Our main result in this section is the following: Theorem 3.3. Assume (1.2)-(1.6) and f ∈ L1 (Ω). Then there exists a unique entropy solution u to problem (1.1). Proof. The proof of this Theorem will be done in three steps. ∗Step 1. A priori estimates. We start with the existence of the weak gradient for every − 1,→ p (.) measurable function u such that Tt (u) ∈ W0 (Ω) for all t > 0. − 1,→ p (.) Proposition 3.4. If u is a measurable function such that Tt (u) ∈ W0 (Ω) for all t > 0, then there exists a unique measurable function v : Ω → RN such that vχ{|u| 0, where χA denotes the characteristic function of a measurable set A. Moreover, if u belongs to W01,1 (Ω), then v coincides with the standard distributional gradient of u. → − 1, P
− 1,→ p (.)
Proof. As Tt (u) ∈ W0 (Ω) ֒→ W0 − (Ω) ⊂ W01,1 (Ω) for all t > 0 since 1 < p− i for all i = 1, ..., N , then by Theorem 1.5 in [1], the result follows. Proposition 3.5. Assume (1.2)-(1.6) and f ∈ L1 (Ω). Let u be an entropy solution of (1.1). If there exists a positive constant M such that N Z X tqi (x) dx ≤ M, for all t > 0, (3.2) i=1
then
N Z X i=1
{|u|>t}
tqi (x) dx ∂ αi (.) >t ∂x u i
≤ kf k1 + M for all t > 0,
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where αi (.) = pi (.)/(qi (.) + 1)., for all i = 1, ..., N. Proof. Take ϕ = 0 in (3.1), we have N Z X i=1
∂ ∂ Tt (u)). Tt (u)dx ≤ ∂xi ∂xi
ai (x,
Ω
Z
f (x)Tt (u)dx.
Ω
We deduces from inequality above that pi (x) N Z X ∂ dx ≤ t kf k1 , for all t > 0. ∂xi Tt (u) Ω i=1
Therefore, defining ψ := Tt (u)/t, we have, for all t > 0, N Z X i=1
Ω
t
pi (x) Z ∂ pi (x) ∂ 1 dx = Tt (u) dx ≤ kf k1 . ∂xi ψ t Ω ∂xi
pi (x)−1
From the above inequality, the definition of αi (.) and (3.2), we have N Z X
tqi (x) dx ∂ αi (.) >t ∂x u
i=1
≤
i
Z
tqi (x) dx ∂ αi (.) >t ∩{|u|≤t} ∂x u
+
i
Z
tqi (x) dx
{|u|>t}
pi (x) ∂ αi (x) αi (x) ∂xi u ≤ tqi (x) dx + M ≤ kf k1 + M, for all t > 0. t {|u|≤t} Z
Proposition 3.6. Assume (1.2)-(1.6) and f ∈ L1 (Ω). Let u be an entropy solution of (1.1), then N
1X h i=1
Z
{|u|≤h}
pi (x) ∂ dx ≤ M ∂xi Th (u)
for every h > 0, with M a positive constant. More precisely, there exists D > 0 such that −
meas {|u| > h} ≤ DP−
1+h −
hP−
.
Proof. Taking ϕ = 0 in the entropy inequality (3.1) and using (1.5), we obtain N Z X i=1
for all h > 0. Next, N Z X i=1
{|u|≤h}
{|u|≤h}
pi (x) ∂ T (u) dx ≤ h kf k1 ≤ M h ∂xi h
pi (x) N Z X ∂ T (u) dx ≤ M h ⇒ ∂xi h i=1
{|u|≤h}
P−− ∂ ∂xi Th (u) dx ≤ C(1 + h).
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We can write the above inequality as
P−− N 1 X
∂ − P
− .
∂xi Th (u) − ≤ C(1 + h) or kTh (u)kW 1,P−− (Ω) ≤ [C(1 + h)] i=1
P−
0
By the Poincar´e inequality in constant exponent, we obtain
kTh (u)k
1 −
− P L − (Ω)
≤ D(1 + h) P− .
The above inequality imply that Z
− P−
|Th (u)|
−
dx ≤ DP− (1 + h),
Ω
from which we obtain −
meas {|u| > h} ≤ DP−
1+h −
hP−
.
∗ Step 2. Uniqueness of entropy solution. The proof of uniqueness of entropy solutions follow the technics by B´enilan et al [3] (see also [25]). Let h > 0 and u, v two entropy solutions of (1.1). We write the entropy inequality (3.1) corresponding to the solution u, with Th v as test function, and to the solution v, with Th u as test function. Upon addition, we get Z N X ∂ ∂ ai (x, u). (u − Th v)dx+ ∂x ∂x i i {|u−Th v|≤t} i=1 Z N X ∂ ∂ a(x, v). (v − Th u)dx ∂x ∂x i i {|v−Th u|≤t} i=1 Z ≤ f (x)(Tt (u − Th v) + Tt (v − Th u))dx. Ω
Define
E1 := {|u − v| ≤ t, |v| ≤ h} , E2 := E1 ∩ {|u| ≤ h} , and E3 := E1 ∩ {|u| > h} .
(3.3)
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We start with the first integral in (3.3). By (1.5), we have Z N X ∂ ∂ a(x, u). (u − Th v)dx = ∂xi ∂xi {|u−Th v|≤t} i=1 Z N X ∂ ∂ u). (u − Th v)dx = a(x, ∂x ∂x i i {|u−Th v|≤t}∩({|v|≤h}∪{|v|>h}) i=1 Z N X ∂ ∂ a(x, u). (u − Th v)dx+ {|u−Th v|≤t,|v|≤h} ∂xi ∂xi i=1 Z N X ∂ ∂ u). (u − Th v)dx a(x, ∂x ∂x i i {|u−Th v|≤t,|v|>h} i=1 Z Z N N X X ∂ ∂ ∂ ∂ = a(x, u). (u − v)dx + a(x, u). udx ∂x ∂x ∂x ∂x i i i i {|u−v|≤t,|v|≤h} i=1 {|u−Th v|≤t,|v|>h} i=1
≥
N X
Z
{|u−v|≤t,|v|≤h} i=1
a(x,
∂ ∂ u). (u − v)dx = ∂xi ∂xi
Z
N X
N X
E1 i=1
∂ ∂ u). (u − v)dx = = a(x, ∂x ∂x i i E1 ∩({|u|≤h}∪{|u|>h}) i=1 Z
+
N X
Z
a(x,
E3 i=1 N X
∂ ∂ u). udx − a(x, ∂xi ∂xi i=1
Z
E3
≥
∂ ∂ u). (u − v)dx = ∂xi ∂xi
Z
N X
E2 i=1
a(x,
Z
N X
Z
a(x,
E2 i=1
a(x,
E3 i=1
∂ ∂ u). (u − v)dx − ∂xi ∂xi
N X
Z
Z
a(x,
∂ ∂ u). (u − v)dx ∂xi ∂xi
N X
a(x,
E2 i=1
∂ ∂ u). (u − v)dx ∂xi ∂xi
∂ ∂ u). (u − v)dx+ ∂xi ∂xi
∂ ∂ u). vdx ∂xi ∂xi N X
E3 i=1
a(x,
∂ ∂ u). vdx. ∂xi ∂xi (3.4)
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Using (1.3) and (2.1), we estimate the last integral in (3.4) as follows Z pi (x)−1 ! Z X N N X ∂ ∂ ∂ ∂ u). vdx ≤ C1 ji (x) + u a(x, ∂xi v dx ∂x ∂x ∂x i i i E E 3 3 i=1 i=1 N ∂ pi (x)−1 X |j|p′ (.) + u ≤C i ∂xi i=1
∂ pi (x)−1 where ∂xi u
p′i (.),{hh}
According to Proposition 3.6, both meas {|u| > h} and meas {|v| > h} tend to zero as h goes to infinity, then by the inequality above, the right-hand side of inequality (3.3) tends to zero as h
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goes to infinity. From this assertion, (3.3), (3.6) and (3.7), we obtain, letting h → +∞, N X
Z
{|u−v|≤t} i=1
(a(x,
∂ ∂ ∂ u) − a(x, v)). (u − v)dx ≤ 0, for all t > 0. ∂xi ∂xi ∂xi
By assertion (1.4), we conclude that We deduce that
∂ ∂ u= v, for all i = 1, .., N a.e. in Ω. ∂xi ∂xi
ku − vk→ − p (.)
N X ∂ ∂ = = 0, ∂xi u − ∂xi v pi (.) i=1
and hence u = v, a.e. in Ω. ∗ Step 3. Existence of entropy solutions. Let (fn )n be a sequence of bounded functions, strongly converging to f ∈ L1 (Ω) and such that kfn k1 ≤ kf k1 , for all n. We consider the problem N X ∂ ∂ − ai (x, un ) = fn in Ω ∂xi ∂xi i=1
(3.8)
(3.9)
un = 0 on ∂Ω.
It follows from Theorem 2.3 (cf. [15]) that problem (3.9) has a unique weak energy solution − 1,→ p (.) un ∈ W0 (Ω) since fn ∈ L∞ (Ω). Our interest is to prove that these approximated solutions un tend, as n goes to infinity, to a measurable function u which is an entropy solution of the limit ∂ un in the Marcinkiewicz problem (1.1). To this end, we derive a priori estimates for un and ∂x i ∗
−
spaces Mq and Mpi q/p¯ where q ∗ and p¯ are defined in (1.6) and (1.7). Let γ > 0, denote Tγ the corresponding truncation function. Note that 1 if |r| < γ DTγ (r) = 0 if |r| > γ. In particular, we have
ai (x, ξ)DTγ (r)ξ ≥ ai (x, ξ)ξχ{|r| 0. {|un |≤γ} ∂xi
N Z X i=1
(3.11)
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Proof. Inserting ϕ = Tγ (un ) into (2.2), we have Z X N Ω i=1
ai (x,
∂ ∂ un )DTγ (un ) un dx = ∂xi ∂xi
Z
fn Tγ (un )dx.
Ω
Using (3.10) and the coercivity condition (1.5), we obtain (3.11). Lemma 3.8. There exists a constant C, not depending on n, such that kun kMq∗ (Ω) ≤ C and
∂un
≤ C, for all i = 1, ..., N.
∂xi p− q/p¯ M i (Ω)
Proof. The result of Lemma 3.8 is a direct consequence of Lemma 3.7 (cf. [2, proof of Lemma 3.3]). In view of Lemma 3.8 and following [2] ( see also [4, Lemma A.2]), we deduce that un is unin formly bounded in Lk0 (Ω) for some k0 < q ∗ with k0 > p− p and ∂u i q/¯ ∂xi is uniformly bounded in − − ki L (Ω) for some ki > 1 with pi − 1 < ki < pi q/¯ p, for all i = 1, ..., N . From this, we get that un is uniformly bounded in the isotropic Sobolev space W01,kmin (Ω), where kmin = min(k1 , ..., kN ). Consequently, we can assume without loss of generality (see also [2, Lemma 3.4]) that as n → 0, un → u a.e in Ω and in Lkmin (Ω) un ⇀ u in W01,kmin (Ω)
∂un ∂u → in L1 (Ω), for all i = 1, ..., N. ∂xi ∂xi ∂u ∂u ai (x, n ) → ai (x, ) a.e in Ω and in L1 (Ω), for all i = 1, ..., N. ∂xi ∂xi
(3.12)
− 1,→ p (.)
Now, fix t > 0, ϕ ∈ W0 (Ω) ∩ L∞ (Ω), and choose Tt (un − ϕ) as a test function in (2.2), with u replaced by un to obtain Z X Z N ∂ ∂ ai (x, un ). Tt (un − ϕ)dx = fn (x)Tt (un − ϕ)dx. ∂xi ∂xi Ω i=1 Ω Note that this choice can be made using a standard density argument. We now pass to the limit in the previous identity. For the right-hand side, the convergence is obvious since fn converges strongly in L1 to f and Tt (un − ϕ) converges weakly-* in L∞ , and a.e., to Tt (u − ϕ). Next, we write the left hand side as Z Z N N X X ∂ ∂ ∂ ∂ ai (x, un ). un dx − ai (x, un ). ϕdx (3.13) ∂xi ∂xi ∂xi ∂xi {|un −ϕ|≤t} i=1 {|un −ϕ|≤t} i=1
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Stanislas OUARO
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By (3.12), the second integral of (3.13) converges to N X
Z
ai (x,
{|u−ϕ|≤t} i=1
For the first integral in (3.13), as
N X i=1
using (3.12) and Fatou’s Lemma Z
N X
∂ ∂ u). ϕdx. ∂xi ∂xi
ai (x,
∂ ∂ un ). un is nonnegative by (1.5), we obtain by ∂xi ∂xi
∂ ∂ u). udx ≤ lim inf n→∞ ∂xi ∂xi
ai (x,
{|u−ϕ|≤t} i=1
N X
Z
{|un −ϕ|≤t} i=1
ai (x,
∂ ∂ un ). un dx. ∂xi ∂xi
Gathering results, we obtain
Z X N Ω i=1
ai (x,
∂ ∂ u). Tt (u − ϕ)dx ≤ ∂xi ∂xi
Z
f (x)Tt (u − ϕ)dx,
Ω
i.e., u is an entropy solution of (1.1).
Received:
June, 2008.
Revised:
november, 2008.
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