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International Journal of Differential Equations
Volume 2013 (2013), Article ID 476781, 14 pages
http://dx.doi.org/10.1155/2013/476781
Research Article

Entropy Solutions for Nonlinear Elliptic Anisotropic Homogeneous Neumann Problem

Laboratoire d'Analyse Mathématique des Equations (LAME), UFR, Sciences Exactes et Appliquées, Université de Ouagadougou, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso

Received 22 June 2012; Accepted 1 February 2013

Academic Editor: Elena Braverman

Copyright © 2013 B. K. Bonzi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove the existence and uniqueness of entropy solution for nonlinear anisotropic elliptic equations with Neumann homogeneous boundary value condition for -data. We prove first, by using minimization techniques, the existence and uniqueness of weak solution when the data is bounded, and by approximation methods, we prove a result of existence and uniqueness of entropy solution.

1. Introduction

Let be an open bounded domain of with smooth boundary. Our aim is to prove the existence and uniqueness of entropy solution for the anisotropic nonlinear elliptic problem of the form where the right-hand side and , are the components of the outer normal unit vector.

For the rest of the functions involved in (1), we are going to enumerate their properties after we make some notations.

For any , we set and we denote For the exponents, , with for every and for all , we put and . Now, we can give the properties of the rest of the functions involved in (1).

We assume that for , the function is Carathéodory and satisfies the following conditions: is the continuous derivative with respect to of the mapping , , that is, such that the following equality and inequalities holds for almost every .

There exists a positive constant such that for almost every and for every , where is a nonnegative function in , with .

There exists a positive constant such that for almost every and for every , with and for almost every and for every .

We also assume that the variable exponents are continuous functions for all such that where .

We introduce the numbers A prototype example, that is, covered by our assumptions is the following anisotropic equation:

Set , where . Then, we get the following equation. Actually, one of the topics from the field of PDEs that continuously gained interest is the one concerning the Sobolev space with variable exponents, or depending on the boundary condition (see [123]). In that context, problems involving the -Laplace operator or the more general operator were intensively studied (see [13]). At the same time, some authors was interested by PDEs involving anisotropic Sobolev spaces with variable exponent when the boundary condition is the homogeneous Dirichlet boundary condition (see [15, 16, 18, 20, 2426]). In that context, the authors have considered the anisotropic -Laplace operator or the more general variable exponent anisotropic operator When the homogeneous Dirichlet boundary condition is replaced by the Neumann boundary condition, one has to work with the anisotropic variable exponent Sobolev space instead of . The main difficulty which appears is that the famous Poincaré inequality does not apply and then it is very difficult to get a priori estimates which are necessary for the proof of the existence result of entropy solutions. Sometimes one can use the Wirtinger inequality which does not apply, in some problems like (1). The first systematic study of anisotropic Neumann problem was done by Fan (see [11]). In a second time, Boureanu and Rădulescu studied an anisotropic nonhomogeneous Neumann problem with obstacle (see [2]). In the two papers, the authors were interested by the existence and multiplicity results of weak solution even if in [2], Boureanu and Rădulescu have showed some conditions under which we can get uniqueness of weak solution. In this paper, we are interested to the existence and uniqueness of entropy solution. For the proof of the existence of entropy solution of (1), we follow [27] and derive a priori estimates for the approximated solutions and the partial derivatives in the Marcinkiewicz spaces and , respectively (see Section 2 or [27, 28] for definition and properties of Marcinkiewicz spaces).

The study of anisotropic problems are motivated, for example, by their applications to the mathematical analysis of a system of nonlinear partial differential equations arising in a population dynamics model describing the spread of an epidemic disease through a heterogeneous habitat.

The paper is organized as follows. In Section 2, we introduce some notations/functional spaces. In Section 3, we prove for the problem (1), the existence and uniqueness of weak solution when the data is bounded, and the existence and uniqueness of entropy solution when the data is in .

2. Preliminaries

In this section, we define Lebesgue, Sobolev, and anisotropic spaces with variable exponent and give some of their properties (see [29] for more details about Lebesgue and Sobolev spaces with variable exponent). 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 , we define the Lebesgue space with variable exponent as the set of all measurable functions for which the convex modular is finite. If the exponent is bounded, that is, if , then the expression defines a norm in , called the Luxembourg norm. The space is a separable Banach space. Moreover, if , then is uniformly convex, hence reflexive, and its dual space is isomorphic to , where . Finally, we have the Hölder type inequality.

Proposition 1 (generalized Hölder inequality, see [10]). (i) For any and , we have
(ii) If ,   for any , then and the imbedding is continuous.

Moreover, the application called the modular of the space is very useful in handling these Lebesgue spaces with variable exponent. Indeed we have the following properties (see [10]). If and then If, in addition, , then

converges to in measure and .

Now, let us introduce the definition of the isotropic Sobolev space with variable exponent, .

We set which is a Banach space equipped with the norm

Now, we present a natural generalization of the variable exponent Sobolev space that will enable us to study the problem (1) with sufficient accuracy.

The anisotropic variable exponent Sobolev space is defined as follows: Endowed with the norm the space is a reflexive Banach space (see [11, Theorems 2.1 and 2.2]).

We have the following result.

Theorem 2 (see [11, Corollary 2.1]). Let be a bounded open set and for all ,   a.e. in . Then, for any with a.e. in such that we have the compact embedding

Next, we define Finally, in this paper, we will use the Marcinkiewicz spaces with constant exponent. Note that the Marcinkiewicz spaces in the variable exponent setting were introduced for the first time by Sanchon and Urbano (see [23]).

Marcinkiewicz spaces contain the measurable functions for which the distribution function satisfies an estimate of the form The space is a Banach space under the norm where denotes the nonincreasing rearrangement of : We will use the following pseudonorm which is equivalent to the norm (see [27]).

We need the following Lemma (see [28, Lemma  A.2]).

Lemma 3. Let . Then, for every measurable function on , we have(i) Moreover,(ii), for every measurable subset .
In particular, with continuous injection and implies .

The following result is due to Troisi (see [30]).

Theorem 4. Let and let Then, there exists a constant depending on if  and also on and if  such that where and .
We will use through the paper, the truncation function at height , that is

We need the following lemma.

Lemma 5. Let be a nonnegative function in . Assume and there exists a constant such that Then, there exists a constant , depending on , such that where .

Proof. Consider the following
Step  1 (). Then, obviously we have , for some positive constant . Indeed, since , according to Proposition 1 there exists a positive constant such that It follows that there exists a positive constant such that
Step  2 (). We get from (37) Not also that Therefore, by using (35), we obtain for , It follows that which is equivalent to Therefore, Since, we get which implies that .
For we have So, We need the following well-known results.

Theorem 6 (see [31, Theorem ]). Let be a reflexive Banach space and let be Gateaux differentiable over the closed set . Then, the following are equivalent.(i) is convex over . (ii)We have where denotes the dual of the space . (iii)The first Gateaux derivative is monotone, that is, (iv)The second Gateaux derivative of exists and it is positive, that is,

Theorem 7 (see [32, Theorem 1.2]). Suppose is a reflexive Banach space with norm , and let be a weakly closed subset of . Suppose is coercive and (sequentially) weakly lower semicontinuous on M with respect to , that is, suppose the following conditions are fulfilled.(i) as .(ii)For any , any subsequence in such that weakly in there holds Then, is bounded from below and attains its infinimun in .

3. Main Results

In the sequel, we denote and .

3.1. Weak Solutions

Let us define first the notion of weak solution.

Definition 8. Let be a measurable function, we say that is a weak solution of problem (1) if belongs to and satisfies the following equation: for every .
We associate to problem (1) the energy functional , defined by To simplify our writing, we denote by the functional We recall the following result (see [15, Lemma 3.4]).

Lemma 9. The functional is well-defined on . In addition, is of class and for all .

Due to Lemma 9, a standard calculus leads to the facts that is well-defined on and with the derivative given by for all . Obviously, the weak solutions of (1) are the critical points of ; so by means of Theorem 7, we intend to prove the existence of critical points in order to deduce the existence of weak solutions.

Theorem 10. Assume (4)–(8) and . Then, there exists a unique weak solution of problem (1).

Let us start the proof by establishing some useful lemmas.

Lemma 11. If hypotheses (4)–(8) are fulfilled, then the functional is coercive.

Proof. Let be such that . Using (7), we deduce that We make the following notations: We then have Using (19), (20), and (21), we have By the generalized mean inequality or the Jensen’s inequality applied to the convex function , we get thus,
Case  1 (). We have Therefore,
Case  2  (). Then , and we get So, we obtain Then, letting goes to infinity in (66) and (68), we conclude that reaches infinity. Thus, is coercive.

Lemma 12. The functional is weakly lower semicontinuous.

Proof. By [33, Corollary ], it is enough to show that is lower semicontinuous. To this aim, fix and . Since for every is monotone, Theorem 6 yields Since the map is convex, again by Theorem 6, we have then (69) becomes Consider the second term in the right-hand side of (71). By (5) and Hölder type inequality, we have For the fourth term in the right-hand side of (71), we have The third term in the right-hand side of (71) gives by using Hölder type inequality Gathering these inequalities, it follows that for every such that . Thus, is lower semicontinuous.

Proof of Theorem 10. Consider the following
Step  1. Existence of weak solutions. The proof follows directly from Lemmas 11 and 12 and Theorem 7.
Step  2. Uniqueness of weak solution. Let be two weak solutions of problem (1). Choosing a test function in (54), for the weak solution and for the weak solution , we get Summing up (76) and (77), we obtain Thus, by the monotonicity of the functions and , we deduce that almost everywhere.

3.2. Entropy Solutions

First of all, we define a space in which we will look for entropy solutions. We define the space as the set of every measurable function which satisfies for every .

Lemma 13 (see [34, 35]). Let . Then, there exists a unique measurable function such that where denotes the characteristic function of a measurable set . The functions are called the weak partial gradients of and are still denoted . Moreover, if belongs to , then and coincides with the standard distributional gradient of , that is, .

Definition 14. We define the space as the set of function such that there exists a sequence satisfying(a) a.e. in ,(b) in , for all .

Definition 15. A measurable function is an entropy solution of (1) if and for every , for all .

Our main result in this section is the following.

Theorem 16. Assume (4)–(8) and . Then, there exists a unique entropy solution to problem (1).
Proof. The proof of this Theorem will be done in three steps.

Step  1 (a priori estimates).

Lemma 17. Assume (4)–(8) and . Let be an entropy solution of (1). If there exists a positive constant such that then where , for all .

Proof. Take in (80), we have Since the second term in the previous inequality is nonnegative, it follows that
According to (7), we deduce that Therefore, defining , we have for all , From the previous inequality, the definition of and (81), we have

Lemma 18. Assume (4)–(8) and . Let be an entropy solution of (1), then for every , with a positive constant. Moreover, we have and there exists a constant which depends on and such that

Proof. Taking in the entropy inequality (80) and using (7), we obtain for all . This yields As , we get from the previous inequality by using Fatou’s lemma Now, since we have We deduce that Indeed, From aforementioned, we get Therefore, which implies

Lemma 19. If is an entropy solution of (1) then there exists a constant such that

Proof. Taking in the entropy inequality (80) and using (7), we get Note that Therefore, we deduce according to (101) that

Lemma 20. If is an entropy solution of (1) then where , , .

Proof. Taking as a test function in the entropy inequality (80), we get It follows by using (7) that Therefore,

Lemma 21. If is an entropy solution of (1) then where , .

Proof. By Lemma 18, we deduce that and as , it follows by using the Lebesgue dominated convergence theorem that The proof of the following lemma can be found in [1].

Lemma 22.  Assume (4)–(8) and . Let be an entropy solution of (1), then where is a positive constant which depends on and .

Step  2 (uniqueness of entropy solution). The proof of the uniqueness of entropy solutions follows the same techniques by Ouaro [20] (see also [35]). Indeed, let and be two entropy solutions of (1). We write the entropy inequality (54) corresponding to the solution , with as test function, and to the solution , with as test function. Upon addition, we get Define