We give the characterization of the limiting behavior of solutions of elliptic equations driven by the -Laplacian operator with Neumann boundary conditions posed in a family of thin domains.

1. Introduction

The investigation of parabolic and elliptic equations on thin domains has received considerable attention over the last twenty years. Such equations can appear motivated by homogenization problems in thin structures as in [17], as well as in the parabolic counterpart, associated with the continuity of global attractors for dissipative equations as in [1, 815]. Whatever the motivations that appear, the key point in the study of any kind of perturbation problem is to find the limiting one. In this specific domain perturbation problem (thin domains), it means to find an equation posed in a lower dimensional domain in order to compare the perturbed problems with. Our contribution in this short note goes in this direction. We give the characterization of the limiting problem of a family of elliptic equations driven by the -Laplacian operator. This can be used, for example, in the study of the asymptotic behavior (attractors) of dissipative equations governed by the -Laplacian on thin domains, which is associated with localized large diffusion phenomena, see, for example, [16]. This is the first step in order to consider other aspects as the asymptotic dynamics (attractors). For the best of our knowledge this is an untouched topic in the literature and can be the starting point for investigation of quasi-linear parabolic equations on thin domains which is relevant in a variety of physical phenomena as non-Newtonian fluids as well as in flow through porous media.

In order to set up the problem, let be a smooth bounded domain in , , and a positive function; will represent a small positive parameter which will converge to zero. We consider the family of domains defined by The aim of this paper is to characterize the limiting problem () for the family of elliptic equations where , , , denotes the -Laplacian operator and denotes the outward unitary normal vector field to .

Definition 1. Given , , one says that , is a solution of (2) if for all .

We recall that by [17, Theorem 2.1] and [17, Theorem 2.3] (3) has a unique solution .

In the analysis of the limiting behavior of , it will be useful to introduce the domain which is independent of and is obtained from by the change of coordinates

Such change of coordinates induces an isomorphism from onto by with partial derivatives related by

In this new system of coordinates, (2) is written as where , , and denotes the unit outward normal vector field to .

Noticing that is a solution of (2) if and only if is a solution of (7), the rest of this paper is dedicated to the study of the limiting behavior of the solutions of (7); that is, functions such that for all .

Due to the nature of this specific domain perturbation, solutions of (7) tend not to depend “so much” on the variable as . This suggests comparing such solutions with solutions of the following equation: for some appropriate , where denotes the unit outward normal vector field to .

Again by [17, Theorem 2.1] and [17, Theorem 2.3] we can derive a unique solution of (10).

The paper is organized as follows. In Section 2 we set up the appropriate functional framework which will be used to compare the problems (7) and (10), and in the subsequent Section 3, we formulate and prove the convergence results.

2. Preliminaries

Stressing for the fact that the domains vary in accordance with the small parameter , collapsing themselves to a lower dimensional subset as goes to , we perform a dilatation on the Lebesgue measure in in order to preserve the relative capacity of measurable subsets of . Thus, we consider the Lebesgue space, , endowed with the equivalent norm and the Sobolev space, , endowed with the equivalent norm

We also consider equivalent norms in and in given, respectively, by

It is immediate from these definitions that and there exist positive constants , such that

Finally, since we need to compare functions defined in different domains, for example, and , is natural to introduce the following operators.

Average projector:

Extension operator:

3. Convergence

Given , , if is a solution of (2) we have by Hölder's inequality that This shows that which gives us the following a priori estimate for solutions of (7): where and .

Such a priori estimate is the essence of the following lemma.

Lemma 2. Let , , such that in . If is a solution of (7), there exists solution of (10) with , such that up to subsequence

Proof. It follows from (20) that , for some constant independent of . Since is a reflexive space and compactly, taking subsequence if necessary, there exists such that , weakly in and strongly in . Moreover, noticing that is bounded, one has that . Therefore in , which means that almost everywhere in . This implies the existence of , such that , almost everywhere in .
On the other hand, noticing that whenever and since , it follows from the weak convergence in that for all .

As a consequence of the following theorem and inspired by [8, 15], we obtain the convergence of the family of solutions in the norm .

Theorem 3. Let , , , and be as in Lemma 2. Then

Proof. It follows from the weak convergence in (obtained in Lemma 2) that Recalling that for , is an equivalent norm in , this proves the statement.

Corollary 4. Let and be as in Theorem 3. Then

Proof. According to Theorem 3 which implies that .

Remark 5. We would like to recall that Hale and Raugel in [12] obtained in the case as the limiting problem for the similar equation the problem
After the previous considerations we point out the robustness of the structure of this limiting problem in the following sense: considering as a parameter as well as , allowing and , independent of the order of the convergence, we obtain the same limiting problem, namely, (28). We summarize that in the following commutative diagram:

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.


The author thanks Professor Olímpio Myagaki for stimulating conversations and useful remarks. The author also thanks the referees for their suggestions which led to an improvement of the text. This work was partially supported by FAPESP no. 2012/06753-8, FUNDUNESP no. 0135812, and FUNDUNESP-PROPe no. 0019/008/13, Brazil.