Behavior of the p-Laplacian on Thin Domains

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 [1–7], as well as in the parabolic counterpart, associated with the continuity of global attractors for dissipative equations as in [1, 8–15].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 pLaplacian operator. This can be used, for example, in the study of the asymptotic behavior (attractors) of dissipative equations governed by the p-Laplacian on thin domains, which is associatedwith 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 nonNewtonian fluids as well as in flow through porous media. In order to set up the problem, letω be a smooth bounded domain in R, n ≥ 1, and g ∈ C2(ω;R) a positive function; ε will represent a small positive parameter which will converge to zero.We consider the family of domainsΩ ⊂ Rn+1 defined by


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 [1][2][3][4][5][6][7], as well as in the parabolic counterpart, associated with the continuity of global attractors for dissipative equations as in [1,[8][9][10][11][12][13][14][15].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 R  ,  ≥ 1, and  ∈  2 (; R) a positive function;  will represent a small positive parameter which will converge to zero.We consider the family of domains Ω  ⊂ R +1 defined by The aim of this paper is to characterize the limiting problem ( = 0) for the family of elliptic equations where  > 2,   ∈   (Ω  ), (1/) + (1/) = 1, Δ   := div(|∇| −2 ∇) denotes the -Laplacian operator and   denotes the outward unitary normal vector field to Ω  .
In the analysis of the limiting behavior of   , it will be useful to introduce the domain Ω :=  × (0, 1) which is International Journal of Differential Equations independent of  and is obtained from Ω  by the change of coordinates T  : Ω → Ω  (, )  → (,  () ) .
Such change of coordinates induces an isomorphism from  , (Ω  ) onto  , (Ω) 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 Ω.
Noticing that  ∈  1, (Ω  ) is a solution of (2) if and only if V := Φ() ∈  1, (Ω) 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 for all  ∈  1, (Ω).
Due to the nature of this specific domain perturbation, solutions of (7) tend not to depend "so much" on the variable  as  ≈ 0. This suggests comparing such solutions with solutions of the following equation: for some appropriate ĥ ∈   (), where ] denotes the unit outward normal vector field to .
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.

Preliminaries
Stressing for the fact that the domains Ω  vary in accordance with the small parameter , collapsing themselves to a lower dimensional subset as  goes to 0, we perform a dilatation on the Lebesgue measure in R +1 in order to preserve the relative capacity of measurable subsets of Ω  .Thus, we consider the Lebesgue space,   (Ω  ), endowed with the equivalent norm 1/ (11) and the Sobolev space,  1, (Ω  ), endowed with the equivalent norm We also consider equivalent norms in   (Ω) and in  1, (Ω) given, respectively, by It is immediate from these definitions that and there exist positive constants  1 ,  2 such that Finally, since we need to compare functions defined in different domains, for example, Ω and , is natural to introduce the following operators.
Average projector: Extension operator:
As a consequence of the following theorem and inspired by [8,15], we obtain the convergence of the family of solutions V  in the norm ||| ⋅ |||  .Theorem 3. Let ℎ  , V  , ℎ 0 , and V 0 be as in Lemma 2. Then which implies that lim  → 0 (1/  )|∇  V  |  = 0.
Remark 5. We would like to recall that Hale and Raugel in [12] obtained in the case  = 2 as the limiting problem for the similar equation the problem − 1  div (∇V) + V = ĥ, in , V ] = 0, on . (28) 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  → 2 and  → 0, independent of the order of the convergence, we obtain the same limiting problem, namely, (28).We summarize that in the following commutative diagram: Equation