Behavior of the -Laplacian on Thin Domains

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 [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 -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