International Journal of Differential Equations

Volume 2015, Article ID 392479, 11 pages

http://dx.doi.org/10.1155/2015/392479

## On the Convergence of a Nonlinear Boundary-Value Problem in a Perforated Domain

Department of Higher Mathematics, Gubkin Russian State University of Oil and Gas, Leninsky Prospect 65-1, Moscow 119991, Russia

Received 16 July 2015; Accepted 7 September 2015

Academic Editor: Elena I. Kaikina

Copyright © 2015 Yulia Koroleva. 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 consider a family with respect to a small parameter of nonlinear boundary-value problems as well as the corresponding spectral problems in a domain perforated periodically along a part of the boundary. We prove the convergence of solution of the original problems to the solution of the respective homogenized problem in this domain.

#### 1. Introduction

The paper is devoted to study of convergence of nonlinear boundary-value problems in a domain perforated along the boundary. There exist a lot of literatures, where boundary-value problems in perforated domains were studied. We refer to works [1–33]. In these papers and monographs the authors studied different kinds of perforation for linear as well as for nonlinear differential operators. Usually it is considered a family of problems depending on small parameter that characterizes the size of perforation. The main goal of the research is to find a homogenized (limit) model which is close to originally considered problems posed in the perforated domain. The general technique of homogenization method can be found in [23, 24, 27, 28].

The present paper will deal with convergence of boundary-value problems in perforated domains for nonlinear -Laplace operator. Some problems for nonlinear operators were homogenized, for example, in [1, 20, 21, 30–33]. We consider a family of boundary-value problems in -dimensional domain, , which is periodically perforated along the boundary by small sets. It is assumed that the diameter of each set and the distance between them have the same order. In our problem we suppose that the Dirichlet condition holds on the boundary of cavities, while the Naumann boundary condition is fulfilled on the boundary of the domain. We derive the limit (homogenized) problem for the original problems when the small parameter characterizing the size of perforation tends to zero. Moreover, we establish the strong convergence in of the solutions for the considered problems to the corresponding solution of the limit problem. In addition we have obtained an estimate of the solution in a neighborhood of the eigenvalue of a corresponding spectral problem.

One of our goals is to prove the asymptotic behavior for the eigenvalue problem for -Laplace operator in our perforated domain. Many authors considered spectral problems for -Laplace operator; see, for example, [33–38]. These papers contain the results on qualitative properties of the -Laplace spectral problems, convergence of eigenvalue problems, and some estimates for the difference between considered eigenvalues. The applications of our problem do not require the knowledge about the full spectrum of eigenvalues. Therefore we have proved the homogenization theorems only for the first eigenelement of the spectral problem in perforated domain. More precisely, we have proved that the first eigenelement of the spectral problems converges to the corresponding eigenelement of the spectral limit problem. An analogous problem for linear elliptic operators for the two-dimensional domain was considered in [15] and for dimension three in [10].

The crucial point in our analysis is the validity of the Friedrichs inequality for functions in perforated domains. We prove this nontrivial result which is of an independent interest. Some papers devoted to this inequality in domains with microinhomogeneous structure are [8–10, 18, 19].

#### 2. Preliminaries and the Main Results

Let , , be a domain with boundary . We assume that is piece-wise smooth and consists of the parts , , where , , are orthogonal to and belong to the planes and correspondingly, and is a smooth surface. In the sequel is a small parameter, , .

Consider the set belonging to the ball and having a smooth boundary. If one multiplies each coordinate of with parameter and does integer translations of this set along , we obtain the set denoted by Let . Define the perforated domain as See the illustration for cut of on Figure 1.