Convergence for elliptic equations in periodic perforated domains

Abstract

Convergence for the solutions of elliptic equations in periodic perforated domains is concerned. Let epsilon denote the size ratio of the holes of a periodic perforated domain to the whole domain. It is known that, by energy method, the gradient of the solutions of elliptic equations is bounded uniformly in epsilon in L-2 space. Also, when epsilon approaches 0, the elliptic solutions converge to a solution of some simple homogenized elliptic equation. In this work, above results are extended to general W-1,W-p space for p > 1. More precisely, a uniform W-1,W-p estimate in epsilon for p is an element of (1, infinity] and a W-1,W-p convergence result for p is an element of (n/n-2, infinity] for the elliptic solutions in periodic perforated domains are derived. Here n is the dimension of the space domain. One also notes that the L-p norm of the second order derivatives of the elliptic solutions in general cannot be bounded uniformly in epsilon. (c) 2013 Elsevier Inc. All rights reserved.