Title: Convergence for elliptic equations in periodic perforated domains
Authors: Yeh, Li-Ming
Department of Applied Mathematics
Keywords: Periodic perforated domain;Homogenized elliptic equation
Issue Date: 1-Oct-2013
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.
URI: http://dx.doi.org/10.1016/j.jde.2013.05.023
ISSN: 0022-0396
DOI: 10.1016/j.jde.2013.05.023
Volume: 255
Issue: 7
Begin Page: 1734
End Page: 1783
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