POINTWISE ESTIMATE FOR ELLIPTIC EQUATIONS IN PERIODIC PERFORATED DOMAINS

Abstract

Pointwise estimate for the solutions of elliptic equations in periodic perforated domains is concerned. Let epsilon denote the size ratio of the period of a periodic perforated domain to the whole domain. It is known that even if the given functions of the elliptic equations are bounded uniformly in epsilon, the C-1,C- alpha norm and the W-2,W- p norm of the elliptic solutions may not be bounded uniformly in epsilon. It is also known that when epsilon closes to 0, the elliptic solutions in the periodic perforated domains approach a solution of some homogenized elliptic equation. In this work, the Holder uniform bound in epsilon and the Lipschitz uniform bound in epsilon for the elliptic solutions in perforated domains are proved. The L-infinity and the Lipschitz convergence estimates for the difference between the elliptic solutions in the perforated domains and the solution of the homogenized elliptic equation are derived.