Pointwise estimate for elliptic equations in periodic perforated domains

Li-Ming  Yeh

doi:10.3934/cpaa.2015.14.1961

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2015, Volume 14, Issue 5: 1961-1986. Doi: 10.3934/cpaa.2015.14.1961

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Pointwise estimate for elliptic equations in periodic perforated domains

Li-Ming Yeh^1,

1.
Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 30050

Received: October 31, 2014

Revised: January 31, 2015

Published: August 2015

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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 $C^{1,\alpha}$ norm and the $W^{2,p}$ 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 Hölder uniform bound in $\epsilon$ and the Lipschitz uniform bound in $\epsilon$ for the elliptic solutions in perforated domains are proved. The $L^\infty$ 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.

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Mathematics Subject Classification: Primary: 35J05, 35J15, 35J25.

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