American Institute of Mathematical Sciences

September  2015, 14(5): 1961-1986. doi: 10.3934/cpaa.2015.14.1961

Pointwise estimate for elliptic equations in periodic perforated domains

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

Received  November 2014 Revised  February 2015 Published  June 2015

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.
Citation: Li-Ming Yeh. Pointwise estimate for elliptic equations in periodic perforated domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1961-1986. doi: 10.3934/cpaa.2015.14.1961
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