Homogenization of elliptic problems with quadratic growth and nonhomogenous Robin conditions in perforated domains

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• 作者：Imen Chourabi ; Patrizia Donato
• 关键词：Homogenization ; Elliptic problems ; Quadratic growth ; Nonhomogeneous Robin boundary conditions ; Perforated domains
• 刊名：Chinese Annals of Mathematics - Series B
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：37
• 期：6
• 页码：833-852
• 全文大小：494 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Applications of Mathematics
  Chinese Library of Science
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1860-6261
• 卷排序：37

文摘

This paper deals with the homogenization of a class of nonlinear elliptic problems with quadratic growth in a periodically perforated domain. The authors prescribe a Dirichlet condition on the exterior boundary and a nonhomogeneous nonlinear Robin condition on the boundary of the holes. The main difficulty, when passing to the limit, is that the solution of the problems converges neither strongly in L2(Ω) nor almost everywhere in Ω. A new convergence result involving nonlinear functions provides suitable weak convergence results which permit passing to the limit without using any extension operator. Consequently, using a corrector result proved in [Chourabi, I. and Donato, P., Homogenization and correctors of a class of elliptic problems in perforated domains, Asymptotic Analysis, 92(1), 2015, 1–43, DOI: 10.3233/ASY-151288], the authors describe the limit problem, presenting a limit nonlinearity which is different for the two cases, that of a Neumann datum with a nonzero average and with a zero average.