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

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Abstract

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 L 2(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.

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Authors and Affiliations

  1. Department of Mathematics Faculty of Sciences of Monastir, Université de Monastir, UR Analysis and Control of PDE, UR13ES64, 5019, Monastir, Tunisia

    Imen Chourabi

  2. Laboratoire de Mathématiques Raphaël Salem, UMR CNRS 6085, Normandie Université, Université de Rouen, Avenue de l’Université, BP 12, 76801, Saint Etienne de Rouvray, France

    Patrizia Donato

Authors
  1. Imen Chourabi
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  2. Patrizia Donato
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Correspondence to Imen Chourabi.

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Chourabi, I., Donato, P. Homogenization of elliptic problems with quadratic growth and nonhomogenous Robin conditions in perforated domains. Chin. Ann. Math. Ser. B 37, 833–852 (2016). https://doi.org/10.1007/s11401-016-1008-y

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  • DOI: https://doi.org/10.1007/s11401-016-1008-y

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