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ESAIM: COCV
DOI: 10.1051/cocv:2008021

Asymptotic behavior of nonlinear systems in varying domains with boundary conditions on varying sets

Carmen Calvo-Jurado1, Juan Casado-Díaz2 and Manuel Luna-Laynez2

1  Dpto. de Matemáticas, Escuela Politécnica, Avenida de la Universidad s/n, 10071 Cáceres, Spain; ccalvo@unex.es
2  Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Fac. de Matemáticas, C. Tarfia s/n, 41012 Sevilla, Spain; jcasadod@us.es; mllaynez@us.es


(Received March 15, 2007. Revised April 10, 2007 and June 26, 2007. Published online March 6, 2008.)

Abstract
For a fixed bounded open set $\Omega\subset\mathbb{R} ^N$, a sequence of open sets $\Omega_n\subset\Omega$ and a sequence of sets $\Gamma_n\subset\partial\Omega\cap\partial\Omega_n$, we study the asymptotic behavior of the solution of a nonlinear elliptic system posed on $\Omega_n$, satisfying Neumann boundary conditions on $\Gamma_n$ and Dirichlet boundary conditions on $\partial\Omega_n\setminus\Gamma_n$. We obtain a representation of the limit problem which is stable by homogenization and we prove that this representation depends on $\Omega_n$ and $\Gamma_n$ locally.


Mathematics Subject Classification. 35B40

Key words: Homogenization, varying domains, nonlinear problems


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