PORTUGALIAE MATHEMATICA Vol. 54, No. 1, pp. 5172 (1997) 

Homogenization of Elliptic Equations with Quadratic Growth in Periodically Perforated Domains: The Case of Unbounded SolutionsGiuseppe Cardone and Antonio GaudielloAbstract: This paper is devoted to the homogenization of the following non linear problem $$ \cases{\dps{\Div\biggl(A\biggl(\frac{x}{\varepsilon}\biggr) Du_{\varepsilon}\biggr)+H\biggl(\frac{x}{\varepsilon},u_{\varepsilon}, Du_{\varepsilon}\biggr)=f}& in $\Omega_{\varepsilon}$,\cr \dps{\biggl(A\biggl(\frac{x}{\varepsilon}\biggr)Du_{\varepsilon}\biggr) \underline\mu=0}& on $\partial T_{\varepsilon}$,\cr u_{\varepsilon}=0& on $\partial\Omega$,\cr \dps{u_{\varepsilon}\in H^{1}(\Omega_{\varepsilon}), H\biggl(\frac{x}{\varepsilon},u_{\varepsilon},Du_{\varepsilon}\biggr)\in L^{1}(\Omega_{\varepsilon}), H\biggl(\frac{x}{\varepsilon}, u_{\varepsilon},Du_{\varepsilon}\biggr)u_{\varepsilon}\in L^{1}(\Omega_{\varepsilon})},} $$ where $\Omega_{\varepsilon}=\OmegaT_{\varepsilon}$ is obtained by removing from a bounded open set $\Omega$ of $\R^{n}$ a closed set $T_{\varepsilon}$ of $\varepsilon$periodic balls of size $\varepsilon$, $H(y,s,\xi)$ is $]0,1[^{n}$periodic in $y$, has the same sign as $s$ and has a quadratic growth with respect to $\xi$, and $f$ belongs to $L^{2}(\Omega)$. (The corresponding problem with bounded solutions has been treated by P. Donato, A. Gaudiello and L. Sgambati in [11]). Full text of the article:
Electronic version published on: 29 Mar 2001. This page was last modified: 27 Nov 2007.
© 1997 Sociedade Portuguesa de Matemática
