Homogenization of Elliptic Equations with Quadratic Growth in Periodically Perforated Domains: The Case of Unbounded Solutions

Giuseppe Cardone and Antonio Gaudiello

Abstract: 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 $\Omega_{\varepsilon}=\Omega-T_{\varepsilon}$ 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]).
We prove that the linear part gives the homogenized matrix of the linear part and the nonlinear one changes into $H^{0}(u,Du)$, where $H^{0}$ is defined by $$ H^{0}(s,\xi)=\int_{]0,1[^{n}-\overline T}H(y,s,C(y)\xi)\,dy \forall\,(s,\xi)\in\R\times\R^{n}, $$ with $C(\frac{x}{\varepsilon})$ the corrector matrices of the linear problem and $T$ the reference hole.

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