Some homogenization and corrector results for nonlinear monotone operators

Abstract

This paper deals with the limit behaviour of the solutions of quasi-linear equations of the form \ $\ds -\limfunc{div}\left(a\left(x, x/{\varepsilon _h},Du_h\right)\right)=f_h$ on $Ω$ with Dirichlet boundary conditions. The sequence $(\varepsilon _h)$ tends to $0$ and the map $a(x,y,ξ)$ is periodic in $y$, monotone in $ξ$ and satisfies suitable continuity conditions. It is proved that $u_h\rightarrow u$ weakly in $H_0^{1,2}(Ω)$, where $u$ is the solution of a homogenized problem \ $-\limfunc{div}(b(x,Du))=f$ on $Ω$. We also prove some corrector results, i.e. we find $(P_h)$ such that $Du_h-P_h(Du)\rightarrow 0$ in $L^2(Ω,R^n)$.