Correctors for some nonlinear monotone operators

Abstract

In this paper we study homogenization of quasi-linear partial differential equations of the form $-\mbox{div}\left( a\left( x,x/\varepsilon _h,Du_h\right) \right) =f_h$ on $\Omega $ with Dirichlet boundary conditions. Here the sequence $\left( \varepsilon _h\right) $ tends to $0$ as $h\rightarrow \infty $ and the map $a\left( x,y,\xi \right) $ is periodic in $y,$ monotone in $\xi $ and satisfies suitable continuity conditions. We prove that $u_h\rightarrow u$ weakly in $W_0^{1,p}\left( \Omega \right) $ as $h\rightarrow \infty ,$ where $u$ is the solution of a homogenized problem of the form $-\mbox{div}\left( b\left( x,Du\right) \right) =f$ on $\Omega .$ We also derive an explicit expression for the homogenized operator $b$ and prove some corrector results, i.e. we find $\left( P_h\right) $ such that $Du_h-P_h\left( Du\right) \rightarrow 0$ in $L^p\left( \Omega, \mathbf{R}^n\right)$.