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 $\left({\epsilon }_h\right)$ tends to $0$ as $h\to \infty$ and the map $a\left(x,y,\xi \right)$ is periodic in $y,$ monotone in $\xi$ and satisfies suitable continuity conditions. We prove that $u_h\to u$ weakly in $W_0^{1,p}\left(\Omega \right)$ as $h\to \infty ,$ 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 $D{u}_h-P_h\left(Du\right)\to 0$ in $L^p\left(\Omega ,{\mathbf{R}}^n\right)$.

References in corpus (1)

• Some homogenization and corrector results for nonlinear monotone operators

Cited by in corpus (1)

• On weak convergence of locally periodic functions