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 $\Omega$ with Dirichlet boundary conditions. The sequence $({\epsilon }_h)$ tends to $0$ and the map $a(x,y,\xi )$ is periodic in $y$, monotone in $\xi$ and satisfies suitable continuity conditions. It is proved that $u_h\to u$ weakly in $H_0^{1,2}(\Omega )$, where $u$ is the solution of a homogenized problem \ $-\limfunc{div}(b(x,Du))=f$ on $\Omega$. We also prove some corrector results, i.e. we find $(P_h)$ such that $D{u}_h-P_h(Du)\to 0$ in $L^2(\Omega ,R^n)$.

References in corpus (1)

• Some homogenization and corrector results for nonlinear monotone operators

Cited by in corpus (2)

• Some homogenization and corrector results for nonlinear monotone operators
• Correctors for some nonlinear monotone operators