$G$-convergence and homogenization of nonlinear elliptic operators in divergence form with variable domain

A. A. Kovalevsky

Izvestiya. Mathematics , Tome 44 (1995) no. 3, pp. 431-460

Résumé

The concepts of $G$-convergence and strong $G$-convergence of a sequence of elliptic operators $A_s\colon W^{1,m}(\Omega_s)\to(W^{1,m}(\Omega_s))^*$ are studied, where $\Omega_s$, $s=1,2,\dots$, are perforated domains contained in a bounded domain $\Omega\subset\mathbf R^n$. It is established that $G$-convergence of the operators $A_s$ is accompanied by convergence of solutions of certain equations and variational inequalities connected with the operators $A_s$ and a theorem on selection from the sequence $\{A_s\}$ of a strongly $G$-convergent subsequence. It is shown that under the condition of periodicity of the perforation of domains $\Omega_s$ and certain assumptions regarding the coefficients of the operators $A_s$, strong $G$-convergence of $\{A_s\}$ to an operator $A\colon W^{1,m}(\Omega)\to(W^{1,m}(\Omega))^*$ holds with effectively computable coefficients.