On theg-convergence of nonlinear elliptic operators related to the dirichlet problem in variable domains

Abstract

A notion of $G$-convergence of operators $A_s :\; W_s \rightarrow W_s^*$ to the operator $A:\; W \rightarrow W^*$ is introduced and studied under certain connection conditions for the Banach spaces $W_s,\; s = 1, 2, ... ,$ and the Banach space $W$. It has been established that the connection conditions for abstract space are satisfied by the Sobolev spaces $\overset{\circ}{W}^{k, m}(\Omega_s),\quad \overset{\circ}{W}^{k, m}(\Omega)$ ($\{\Omega_s\}$ is a sequence of perforated domains contained in a bounded domain $\Omega \subset \mathbb{R}^n$). Hence, the results obtained for abstract operators can be applied to the operators of Dirichlet problems in the domains $\Omega_s$.