The Norm Resolvent Convergence for Elliptic Operators in Multi-Dimensional Domains with Small Holes

Abstract

We consider a second order elliptic operator with variable coefficients in a multidimensional domain with a small hole and some classical boundary condition on the hole boundary. We show that the resolvent of this operator converges to the resolvent of the limit operator in the domain without holes in the sense of the norm of bounded operators acting from L₂ to \( {W}_2^1 \). For the convergence rate we obtain sharp estimates relative to the smallness order.