Operator-Norm Resolvent Asymptotic Analysis of Continuous Media with High-Contrast Inclusions

Using a generalization of the classical notion of Weyl $m$-function and related formulas for the resolvents of boundary-value problems, we analyze the asymptotic behavior of solutions to a “transmission problem” for a high-contrast inclusion in a continuous medium, for which we prove the operator-norm resolvent convergence to a limit problem of “electrostatic” type. In particular, our results imply the convergence of the spectra of high-contrast problems to the spectrum of the limit operator, with order-sharp convergence estimates. The approach developed in the paper is of a general nature and can thus be successfully applied in the study of other problems of the same type.