Operator-norm resolvent asymptotic analysis of continuous media with high-contrast inclusions

TL;DR

This paper investigates the asymptotic behavior of solutions to high-contrast transmission problems in continuous media, demonstrating operator-norm resolvent convergence and spectral convergence to a limit electrostatic problem.

Contribution

It introduces a generalized Dirichlet-to-Neumann map approach to analyze resolvent asymptotics for high-contrast inclusions, providing sharp convergence estimates.

Findings

Operator-norm resolvent convergence to a limit electrostatic problem.

Spectral convergence of high-contrast problems to the limit spectrum.

Order-sharp estimates for convergence rates.

Abstract

Using a generalisation of the classical notion of Dirichlet-to-Neumann map and the related formulae for the resolvents of boundary-value problems, we analyse the asymptotic behaviour 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 for functions that are constant on the inclusion. 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.