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

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.

Functional model for generalised resolvents and its application to time-dispersive media

“…The need to quantify this effect for various classes of boundary value problems (BVP), which ultimately aims at addressing the effect of the underlying miscrosopic resonance on the overall behaviour of a class of physical systems, has also motivated the development of functional analytic frameworks for the analysis of wave scattering and effects of length-scale interactions for parameterdependent BVP, see [19,20,21,22]. The approach of the latter works was inspired by a treatment of BVP going back to the so-called Birman-Kreȋn-Vishik methodology [9,40,41,72] and its recent development by Ryzhov [64], rooted in an earlier construction of the functional model of perturbation theory by one of the authors [44,45].…”