Asymptotic analysis of operator families and applications to resonant media

TL;DR

This paper reviews operator-theoretic methods for analyzing the asymptotic behavior of resolvents in boundary-value problems for PDEs, connecting mathematical frameworks to practical physics and engineering applications.

Contribution

It provides an overview of recent operator-theoretic tools and their applications to the asymptotic analysis of parameter-dependent PDE operators.

Findings

Link between operator theory and physical applications clarified

Asymptotic behavior of resolvents characterized

Connections to functional frameworks established

Abstract

We give an overview of operator-theoretic tools that have recently proved useful in the analysis of boundary-value and transmission problems for second-order partial differential equations, with a view to addressing, in particular, the asymptotic behaviour of resolvents of physically motivated parameter-dependent operator families. We demonstrate the links of this rich area, on the one hand, to functional frameworks developed by S. N. Naboko and his students, and on the other hand, to concrete applications of current interest in the physics and engineering communities.