On the existence and stability of modified Maxwell Steklov eigenvalues

TL;DR

This paper proves the existence of infinitely many eigenvalues for modified Maxwell Steklov problems under realistic, minimal regularity assumptions and establishes their stability with respect to material parameter changes.

Contribution

It introduces a new technique to prove eigenvalue existence for nonselfadjoint problems with minimal regularity assumptions, advancing understanding of inverse scattering methods.

Findings

Proved existence of infinitely many eigenvalues under realistic assumptions.

Established stability of eigenvalues with respect to material changes.

Provided minimal regularity conditions for domain and coefficients.

Abstract

In recent decades qualitative inverse scattering methods with eigenvalues as target signatures received much attention. To understand those methods a knowledge on the properties of the related eigenvalue problems is essential. However, even the existence of eigenvalues for such (nonselfadjoint) problems is a challenging question and existing results for absorbing media are usually established under unrealistic assumptions or a smoothing of the eigenvalue problem. We present a technique to prove the existence of infinitely many eigenvalues for such problems under realistic assumptions. In particular we consider the class of scalar and modified Maxwell nonselfadjoint Steklov eigenvalue problems. In addition, we present stability results for the eigenvalues with respect to changes in the material parameters. In distinction to existing results the analysis of the present article requires only minimal regularity assumptions. By that we mean that the regularity of the domain is not required to be better than Lipschitz, and the material coefficients are only assumed to be piece-wise $W^{1,\infty}$. Also the stability estimates for eigenvalues are obtained solely in $L^p$-norms ($p<\infty$) of the material perturbations.