Spectral enclosures for the damped elastic wave equation

TL;DR

This paper explores the spectral characteristics of the damped elastic wave equation, establishing bounds on eigenvalue locations related to damping properties, thus advancing understanding of wave behavior in elastic media with damping.

Contribution

It introduces a novel connection between the eigenvalue problem of damped elastic waves and non-self-adjoint Lamé operators, providing quantitative spectral bounds.

Findings

Eigenvalues are bounded in terms of damping coefficient norms

A correspondence between damped elastic wave and Lamé operators is established

Quantitative spectral bounds are derived

Abstract

In this paper we investigate spectral properties of the damped elastic wave equation. Deducing a correspondence between the eigenvalue problem of this model and the one of Lam\'e operators with non self-adjoint perturbations, we provide quantitative bounds on the location of the point spectrum in terms of suitable norms of the damping coefficient.