Asymptotic Expansions of The Traces of the Thermoelastic Operators

TL;DR

This paper derives detailed asymptotic expansions for the traces of thermoelastic operators on Riemannian manifolds, revealing geometric insights and establishing spectral uniqueness of geodesic balls.

Contribution

It provides a method to compute all coefficients of the asymptotic expansion, linking spectral data to geometric properties and proving spectral uniqueness of geodesic balls.

Findings

Explicit formulas for the first two coefficients involving volume and boundary

A new effective method for calculating asymptotic expansion coefficients

Proof that geodesic balls are uniquely determined by their thermoelastic spectrum

Abstract

We obtain the asymptotic expansions of the traces of the thermoelastic operators with the Dirichlet and Neumann boundary conditions on a Riemannian manifold, and give an effective method to calculate all the coefficients of the asymptotic expansions. These coefficients provide precise geometric information. In particular, we explicitly calculate the first two coefficients concerning the volumes of the manifold and its boundary. As an application, by combining our results with the isoperimetric inequality we show that an $n$-dimensional geodesic ball is uniquely determined up to isometry by its thermoelastic spectrum among all bounded thermoelastic bodies with boundary.