Geometric invariants of spectrum of the Navier-Lam\'{e} operator

TL;DR

This paper computes spectral invariants of the Navier-Lamé operator on elastic bodies, linking eigenvalues to geometric properties like volume and surface area, and shows the spectrum uniquely determines a ball among elastic bodies.

Contribution

It explicitly calculates the first two spectral coefficients of the Navier-Lamé operator and demonstrates their geometric significance, also establishing spectral uniqueness for the ball.

Findings

The first two spectral coefficients relate to volume and surface area.

The method can compute all higher-order coefficients explicitly.

A ball is uniquely determined by its spectrum among elastic bodies.

Abstract

For a compact connected Riemannian $n$-manifold $(\Omega,g)$ with smooth boundary, we explicitly calculate the first two coefficients $a_0$ and $a_1$ of the asymptotic expansion of $\sum_{k=1}^\infty e^{-t \tau_k^\mp}= a_0t^{-n/2} \mp a_1 t^{-(n-1)/2}+a_2^\mp t^{-(n-2)/2} +\cdots+ a_m^\mp t^{-(n-m)/2} +O(t^{-(n-m-1)/2})$ as $t\to 0^+$, where $\tau^-_k$ (respectively, $\tau^+_k$) is the $k$-th Navier-Lam\'{e} eigenvalue on $\Omega$ with Dirichlet (respectively, Neumann) boundary condition. These two coefficients provide precise information for the volume of the elastic body $\Omega$ and the surface area of the boundary $\partial \Omega$ in terms of the spectrum of the Navier-Lam\'{e} operator. This gives an answer to an interesting and open problem mentioned by Avramidi in \cite{Avr10}. More importantly, our method is valid to explicitly calculate all the coefficients $a_l^\mp$, $2\le l\le m$, in the above asymptotic expansion. As an application, we show that an $n$-dimensional ball is uniquely determined by its Navier-Lam\'{e} spectrum among all bounded elastic bodies with smooth boundary.