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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 $\sum_{k=1}^\infty e^{-t \tau_k^\mp}= a_0t^{-n/2} \mp a_1 t^{-(n-1)/2} +O(t^{1-n/2})$ as $t\to 0^+$, where $\tau^-_k$ (respectively, $\tau^+_k$) is the $k$-th Navier-Lame 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-Lame operator. As an application, we show that an $n$-dimensional ball is uniquely determined by its Navier-Lame spectrum among all bounded elastic bodies with smooth boundaries.

Geometric Invariants of Spectrum of the Navier–Lamé Operator