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

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.