The vibrational frequencies of the elastic body and its geometric quantities

For a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion of the trace of the strongly continuous semigroup associated with the Navier-Lam'{e} operator on $\mathrm{\Omega }$ as $t\to 0^{+}$. These coefficients (i.e., spectral invariants) provide precise information for the volume of the elastic body $\mathrm{\Omega }$ and the surface area of the boundary $\mathrm{\partial }\mathrm{\Omega }$ in terms of the spectrum of the Navier-Lam'{e} problem. As an application, we show that an $n$-dimensional ball is uniquely determined by its Navier-Lam'{e} spectrum among all bounded elastic body with smooth boundary.