The vibrational frequencies of the elastic body and its geometric quantities

Abstract

For a bounded domain $\Omega\subset {\Bbb R}^n$ 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 $\Omega$ as $t\to 0^+$. These coefficients (i.e., spectral invariants) 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} 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.