Geometric Invariants of Spectrum of the Navier–Lamé 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 asymptotic expansion $$\sum _{k=1}^\infty \mathrm{{e}}^{-t \tau _k^{\mp }}= a_0t^{-n/2} {\mp } a_1 t^{-(n-1)/2} +O(t^{1-n/2})$$ as $$t\rightarrow 0^+$$ , where $$\tau ^-_k$$ (respectively, ^+_k$$ ) is k-th Navier–Lamé eigenvalue on $$\Omega $$ Dirichlet Neumann) boundary condition. These provide precise information for volume elastic body surface area $$\partial \Omega in terms spectrum operator. This gives an answer to interesting open problem mentioned by Avramidi (Non-Laplace type operators manifolds analysis, geometry topology elliptic operators. World Sci. Publ., Hackensack, pp. 107–140, 2006). As application, show that n-dimensional ball uniquely determined its among all bounded bodies boundary.