Geometric Invariants of Spectrum of the Navier–Lamé Operator

Abstract

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 $$\sum _{k=1}^\infty \mathrm{{e}}^{-t \tau _k^{\mp }}= a_0t^{-n/2} {\mp } a_1 t^{-(n-1)/2} +O(t^{1-n/2})$$ as $$t\rightarrow 0^+$$ , where $$\tau ^-_k$$ (respectively, $$\tau ^+_k$$ ) is the k-th Navier–Lamé 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–Lamé operator. This gives an answer to an interesting and open problem mentioned by Avramidi in (Non-Laplace type operators on manifolds with boundary, analysis, geometry and topology of elliptic operators. World Sci. Publ., Hackensack, pp. 107–140, 2006). As an application, we show that an n-dimensional ball is uniquely determined by its Navier–Lamé spectrum among all bounded elastic bodies with smooth boundary.