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 ${\sum }_{k=1}^{\infty }{e}^{-t{\tau }_k^{\mp }}=a_0{t}^{-n/2}\mp a_1{t}^{-(n-1)/2}+a_2^{\mp }{t}^{-(n-2)/2}+\cdots +a_m^{\mp }{t}^{-(n-m)/2}+O(t^{-(n-m-1)/2})$ as $t\to 0^{+}$, where ${\tau }_k^{-}$ (respectively, ${\tau }_k^{+}$) is the $k$-th Navier-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 \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-Lamé spectrum among all bounded elastic bodies with smooth boundary.

arXiv comment22 pages.The double manifold and the Green functions are explained in detail based on arXiv:2007.09730v2. This is a new work for eigenvalues of Navier-Lame operator on a Riemannian manifold based on arXiv:1512.07552

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Cited by in corpus (1)

• Spectral Invariants of the Magnetic Dirichlet-to-Neumann Map on Riemannian Manifolds