Determination of isometric real-analytic metric and spectral invariants for elastic Dirichlet-to-Neumann map on Riemannian manifolds

Abstract

In this paper, the elastic Dirichlet-to-Neumann map ${\Xi }_g$ is studied for the stationary elasticity system in a compact Riemannian manifold $(\Omega ,g)$ with smooth boundary $\partial \Omega$. By overcoming methodological difficulties, we explicitly get matrix-valued full symbol for the elastic Dirichlet-to-Neumann map ${\Xi }_g$. We prove that for a strong convex or extendable real-analytic manifold with boundary, the elastic Dirichlet-to-Neumann map ${\Xi }_g$ uniquely determines the metric $g$ of $\Omega$ in the sense of isometry, thereby solving an open problem for the uniqueness of the metric under real-analytic setting. Furthermore, by calculating the symbol representation of the resolvent operator $(\Xi -\tau I{)}^{-1}$ we can explicitly obtain all coefficients $a_0,a_1\cdots ,a_{n-1}$ of the asymptotic expansion ${\sum }_{k=1}^{\infty }{e}^{-t{\tau }_k}\sim {\sum }_{m=0}^{n-1}{a}_m{t}^{m+1-n}+o(1)$ as $t\to 0^{+}$, where ${\tau }_k$ is the $k$-th eigenvalue of the elastic Dirichlet-to-Neumann map ${\Xi }_g$ (i.e., $k$-th elastic Steklov eigenvalue). These coefficients (spectral invariants) provide important geometric information for the manifold, which give an answer to another open problem for the elastic Steklov spectral asymptotics.

56 pages, only add some detailed explanations for the elastic Dirichlet-to-Neumann map in Section 1 and the Section 2

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