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

Genqian Liu|arXiv (Cornell University)|Aug 14, 2019

Numerical methods in inverse problems44 references4 citations

TL;DR

This paper establishes the isometric uniqueness of the Riemannian metric on a real-analytic compact manifold with boundary via the elastic Dirichlet-to-Neumann map, resolving an open problem in spectral geometry. It further derives explicit spectral invariants through the asymptotic expansion of the heat trace, providing geometric information from the elastic Steklov eigenvalues.

ABSTRACT

In this paper, the elastic Dirichlet-to-Neumann map $Ξ_g$ is studied for the stationary elasticity system in a compact Riemannian manifold $(Ω,g)$ with smooth boundary $\partial Ω$. By overcoming methodological difficulties, we explicitly get matrix-valued full symbol for the elastic Dirichlet-to-Neumann map $Ξ_g$. We prove that for a strong convex or extendable real-analytic manifold with boundary, the elastic Dirichlet-to-Neumann map $Ξ_g$ uniquely determines the metric $g$ of $Ω$ 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 $(Ξ-τI)^{-1}$ we can explicitly obtain all coefficients $a_0, a_1 \cdots, a_{n-1}$ of the asymptotic expansion $\sum_{k=1}^\infty e^{-t τ_k}\sim \sum_{m=0}^{n-1} a_m t^{m+1-n} +o(1)$ as $t o 0^+$, where $τ_k$ is the $k$-th eigenvalue of the elastic Dirichlet-to-Neumann map $Ξ_g$ (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.

Motivation & Objective

• To resolve the open problem of whether the elastic Dirichlet-to-Neumann map uniquely determines the Riemannian metric up to isometry on real-analytic manifolds with boundary.
• To explicitly compute the spectral invariants arising from the asymptotic expansion of the heat trace associated with the elastic Steklov eigenvalues.
• To establish a complete symbol representation of the elastic Dirichlet-to-Neumann map and its resolvent for real-analytic manifolds.
• To extend the classical Calderón problem to the elastic setting by proving uniqueness under strong convexity or extendability and real-analyticity assumptions.

Proposed method

• Derives the matrix-valued full symbol of the elastic Dirichlet-to-Neumann map using local boundary normal coordinates and pseudodifferential operator calculus.
• Applies complex contour integration and residue calculus to compute the heat trace coefficients via the resolvent of the map.
• Uses the parametrix construction and spectral asymptotics to extract coefficients in the expansion of the trace of the heat kernel.
• Employs the Tauberian theorem to derive the Weyl-type law for the counting function of elastic Steklov eigenvalues.
• Calculates the full symbol of the resolvent (Ξg − τI)−1 and uses it to extract the coefficients am in the asymptotic expansion of the heat trace.
• Applies curvature decomposition and trace formulas to express the coefficients am in terms of geometric invariants such as scalar and Ricci curvatures of the manifold and its boundary.

Experimental results

Research questions

• RQ1 Can the elastic Dirichlet-to-Neumann map uniquely determine the Riemannian metric of a compact real-analytic manifold with boundary up to isometry?
• RQ2 What geometric information is encoded in the asymptotic expansion of the heat trace associated with the elastic Steklov eigenvalues?
• RQ3 How can the spectral invariants of the elastic Dirichlet-to-Neumann map be explicitly computed from the symbol of the operator?
• RQ4 To what extent does the elastic Dirichlet-to-Neumann map retain the uniqueness and spectral properties of the classical Dirichlet-to-Neumann map in the context of elasticity?
• RQ5 What is the precise form of the first few coefficients in the short-time asymptotic expansion of the heat trace for the elastic Steklov problem?

Key findings

• The elastic Dirichlet-to-Neumann map uniquely determines the Riemannian metric g on a strong convex or extendable real-analytic manifold with boundary up to isometry, solving Problem A.
• The first two coefficients a0 and a1 in the asymptotic expansion of the heat trace are explicitly computed and expressed in terms of the boundary curvature, the dimension, and Lamé parameters.
• The coefficient a2(x′) is shown to depend on the scalar and Ricci curvatures of both the domain Ω and its boundary ∂Ω, with explicit constants depending on n, µ, and λ.
• A Weyl-type law for the counting function N(τ) of elastic Steklov eigenvalues is derived, showing N(τ) ∼ C τ^{n−1} as τ → ∞, with C depending on the volume of ∂Ω and geometric parameters.
• All coefficients am for 0 ≤ m ≤ n−1 in the asymptotic expansion of the heat trace are explicitly computable via the symbol of the resolvent operator (Ξg − τI)−1.
• The method enables the extraction of all spectral invariants am from the eigenvalues {τk}, providing a complete spectral geometric characterization of the elastic boundary problem.