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Xiaoming Tan

Publications and source records attributed to Xiaoming Tan.

8 recordsLinked to original sources
arXiv2025

Spectral invariants of the Dirichlet-to-Neumann map associated to the Witten-Laplacian with potential

For a compact connected Riemannian manifold with smooth boundary, we establish an effective procedure, by which we can calculate all the coefficients of the spectral asymptotic formula of the Dirichlet-to-Neumann map associated to the Witten-Laplacian with potential. In particular, by computing the full symbol of the Dirichlet-to-Neumann map we explicitly give the first four coefficients. They are spectral invariants, which provide precise information concerning the volume, curvatures, drifting function and potential.

math.DG↗
arXiv2023

Determining coefficients of thermoelastic system from boundary information

Given a compact Riemannian manifold $(M,g)$ with smooth boundary $\partial M$, we give an explicit expression for full symbol of the thermoelastic Dirichlet-to-Neumann map $Λ_g$ with variable coefficients $λ,μ,α,β\in C^{\infty}(\bar{M})$. We prove that $Λ_g$ uniquely determines partial derivatives of all orders of the coefficients on the boundary. Moreover, for a nonempty open subset $Γ\subset\partial M$, suppose that the manifold and the coefficients are real analytic up to $Γ$, we show that $Λ_g$ uniquely determines the coefficients on the whole manifold $\bar{M}$.

math.AP↗
arXiv2022

Determining Lamé coefficients by elastic Dirichlet-to-Neumann map on a Riemannian manifold

For the Lamé operator $\mathcal{L}_{λ,μ}$ with variable coefficients $λ$ and $μ$ on a smooth compact Riemannian manifold $(M,g)$ with smooth boundary $\partial M$, we give an explicit expression for full symbol of the elastic Dirichlet-to-Neumann map $Λ_{λ,μ}$. We show that $Λ_{λ,μ}$ uniquely determines partial derivatives of all orders of the Lamé coefficients $λ$ and $μ$ on $\partial M$. Moreover, for a nonempty open subset $Γ\subset\partial M$, suppose that the manifold and the Lamé coefficients are real analytic up to $Γ$, we prove that $Λ_{λ,μ}$ uniquely determines the Lamé coefficients on the whole manifold $\bar{M}$.

math.SP↗
arXiv2022

Asymptotic Expansion of the Heat Trace of the Thermoelastic Dirichlet-to-Neumann map

This paper is devoted to study the asymptotic expansion of the heat trace of the Dirichlet-to-Neumann map for the thermoelastic equation on a Riemannian manifold with doundary. By providing a method we can obtain all the coefficients of the asymptotic expansion. In particular, we explicitly give the first two coefficients involving the volume and the total mean curvature of the boundary.

math.AP↗
arXiv2022

Asymptotic Expansions of The Traces of the Thermoelastic Operators

We obtain the asymptotic expansions of the traces of the thermoelastic operators with the Dirichlet and Neumann boundary conditions on a Riemannian manifold, and give an effective method to calculate all the coefficients of the asymptotic expansions. These coefficients provide precise geometric information. In particular, we explicitly calculate the first two coefficients concerning the volumes of the manifold and its boundary. As an application, by combining our results with the isoperimetric inequality we show that an $n$-dimensional geodesic ball is uniquely determined up to isometry by its thermoelastic spectrum among all bounded thermoelastic bodies with boundary.

math.SP↗
arXiv2021

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

This paper is devoted to investigate the heat trace asymptotic expansion corresponding to the magnetic Steklov eigenvalue problem on Riemannian manifolds with boundary. We establish an effective procedure, by which we can calculate all the coefficients $a_0$, $a_1$, $\dots$, $a_{n-1}$ of the heat trace asymptotic expansion. In particular, we explicitly give the expressions for the first four coefficients. These coefficients are spectral invariants which provide precise information concerning the volume and curvatures of the boundary of the manifold and some physical quantities by the magnetic Steklov eigenvalues.

math.AP↗