# Asymptotic Expansions of The Traces of the Thermoelastic Operators

Genqian Liu  School of Mathematics and Statistics, Beijing Institute of
Technology, Beijing 100081, China<EMAIL_ADDRESS>and  Xiaoming Tan  School
of Mathematics and Statistics, Beijing Institute of Technology, Beijing
100081, China<EMAIL_ADDRESS>
###### Abstract.

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.

###### Key words and phrases:

Thermoelastic operator; Thermoelastic eigenvalues; Asymptotic expansion;
Spectral invariants; Spectral rigidity

###### 2020 Mathematics Subject Classification:

35P20, 58J50, 35S05, 58J40, 74F05

##  1\. Introduction

Let $\Omega$ be a smooth compact Riemannian manifold of dimension $n$ with
smooth boundary $\partial\Omega$. We consider $\Omega$ as a homogeneous,
isotropic, thermoelastic body. Assume that the Lamé coefficients $\lambda$,
$\mu$ and the heat conduction coefficient $\alpha$ of the thermoelastic body
are constants which satisfy $\mu>0$, $\lambda+\mu\geqslant 0$ and $\alpha>0$.
We denote by
$\operatorname{grad},\,\operatorname{div},\,\Delta_{g},\,\Delta_{B}$ and
$\operatorname{Ric}$, respectively, the gradient operator, the divergence
operator, the Laplace–Beltrami operator, the Bochner Laplacian and the Ricci
tensor. We define the thermoelastic operator

(1.1)
$\displaystyle\mathcal{L}_{g}:=\begin{pmatrix}\mu\Delta_{B}+(\lambda+\mu)\operatorname{grad}\operatorname{div}+\mu\operatorname{Ric}+\rho\omega^{2}&-\beta\operatorname{grad}\\\
i\omega\theta_{0}\beta\operatorname{div}&\alpha\Delta_{g}+i\omega\gamma\end{pmatrix},$

where the coefficient $\beta$ depends on $\lambda,\,\mu$ and the linear
expansion coefficient, $\gamma$ is the specific heat per unit volume,
$\theta_{0}$ is the reference temperature, $\rho$ is the density of the
thermoelastic body, $\omega$ is the angular frequency and $i=\sqrt{-1}$. We
denote by $\textbf{{u}}(x)\in(C^{\infty}(\Omega))^{n}$ and $\theta(x)\in
C^{\infty}(\Omega)$ the displacement and the temperature variation,
respectively. By the methods of [27, 34], we can write the steady
thermoelastic equation as follows

(1.2) $\displaystyle\mathcal{L}_{g}\textbf{{U}}=0\quad\text{in}\ \Omega,$

where $\textbf{{U}}=(\textbf{{u}},\theta)^{t}$, the superscript $t$ denotes
the transpose. Equation (1.2) is an extension of the classical elastic
equation. In particular, when $\Omega$ is a bounded domain in $\mathbb{R}^{n}$
and the temperature of the medium is not taken into consideration, equation
(1.2) reduces to the classical elastic equation (see pp. 35–36 of [27]).

In the case of classical elasticity, assume that the temperature of the medium
is the same at all points and is not changed during deformation. Nevertheless,
in reality, most materials are not pure elastic, a temperature change in the
elastic body will generate additional strain and stress. Therefore,
deformation is followed by temperature variation and, conversely, temperature
variation is followed by deformation of the thermoelastic body due to thermal
expansion. Even if there is no external stresses or mass forces, some heat
sources change the temperature of the thermoelastic body, there will be
deformation. The significance of the equation (1.2) consists not only in the
important independent role of this state, frequently occurring in engineering,
but also in the fact that their investigation opens up the way to the study
the general type (see p. 528 of [27]).

In this paper, we study the asymptotic expansions of the taces of the
thermoelastic operators on $\Omega$. We denote by $\mathcal{L}_{g}^{-}$ and
$\mathcal{L}_{g}^{+}$ the thermoelastic operators with the Dirichlet and
Neumann boundary conditions, respectively. Here the Neumann boundary condition
is defined by, for any $\textbf{{U}}\in(C^{1}(\Omega))^{n+1}$,

$\displaystyle
N(\textbf{{U}}):=\begin{pmatrix}\lambda\nu\operatorname{div}+\mu\nu
S&-\beta\nu\\\ 0&\alpha\partial_{\nu}\end{pmatrix}\textbf{{U}}\quad\text{on}\
\partial\Omega,$

where $\nu$ is the outward unit normal vector to $\partial\Omega$ and $S$ is
the stress tensor (also called the deformation tensor) of type $(1,1)$ defined
by $S\textbf{{u}}:=\nabla\textbf{{u}}+\nabla\textbf{{u}}^{t}$ for any
$\textbf{{u}}\in(C^{1}(\Omega))^{n}$ (see p. 562 of [53]). According to the
theory of elliptic operators, $\mathcal{L}_{g}^{-}$ (respectively,
$\mathcal{L}_{g}^{+}$) is an unbounded, self-adjoint and negative
(respectively, non-positive) operator in $(H_{0}^{1}(\Omega))^{n+1}$
(respectively, $(H^{1}(\Omega))^{n+1}$), it follows from [3, 20, 28, 34, 43]
that there exists a sequence $\\{\tau^{-}_{k}\\}_{k\geqslant 1}$
(respectively, $\\{\tau^{+}_{k}\\}_{k\geqslant 1}$) such that the following
eigenvalue problem

(1.3)
$\displaystyle\begin{cases}\mathcal{L}_{g}\textbf{{U}}^{-}_{k}+\tau^{-}_{k}\textbf{{U}}^{-}_{k}=0&\text{in}\
\Omega,\\\ \textbf{{U}}^{-}_{k}=0&\text{on}\ \partial\Omega\end{cases}$

(respectively,

(1.4)
$\displaystyle\begin{cases}\mathcal{L}_{g}\textbf{{U}}^{+}_{k}+\tau^{+}_{k}\textbf{{U}}^{+}_{k}=0&\text{in}\
\Omega,\\\ N(\textbf{{U}}^{+}_{k})=0&\text{on}\ \partial\Omega)\end{cases}$

has a discrete spectrum
$0<\tau^{-}_{1}<\tau^{-}_{2}\leqslant\cdots\leqslant\tau^{-}_{k}\leqslant\cdots\to+\infty$
(respectively,
$0\leqslant\tau^{+}_{1}<\tau^{+}_{2}\leqslant\cdots\leqslant\tau^{+}_{k}\leqslant\cdots\to+\infty$),
where $\textbf{{U}}^{-}_{k}\in(H_{0}^{1}(\Omega))^{n+1}$ (respectively,
$\textbf{{U}}^{+}_{k}\in(H^{1}(\Omega))^{n+1}$) is the eigenvector
corresponding to the Dirichlet eigenvalue $\tau^{-}_{k}$ (respectively,
Neumann eigenvalue $\tau^{+}_{k}$).

Let us point out that the thermoelastic operator is a non-Laplace type
operator. Contrary to the Laplace type operators, there are no systematic
effective methods to explicitly calculate the coefficients of the asymptotic
expansions of the traces for non-Laplace type operators. In 1971, Greiner [17]
indicated that “the problem of interpreting these coefficients geometrically
remains open”. In this sense, it is just beginning that the study of geometric
aspects of spectral asymptotics for non-Laplace type operators, the
corresponding methodology is still underdeveloped in comparison with the
theory of the Laplace type operator. Thus, the geometric aspect of the
spectral asymptotics of non-Laplace type operators remains an open problem
(see [3]).

Due to the fact that the Riemannian structure on a manifold is determined by a
Laplace type operator, the systematic explicit calculations of coefficients of
heat kernels for Laplace type operators are now well understood, see Gilkey
[16] and many others (see [2, 5, 15, 14, 19, 18, 23, 31, 55] and references
therein). For the classical boundary conditions, like Dirichlet, Neumann,
Robin, and mixed combination thereof on vector bundles, the coefficients of
the asymptotic expansions have been explicitly computed up to the first five
terms (see, for example, [7, 8, 24]). For other types of operators, the first
author of this paper gave the first two coefficients of the asymptotic
expansions of the traces of the Navier–Lamé operator [34] and the Stokes
operator [30]. The first author of this paper also gave the first four
coefficients of the asymptotic expansions of heat traces for the classical
Dirichlet-to-Neumann map [32] (see Polterovich and Sher [44] for the first
three coefficients) and the polyharmonic Steklov operator [33], and gave the
first two coefficients of the asymptotic expansions of the heat traces for
elastic Dirichlet-to-Neumann map [35] as well. Recently, the authors of this
paper have obtained the first four coefficients of the asymptotic expansion of
heat trace for the magnetic Dirichlet-to-Neumann map associated with the
magnetic Schrödinger operator [36]. To the best of our knowledge, it has not
previously been systematically applied in the context of the thermoelastic
operators with the Dirichlet and Neumann boundary conditions.

The main result of this paper is the following theorem.

######  Theorem 1.1.

Let $\Omega$ be a smooth compact Riemannian manifold of dimension $n$ with
smooth boundary $\partial\Omega$. Assume that $\alpha>0$, $\mu>0$ and
$\lambda+\mu\geqslant 0$. Let $\mathcal{L}_{g}^{-}$ and $\mathcal{L}_{g}^{+}$
be the thermoelastic operators with the Dirichlet and Neumann boundary
conditions, respectively, and $\\{\tau^{\mp}_{k}\\}_{k\geqslant 1}$ satisfy
the eigenvalue equations
$\mathcal{L}_{g}^{\mp}\textbf{{U}}^{\mp}_{k}+\tau^{\mp}_{k}\textbf{{U}}^{\mp}_{k}=0$.
Then

$\displaystyle\sum_{k=1}^{\infty}e^{-t\tau^{\mp}_{k}}=\operatorname{Tr}(e^{t\mathcal{L}_{g}^{\mp}})=t^{-n/2}\bigl{(}a_{0}+a_{1}^{\mp}t^{1/2}+\cdots+a_{n}^{\mp}t^{n/2}+O(t^{(n+1)/2})\bigr{)}\quad\text{as}\
t\to 0^{+}.$

In particular,

$\displaystyle a_{0}$
$\displaystyle=\frac{\operatorname{vol}(\Omega)}{(4\pi)^{n/2}}\biggl{(}\frac{n-1}{\mu^{n/2}}+\frac{1}{(\lambda+2\mu)^{n/2}}+\frac{1}{\alpha^{n/2}}\biggr{)},$
$\displaystyle a_{1}^{\mp}$
$\displaystyle=\mp\frac{1}{4}\frac{\operatorname{vol}(\partial\Omega)}{(4\pi)^{(n-1)/2}}\biggr{(}\frac{n-1}{\mu^{(n-1)/2}}+\frac{1}{(\lambda+2\mu)^{(n-1)/2}}+\frac{1}{\alpha^{(n-1)/2}}\biggr{)},$

where $\operatorname{vol}(\Omega)$ and $\operatorname{vol}(\partial\Omega)$
denote the $n$-dimensional volume of $\Omega$ and the $(n-1)$-dimensional
volume of $\partial\Omega$, respectively.

######  Remark 1.2.

Theorem 1.1 shows that both the volume $\operatorname{vol}(\Omega)$ and
$\operatorname{vol}(\partial\Omega)$ are spectral invariants, they can also be
obtained by the thermoelastic eigenvalues with respect to the Dirichlet or
Neumann boundary condition. This gives an answer to an interesting and open
problem mentioned in [3, 17, 22]. Roughly speaking, one can “hear” the volume
of the domain and the surface area of its boundary by “hearing” all the
pitches of the vibration of a thermoelastic body. This result is a
generalization of the corresponding result for the elastic Navier–Lamé
operator $($see [34]$)$.

######  Remark 1.3.

By applying the Tauberian theorem $($see Theorem $15.3$ on p. $30$ of [26] or
p. $107$ of [52]$)$, we can easily obtain the Weyl-type law for the counting
function $N^{\mp}(\tau):=\\#\\{k|\tau_{k}^{\mp}\leqslant\tau\\}$ of
thermoelastic Steklov eigenvalues:

$\displaystyle N^{\mp}(\tau)=$
$\displaystyle\frac{\operatorname{vol}(\Omega)}{(4\pi)^{n/2}\Gamma(1+n/2)}\biggl{(}\frac{n-1}{\mu^{n/2}}+\frac{1}{(\lambda+2\mu)^{n/2}}+\frac{1}{\alpha^{n/2}}\biggr{)}\tau^{n/2}+o(\tau^{n/2})\quad\text{as}\
\tau\to+\infty.$

As an application of Theorem 1.1, we have the following spectral rigidity
result:

######  Corollary 1.4.

Let $\mathcal{M}$ be a Riemannian manifold of dimension $n$,
$\Omega\subset\mathcal{M}$ be a compact domain with smooth boundary and
$B^{n}\subset\mathcal{M}$ be an $n$-dimensional geodesic ball. Asuume that the
following two conditions hold:

  1. $(a)$

The thermoelastic spectrum of $\Omega$ with respect to the Dirichlet or
Neumann boundary condition is equal to that of $B^{n}$;

  2. $(b)$

The isoperimetric inequality holds on $\mathcal{M}$, $($i.e., let
$\tilde{\Omega}\subset\mathcal{M}$ be a compact subset with boundary. If
$\operatorname{vol}(\tilde{\Omega})=\operatorname{vol}(B^{n})$, then
$\operatorname{vol}(\partial\tilde{\Omega})\geqslant\operatorname{vol}(\partial
B^{n})$, equality holds if and only if $\tilde{\Omega}$ is isometric to
$B^{n}$$)$.

Then $\Omega$ is isometric to $B^{n}$.

######  Remark 1.5.

There are many cases such that condition $(b)$ holds, we only list some of
them as follows:

  1. (i)

$\mathcal{M}$ is a space form, $($i.e., a complete Riemannian manifold with
constant sectional curvature$)$, $($see [41, 45, 46, 47, 48, 49]$)$;

  2. (ii)

$\mathcal{M}\subset\mathbb{R}^{n+m}\ (m=1,2)$ is a compact $n$-dimensional
minimal submanifold with boundary $($see [9]$)$;

  3. (iii)

$\mathcal{M}$ is a complete noncompact manifold with nonnegative Ricci
curvature $($see [10]$)$;

  4. (iv)

$\mathcal{M}$ is a compact $n$-dimensional minimal submanifold with boundary
in a complete noncompact manifold of dimension $n+m\ (m=1,2)$ with nonnegative
sectional curvature $($see [10]$)$;

  5. (v)

$\mathcal{M}$ is a Cartan–Hadamard manifold of dimension $n$ $($i.e., a
complete, simply connected Riemannian manifold with nonpositive sectional
curvature $K)$ for $n=2$ $($see [56, 4]$)$, $n=3$ $($see [25]$)$, $n=4$ $($see
[12]$)$. Generally, the isoperimetric inequality also holds when $K\leqslant
k\leqslant 0$ for $n=2$ $($see [6]$)$, $n=3$ $($see [25]$)$.

The main ideas of this paper are as follows. Firstly, we denote by
$(e^{t\mathcal{L}_{g}^{\mp}})_{t\geqslant 0}$ the parabolic semigroups
generated by $\mathcal{L}_{g}^{\mp}$. More precisely,
$\textbf{{V}}^{\mp}(t,x)=e^{t\mathcal{L}_{g}^{\mp}}\textbf{{V}}(x)$ solve the
following initial-boundary problems, respectively,

$\displaystyle\begin{cases}\frac{\partial\textbf{{V}}^{-}}{\partial
t}-\mathcal{L}_{g}\textbf{{V}}^{-}=0&\text{in}\ (0,+\infty)\times\Omega,\\\
\textbf{{V}}^{-}=0\quad&\text{on}\ (0,+\infty)\times\partial\Omega,\\\
\textbf{{V}}^{-}(0,x)=\textbf{{V}}_{0}&\text{on}\
\\{0\\}\times\Omega\end{cases}$

and

$\displaystyle\begin{cases}\frac{\partial\textbf{{V}}^{+}}{\partial
t}-\mathcal{L}_{g}\textbf{{V}}^{+}=0&\text{in}\ (0,+\infty)\times\Omega,\\\
N(\textbf{{V}}^{+})=0\quad&\text{on}\ (0,+\infty)\times\partial\Omega,\\\
\textbf{{V}}^{+}(0,x)=\textbf{{V}}_{0}&\text{on}\
\\{0\\}\times\Omega.\end{cases}$

Let $\\{\textbf{{U}}^{\mp}_{k}\\}_{k\geqslant 1}$ be the orthonormal
eigenvectors corresponding to the eigenvalues
$\\{\tau^{\mp}_{k}\\}_{k\geqslant 1}$, then the integral kernels
$\textbf{{K}}^{\mp}(t,x,y)$ are given by

$\displaystyle\textbf{{K}}^{\mp}(t,x,y)=\sum_{k=1}^{\infty}e^{-t\tau^{\mp}_{k}}\textbf{{U}}^{\mp}_{k}(x)\otimes\textbf{{U}}^{\mp}_{k}(y).$

Thus,

$\displaystyle\int_{\Omega}\operatorname{Tr}\textbf{{K}}^{\mp}(t,x,y)\,dV=\sum_{k=1}^{\infty}e^{-t\tau^{\mp}_{k}}$

are spectral invariants.

Furthermore, to investigate the geometric aspects of the spectrum and the
asymptotic expansion of the integrals, we combine calculus of symbols (see
[16, 19, 50]) and “method of images” (see [34, 37]). Let
$M=\Omega\cup\partial\Omega\cup\Omega^{*}$ be the closed double of $\Omega$,
and $L_{g}$ the double to $M$ of the operator $\mathcal{L}_{g}$. Then $L_{g}$
generates a strongly continuous semigroup $(e^{tL_{g}})_{t\geqslant 0}$ on
$L^{2}(M)$ with integral kernel $\textbf{{K}}(t,x,y)$. Clearly,
$\textbf{{K}}^{\mp}(t,x,y)=\textbf{{K}}(t,x,y)\mp\textbf{{K}}(t,x,y^{*})$ for
$x,y\in\bar{\Omega}$, where $y^{*}$ is the double of $y\in\Omega$. Since

$\displaystyle e^{tL_{g}}f(x)=\frac{1}{2\pi
i}\int_{\mathcal{C}}e^{-t\tau}(\tau I+L_{g})^{-1}f(x)\,d\tau,$

where $\mathcal{C}$ is a suitable curve in the complex plane in the positive
direction around the spectrum of $L_{g}$. It follows that

$\displaystyle\textbf{{K}}(t,x,y)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{i(x-y)\cdot\xi}\bigg{(}\frac{1}{2\pi
i}\int_{\mathcal{C}}e^{-t\tau}\sigma\big{(}(\tau
I+L_{g})^{-1}\big{)}\,d\tau\bigg{)}\,d\xi,$

where $\sigma\big{(}(\tau I+L_{g})^{-1}\big{)}$ is the full symbol of $(\tau
I+L_{g})^{-1}$. In particular, we get $\operatorname{Tr}\textbf{{K}}(t,x,x)$
and $\operatorname{Tr}\textbf{{K}}(t,x,x^{*})$ for any $x\in\Omega$. Finally,
we explicitly calculate the first two coefficients $a_{0}$ and $a_{1}$ of the
asymptotic expansion

$\displaystyle\int_{\Omega}\operatorname{Tr}\textbf{{K}}^{\mp}(t,x,x)\,dV=a_{0}t^{-n/2}\mp
a_{1}t^{-(n-1)/2}+O(t^{1-n/2})\quad\text{as}\ t\to 0^{+},$

where $a_{0}$ and $a_{1}$ are given in Theorem 1.1.

This paper is organized as follows. In Section 2 we give the expression of the
thermoelastic operator in local coordinates and derive the full symbol of the
resolvent operator $(\tau I+\mathcal{L}_{g})^{-1}$. Section 3 is devoted to
prove Theorem 1.1 and Corollary 1.4.

##  2\. Full Symbol of The Resolvent Operator

In the local coordinates $\\{x_{i}\\}_{i=1}^{n}$, we denote by
$\bigl{\\{}\frac{\partial}{\partial x_{i}}\bigr{\\}}_{i=1}^{n}$ a natural
basis for the tangent space $T_{x}\Omega$ at the point $x\in\Omega$. In what
follows, we will use the Einstein summation convention, the Roman indices run
from 1 to $n$, unless otherwise specified. Then the Riemannian metric is given
by $g=g_{ij}\,dx_{i}\,dx_{j}$. Let $\nabla$ be the Levi-Civita connection on
$\Omega$, $\nabla_{i}=\nabla_{\frac{\partial}{\partial x_{i}}}$ and
$\nabla^{i}=g^{ij}\nabla_{j}$. The gradient operator is denoted by

(2.1) $\operatorname{grad}f=\nabla^{i}f\frac{\partial}{\partial
x_{i}}=g^{ij}\frac{\partial f}{\partial x_{i}}\frac{\partial}{\partial
x_{j}},\quad f\in C^{\infty}(\Omega),$

where $(g^{ij})=(g_{ij})^{-1}$. The divergence operator is denoted by

(2.2) $\operatorname{div}\textbf{{u}}=\nabla_{i}u^{i}=\frac{\partial
u^{i}}{\partial
x_{i}}+\Gamma^{j}_{ij}u^{i},\quad\textbf{{u}}=u^{i}\frac{\partial}{\partial
x_{i}}\in\mathfrak{X}(\Omega),$

where the Christoffel symbols
$\Gamma^{k}_{ij}=\frac{1}{2}g^{kl}\bigl{(}\frac{\partial g_{jl}}{\partial
x_{i}}+\frac{\partial g_{il}}{\partial x_{j}}-\frac{\partial g_{ij}}{\partial
x_{l}}\bigr{)}$. It follows from (2.1) and (2.2) that

(2.3)
$\displaystyle\operatorname{grad}\operatorname{div}\textbf{{u}}=g^{ij}\Bigl{(}\frac{\partial^{2}u^{k}}{\partial
x_{j}\partial x_{k}}+\Gamma^{l}_{kl}\frac{\partial u^{k}}{\partial
x_{j}}+\frac{\partial\Gamma^{l}_{kl}}{\partial
x_{j}}u^{k}\Bigr{)}\frac{\partial}{\partial x_{i}}.$

Recall that the components of the curvature tensor

(2.4) $\displaystyle R^{l}_{ijk}=\frac{\partial\Gamma^{l}_{jk}}{\partial
x_{i}}-\frac{\partial\Gamma^{l}_{ik}}{\partial
x_{j}}+\Gamma^{h}_{jk}\Gamma^{l}_{ih}-\Gamma^{h}_{ik}\Gamma^{l}_{jh},$

and $R_{ijkl}=g_{lm}R^{m}_{ijk}$, the components of the the Ricci tensor

(2.5) $\displaystyle R_{ij}=g^{kl}R_{iklj}.$

The Bochner Laplacian is given by

$\displaystyle\Delta_{B}\textbf{{u}}=(\nabla^{j}\nabla_{j}u^{i})\frac{\partial}{\partial
x_{i}},\quad\textbf{{u}}=u^{i}\frac{\partial}{\partial
x_{i}}\in\mathfrak{X}(\Omega).$

Then combining (2.4) and (2.5), we compute

(2.6) $\displaystyle(\Delta_{B}\textbf{{u}})^{i}$
$\displaystyle=\Delta_{g}u^{i}+g^{kl}\Bigl{(}2\Gamma^{i}_{jk}\frac{\partial
u^{j}}{\partial x_{l}}+\frac{\partial\Gamma^{i}_{jk}}{\partial
x_{l}}u^{j}+(\Gamma^{i}_{hl}\Gamma^{h}_{jk}-\Gamma^{h}_{kl}\Gamma^{i}_{jh})u^{j}\Bigr{)}$
$\displaystyle=\Delta_{g}u^{i}-\operatorname{Ric}(\textbf{{u}})^{i}+g^{kl}\Bigl{(}2\Gamma^{i}_{jk}\frac{\partial
u^{j}}{\partial x_{l}}+\frac{\partial\Gamma^{i}_{kl}}{\partial
x_{j}}u^{j}\Bigr{)},$

where $\operatorname{Ric}(\textbf{{u}})^{i}=g^{ij}R_{jk}u^{k}$ and the
Laplace–Beltrami operator is given by

(2.7) $\Delta_{g}f=g^{ij}\Bigl{(}\frac{\partial^{2}f}{\partial x_{i}\partial
x_{j}}-\Gamma^{k}_{ij}\frac{\partial f}{\partial x_{k}}\Bigr{)},\quad f\in
C^{\infty}(\Omega).$

For the sake of simplicity, we denote by $I_{n}$ the $n\times n$ identity
matrix and

$\displaystyle\begin{pmatrix}(a^{j}_{k})&(b^{j})\\\
(c_{k})&d\end{pmatrix}:=\begin{pmatrix}a^{1}_{1}&\dots&a^{1}_{n}&b^{1}\\\
\vdots&\ddots&\vdots&\vdots\\\ a^{n}_{1}&\dots&a^{n}_{n}&b^{n}\\\
c_{1}&\dots&c_{n}&d\\\ \end{pmatrix}.$

According to (2.1), (2.2), (2.3), (2.6) and (2.7) (cf. [34]), we can write
(1.1) in local coordinates as follows:

$\displaystyle\mathcal{L}_{g}$ $\displaystyle=\begin{pmatrix}\mu I_{n}&0\\\
0&\alpha\end{pmatrix}g^{ml}\frac{\partial^{2}}{\partial x_{m}\partial
x_{l}}+(\lambda+\mu)\begin{pmatrix}\big{(}g^{jm}\frac{\partial^{2}}{\partial
x_{k}\partial x_{m}}\big{)}&0\\\ 0&0\end{pmatrix}-\begin{pmatrix}\mu
I_{n}&0\\\ 0&\alpha\end{pmatrix}g^{ml}\Gamma^{s}_{ml}\frac{\partial}{\partial
x_{s}}$
$\displaystyle\quad+2\mu\begin{pmatrix}\big{(}g^{ml}\Gamma^{j}_{mk}\frac{\partial}{\partial
x_{l}}\big{)}&0\\\
0&0\end{pmatrix}+(\lambda+\mu)\begin{pmatrix}\big{(}g^{jm}\Gamma^{l}_{kl}\frac{\partial}{\partial
x_{m}}\big{)}&0\\\
0&0\end{pmatrix}+\beta\begin{pmatrix}0&-\big{(}g^{jk}\frac{\partial}{\partial
x_{k}}\big{)}\\\ i\omega\theta_{0}\big{(}\frac{\partial}{\partial
x_{k}}\big{)}&0\end{pmatrix}$
$\displaystyle\quad+(\lambda+\mu)\begin{pmatrix}\big{(}g^{jm}\frac{\partial\Gamma^{l}_{kl}}{\partial
x_{m}}\big{)}&0\\\
0&0\end{pmatrix}+\mu\begin{pmatrix}\big{(}g^{ml}\frac{\partial\Gamma^{j}_{ml}}{\partial
x_{k}}\big{)}&0\\\ 0&0\end{pmatrix}+\begin{pmatrix}\rho\omega^{2}I_{n}&0\\\
i\omega\beta\theta_{0}(\Gamma^{l}_{kl})&i\omega\gamma\end{pmatrix}.$

Moreover, we have

$\displaystyle\mathcal{L}_{g}\textbf{{U}}(x)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{ix\cdot\xi}\textbf{{C}}_{g}(x,\xi)\hat{\textbf{{U}}}(\xi)\,d\xi,$

where

$\displaystyle\textbf{{C}}_{g}(x,\xi)$
$\displaystyle=-|\xi|^{2}\begin{pmatrix}\mu I_{n}&0\\\
0&\alpha\end{pmatrix}-(\lambda+\mu)\begin{pmatrix}(g^{jm}\xi_{k}\xi_{m})&0\\\
0&0\end{pmatrix}-ig^{ml}\Gamma^{s}_{ml}\xi_{s}\begin{pmatrix}\mu I_{n}&0\\\
0&\alpha\end{pmatrix}$
$\displaystyle\quad+2i\mu\begin{pmatrix}(g^{ml}\Gamma^{j}_{mk}\xi_{l})&0\\\
0&0\end{pmatrix}+i(\lambda+\mu)\begin{pmatrix}(g^{jm}\Gamma^{l}_{kl}\xi_{m})&0\\\
0&0\end{pmatrix}-\beta\begin{pmatrix}0&i(g^{jk}\xi_{k})\\\
\omega\theta_{0}(\xi_{k})&0\end{pmatrix}$
$\displaystyle\quad+(\lambda+\mu)\begin{pmatrix}\big{(}g^{jm}\frac{\partial\Gamma^{l}_{kl}}{\partial
x_{m}}\big{)}&0\\\
0&0\end{pmatrix}+\mu\begin{pmatrix}\big{(}g^{ml}\frac{\partial\Gamma^{j}_{ml}}{\partial
x_{k}}\big{)}&0\\\ 0&0\end{pmatrix}+\begin{pmatrix}\rho\omega^{2}I_{n}&0\\\
i\omega\beta\theta_{0}(\Gamma^{l}_{kl})&i\omega\gamma\end{pmatrix},$

and $|\xi|^{2}=g^{ij}\xi_{i}\xi_{j}$. For each $\tau\in\mathbb{C}$, we denote
$\tau
I_{n}+\textbf{{C}}_{g}=\textbf{{c}}_{2}+\textbf{{c}}_{1}+\textbf{{c}}_{0}$,
where

(2.8) $\displaystyle\textbf{{c}}_{2}$
$\displaystyle=\begin{pmatrix}(\tau-\mu|\xi|^{2})I_{n}&0\\\
0&\tau-\alpha|\xi|^{2}\end{pmatrix}-(\lambda+\mu)\begin{pmatrix}(g^{jm}\xi_{k}\xi_{m})&0\\\
0&0\end{pmatrix},$ (2.9) $\displaystyle\textbf{{c}}_{1}$
$\displaystyle=-ig^{ml}\Gamma^{s}_{ml}\xi_{s}\begin{pmatrix}\mu I_{n}&0\\\
0&\alpha\end{pmatrix}+2i\mu\begin{pmatrix}(g^{ml}\Gamma^{j}_{mk}\xi_{l})&0\\\
0&0\end{pmatrix}+i(\lambda+\mu)\begin{pmatrix}(g^{jm}\Gamma^{l}_{kl}\xi_{m})&0\\\
0&0\end{pmatrix}$
$\displaystyle\quad-\beta\begin{pmatrix}0&i(g^{jk}\xi_{k})\\\
\omega\theta_{0}(\xi_{k})&0\end{pmatrix},$ (2.10)
$\displaystyle\textbf{{c}}_{0}$
$\displaystyle=(\lambda+\mu)\begin{pmatrix}\big{(}g^{jm}\frac{\partial\Gamma^{l}_{kl}}{\partial
x_{m}}\big{)}&0\\\
0&0\end{pmatrix}+\mu\begin{pmatrix}\big{(}g^{ml}\frac{\partial\Gamma^{j}_{ml}}{\partial
x_{k}}\big{)}&0\\\ 0&0\end{pmatrix}+\begin{pmatrix}\rho\omega^{2}I_{n}&0\\\
i\omega\beta\theta_{0}(\Gamma^{l}_{kl})&i\omega\gamma\end{pmatrix}.$

Now we calculate the full symbol of the operator $(\tau
I+\mathcal{L}_{g})^{-1}$. Let $B(x,\xi,\tau)$ be a two-sided parametrix for
$\tau I+\mathcal{L}_{g}$, that is, $B(x,\xi,\tau)$ is a pseudodifferential
operator of order $-2$ with parameter $\tau$ for which

$\displaystyle B(\tau)(\tau I+\mathcal{L}_{g})=I\mod OPS^{-\infty},$
$\displaystyle(\tau I+\mathcal{L}_{g})B(\tau)=I\mod OPS^{-\infty}.$

Recall that if $P_{1}$ and $P_{2}$ are two pseudodifferential operators with
full symbols $p_{1}$ and $p_{2}$, respectively, then the full symbol
$\sigma(P_{1}P_{2})$ is given by (see p. 11 of [52], or p. 71 of [21], see
also [19, 54])

$\displaystyle\sigma(P_{1}P_{2})\sim\sum_{J}\frac{1}{J!}\partial_{\xi}^{J}p_{1}D_{x}^{J}p_{2},$

where the sum is over all multi-indices $J$. Let
$\sigma(B(x,\xi,\tau))\sim\sum_{j\leqslant-2}\textbf{{b}}_{j}(x,\xi,\tau)$ be
the full symbol of $B(x,\xi,\tau)$. We then get

$\displaystyle I_{n}$
$\displaystyle=\sum_{0=j+|J|+2-k}\frac{1}{J!}\partial_{\xi}^{J}\textbf{{c}}_{k}D_{x}^{J}\textbf{{b}}_{-2-j}=\textbf{{c}}_{2}\textbf{{b}}_{-2},$
$\displaystyle 0$ $\displaystyle=\sum_{1\leqslant
l=j+|J|+2-k}\frac{1}{J!}\partial_{\xi}^{J}\textbf{{c}}_{k}D_{x}^{J}\textbf{{b}}_{-2-j}$
$\displaystyle=\textbf{{c}}_{2}\textbf{{b}}_{-2-l}+\sum_{\begin{subarray}{c}l=j+|J|+2-k\\\
j<l\end{subarray}}\frac{1}{J!}\partial_{\xi}^{J}\textbf{{c}}_{k}D_{x}^{J}\textbf{{b}}_{-2-j},\quad
l\geqslant 1.$

Therefore,

$\displaystyle\textbf{{b}}_{-2}$ $\displaystyle=\textbf{{c}}_{2}^{-1},$
$\displaystyle\textbf{{b}}_{-2-l}$
$\displaystyle=-\textbf{{c}}_{2}^{-1}\sum_{\begin{subarray}{c}l=j+|J|+2-k\\\
j<l\end{subarray}}\frac{1}{J!}\partial_{\xi}^{J}\textbf{{c}}_{k}D_{x}^{J}\textbf{{b}}_{-2-j},\quad
l\geqslant 1.$

In particular, by (2.8) we have

$\displaystyle\textbf{{b}}_{-2}=\begin{pmatrix}\frac{1}{\tau-\mu|\xi|^{2}}I_{n}&0\\\
0&\frac{1}{\tau-\alpha|\xi|^{2}}\end{pmatrix}+\frac{\lambda+\mu}{(\tau-\mu|\xi|^{2})(\tau-(\lambda+2\mu)|\xi|^{2})}\begin{pmatrix}(g^{jm}\xi_{k}\xi_{m})&0\\\
0&0\end{pmatrix},$

and

(2.11)
$\displaystyle\operatorname{Tr}\textbf{{b}}_{-2}=\frac{n-1}{\tau-\mu|\xi|^{2}}+\frac{1}{\tau-\alpha|\xi|^{2}}+\frac{1}{\tau-(\lambda+2\mu)|\xi|^{2}}.$

##  3\. Coefficients of the Asymptotic Expansions

###### Proof of Theorem 1.1.

According to the theory of elliptic operators (see [38, 39, 40, 42, 51]), we
see that the thermoelastic operator $\mathcal{L}_{g}$ can generate strongly
continuous semigroups $(e^{t\mathcal{L}_{g}^{\mp}})_{t\geqslant 0}$ with
respect to the Dirichlet and Neumann boundary conditions, respectively.
Moreover, there exist matrix-valued functions (integral kernels)
$\textbf{{K}}^{\mp}(t,x,y)$ such that (see [11] or p. 4 of [13])

$\displaystyle
e^{t\mathcal{L}_{g}^{\mp}}\textbf{{V}}_{0}(x)=\int_{\Omega}\textbf{{K}}^{\mp}(t,x,y)\textbf{{V}}_{0}(y)\,dy,\quad\textbf{{V}}_{0}(x)\in(L^{2}(\Omega))^{n+1}.$

Let $\\{\textbf{{U}}^{\mp}_{k}\\}_{k\geqslant 1}$ be the orthonormal
eigenvectors corresponding to the eigenvalues
$\\{\tau^{\mp}_{k}\\}_{k\geqslant 1}$, then the integral kernels
$\textbf{{K}}^{\mp}(t,x,y)$ are given by

$\displaystyle\textbf{{K}}^{\mp}(t,x,y)=e^{t\mathcal{L}_{g}^{\mp}}\delta(x-y)=\sum_{k=1}^{\infty}e^{-t\tau^{\mp}_{k}}\textbf{{U}}^{\mp}_{k}(x)\otimes\textbf{{U}}^{\mp}_{k}(y).$

This implies that

$\displaystyle\int_{\Omega}\operatorname{Tr}\textbf{{K}}^{\mp}(t,x,y)\,dV=\sum_{k=1}^{\infty}e^{-t\tau^{\mp}_{k}}$

are spectral invariants.

We will combine calculus of symbols (see [50]) and “method of images” (see
[34, 37]), to deal with the asymptotic expansions of traces of the integral
kernels. Let $M=\Omega\cup\partial\Omega\cup\Omega^{*}$ be the closed double
of $\Omega$, and $L_{g}$ the double to $M$ of the operator $\mathcal{L}_{g}$.
The coefficients of the operator $L_{g}$ jump as $x$ crosses $\partial\Omega$,
but $\frac{\partial\textbf{{U}}}{\partial t}=L_{g}\textbf{{U}}$ still has a
nice fundamental solution (integral kernel) $\textbf{{K}}(t,x,y)$ of class
$C^{\infty}((0,+\infty)\times(M\setminus\partial\Omega)\times(M\setminus\partial\Omega))\cap
C^{1}((0,+\infty)\times M\times M)$, approximable even on $\partial\Omega$,
and the integral kernels $\textbf{{K}}^{\mp}(t,x,y)$ can be expressed on
$(0,+\infty)\times\Omega\times\Omega$ as

$\displaystyle\textbf{{K}}^{\mp}(t,x,y)=\textbf{{K}}(t,x,y)\mp\textbf{{K}}(t,x,y^{*}),$

where $y^{*}$ is the double of $y\in\Omega$. Since $(e^{tL_{g}})_{t\geqslant
0}$ can also be represented as

$\displaystyle e^{tL_{g}}=\frac{1}{2\pi i}\int_{\mathcal{C}}e^{-t\tau}(\tau
I+L_{g})^{-1}\,d\tau,$

where $\mathcal{C}$ is a suitable curve in the complex plane in the positive
direction around the spectrum of $L_{g}$, that is, $\mathcal{C}$ is a contour
around the positive real axis. It follows that, for any $x,y\in M$,

$\displaystyle\textbf{{K}}(t,x,y)=e^{tL_{g}}\delta(x-y)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{i(x-y)\cdot\xi}\bigg{(}\frac{1}{2\pi
i}\int_{\mathcal{C}}e^{-t\tau}\sigma\big{(}(\tau
I+L_{g})^{-1}\big{)}\,d\tau\bigg{)}\,d\xi.$

In particular, for any $x\in\Omega$,

$\displaystyle\textbf{{K}}(t,x,x)$
$\displaystyle=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\bigg{(}\frac{1}{2\pi
i}\int_{\mathcal{C}}e^{-t\tau}\sum_{l\geqslant
0}\textbf{{b}}_{-2-l}\,d\tau\bigg{)}\,d\xi,$
$\displaystyle\textbf{{K}}(t,x,x^{*})$
$\displaystyle=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{i(x-x^{*})\cdot\xi}\bigg{(}\frac{1}{2\pi
i}\int_{\mathcal{C}}e^{-t\tau}\sum_{l\geqslant
0}\textbf{{b}}_{-2-l}\,d\tau\bigg{)}\,d\xi.$

By applying the residue theorem to (2.11), we get

$\displaystyle\frac{1}{2\pi
i}\int_{\mathcal{C}}e^{-t\tau}\operatorname{Tr}\textbf{{b}}_{-2}\,d\tau=(n-1)e^{-t\mu|\xi|^{2}}+e^{-t\alpha|\xi|^{2}}+e^{-t(\lambda+2\mu)|\xi|^{2}}.$

We use the geodesic normal coordinate system centered at $x\in\Omega$, then
$g_{ij}(x)=\delta_{ij}$ and $\Gamma^{k}_{ij}(x)=0$. It follows that

$\displaystyle\int_{\Omega}\biggl{[}\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\bigg{(}\frac{1}{2\pi
i}\int_{\mathcal{C}}e^{-t\tau}\operatorname{Tr}\textbf{{b}}_{-2}\,d\tau\bigg{)}\,d\xi\biggr{]}\,dV=\frac{\operatorname{vol}(\Omega)}{(4\pi
t)^{n/2}}\biggl{(}\frac{n-1}{\mu^{n/2}}+\frac{1}{\alpha^{n/2}}+\frac{1}{(\lambda+2\mu)^{n/2}}\biggr{)}.$

For given $\varepsilon>0$, denote by $U_{\varepsilon}(\partial\Omega)=\\{z\in
M|\text{dist}(z,\partial\Omega)<\varepsilon\\}$ the $\varepsilon$-neighborhood
of $\partial\Omega$ in $M$.

(i) For any $x\in\Omega\setminus U_{\varepsilon}(\partial\Omega)$, we have

$\displaystyle\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{i(x-x^{*})\cdot\xi}\bigg{(}\frac{1}{2\pi
i}\int_{\mathcal{C}}e^{-t\tau}\operatorname{Tr}\textbf{{b}}_{-2}\,d\tau\bigg{)}\,d\xi$
$\displaystyle\quad=\frac{1}{(4\pi
t)^{n/2}}\biggl{(}\frac{n-1}{\mu^{n/2}}e^{-\frac{|x-x^{*}|^{2}}{4\mu
t}}+\frac{1}{\alpha^{n/2}}e^{-\frac{|x-x^{*}|^{2}}{4\alpha
t}}+\frac{1}{(\lambda+2\mu)^{n/2}}e^{-\frac{|x-x^{*}|^{2}}{4(\lambda+2\mu)t}}\biggr{)},$

which tends to zero as $t\to 0^{+}$ since $|x-x^{*}|\geqslant\varepsilon$.
Integrating the above result over $\Omega\setminus
U_{\varepsilon}(\partial\Omega)$, we get $O(t^{1-n/2})$ as $t\to 0^{+}$.

For $l\geqslant 1$, it can be verified that
$\operatorname{Tr}\textbf{{b}}_{-2-l}$ is a sum of finitely many terms, each
of which has the form

$\displaystyle\frac{r_{k}(x,\xi)}{(\tau-\mu|\xi|^{2})^{j}(\tau-\alpha|\xi|^{2})^{m}(\tau-(\lambda+2\mu)|\xi|^{2})^{p}},$

where $k-2j-2m-2p=-2-l$, and $r_{k}(x,\xi)$ is the symbol independent of
$\tau$ and homogeneous of degree $k$. Hence

$\displaystyle\int_{\Omega}\biggl{[}\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\bigg{(}\frac{1}{2\pi
i}\int_{\mathcal{C}}e^{-t\tau}\sum_{l\geqslant
1}\operatorname{Tr}\textbf{{b}}_{-2-l}\,d\tau\bigg{)}\,d\xi\biggr{]}dV$
$\displaystyle=O(t^{1-n/2})\quad\text{as}\ t\to 0^{+},$
$\displaystyle\int_{\Omega}\biggl{[}\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{i(x-x^{*})\cdot\xi}\bigg{(}\frac{1}{2\pi
i}\int_{\mathcal{C}}e^{-t\tau}\sum_{l\geqslant
1}\operatorname{Tr}\textbf{{b}}_{-2-l}\,d\tau\bigg{)}\,d\xi\biggr{]}dV$
$\displaystyle=O(t^{1-n/2})\quad\text{as}\ t\to 0^{+}.$

Combining the above results, we observe

$\displaystyle\int_{\Omega}\operatorname{Tr}\textbf{{K}}(t,x,x)\,dV=\frac{\operatorname{vol}(\Omega)}{(4\pi
t)^{n/2}}\biggl{(}\frac{n-1}{\mu^{n/2}}+\frac{1}{\alpha^{n/2}}+\frac{1}{(\lambda+2\mu)^{n/2}}\biggr{)}+O(t^{1-n/2})\quad\text{as}\
t\to 0^{+}.$

(ii) We consider the case when $x\in\Omega\cap
U_{\varepsilon}(\partial\Omega)$. Inspired by [34, 37], we pick a self-double
patch $W$ of $M$ (such that $W\subset U_{\varepsilon}(\partial\Omega)$)
covering a patch $W\cap\partial\Omega$ endowed with local coordinates $x$ such
that $\varepsilon>x_{n}>0$ in $W\cap\Omega$, $x_{n}=0$ on $W\subset
U_{\varepsilon}(\partial\Omega)$, $x_{n}(x^{*})=-x_{n}(x)$, and the positive
$x_{n}$-direction is perpendicular to $\partial\Omega$ (see Figure 1).

$\Omega^{*}$$x_{n}>0$$x^{*}$$x$$\Omega$$\partial\Omega$ Figure 1.

This has the effect that

$\displaystyle g_{jk}(x^{*})$ $\displaystyle=-g_{jk}(x)\quad\text{for}\ j<k=n\
\text{or}\ k<j=n,$ $\displaystyle=g_{jk}(x)\quad\text{for}\ j,k=n\ \text{or}\
j=k=n,$ $\displaystyle g_{jk}(x)$ $\displaystyle=0\quad\text{for}\ j<k=n\
\text{or}\ k<j=n\ \text{on}\ \partial\Omega,$
$\displaystyle\sqrt{\operatorname{det}g/g_{nn}}\,dx_{1}\cdots dx_{n-1}$
$\displaystyle=\text{the element of Riemannian surface area on}\
\partial\Omega.$

We choose the geodesic normal coordinate system on $\partial\Omega$ (see [29]
or p. 532 of [52]) such that the metric has the form
$g=\sum_{\alpha,\beta=1}^{n-1}g_{\alpha\beta}\,dx_{\alpha}\,dx_{\beta}+dx_{n}^{2}$.
Then we see that for
$x\in\\{z\in\Omega|\text{dist}(z,\partial\Omega)<\varepsilon\\}$,

$\displaystyle
x-x^{*}=\big{(}0,\dots,0,x_{n}-(-x_{n})\big{)}=(0,\dots,0,2x_{n}).$

We have

$\displaystyle\int_{W\cap\Omega}\biggl{[}\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{i(x-x^{*})\cdot\xi}\bigg{(}\frac{1}{2\pi
i}\int_{\mathcal{C}}e^{-t\tau}\operatorname{Tr}\textbf{{b}}_{-2}\,d\tau\bigg{)}\,d\xi\biggr{]}dV$
$\displaystyle\quad=\frac{1}{(4\pi
t)^{n/2}}\int_{0}^{\varepsilon}dx_{n}\int_{W\cap\partial\Omega}\biggl{(}\frac{n-1}{\mu^{n/2}}e^{-\frac{x_{n}^{2}}{\mu
t}}+\frac{1}{\alpha^{n/2}}e^{-\frac{x_{n}^{2}}{\alpha
t}}+\frac{1}{(\lambda+2\mu)^{n/2}}e^{-\frac{x_{n}^{2}}{(\lambda+2\mu)t}}\biggr{)}dx^{\prime}$
$\displaystyle\quad=\frac{1}{(4\pi
t)^{n/2}}\int_{0}^{+\infty}dx_{n}\int_{W\cap\partial\Omega}\biggl{(}\frac{n-1}{\mu^{n/2}}e^{-\frac{x_{n}^{2}}{\mu
t}}+\frac{1}{\alpha^{n/2}}e^{-\frac{x_{n}^{2}}{\alpha
t}}+\frac{1}{(\lambda+2\mu)^{n/2}}e^{-\frac{x_{n}^{2}}{(\lambda+2\mu)t}}\biggr{)}dx^{\prime}$
$\displaystyle\qquad-\frac{1}{(4\pi
t)^{n/2}}\int_{\varepsilon}^{+\infty}dx_{n}\int_{W\cap\partial\Omega}\biggl{(}\frac{n-1}{\mu^{n/2}}e^{-\frac{x_{n}^{2}}{\mu
t}}+\frac{1}{\alpha^{n/2}}e^{-\frac{x_{n}^{2}}{\alpha
t}}+\frac{1}{(\lambda+2\mu)^{n/2}}e^{-\frac{x_{n}^{2}}{(\lambda+2\mu)t}}\biggr{)}dx^{\prime}$
$\displaystyle\quad=\frac{1}{4}\frac{\operatorname{vol}(W\cap\partial\Omega)}{(4\pi
t)^{(n-1)/2}}\biggr{(}\frac{n-1}{\mu^{(n-1)/2}}+\frac{1}{(\lambda+2\mu)^{(n-1)/2}}+\frac{1}{\alpha^{(n-1)/2}}\biggr{)}$
$\displaystyle\qquad-\frac{1}{(4\pi
t)^{n/2}}\int_{W\cap\partial\Omega}dx^{\prime}\int_{\varepsilon}^{+\infty}\biggl{(}\frac{n-1}{\mu^{n/2}}e^{-\frac{x_{n}^{2}}{\mu
t}}+\frac{1}{\alpha^{n/2}}e^{-\frac{x_{n}^{2}}{\alpha
t}}+\frac{1}{(\lambda+2\mu)^{n/2}}e^{-\frac{x_{n}^{2}}{(\lambda+2\mu)t}}\biggr{)}dx_{n}.$

Note that, for any fixed $\varepsilon>0$, the last integral is $O(t^{1-n/2})$
as $t\to 0^{+}$. Hence

$\displaystyle\int_{W\cap\Omega}\biggl{[}\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{i(x-x^{*})\cdot\xi}\bigg{(}\frac{1}{2\pi
i}\int_{\mathcal{C}}e^{-t\tau}\operatorname{Tr}\textbf{{b}}_{-2}\,d\tau\bigg{)}\,d\xi\biggr{]}dV$
$\displaystyle\quad=\frac{1}{4}\frac{\operatorname{vol}(W\cap\partial\Omega)}{(4\pi
t)^{(n-1)/2}}\biggr{(}\frac{n-1}{\mu^{(n-1)/2}}+\frac{1}{(\lambda+2\mu)^{(n-1)/2}}+\frac{1}{\alpha^{(n-1)/2}}\biggr{)}+O(t^{1-n/2})\quad{\rm
as}\ t\to 0^{+}.$

Similarly, we obtain

$\displaystyle\int_{W\cap\Omega}\operatorname{Tr}\textbf{{K}}(t,x,x^{*})\,dV=$
$\displaystyle\frac{1}{4}\frac{\operatorname{vol}(W\cap\partial\Omega)}{(4\pi
t)^{(n-1)/2}}\biggr{(}\frac{n-1}{\mu^{(n-1)/2}}+\frac{1}{(\lambda+2\mu)^{(n-1)/2}}+\frac{1}{\alpha^{(n-1)/2}}\biggr{)}$
$\displaystyle+O(t^{1-n/2})\quad{\rm as}\ t\to 0^{+}.$

Therefore

$\displaystyle\int_{W\cap\Omega}\operatorname{Tr}\textbf{{K}}^{\mp}(t,x,x)\,dV=\int_{W\cap\Omega}\operatorname{Tr}\textbf{{K}}(t,x,x)\,dV\mp\int_{W\cap\Omega}\operatorname{Tr}\textbf{{K}}(t,x,x^{*})\,dV$
$\displaystyle\quad=\frac{\operatorname{vol}(W\cap\Omega)}{(4\pi
t)^{n/2}}\biggr{(}\frac{n-1}{\mu^{n/2}}+\frac{1}{(\lambda+2\mu)^{n/2}}+\frac{1}{\alpha^{n/2}}\biggr{)}$
$\displaystyle\qquad\mp\frac{1}{4}\frac{\operatorname{vol}(W\cap\partial\Omega)}{(4\pi
t)^{(n-1)/2}}\biggr{(}\frac{n-1}{\mu^{(n-1)/2}}+\frac{1}{(\lambda+2\mu)^{(n-1)/2}}+\frac{1}{\alpha^{(n-1)/2}}\biggr{)}+O(t^{1-n/2})\quad\text{as}\
t\to 0^{+}.$

Similarly, we can also get the other coefficients $a_{k}^{\mp}$ for
$2\leqslant k\leqslant n$ by this method. ∎

###### Proof of Corollary 1.4.

Theorem 1.1 shows that $a_{0}$ and $a_{1}^{\mp}$ are spectral invariants.
Since the thermoelastic spectrum of $\Omega$ with respect to the Dirichlet or
Neumann boundary condition is equal to that of the geodesic ball
$B^{n}\subset\mathcal{M}$, we observe that

(3.1)
$\displaystyle\operatorname{vol}(\Omega)=\operatorname{vol}(B^{n}),\quad\operatorname{vol}(\partial\Omega)=\operatorname{vol}(\partial
B^{n}).$

In addition, the isoperimetric inequality holds on $\mathcal{M}$ by condition
$(b)$. Then

(3.2)
$\displaystyle\operatorname{vol}(\partial\Omega)\geqslant\operatorname{vol}(\partial
B^{n}),$

equality holds if and only if $\Omega$ is isometric to $B^{n}$. Therefore, we
immediately deduce that $\Omega$ is isometric to $B^{n}$ by combining (3.1)
and (3.2). ∎

## Acknowledgements

This research was supported by NNSF of China (11671033/A010802) and NNSF of
China
(11171023/A010801).

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