Determining the Coefficients of the Thermoelastic System from Boundary Information

Abstract

Given a compact Riemannian manifold $(M,g)$ with smooth boundary $\partial M$, we give an explicit expression for the full symbol of the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$ with variable coefficients $\lambda ,\mu ,\alpha ,\beta \in C^{\infty }\left(\overline{M}\right)$. We prove that ${\mathsf{\Lambda }}_g$ uniquely determines partial derivatives of all orders of these coefficients on the boundary $\partial M$. Moreover, for a nonempty smooth open subset $\mathsf{\Gamma }\subset \partial M$, suppose that the manifold and these coefficients are real analytic up to $\mathsf{\Gamma }$. We show that ${\mathsf{\Lambda }}_g$ uniquely determines these coefficients on the whole manifold $\overline{M}$.

1. Introduction

In this paper, we will study the thermoelastic Calderón problem, i.e., whether one can uniquely determine the Lamé coefficients $\lambda ,\mu$, and the other two physical coefficients $\alpha ,\beta$ of a thermoelastic body by boundary information. Let $(M,g)$ be a compact Riemannian manifold of dimension n with smooth boundary $\partial M$. We consider the manifold M as an inhomogeneous, isotropic, thermoelastic body. Assume that the coefficient $\beta \in C^{\infty }\left(\overline{M}\right)$, the Lamé coefficients $\lambda ,\mu \in C^{\infty }\left(\overline{M}\right)$, and the heat conduction coefficient $\alpha \in C^{\infty }\left(\overline{M}\right)$ of the thermoelastic body satisfy

$$\begin{array}{c}\hfill \mu >0,\lambda +\mu ⩾0,\alpha >0.\end{array}$$

(1)

1.1. Thermoelastic Operator

We denote by $grad,div,{\Delta }_g,{\Delta }_B$, and Ric, respectively, the gradient operator, the divergence operator, the Laplace–Beltrami operator, the Bochner Laplacian, and the Ricci tensor with respect to the metric g. For the displacement vector field $\mathit{u}\in {\left[C^{\infty }\left(M\right)\right]}^n$ and the temperature variation $\theta \in C^{\infty }\left(M\right)$, we define the thermoelastic operator $T_g$ with variable coefficients as (cf. [1,2,3,4])

$$T_g\left[\begin{array}{c}\mathit{u}\\ \theta \end{array}\right]:=\left[\begin{array}{cc}{L}_g+\rho {\omega }^2& -\beta grad\\ i\omega {\theta }_0\beta div& \alpha {\Delta }_g+i\omega \gamma \end{array}\right]\left[\begin{array}{c}\mathit{u}\\ \theta \end{array}\right],$$

(2)

where the Lamé operator $L_g$ with variable coefficients is defined by (see [4])

$$\begin{array}{cc}\hfill L_g\mathit{u}& :=\mu {\Delta }_B^{}\mathit{u}+(\lambda +\mu )graddiv\mathit{u}+\mu Ric\left(\mathit{u}\right)\hfill \\ \hfill & +(grad\lambda )div\mathit{u}+\left(S\mathit{u}\right)(grad\mu ).\hfill \end{array}$$

(3)

Here the strain tensor S (also called the deformation tensor) of type $(1,1)$ is defined by (see [5], p. 562)

$$\begin{array}{c}\hfill S\mathit{u}:=\nabla \mathit{u}+{(\nabla \mathit{u})}^t,\end{array}$$

(4)

where the superscript t denotes the transpose. The coefficient $\beta \in C^{\infty }\left(\overline{M}\right)$ depends on the Lamé coefficients and the linear expansion coefficient of the thermoelastic body, $\gamma$ 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 $i=\sqrt{-1}$.

In particular, the Lamé operator with constant coefficients has the form $L\mathit{u}=\mu \Delta \mathit{u}+(\lambda +\mu )\nabla (\nabla ·\mathit{u})$ in Euclidean bounded domains (see [1,6]).

1.2. Thermoelastic Calderón Problem

We first consider the following Dirichlet boundary value problem for the thermoelastic system

$$\left\{\begin{array}{cc}{T}_g\mathit{U}=0\hfill & \mathrm{in}M,\hfill \\ \mathit{U}=\mathit{V}\hfill & \mathrm{on}\partial M,\hfill \end{array}\right.$$

(5)

where $\mathit{U}={(\mathit{u},\theta )}^t$. Problem (5) is an extension of the boundary value problem for classical elastic system. Particularly, when M is a bounded Euclidean domain and the temperature is not taken into consideration, problem (5) reduces to the corresponding problem for classical elastic system.

For any vector $\mathit{V}\in {\left[H^{1/2}(\partial M)\right]}^{n+1}$, there is a unique solution $\mathit{U}\in {\left[H^1\left(M\right)\right]}^{n+1}$ solving problem (5) by the theory of elliptic operators. Therefore, we define the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g:{\left[H^{1/2}(\partial M)\right]}^{n+1}\to {\left[H^{-1/2}(\partial M)\right]}^{n+1}$ associated with the thermoelastic operator $T_g$ as (see [3])

$$\begin{array}{c}\hfill {\mathsf{\Lambda }}_g\left(\mathit{U}{|}_{\partial M}\right):=\left[\begin{array}{cc}\lambda \nu div+\mu \nu S& -\beta \nu \\ 0& \alpha {\partial }_{\nu }\end{array}\right]\mathit{U}\mathrm{on}\partial M,\end{array}$$

(6)

where $\nu$ is the outward unit normal vector to the boundary $\partial M$. The thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$ is an elliptic, self-adjoint pseudodifferential operator of order one defined on the boundary. For the studies about other types of Dirichlet-to-Neumann map, we also refer the reader to [3,7,8,9] and references therein.

In this paper, we will study the thermoelastic Calderón problem on a Riemannian manifold, which determines the coefficients $\lambda ,\mu ,\alpha ,\beta \in C^{\infty }\left(\overline{M}\right)$ by the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$. By giving explicit expressions for ${\mathsf{\Lambda }}_g$ and its full symbol $\sigma \left({\mathsf{\Lambda }}_g\right)$, we show that ${\mathsf{\Lambda }}_g$ uniquely determines the coefficients $\lambda ,\mu ,\alpha ,\beta$.

We briefly recall some uniqueness results for the classical Calderón problem and the elastic Calderón problem. The classical Calderón problem [10] is concerned with whether one can uniquely determine the electrical conductivity of a medium by making voltage and current measurements at the boundary of the medium. This problem has been studied for decades. For a bounded Euclidean domain $\mathsf{\Omega }\subset {\mathbb{R}}^n$ with smooth boundary $\partial \mathsf{\Omega }$, $n⩾2$, Kohn and Vogelius [11] proved a famous uniqueness result on the boundary for $C^{\infty }$-conductivities, that is, if ${\mathsf{\Lambda }}_{{\gamma }_1}={\mathsf{\Lambda }}_{{\gamma }_2}$, then, $\frac{{\partial }^{\left|J\right|}{\gamma }_1}{\partial x^J}{|}_{\partial \mathsf{\Omega }}=\frac{{\partial }^{\left|J\right|}{\gamma }_2}{\partial x^J}{|}_{\partial \mathsf{\Omega }}$ for all multi-indices $J\in {\mathbb{N}}^n$. This settled the uniqueness problem on the boundary in the real analytic category. They extended the uniqueness result to piecewise real analytic conductivities in [12]. In dimensions $n⩾3$, in a celebrated paper [13], Sylvester and Uhlmann proved the uniqueness of the $C^{\infty }$-conductivities by constructing the complex geometrical optics solutions. The classical Calderón problem has attracted much attention for decades (see, for example, [14,15,16,17,18] in two dimensional cases, and [19,20,21,22] in higher dimensional cases). We also refer the reader to the survey articles [23,24] for the classical Calderón problem and related topics.

For the elastic Calderón problem, partial uniqueness results for determination of Lamé coefficients from boundary measurements were obtained. For a bounded Euclidean domain $\mathsf{\Omega }\subset {\mathbb{R}}^n$ with smooth boundary $\partial \mathsf{\Omega }$, Nakamura and Uhlmann [25] proved that one can determine the full Taylor series of Lamé coefficients on the boundary in all dimensions $n⩾2$ and for a generic anisotropic elastic tensor in two dimensions. In [26], Imanuvilov and Yamamoto also proved the global uniqueness of the Lamé coefficients $\lambda ,\mu \in C^{10}\left(\overline{\mathsf{\Omega }}\right)$. In three-dimensional Euclidean domains, Nakamura and Uhlmann [27] and Eskin and Ralston [28] proved the global uniqueness of Lamé coefficients provided that $\nabla \mu$ is small in a suitable norm. However, in dimensions $n⩾3$, the global uniqueness of the Lamé coefficients $\lambda ,\mu \in C^{\infty }\left(\overline{\mathsf{\Omega }}\right)$ without the smallness assumption ($\parallel \nabla \mu \parallel <{\epsilon }_0$ for some small positive ${\epsilon }_0$) remains an open problem (see [29], p. 210). We also refer the reader to [30,31,32,33] for the elastic Calderón problem.

Recently, Tan and Liu [4] gave an explicit expression for the full symbol of the elastic Dirichlet-to-Neumann map on a Riemannian manifold M, and showed that the elastic Dirichlet-to-Neumann map uniquely determines partial derivatives of all orders of the Lamé coefficients on the boundary. Moreover, for a nonempty open subset $\mathsf{\Gamma }\subset \partial M$, suppose that the manifold and the Lamé coefficients are real analytic up to $\mathsf{\Gamma }$, they proved that the elastic Dirichlet-to-Neumann map uniquely determines the Lamé coefficients on the whole manifold $\overline{M}$.

In mathematics, physics, and engineering, there are lots of inverse problems have been studied for decades. Here we do not list all the references about these topics. We refer the reader to [34,35,36,37,38] for Maxwell’s equations, to [39,40,41,42,43,44,45,46,47,48,49] for incompressible fluid, Schrödinger operator, elastic operator, and the related problems. For the studies about other types of Dirichlet-to-Neumann map, we also refer the reader to [3,7,8,9,50,51] and references therein.

Before we state the main results of this paper, we recall some basic concepts about boundary normal coordinates, pseudodifferential operators and symbols.

1.3. Boundary Normal Coordinates

We briefly introduce the construction of geodesic coordinates with respect to the boundary $\partial M$ (see [21], [52], p. 532).

For each boundary point $x^{\prime }\in \partial M$, let ${\gamma }_{x^{\prime }}:[0,\epsilon )\to \overline{M}$ denote the unit-speed geodesic starting at $x^{\prime }$ and normal to $\partial M$. If $x^{\prime }:=\{x_1,\dots ,x_{n-1}\}$ are any local coordinates for $\partial M$ near $x_0\in \partial M$, we can extend them smoothly to functions on a neighborhood of $x_0$ in $\overline{M}$ by letting them be constant along each normal geodesic ${\gamma }_{x^{\prime }}$. If we then define $x_n$ to be the parameter along each ${\gamma }_{x^{\prime }}$, it follows easily that $\{x_1,\dots ,x_n\}$ form coordinates for $\overline{M}$ in some neighborhood of $x_0$, which we call the boundary normal coordinates determined by $\{x_1,\dots ,x_{n-1}\}$. In these coordinates $x_n>0$ in M, and the boundary $\partial M$ is locally characterized by $x_n=0$. A standard computation shows that the metric has the form $g=g_{\alpha \beta }d{x}_{\alpha }d{x}_{\beta }+d{x}_n^2$.

1.4. Pseudodifferential Operators and Symbols

We recall some concepts of pseudodifferential operators and their symbols (cf. [52], Chapter 7).

Assuming $U\subset {\mathbb{R}}^n$ and $m\in \mathbb{R}$, we define $S_{1,0}^m=S_{1,0}^m(U,{\mathbb{R}}^n)$ to consist of $C^{\infty }$-functions $p(x,\xi )$ satisfying for every compact set $V\subset U$,

$$\begin{array}{c}\hfill |D_x^{\beta }{D}_{\xi }^{\alpha }p(x,\xi )|⩽C_{V,\alpha ,\beta }{\langle \xi \rangle }^{m-\left|\alpha \right|},x\in V,\xi \in {\mathbb{R}}^n\end{array}$$

for all $\alpha ,\beta \in {\mathbb{N}}^n$, where $D^{\alpha }=D^{{\alpha }_1}\cdots D^{{\alpha }_n}$, $D_j=-i\frac{\partial }{\partial x_j}$ and $\langle \xi \rangle ={(1+|\xi |}^2{)}^{1/2}$. The elements of $S_{1,0}^m$ are called symbols of order m. It is clear that $S_{1,0}^m$ is a Fréchet space with semi-norms given by

$$\begin{array}{c}\hfill {\parallel p\parallel }_{V,\alpha ,\beta }:=\underset{x\in V}{sup}\left|\left(D_x^{\beta }{D}_{\xi }^{\alpha }p(x,\xi )\right){(1+|\xi \left|\right)}^{-m+\left|\alpha \right|}\right|.\end{array}$$

Let $p(x,\xi )\in S_{1,0}^m$ and $\widehat{u}\left(\xi \right)={\int }_{{\mathbb{R}}^n}{e}^{-iy·\xi }u\left(y\right)dy$ be the Fourier transform of u. A pseudodifferential operator in an open set U is essentially defined by a Fourier integral operator

$$\begin{array}{c}\hfill P(x,D)u\left(x\right)=\frac{1}{{\left(2\pi \right)}^n}{\int }_{{\mathbb{R}}^n}p(x,\xi )e^{ix·\xi }\widehat{u}\left(\xi \right)d\xi \end{array}$$

for $u\in C_0^{\infty }\left(U\right)$. In such a case, we say the associated operator $P(x,D)$ belongs to $OP{S}^m$. We denote $OP{S}^{-\infty }={\bigcap }_{m}OP{S}^m$. If there are smooth $p_{m-j}(x,\xi )$, homogeneous in $\xi$ of degree $m-j$ for $\left|\xi \right|⩾1$, that is, $p_{m-j}(x,r\xi )=r^{m-j}{p}_{m-j}(x,\xi )$ for $r>0$, and if

$$\begin{array}{c}\hfill p(x,\xi )\sim \sum _{j⩾0}{p}_{m-j}(x,\xi )\end{array}$$

(7)

in the sense that

$$\begin{array}{c}\hfill p(x,\xi )-\sum _{j=0}^N{p}_{m-j}(x,\xi )\in S_{1,0}^{m-N-1}\end{array}$$

for all N, then, we say $p(x,\xi )\in S_{cl}^m$, or just $p(x,\xi )\in S^m$. We denote $S^{-\infty }={\bigcap }_m{S}^m$. We call $p_m(x,\xi )$ the principal symbol of $P(x,D)$. We say $P(x,D)\in OP{S}^m$ is elliptic of order m if on each compact $V\subset U$ there are constants $C_V$ and $r<\infty$ such that

$$\begin{array}{c}\hfill {|p(x,\xi )}^{-1}|⩽C_V{\langle \xi \rangle }^{-m},\left|\xi \right|⩾r.\end{array}$$

We can now define a pseudodifferential operator on a manifold M. In particular,

$$\begin{array}{c}\hfill P:C_0^{\infty }\left(M\right)\to C^{\infty }\left(M\right)\end{array}$$

belongs to $OP{S}_{1,0}^m\left(M\right)$ if the kernel of P is smooth off the diagonal in $M\times M$ and if for any coordinate neighborhood $U\subset M$ with $\mathsf{\Phi }:U\to \mathcal{O}$ a diffeomorphism onto an open subset $\mathcal{O}\subset {\mathbb{R}}^n$, the map $\tilde{P}:C_0^{\infty }\left(\mathcal{O}\right)\to C^{\infty }\left(\mathcal{O}\right)$ given by

$$\begin{array}{c}\hfill \tilde{P}u:=P(u\circ \mathsf{\Phi })\circ {\mathsf{\Phi }}^{-1}\end{array}$$

belongs to $OP{S}_{1,0}^m\left(\mathcal{O}\right)$. We refer the reader to [53,54,55] for more details.

1.5. The Main Results of This Paper

For the sake of simplicity, we denote by $i=\sqrt{-1}$, ${\xi }^{\prime }=({\xi }_1,\cdots ,{\xi }_{n-1})$, ${\xi }^{\alpha }=g^{\alpha \beta }{\xi }_{\beta }$, $|{\xi }^{\prime }|=\sqrt{{\xi }^{\alpha }{\xi }_{\alpha }}$, $I_n$ the $n\times n$ identity matrix,

$$\left[a_{\beta }^{\alpha }\right]:=\left[\begin{array}{ccc}{a}_1^1& \cdots & a_{n-1}^1\\ ⋮& \ddots & ⋮\\ a_1^{n-1}& \cdots & a_{n-1}^{n-1}\end{array}\right],\left[a_k^j\right]:=\left[\begin{array}{ccc}{a}_1^1& \cdots & a_n^1\\ ⋮& \ddots & ⋮\\ a_1^n& \cdots & a_n^n\end{array}\right],$$

and

$$\left[\begin{array}{cc}\left[a_k^j\right]& \left[b^j\right]\\ \left]c_k\right]& d\end{array}\right]:=\left[\begin{array}{ccc}\begin{array}{c}\hfill \left[a_{\beta }^{\alpha }\right]\end{array}& \begin{array}{c}\hfill \left[a_n^{\alpha }\right]\end{array}& \left[b^{\alpha }\right]\\ \left[a_{\beta }^n\right]& a_n^n& b^n\\ [c_{\beta }]& c_n& d\end{array}\right]=\left[\begin{array}{cccc}\begin{array}{c}\hfill a_1^1\end{array}& \cdots & a_n^1& b^1\\ ⋮& \ddots & ⋮& ⋮\\ a_1^n& \cdots & a_n^n& b^n\\ c_1& \cdots & c_n& d\end{array}\right],$$

where $1⩽\alpha ,\beta ⩽n-1$, and $1⩽j,k⩽n$.

The main results of this paper are the following three theorems.

Theorem 1. 

Let $(M,g)$ be a compact Riemannian manifold of dimension n with smooth boundary $\partial M$. Assume that the coefficient $\beta \in C^{\infty }\left(\overline{M}\right)$, the Lamé coefficients $\lambda ,\mu \in C^{\infty }\left(\overline{M}\right)$, and the heat conduction coefficient $\alpha \in C^{\infty }\left(\overline{M}\right)$ satisfy $\mu >0,\lambda +\mu ⩾0$, and $\alpha >0$. Let $\sigma \left({\mathsf{\Lambda }}_g\right)\sim {\sum }_{j⩽1}{p}_j(x,{\xi }^{\prime })$ be the full symbol of the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$. Then, in boundary normal coordinates,

$$\begin{array}{cc}\hfill p_1(x,{\xi }^{\prime })& =\left[\begin{array}{ccc}\mu |{\xi }^{\prime }|I_{n-1}+\frac{\mu (\lambda +\mu )}{(\lambda +3\mu )|{\xi }^{\prime }|}\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& -\frac{2i{\mu }^2}{\lambda +3\mu }\left[{\xi }^{\alpha }\right]& 0\\ \frac{2i{\mu }^2}{\lambda +3\mu }\left[{\xi }_{\beta }\right]& \frac{2\mu (\lambda +2\mu )}{\lambda +3\mu }\left|{\xi }^{\prime }\right|& 0\\ 0& 0& \alpha |{\xi }^{\prime }|\end{array}\right],\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill p_0(x,{\xi }^{\prime })& =\left[\begin{array}{ccc}\mu I_{n-1}& 0& 0\\ 0& \lambda +2\mu & 0\\ 0& 0& \alpha \end{array}\right]q_0(x,{\xi }^{\prime })-\left[\begin{array}{ccc}0& 0& 0\\ \lambda \left[{\mathsf{\Gamma }}_{\alpha \beta }^{\alpha }\right]& \lambda {\mathsf{\Gamma }}_{\alpha n}^{\alpha }& -\beta \\ 0& 0& 0\end{array}\right],\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill p_{-m}(x,{\xi }^{\prime })& =\left[\begin{array}{ccc}\mu I_{n-1}& 0& 0\\ 0& \lambda +2\mu & 0\\ 0& 0& \alpha \end{array}\right]q_{-m}(x,{\xi }^{\prime }),m⩾1,\hfill \end{array}$$

(10)

where $q_{-m}(x,{\xi }^{\prime })$, $m⩾0$, are the remain symbols of a pseudodifferential operator (see (42) in Section 2).

By studying the full symbol of the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$, we prove the following result:

Theorem 2. 

Let $(M,g)$ be a compact Riemannian manifold of dimension n with smooth boundary $\partial M$. Assume that the coefficient $\beta \in C^{\infty }\left(\overline{M}\right)$, the Lamé coefficients $\lambda ,\mu \in C^{\infty }\left(\overline{M}\right)$, and the heat conduction coefficient $\alpha \in C^{\infty }\left(\overline{M}\right)$ satisfy $\mu >0,\lambda +\mu ⩾0$, and $\alpha >0$. Then, the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$ uniquely determines $\frac{{\partial }^{\left|J\right|}\lambda }{\partial x^J}$, $\frac{{\partial }^{\left|J\right|}\mu }{\partial x^J}$, $\frac{{\partial }^{\left|J\right|}\alpha }{\partial x^J}$, and $\frac{{\partial }^{\left|J\right|}\beta }{\partial x^J}$ on the boundary for all multi-indices J.

The uniqueness result in Theorem 2 can be extended to the whole manifold for the real analytic setting.

Theorem 3. 

Let $(M,g)$ be a compact Riemannian manifold of dimension n with smooth boundary $\partial M$, and let $\mathsf{\Gamma }\subset \partial M$ be a nonempty open subset. Suppose that the manifold is real analytic up to Γ, and the coefficients $\lambda ,\mu ,\alpha ,\beta$ are also real analytic up to Γ and satisfy $\mu >0$, $\lambda +\mu ⩾0$, and $\alpha >0$. Then, the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$ uniquely determines λ, μ, α, and β on $\overline{M}$.

Theorem 3 shows that the global uniqueness of real analytic coefficients on a real analytic Riemannian manifold. To the best of our knowledge, this is the first global uniqueness result for variable coefficients in thermoelasticity on a Riemannian manifold. It is clear that Theorem 3 also holds for any real analytic bounded Euclidean domain with smooth boundary.

1.6. The Main Ideas of This Paper

The main ideas of this paper are as follows. First, Liu [2] established a method such that one can calculate the full symbol of the elastic Dirichlet-to-Neumann map with constant coefficients. In [4], the full symbol of the elastic Dirichlet-to-Neumann map with variable coefficients was obtained. The full symbol of the thermoelastic Dirichlet-to-Neumann map with constant coefficients was obtained in [3]. Combining the methods and the results in [2,3,4], we can deal with the case for variable coefficients in thermoelasticity.

In boundary normal coordinates, there is a factorization for the thermoelastic operator $T_g$ as follows:

$$\begin{array}{c}\hfill A^{-1}{T}_g=I_{n+1}\frac{{\partial }^2}{\partial x_n^2}+B\frac{\partial }{\partial x_n}+C=\left(I_{n+1}\frac{\partial }{\partial x_n}+B-Q\right)\left(I_{n+1}\frac{\partial }{\partial x_n}+Q\right),\end{array}$$

where B, C are two differential operators, and $Q=Q(x,{\partial }_{x^{\prime }})$ is a pseudodifferential operator. As a result, we obtain the equation

$$\begin{array}{c}\hfill Q^2-BQ-\left[\frac{\partial }{\partial x_n},Q\right]+C=0,\end{array}$$

where $[\frac{\partial }{\partial x_n},Q]$ is the commutator. The corresponding full symbol equation of the above equation is

$$\begin{array}{c}\hfill \sum _J\frac{{(-i)}^{\left|J\right|}}{J!}{\partial }_{{\xi }^{\prime }}^{J}q{\partial }_{x^{\prime }}^{J}q-\sum _J\frac{{(-i)}^{\left|J\right|}}{J!}{\partial }_{{\xi }^{\prime }}^{J}b{\partial }_{x^{\prime }}^{J}q-\frac{\partial q}{\partial x_n}+c=0,\end{array}$$

(11)

which is an $(n+1)\times (n+1)$ matrix equation, where the sum is over all multi-indices J, ${\xi }^{\prime }=({\xi }_1,\cdots ,{\xi }_{n-1})$, and $x^{\prime }=(x_1,\cdots ,x_{n-1})$. Here b, c, and q are the full symbols of the operators B, C, and Q, respectively.

Note that the computations of the full symbols of matrix-valued pseudodifferential operators (i.e., solving the above full symbol Equation (11)) are quite difficult tasks. There are two major difficulties:

(i)

How to solve the unknown matrix $q_1$ from the following matrix equation?

$$\begin{array}{c}\hfill q_1^2-b_1{q}_1+c_2=0,\end{array}$$

(12)

where $q_1$, $b_1$, and $c_2$ are the principal symbols of Q, B, and C, respectively.

(ii)

How to solve the unknown matrix $q_{-m-1}$, $m⩾-1$, from the following Sylvester equation?

$$\begin{array}{c}\hfill (q_1-b_1)q_{-m-1}+q_{-m-1}{q}_1=E_{-m},m⩾-1,\end{array}$$

(13)

where $q_{-m-1}$, $m⩾-1$, are the remain symbols, and $E_{-m}$, $m⩾-1$, are given by (38)–(40).

For the first part of the problem, generally, the quadratic matrix equation of the form

$$\begin{array}{c}\hfill X^2+U_{1}X+V_1=0\end{array}$$

(14)

can not be solved exactly, where X is an unknown matrix, $U_1$ and $V_1$ are given matrices. In other words, there is not a general formula of the solution represented by the coefficients of matrix equation (14). Fortunately, in our setting, $b_1$ and $c_2$ can be represented as special block matrices. This implies that the $q_1$ can also be represented as a block matrix, which is a linear combination of $I_{n+1}$ and a special matrix $F_1$ with the property $F_1^2=0$ (see (56) in Section 2). Then, by solving a system of the coefficients, we can obtain the explicit expression for $q_1$ (see (41)).

For the second part of the problem, the matrix equation of the form

$$\begin{array}{c}\hfill U_{2}X+X{V}_2=Y\end{array}$$

(15)

is called the Sylvester equation (see [56], Chapter 9), where $X$ is an unknown matrix, $U_2$, $V_2$, and $Y$ are given matrices. Let

$$\begin{array}{c}\hfill \mathcal{A}=\left[\begin{array}{cccc}{a}_1^1& a_2^1& \cdots & a_n^1\\ a_1^2& a_2^2& \cdots & a_n^2\\ ⋮& ⋮& \ddots & ⋮\\ a_1^n& a_2^n& \cdots & a_n^n\end{array}\right].\end{array}$$

The vectorization $vec\mathcal{A}$ of the matrix $\mathcal{A}$ is a column vector, which is defined by (see [56], Chapter 9)

$$\begin{array}{c}\hfill vec\mathcal{A}:={(a_1^1,a_1^2,\cdots ,a_1^n,a_2^1,a_2^2,\cdots ,a_2^n,\cdots ,a_n^1,a_n^2,\cdots ,a_n^n)}^t.\end{array}$$

(16)

The Kronecker product $\mathcal{A}\otimes \mathcal{B}$ of two matrices $\mathcal{A}$ and $\mathcal{B}$ is defined by (see [56], Chapter 9)

$$\begin{array}{c}\hfill \mathcal{A}\otimes \mathcal{B}:=\left[\begin{array}{cccc}{a}_1^1\mathcal{B}& a_2^1\mathcal{B}& \cdots & a_n^1\mathcal{B}\\ a_1^2\mathcal{B}& a_2^2\mathcal{B}& \cdots & a_n^2\mathcal{B}\\ ⋮& ⋮& \ddots & ⋮\\ a_1^n\mathcal{B}& a_2^n\mathcal{B}& \cdots & a_n^n\mathcal{B}\end{array}\right].\end{array}$$

(17)

There are some properties of Kronecker product and vectorization as follows (see [56], Chapter 9):

$$\begin{array}{cc}\hfill (\mathcal{A}+\mathcal{B})\otimes \mathcal{C}& =\mathcal{A}\otimes \mathcal{C}+\mathcal{B}\otimes \mathcal{C},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \mathcal{C}\otimes (\mathcal{A}+\mathcal{B})& =\mathcal{C}\otimes \mathcal{A}+\mathcal{C}\otimes \mathcal{B},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {(\mathcal{A}\otimes \mathcal{B})}^{-1}& ={\mathcal{A}}^{-1}\otimes {\mathcal{B}}^{-1},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill (\mathcal{A}\otimes \mathcal{B})(\mathcal{C}\otimes \mathcal{D})& =\mathcal{A}\mathcal{C}\otimes \mathcal{B}\mathcal{D},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill vec\left(\mathcal{A}\mathcal{B}\right)& =(I_n\otimes \mathcal{A})vec\mathcal{B},\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill vec\left(\mathcal{B}\mathcal{C}\right)& =({\mathcal{C}}^t\otimes I_n)vec\mathcal{B},\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill vec\left(\mathcal{A}\mathcal{B}\mathcal{C}\right)& =({\mathcal{C}}^t\otimes \mathcal{A})vec\mathcal{B}.\hfill \end{array}$$

(24)

It follows from (15), (22), and (23) that

$$\begin{array}{cc}\hfill vec\mathcal{Y}& =vec(U_{2}X+X{V}_2)\hfill \\ \hfill & =\left((I_n\otimes U_2)+(V_2^t\otimes I_n)\right)vec\left(X\right)\hfill \\ \hfill & :=GvecX.\hfill \end{array}$$

(25)

Therefore, (15) has a unique solution if and only if G is invertible and

$$\begin{array}{c}\hfill vecX=G^{-1}vec\mathcal{Y}.\end{array}$$

Thus, we can obtain X from $vecX$. Finally, we obtain the symbols $q_j$, $j⩽1$, of the pseudodifferential operator Q. Finally, using this method, we solve (13) and obtain the symbols $q_{-m-1}$ for $m⩾-1$ This implies that we obtain $Q(x,{\partial }_{x^{\prime }})$ (modulo a smoothing operator) on the boundary.

Next, we flatten the boundary and induce a Riemannian metric in a neighborhood of the boundary, and give a local representation for the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$ with variable coefficients in boundary normal coordinates, that is,

$$\begin{array}{c}\hfill {\mathsf{\Lambda }}_g=A\left(-\frac{\partial }{\partial x_n}\right)-K \mathrm{on} \partial M,\end{array}$$

where A and K are two matrices. Note that, in boundary normal coordinates, the operator $\frac{\partial }{\partial x_n}{|}_{\partial M}$ can be represented as a pseudodifferential operator (modulo a smoothing operator) of order one in $x^{\prime }=(x_1,\dots ,x_{n-1})$ depending smoothly on $x_n$. Therefore, we have

$$\begin{array}{c}\hfill {\mathsf{\Lambda }}_g=(AQ-K){|}_{\partial M}\end{array}$$

modulo a smoothing operator (see (72) in Section 2).

Finally, we obtain the full symbol of the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$, which contain the information about the coefficients $\lambda ,\mu ,\alpha ,\beta$, and their derivatives on the boundary. Thus, we can prove that they can be uniquely determined by the thermoelastic Dirichlet-to-Neumann map. Furthermore, we prove that the coefficients can be uniquely determined on the whole manifold $\overline{M}$ by the thermoelastic Dirichlet-to-Neumann map provided the manifold and these coefficients are real analytic.

This paper is organized as follows. In Section 2, we derive a factorization of thermoelastic operator $T_g$ with variable coefficients, and compute the full symbol of the pseudodifferential operator Q, we then give the explicit expression of the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$ in boundary normal coordinates. In Section 3, we prove Theorem 1 and Theorem 2 for boundary determination. Finally, Section 4 is devoted to proving Theorem 3 for global uniqueness.

2. Symbols of the Pseudodifferential Operators

Let $(M,g)$ be a compact Riemannian manifold of dimension n with smooth boundary $\partial M$. In the local coordinates ${\left\{x_j\right\}}_{j=1}^n$, we denote by ${\left\{\frac{\partial }{\partial x_j}\right\}}_{j=1}^n$ and ${\left\{d{x}_j\right\}}_{j=1}^n$, respectively, the natural basis for the tangent space $T_{x}M$ and the cotangent space $T_x^{\ast }M$ at the point $x\in M$. We will use the Einstein summation convention. The Greek indices run from 1 to $n-1$, whereas the Roman indices run from 1 to n, unless otherwise specified. Then, the Riemannian metric g is given by $g=g_{jk}d{x}_j\otimes d{x}_k$.

Let ${\nabla }_j={\nabla }_{\frac{\partial }{\partial x_j}}$ be the covariant derivative with respect to $\frac{\partial }{\partial x_j}$ and ${\nabla }^j=g^{jk}{\nabla }_k$. Then, for displacement vector field $\mathit{u}$, we denote by div the divergence operator, i.e.,

$$\begin{array}{c}\hfill div\mathit{u}={\nabla }_j{u}^j=\frac{\partial u^j}{\partial x_j}+{\mathsf{\Gamma }}_{jk}^j{u}^k,\mathit{u}=u^j\frac{\partial }{\partial x_j}\in \mathfrak{X}\left(M\right).\end{array}$$

(26)

Here the Christoffel symbols

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_{jk}^m=\frac{1}{2}{g}^{ml}\left(\frac{\partial g_{jl}}{\partial x_k}+\frac{\partial g_{kl}}{\partial x_j}-\frac{\partial g_{jk}}{\partial x_l}\right),\end{array}$$

and $\left[g^{jk}\right]={\left[g_{jk}\right]}^{-1}$. For smooth function $f\in C^{\infty }\left(M\right)$, the gradient operator is given by

$$\begin{array}{c}\hfill gradf=\left({\nabla }^{j}f\right)\frac{\partial }{\partial x_j}=g^{jk}\frac{\partial f}{\partial x_j}\frac{\partial }{\partial x_k},f\in C^{\infty }\left(M\right).\end{array}$$

(27)

The Laplace–Beltrami operator is given by

$${\Delta }_{g}f=g^{jk}\left(\frac{{\partial }^{2}f}{\partial x_j\partial x_k}-{\mathsf{\Gamma }}_{jk}^l\frac{\partial f}{\partial x_l}\right),f\in C^{\infty }\left(M\right).$$

(28)

The Lamé operator (3) with variable coefficients can be rewritten as (see [4])

$$\begin{array}{cc}\hfill {\left(L_g\mathit{u}\right)}^j& =\mu {\Delta }_g{u}^j+(\lambda +\mu ){\nabla }^j{\nabla }_k{u}^k+\left({\nabla }^j\lambda \right){\nabla }_k{u}^k+\left({\nabla }^k\mu \right)({\nabla }_k{u}^j+{\nabla }^j{u}_k)\hfill \\ \hfill & +\mu g^{kl}\left(2{\mathsf{\Gamma }}_{km}^j\frac{\partial u^m}{\partial x_l}+\frac{\partial {\mathsf{\Gamma }}_{kl}^j}{\partial x_m}{u}^m\right),j=1,2,\cdots ,n.\hfill \end{array}$$

(29)

In boundary normal coordinates, we write the Laplace–Beltrami operator as

$$\begin{array}{cc}\hfill {\Delta }_g& =\frac{{\partial }^2}{\partial x_n^2}+{\mathsf{\Gamma }}_{\alpha n}^{\alpha }\frac{\partial }{\partial x_n}+g^{\alpha \beta }\frac{{\partial }^2}{\partial x_{\alpha }\partial x_{\beta }}+\left(g^{\alpha \beta }{\mathsf{\Gamma }}_{\gamma \alpha }^{\gamma }+\frac{\partial g^{\alpha \beta }}{\partial x_{\alpha }}\right)\frac{\partial }{\partial x_{\beta }}.\hfill \end{array}$$

(30)

Combining this and (2), (3), (26)–(29), we deduce that (cf. [3,4])

$$\begin{array}{c}\hfill A^{-1}{T}_g=I_{n+1}\frac{{\partial }^2}{\partial x_n^2}+B\frac{\partial }{\partial x_n}+C,\end{array}$$

(31)

where

$$\begin{array}{cc}\hfill A& =\left[\begin{array}{ccc}\mu I_{n-1}& 0& 0\\ 0& \lambda +2\mu & 0\\ 0& 0& \alpha \end{array}\right],\hfill \end{array}$$

(32)

$B=B_1+B_0$, $C=C_2+C_1+C_0$, and

$$\begin{array}{cc}\hfill B_1& =(\lambda +\mu )\left[\begin{array}{ccc}0& \frac{1}{\mu }\left[g^{\alpha \beta }\frac{\partial }{\partial x_{\beta }}\right]& 0\\ \frac{1}{\lambda +2\mu }\left[\frac{\partial }{\partial x_{\beta }}\right]& 0& 0\\ 0& 0& 0\end{array}\right],\hfill \\ \hfill B_0& =\left[\begin{array}{ccc}{\mathsf{\Gamma }}_{\alpha n}^{\alpha }{I}_{n-1}+2\left[{\mathsf{\Gamma }}_{n\beta }^{\alpha }\right]& 0& 0\\ \frac{\lambda +\mu }{\lambda +2\mu }\left[{\mathsf{\Gamma }}_{\alpha \beta }^{\alpha }\right]& {\mathsf{\Gamma }}_{n\alpha }^{\alpha }& -\frac{\beta }{\lambda +2\mu }\\ 0& \frac{i\omega \beta {\theta }_0}{\alpha }& {\mathsf{\Gamma }}_{n\alpha }^{\alpha }\end{array}\right]+\left[\begin{array}{ccc}\frac{1}{\mu }\frac{\partial \mu }{\partial x_n}{I}_{n-1}& \frac{1}{\mu }\left[{\nabla }^{\alpha }\lambda \right]& 0\\ \frac{1}{\lambda +2\mu }\left[\frac{\partial \mu }{\partial x_{\beta }}\right]& \frac{1}{\lambda +2\mu }\frac{\partial (\lambda +2\mu )}{\partial x_n}& 0\\ 0& 0& 0\end{array}\right],\hfill \\ \hfill C_2& =\left[\begin{array}{ccc}\left(g^{\alpha \beta }\frac{{\partial }^2}{\partial x_{\alpha }\partial x_{\beta }}\right)I_{n-1}+\frac{\lambda +\mu }{\mu }\left[g^{\alpha \gamma }\frac{{\partial }^2}{\partial x_{\gamma }\partial x_{\beta }}\right]& 0& 0\\ 0& \frac{\mu }{\lambda +2\mu }{g}^{\alpha \beta }\frac{{\partial }^2}{\partial x_{\alpha }\partial x_{\beta }}& 0\\ 0& 0& g^{\alpha \beta }\frac{{\partial }^2}{\partial x_{\alpha }\partial x_{\beta }}\end{array}\right],\hfill \\ \hfill C_1& =\left[\begin{array}{ccc}\left((g^{\alpha \beta }{\mathsf{\Gamma }}_{\alpha \gamma }^{\gamma }+\frac{\partial g^{\alpha \beta }}{\partial x_{\alpha }})\frac{\partial }{\partial x_{\beta }}\right)I_{n-1}& 0& 0\\ 0& \frac{\mu }{\lambda +2\mu }\left(g^{\alpha \beta }{\mathsf{\Gamma }}_{\alpha \gamma }^{\gamma }+\frac{\partial g^{\alpha \beta }}{\partial x_{\alpha }}\right)\frac{\partial }{\partial x_{\beta }}& 0\\ 0& 0& \left(g^{\alpha \beta }{\mathsf{\Gamma }}_{\alpha \gamma }^{\gamma }+\frac{\partial g^{\alpha \beta }}{\partial x_{\alpha }}\right)\frac{\partial }{\partial x_{\beta }}\end{array}\right]\hfill \\ \hfill & +\frac{\lambda +\mu }{\mu }\left[\begin{array}{ccc}\left[g^{\alpha \gamma }{\mathsf{\Gamma }}_{\rho \beta }^{\rho }\frac{\partial }{\partial x_{\gamma }}\right]& \left[g^{\alpha \gamma }{\mathsf{\Gamma }}_{\rho n}^{\rho }\frac{\partial }{\partial x_{\gamma }}\right]& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{ccc}2\left[g^{\gamma \rho }{\mathsf{\Gamma }}_{\rho \beta }^{\alpha }\frac{\partial }{\partial x_{\gamma }}\right]& 2\left[g^{\gamma \rho }{\mathsf{\Gamma }}_{\rho n}^{\alpha }\frac{\partial }{\partial x_{\gamma }}\right]& -\frac{\beta }{\mu }\left[g^{\alpha \beta }\frac{\partial }{\partial x_{\beta }}\right]\\ \frac{2\mu }{\lambda +2\mu }\left[g^{\gamma \rho }{\mathsf{\Gamma }}_{\rho \beta }^n\frac{\partial }{\partial x_{\gamma }}\right]& 0& 0\\ \frac{i\omega \beta {\theta }_0}{\alpha }\left[\frac{\partial }{\partial x_{\beta }}\right]& 0& 0\end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{ccc}\frac{1}{\mu }\left({\nabla }^{\alpha }\mu \frac{\partial }{\partial x_{\alpha }}\right)I_{n-1}+\frac{1}{\mu }\left[{\nabla }^{\alpha }\lambda \frac{\partial }{\partial x_{\beta }}+g^{\alpha \gamma }\frac{\partial \mu }{\partial x_{\beta }}\frac{\partial }{\partial x_{\gamma }}\right]& \frac{1}{\mu }\frac{\partial \mu }{\partial x_n}\left[g^{\alpha \beta }\frac{\partial }{\partial x_{\beta }}\right]& 0\\ \frac{1}{\lambda +2\mu }\frac{\partial \lambda }{\partial x_n}\left[\frac{\partial }{\partial x_{\beta }}\right]& \frac{1}{\lambda +2\mu }{\nabla }^{\alpha }\mu \frac{\partial }{\partial x_{\alpha }}& 0\\ 0& 0& 0\end{array}\right],\hfill \\ \hfill C_0& =(\lambda +\mu )\left[\begin{array}{ccc}\frac{1}{\mu }\left[g^{\alpha \gamma }\frac{\partial {\mathsf{\Gamma }}_{\rho \beta }^{\rho }}{\partial x_{\gamma }}\right]& \frac{1}{\mu }\left[g^{\alpha \gamma }\frac{\partial {\mathsf{\Gamma }}_{\rho n}^{\rho }}{\partial x_{\gamma }}\right]& 0\\ \frac{1}{\lambda +2\mu }\left[\frac{\partial {\mathsf{\Gamma }}_{\alpha \beta }^{\alpha }}{\partial x_n}\right]& \frac{1}{\lambda +2\mu }\frac{\partial {\mathsf{\Gamma }}_{\alpha n}^{\alpha }}{\partial x_n}& 0\\ 0& 0& 0\end{array}\right]+\left[\begin{array}{ccc}\left[g^{ml}\frac{\partial {\mathsf{\Gamma }}_{ml}^{\alpha }}{\partial x_{\beta }}\right]& \left[g^{ml}\frac{\partial {\mathsf{\Gamma }}_{ml}^{\alpha }}{\partial x_n}\right]& 0\\ \frac{\mu }{\lambda +2\mu }\left[g^{ml}\frac{\partial {\mathsf{\Gamma }}_{ml}^n}{\partial x_{\beta }}\right]& \frac{\mu }{\lambda +2\mu }{g}^{ml}\frac{\partial {\mathsf{\Gamma }}_{ml}^n}{\partial x_n}& 0\\ 0& 0& 0\end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{ccc}\frac{\rho {\omega }^2}{\mu }{I}_{n-1}& 0& 0\\ 0& \frac{\rho {\omega }^2}{\lambda +2\mu }& 0\\ \frac{i\omega \beta {\theta }_0}{\alpha }\left[{\mathsf{\Gamma }}_{\alpha \beta }^{\alpha }\right]& \frac{i\omega \beta {\theta }_0}{\alpha }{\mathsf{\Gamma }}_{\alpha n}^{\alpha }& \frac{i\omega \gamma }{\alpha }\end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{ccc}\frac{1}{\mu }\left[\left({\nabla }^{\alpha }\lambda \right){\mathsf{\Gamma }}_{\beta \gamma }^{\gamma }-\frac{\partial \mu }{\partial x_{\gamma }}\frac{\partial g^{\alpha \gamma }}{\partial x_{\beta }}\right]& \frac{1}{\mu }\left[\left({\nabla }^{\alpha }\lambda \right){\mathsf{\Gamma }}_{\beta n}^{\beta }-\frac{\partial \mu }{\partial x_{\beta }}\frac{\partial g^{\alpha \beta }}{\partial x_n}\right]& 0\\ \frac{1}{\lambda +2\mu }\frac{\partial \lambda }{\partial x_n}\left[{\mathsf{\Gamma }}_{\alpha \beta }^{\alpha }\right]& \frac{1}{\lambda +2\mu }\frac{\partial \lambda }{\partial x_n}{\mathsf{\Gamma }}_{\alpha n}^{\alpha }& 0\\ 0& 0& 0\end{array}\right].\hfill \end{array}$$

Let

$$\begin{array}{c}\hfill b(x,{\xi }^{\prime })=b_1(x,{\xi }^{\prime })+b_0(x,{\xi }^{\prime })\end{array}$$

and

$$\begin{array}{c}\hfill c(x,{\xi }^{\prime })=c_2(x,{\xi }^{\prime })+c_1(x,{\xi }^{\prime })+c_0(x,{\xi }^{\prime })\end{array}$$

be the full symbols of B and C, respectively. We denote

$$\begin{array}{c}\hfill {\xi }^{\alpha }=g^{\alpha \beta }{\xi }_{\beta },\left|{\xi }^{\prime }\right|=\sqrt{{\xi }^{\alpha }{\xi }_{\alpha }}.\end{array}$$

Thus, we have

$$\begin{array}{c}{b}_1(x,{\xi }^{\prime })=i(\lambda +\mu )\left[\begin{array}{ccc}0& \frac{1}{\mu }\left[{\xi }^{\alpha }\right]& 0\\ \frac{1}{\lambda +2\mu }\left[{\xi }_{\beta }\right]& 0& 0\\ 0& 0& 0\end{array}\right],\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & b_0(x,{\xi }^{\prime })=B_0,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & c_2(x,{\xi }^{\prime })=-\left[\begin{array}{ccc}|{\xi }^{\prime }{|}^2{I}_{n-1}+\frac{\lambda +\mu }{\mu }\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& 0& 0\\ 0& \frac{\mu }{\lambda +2\mu }{\left|{\xi }^{\prime }\right|}^2& 0\\ 0& 0& |{\xi }^{\prime }{|}^2\end{array}\right],\hfill \\ \hfill & c_1(x,{\xi }^{\prime })=i\left[\begin{array}{ccc}\left({\xi }^{\alpha }{\mathsf{\Gamma }}_{\alpha \beta }^{\beta }+\frac{\partial {\xi }^{\alpha }}{\partial x_{\alpha }}\right)I_{n-1}& 0& 0\\ 0& \frac{\mu }{\lambda +2\mu }\left({\xi }^{\alpha }{\mathsf{\Gamma }}_{\alpha \beta }^{\beta }+\frac{\partial {\xi }^{\alpha }}{\partial x_{\alpha }}\right)& 0\\ 0& 0& {\xi }^{\alpha }{\mathsf{\Gamma }}_{\alpha \beta }^{\beta }+\frac{\partial {\xi }^{\alpha }}{\partial x_{\alpha }}\end{array}\right]\hfill \\ \hfill & +\frac{i(\lambda +\mu )}{\mu }\left[\begin{array}{ccc}\left[{\xi }^{\alpha }{\mathsf{\Gamma }}_{\gamma \beta }^{\gamma }\right]& {\mathsf{\Gamma }}_{\beta n}^{\beta }\left[{\xi }^{\alpha }\right]& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]+\left[\begin{array}{ccc}2i\left[{\xi }^{\gamma }{\mathsf{\Gamma }}_{\gamma \beta }^{\alpha }\right]& 2i\left[{\xi }^{\gamma }{\mathsf{\Gamma }}_{\gamma n}^{\alpha }\right]& -\frac{i\beta }{\mu }\left[{\xi }^{\alpha }\right]\\ \frac{2i\mu }{\lambda +2\mu }\left[{\xi }^{\gamma }{\mathsf{\Gamma }}_{\gamma \beta }^n\right]& 0& 0\\ -\frac{\omega \beta {\theta }_0}{\alpha }\left[{\xi }_{\beta }\right]& 0& 0\end{array}\right]\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & +i\left[\begin{array}{ccc}\frac{1}{\mu }\left({\xi }_{\alpha }{\nabla }^{\alpha }\mu \right)I_{n-1}+\frac{1}{\mu }\left[{\xi }_{\beta }{\nabla }^{\alpha }\lambda +{\xi }^{\alpha }\frac{\partial \mu }{\partial x_{\beta }}\right]& \frac{1}{\mu }\frac{\partial \mu }{\partial x_n}\left[{\xi }^{\alpha }\right]& 0\\ \frac{1}{\lambda +2\mu }\frac{\partial \lambda }{\partial x_n}\left[{\xi }_{\beta }\right]& \frac{1}{\lambda +2\mu }{\xi }_{\alpha }{\nabla }^{\alpha }\mu & 0\\ 0& 0& 0\end{array}\right],\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & c_0(x,{\xi }^{\prime })=C_0.\hfill \end{array}$$

(37)

For the convenience of stating the following proposition, we define

$$\begin{array}{cc}\hfill E_1& :=i\sum _{\alpha }\frac{\partial (q_1-b_1)}{\partial {\xi }_{\alpha }}\frac{\partial q_1}{\partial x_{\alpha }}+b_0{q}_1+\frac{\partial q_1}{\partial x_n}-c_1,\hfill \\ \hfill E_0& :=i\sum _{\alpha }\left(\frac{\partial (q_1-b_1)}{\partial {\xi }_{\alpha }}\frac{\partial q_0}{\partial x_{\alpha }}+\frac{\partial q_0}{\partial {\xi }_{\alpha }}\frac{\partial q_1}{\partial x_{\alpha }}\right)+\frac{1}{2}\sum _{\alpha ,\beta }\frac{{\partial }^2{q}_1}{\partial {\xi }_{\alpha }\partial {\xi }_{\beta }}\frac{{\partial }^2{q}_1}{\partial x_{\alpha }\partial x_{\beta }}\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & -q_0^2+b_0{q}_0+\frac{\partial q_0}{\partial x_n}-c_0,\hfill \end{array}$$

(39)

and

$$\begin{array}{c}\hfill E_{-m}:=b_0{q}_{-m}+\frac{\partial q_{-m}}{\partial x_n}-i\sum _{\alpha }\frac{\partial b_1}{\partial {\xi }_{\alpha }}\frac{\partial q_{-m}}{\partial x_{\alpha }}-\sum _{\begin{array}{c}-m⩽j,k⩽1\\ \left|J\right|=j+k+m\end{array}}\frac{{(-i)}^{\left|J\right|}}{J!}{\partial }_{{\xi }^{\prime }}^J{q}_j{\partial }_{x^{\prime }}^J{q}_k\end{array}$$

(40)

for $m⩾1$.

We then derive the microlocal factorization of the thermoelastic operator $T_g$.

Proposition 4. 

There exists a pseudodifferential operator $Q(x,{\partial }_{x^{\prime }})$ of order one in $x^{\prime }$ depending smoothly on $x_n$ such that

$$\begin{array}{c}\hfill A^{-1}{T}_g=\left(I_{n+1}\frac{\partial }{\partial x_n}+B-Q\right)\left(I_{n+1}\frac{\partial }{\partial x_n}+Q\right)\end{array}$$

modulo a smoothing operator. Moreover, let $q(x,{\xi }^{\prime })\sim {\sum }_{j⩽1}{q}_j(x,{\xi }^{\prime })$ be the full symbol of $Q(x,{\partial }_{x^{\prime }})$. Then, in boundary normal coordinates,

$$\begin{array}{cc}\hfill q_1(x,{\xi }^{\prime })& =|{\xi }^{\prime }|I_{n+1}+\frac{\lambda +\mu }{\lambda +3\mu }{F}_1,\hfill \\ \hfill q_{-m-1}(x,{\xi }^{\prime })& =\frac{1}{2|{\xi }^{\prime }|}{E}_{-m}-\frac{\lambda +\mu }{4(\lambda +3\mu )|{\xi }^{\prime }{|}^2}(F_2{E}_{-m}+E_{-m}{F}_1)\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill & +\frac{{(\lambda +\mu )}^2}{4{(\lambda +3\mu )}^2{\left|{\xi }^{\prime }\right|}^3}{F}_2{E}_{-m}{F}_1,m⩾-1,\hfill \end{array}$$

(42)

where

$$\begin{array}{cc}\hfill F_1& =\left[\begin{array}{ccc}\frac{1}{|{\xi }^{\prime }|}\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& i\left[{\xi }^{\alpha }\right]& 0\\ i\left[{\xi }_{\beta }\right]& -|{\xi }^{\prime }|& 0\\ 0& 0& 0\end{array}\right],\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill F_2& =\left[\begin{array}{ccc}\frac{1}{|{\xi }^{\prime }|}\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& -\frac{i(\lambda +2\mu )}{\mu }\left[{\xi }^{\alpha }\right]& 0\\ -\frac{i\mu }{\lambda +2\mu }\left[{\xi }_{\beta }\right]& -|{\xi }^{\prime }|& 0\\ 0& 0& 0\end{array}\right].\hfill \end{array}$$

(44)

Proof. 

It follows from (31) that

$$\begin{array}{c}\hfill I_{n+1}\frac{{\partial }^2}{\partial x_n^2}+B\frac{\partial }{\partial x_n}+C=\left(I_{n+1}\frac{\partial }{\partial x_n}+B-Q\right)\left(I_{n+1}\frac{\partial }{\partial x_n}+Q\right).\end{array}$$

Equivalently,

$$\begin{array}{c}\hfill Q^2-BQ-\left[I_{n+1}\frac{\partial }{\partial x_n},Q\right]+C=0,\end{array}$$

(45)

where the commutator $\left[I_{n+1}\frac{\partial }{\partial x_n},Q\right]$ is defined by, for any $v\in C^{\infty }\left(M\right)$,

$$\begin{array}{cc}\hfill \left[I_{n+1}\frac{\partial }{\partial x_n},Q\right]v& :=I_{n+1}\frac{\partial }{\partial x_n}\left(Qv\right)-Q\left(I_{n+1}\frac{\partial }{\partial x_n}\right)v\hfill \\ \hfill & =\frac{\partial Q}{\partial x_n}v.\hfill \end{array}$$

Recall that if G₁ and G₂ are two pseudodifferential operators with full symbols $g_1=g_1(x,\xi )$ and $g_1=g_2(x,\xi )$, respectively, then the full symbol $\sigma \left(G_1{G}_2\right)$ of the operator $G_1{G}_2$ is given by (see [52], p. 11, [54], p. 71, and also [53,57])

$$\begin{array}{c}\hfill \sigma G_1{G}_2\sim \sum _J\frac{{(-i)}^{\left|J\right|}}{J!}{\partial }_{\xi }^J{g}_1{\partial }_x^J{g}_2,\end{array}$$

where the sum is over all multi-indices J. Let $q=q(x,{\xi }^{\prime })$ be the full symbol of the operator $Q(x,{\partial }_{x^{\prime }})$, we write $q(x,{\xi }^{\prime })\sim {\sum }_{j⩽1}{q}_j(x,{\xi }^{\prime })$ with $q_j(x,{\xi }^{\prime })$ homogeneous of degree j in ${\xi }^{\prime }$. Hence, we get the following full symbol equation of (45)

$$\sum _J\frac{{(-i)}^{\left|J\right|}}{J!}{\partial }_{{\xi }^{\prime }}^{J}q{\partial }_{x^{\prime }}^{J}q-\sum _J\frac{{(-i)}^{\left|J\right|}}{J!}{\partial }_{{\xi }^{\prime }}^{J}b{\partial }_{x^{\prime }}^{J}q-\frac{\partial q}{\partial x_n}+c=0.$$

(46)

We shall determine $q_j$ recursively so that (46) holds modulo $S^{-\infty }$. Grouping the homogeneous terms of degree two in (46), we have

$$\begin{array}{c}\hfill q_1^2-b_1{q}_1+c_2=0.\end{array}$$

(47)

Note that $c_2$ can be rewritten as (see (35))

$$\begin{array}{c}\hfill c_2=-{\left|{\xi }^{\prime }\right|}^2{I}_{n+1}-\left[\begin{array}{ccc}\frac{\lambda +\mu }{\mu }\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& 0& 0\\ 0& -\frac{\lambda +\mu }{\lambda +2\mu }{\left|{\xi }^{\prime }\right|}^2& 0\\ 0& 0& 0\end{array}\right].\end{array}$$

(48)

In our notations, $\left[{\xi }^{\alpha }\right]={({\xi }^1,\cdots ,{\xi }^{n-1})}^t$ is a column vector, $\left[{\xi }_{\beta }\right]=({\xi }_1,\cdots ,{\xi }_{n-1})$ is a row vector, and $\left[{\xi }^{\alpha }{\xi }_{\beta }\right]$ is an $(n-1)\times (n-1)$ matrix. Then,

$$\begin{array}{cc}\hfill \left[{\xi }^{\alpha }\right]·\left[{\xi }_{\beta }\right]& =\left[{\xi }^{\alpha }{\xi }_{\beta }\right],\hfill \\ \hfill \left[{\xi }_{\beta }\right]·\left[{\xi }^{\alpha }\right]& =|{\xi }^{\prime }{|}^2,\hfill \\ \hfill [{\xi }^{\alpha }{\xi }_{\beta }]·\left[{\xi }^{\alpha }\right]& =|{\xi }^{\prime }{|}^2\left[{\xi }^{\alpha }\right],\hfill \\ \hfill [{\xi }_{\beta }]·\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& =|{\xi }^{\prime }{|}^2\left[{\xi }_{\beta }\right],\hfill \\ \hfill [{\xi }^{\alpha }{\xi }_{\beta }]·\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& =|{\xi }^{\prime }{|}^2\left[{\xi }^{\alpha }{\xi }_{\beta }\right].\hfill \end{array}$$

We find that

$$\begin{array}{cc}\hfill {\left[\begin{array}{ccc}\frac{1}{|{\xi }^{\prime }|}\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& 0& 0\\ 0& |{\xi }^{\prime }|& 0\\ 0& 0& 0\end{array}\right]}^2& ={\left[\begin{array}{ccc}0& \left[{\xi }^{\alpha }\right]& 0\\ {\xi }_{\beta }]& 0& 0\\ 0& 0& 0\end{array}\right]}^2=\left[\begin{array}{ccc}\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& 0& 0\\ 0& |{\xi }^{\prime }{|}^2& 0\\ 0& 0& 0\end{array}\right],\hfill \\ \hfill \left[\begin{array}{ccc}\frac{1}{|{\xi }^{\prime }|}\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& 0& 0\\ 0& |{\xi }^{\prime }|& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{ccc}0& \left[{\xi }^{\alpha }\right]& 0\\ {\xi }_{\beta }]& 0& 0\\ 0& 0& 0\end{array}\right]& =\left[\begin{array}{ccc}0& \left[{\xi }^{\alpha }\right]& 0\\ {\xi }_{\beta }]& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{ccc}\frac{1}{|{\xi }^{\prime }|}\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& 0& 0\\ 0& |{\xi }^{\prime }|& 0\\ 0& 0& 0\end{array}\right]\hfill \\ \hfill & =|{\xi }^{\prime }|\left[\begin{array}{ccc}0& \left[{\xi }^{\alpha }\right]& 0\\ {\xi }_{\beta }]& 0& 0\\ 0& 0& 0\end{array}\right].\hfill \end{array}$$

In view of the special forms of $b_1$ and $c_2$, we set $q_1$ has the form

$$\begin{array}{c}\hfill q_1=\left|{\xi }^{\prime }\right|I_{n+1}+\left[\begin{array}{ccc}{s}_1\frac{1}{|{\xi }^{\prime }|}\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& i{s}_2\left[{\xi }^{\alpha }\right]& 0\\ i{s}_3\left[{\xi }_{\beta }\right]& -s_4\left|{\xi }^{\prime }\right|& 0\\ 0& 0& 0\end{array}\right],\end{array}$$

(49)

where $s_j$, $1⩽j⩽4$, are coefficients to be determined. Substituting (49), (33), and (48) into (47), we get

$$\begin{array}{cc}\hfill 0& =|{\xi }^{\prime }{|}^2{I}_{n+1}+\left[\begin{array}{ccc}(s_1^2-s_2{s}_3)\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& i{s}_2(s_1-s_4)\left|{\xi }^{\prime }\right|\left[{\xi }^{\alpha }\right]& 0\\ i{s}_3(s_1-s_4)\left|{\xi }^{\prime }\right|\left[{\xi }_{\beta }\right]& (s_4^2-s_2{s}_3){\left|{\xi }^{\prime }\right|}^2& 0\\ 0& 0& 0\end{array}\right]\hfill \\ \hfill & +2\left[\begin{array}{ccc}{s}_1\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& i{s}_2\left|{\xi }^{\prime }\right|\left[{\xi }^{\alpha }\right]& 0\\ i{s}_3\left|{\xi }^{\prime }\right|\left[{\xi }_{\beta }\right]& -s_4{\left|{\xi }^{\prime }\right|}^2& 0\\ 0& 0& 0\end{array}\right]-i(\lambda +\mu )\left\{\left[\begin{array}{ccc}0& \frac{1}{\mu }\left|{\xi }^{\prime }\right|\left[{\xi }^{\alpha }\right]& 0\\ \frac{1}{\lambda +2\mu }\left|{\xi }^{\prime }\right|\left[{\xi }_{\beta }\right]& 0& 0\\ 0& 0& 0\end{array}\right]\right.\hfill \\ \hfill & \left.+\left[\begin{array}{ccc}\frac{i{s}_3}{\mu }\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& -\frac{s_4}{\mu }\left|{\xi }^{\prime }\right|\left[{\xi }^{\alpha }\right]& 0\\ \frac{s_1}{\lambda +2\mu }\left|{\xi }^{\prime }\right|\left[{\xi }_{\beta }\right]& \frac{i{s}_2}{\lambda +2\mu }{\left|{\xi }^{\prime }\right|}^2& 0\\ 0& 0& 0\end{array}\right]\right\}-{\left|{\xi }^{\prime }\right|}^2{I}_{n+1}+\left[\begin{array}{ccc}-\frac{\lambda +\mu }{\mu }\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& 0& 0\\ 0& \frac{\lambda +\mu }{\lambda +2\mu }{\left|{\xi }^{\prime }\right|}^2& 0\\ 0& 0& 0\end{array}\right].\hfill \end{array}$$

Therefore, we have the following equations of coefficients:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle s_1^2-s_2{s}_3+2s_1+\frac{\lambda +\mu }{\mu }(s_3-1)=0,}\hfill \\ {\displaystyle s_2(s_1-s_4)+2s_2+\frac{\lambda +\mu }{\mu }(s_4-1)=0,}\hfill \\ {\displaystyle s_3(s_1-s_4)+2s_3-\frac{\lambda +\mu }{\lambda +2\mu }(s_1+1)=0,}\hfill \\ {\displaystyle s_4^2-s_2{s}_3-2s_4+\frac{\lambda +\mu }{\lambda +2\mu }(s_2+1)=0.}\hfill \end{array}\right.\end{array}$$

(50)

Because we have chosen the outer normal vector $\nu$ on the boundary, we should take

$$\begin{array}{c}\hfill s_1⩾0,1-s_4>0.\end{array}$$

(51)

Such a choice implies that the real part of $q_1$ is positive definite. Solving the above equations with the conditions (51) and (1), we then get

$$\begin{array}{c}\hfill s_1=s_2=s_3=s_4=\frac{\lambda +\mu }{\lambda +3\mu }.\end{array}$$

(52)

Let

$$\begin{array}{c}\hfill F_1=\left[\begin{array}{ccc}\frac{1}{|{\xi }^{\prime }|}\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& i\left[{\xi }^{\alpha }\right]& 0\\ i\left[{\xi }_{\beta }\right]& -|{\xi }^{\prime }|& 0\\ 0& 0& 0\end{array}\right].\end{array}$$

(53)

Then, we obtain (41) immediately by combining (49), (52), and (53).

Grouping the homogeneous terms of degree $-m(m⩾-1)$ in (46), we get

$$\begin{array}{c}\hfill (q_1-b_1)q_{-m-1}+q_{-m-1}{q}_1=E_{-m},\end{array}$$

(54)

where $E_{-m}$, $m⩾-1$, are given by (38)–(40). Equation (54) is called the Sylvester equation (see [56], Chapter 9).

Let

$$\begin{array}{c}\hfill F_2=\left[\begin{array}{ccc}\frac{1}{|{\xi }^{\prime }|}\left[{\xi }^{\alpha }{\xi }_{\beta }\right]& -\frac{i(\lambda +2\mu )}{\mu }\left[{\xi }^{\alpha }\right]& 0\\ -\frac{i\mu }{\lambda +2\mu }\left[{\xi }_{\beta }\right]& -|{\xi }^{\prime }|& 0\\ 0& 0& 0\end{array}\right].\end{array}$$

(55)

Then, from (33), (52), (53), and (55), we have

$$\begin{array}{cc}\hfill F_1^2& =F_2^2=0,\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill b_1& =s_1(F_1-F_2),\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill b_1{F}_1& =-s_1{F}_2{F}_1,\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill b_1{F}_2& =s_1{F}_1{F}_2.\hfill \end{array}$$

(59)

By (41) and (57), we get

$$\begin{array}{c}\hfill q_1-b_1=\left|{\xi }^{\prime }\right|I_{n+1}+s_1{F}_2.\end{array}$$

(60)

Recall that, in (17) and (16), ⊗ and vec denote the Kronecker product and the vectorization of matrices, respectively. It follows from (25) that

$$\begin{array}{cc}\hfill vec\left(E_{-m}\right)& =vec((q_1-b_1)q_{-m-1}+q_{-m-1}{q}_1)\hfill \\ \hfill & =Hvec\left(q_{-m-1}\right),m⩾-1,\hfill \end{array}$$

(61)

where

$$\begin{array}{c}\hfill H=(I_{n+1}\otimes (q_1-b_1))+(q_1^t\otimes I_{n+1}).\end{array}$$

(62)

Combining (62) and (60), we obtain

$$\begin{array}{cc}\hfill H& =I_{n+1}\otimes \left(\right|{\xi }^{\prime }|I_{n+1}+s_1{F}_2)+(\left(\right|{\xi }^{\prime }|I_{n+1}+s_1{F}_1^t)\otimes I_{n+1})\hfill \\ \hfill & =2|{\xi }^{\prime }|I_{n+1}\otimes I_{n+1}+s_1(I_{n+1}\otimes F_2+F_1^t\otimes I_{n+1}).\hfill \end{array}$$

(63)

In view of (18)–(21), (56), and H is of order one in ${\xi }^{\prime }$, thus, we set $H^{-1}$ has the form

$$\begin{array}{c}\hfill H^{-1}=\frac{1}{2|{\xi }^{\prime }|}{I}_{n+1}\otimes I_{n+1}+\frac{s_5}{|{\xi }^{\prime }{|}^2}(I_{n+1}\otimes F_2+F_1^t\otimes I_{n+1})+\frac{s_6}{|{\xi }^{\prime }{|}^3}(F_1^t\otimes F_2),\end{array}$$

(64)

where $s_5$ and $s_6$ are coefficients to be determined. From (61), we have

$$\begin{array}{c}\hfill vec\left(q_{-m-1}\right)=H^{-1}vec\left(E_{-m}\right),m⩾-1.\end{array}$$

(65)

Combining (64), (65), and (22)–(24), we obtain, for $m⩾-1$,

$$\begin{array}{c}\hfill q_{-m-1}=\frac{1}{2|{\xi }^{\prime }|}{E}_{-m}+\frac{s_5}{|{\xi }^{\prime }{|}^2}(F_2{E}_{-m}+E_{-m}{F}_1)+\frac{s_6}{|{\xi }^{\prime }{|}^3}{F}_2{E}_{-m}{F}_1.\end{array}$$

(66)

It follows from (21), (56), (63), and (64) that

$$\begin{array}{cc}\hfill I_{{(n+1)}^2}& =H{H}^{-1}\hfill \\ \hfill & =I_{n+1}\otimes I_{n+1}+\frac{2s_5}{|{\xi }^{\prime }|}(I_{n+1}\otimes F_2+F_1^t\otimes I_{n+1})+\frac{2s_6}{|{\xi }^{\prime }{|}^2}(F_1^t\otimes F_2)\hfill \\ \hfill & +\frac{s_1}{2|{\xi }^{\prime }|}(I_{n+1}\otimes F_2+F_1^t\otimes I_{n+1})+\frac{s_1{s}_5}{|{\xi }^{\prime }{|}^2}(I_{n+1}\otimes F_2^2+{(F_1^t)}^2\otimes I_{n+1}\hfill \\ \hfill & +2F_1^t\otimes F_2)+\frac{s_1{s}_6}{|{\xi }^{\prime }{|}^3}(F_1^t\otimes F_2^2+{(F_1^t)}^2\otimes F_2)\hfill \\ \hfill & =I_{n+1}\otimes I_{n+1}+\left(2s_5+\frac{s_1}{2}\right)\frac{1}{|{\xi }^{\prime }|}(I_{n+1}\otimes F_2+F_1^t\otimes I_{n+1})\hfill \\ \hfill & +2(s_6+s_1{s}_5)\frac{1}{|{\xi }^{\prime }{|}^2}(F_1^t\otimes F_2).\hfill \end{array}$$

Note that $I_{n+1}\otimes I_{n+1}=I_{{(n+1)}^2}$. This implies that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle 2s_5+\frac{s_1}{2}=0,}\hfill \\ s_6+s_1{s}_5=0.\hfill \end{array}\right.\end{array}$$

Recall that $s_1=\frac{\lambda +\mu }{\lambda +3\mu }$ by (52). Thus, solving the above equations, we get

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle s_5=-\frac{s_1}{4}=-\frac{\lambda +\mu }{4(\lambda +3\mu )},}\hfill \\ {\displaystyle s_6=\frac{s_1^2}{4}=\frac{{(\lambda +\mu )}^2}{4{(\lambda +3\mu )}^2}.}\hfill \end{array}\right.\end{array}$$

(67)

Substituting (67) into (66), we immediately get (42). □

From Proposition 4, we get the full symbol of the pseudodifferential operator $Q(x,{\partial }_{x^{\prime }})$. This implies that we obtain $Q(x,{\partial }_{x^{\prime }})$ (modulo a smoothing operator) on the boundary.

Proposition 5. 

In boundary normal coordinates, the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$ can be written as

$$\begin{array}{c}\hfill {\mathsf{\Lambda }}_g=A\left(-\frac{\partial }{\partial x_n}\right)-K \mathit{on}\partial M,\end{array}$$

(68)

where A is given by (32), and

$$\begin{array}{cc}\hfill K& =\left[\begin{array}{ccc}0& \mu \left[g^{\alpha \beta }\frac{\partial }{\partial x_{\beta }}\right]& 0\\ \lambda \left[\frac{\partial }{\partial x_{\beta }}+{\mathsf{\Gamma }}_{\alpha \beta }^{\alpha }\right]& \lambda {\mathsf{\Gamma }}_{\alpha n}^{\alpha }& -\beta \\ 0& 0& 0\end{array}\right].\hfill \end{array}$$

(69)

Proof. 

By (4), we have

$$\begin{array}{cc}\hfill {\left(\left(S\mathit{u}\right)\nu \right)}^j={\left(S\mathit{u}\right)}_k^j{\nu }^k& =({\nabla }^j{u}_k+{\nabla }_k{u}^j){\nu }^k.\hfill \end{array}$$

In boundary normal coordinates, we take $\nu ={(0,\cdots ,0,-1)}^t$ and ${\partial }_{\nu }=-{\partial }_{x_n}$. In particular, $u_n=u^n$ since $g_{jn}={\delta }_{jn}$ in boundary normal coordinates. We get

$$\begin{array}{c}\hfill {\left(\left(S\mathit{u}\right)\nu \right)}^j=-({\nabla }^j{u}_n+{\nabla }_n{u}^j).\end{array}$$

Note that ${\mathsf{\Gamma }}_{nk}^n={\mathsf{\Gamma }}_{nn}^k=0$ and $g^{\alpha \beta }{\mathsf{\Gamma }}_{\beta \gamma }^n+{\mathsf{\Gamma }}_{n\gamma }^{\alpha }=0$ in boundary normal coordinates. Thus,

$$\begin{array}{cc}\hfill {\left(\left(S\mathit{u}\right)\nu \right)}^{\alpha }& =-({\nabla }^{\alpha }{u}_n+{\nabla }_n{u}^{\alpha })\hfill \\ \hfill & =-\left[g^{\alpha \beta }\left(\frac{\partial u^n}{\partial x_{\beta }}+{\mathsf{\Gamma }}_{\beta \gamma }^n{u}^{\gamma }\right)+\frac{\partial u^{\alpha }}{\partial x_n}+{\mathsf{\Gamma }}_{n\gamma }^{\alpha }{u}^{\gamma }\right]\hfill \\ \hfill & =-g^{\alpha \beta }\frac{\partial u^n}{\partial x_{\beta }}-\frac{\partial u^{\alpha }}{\partial x_n},\hfill \end{array}$$

(70)

$$\begin{array}{cc}\hfill {\left(\left(S\mathit{u}\right)\nu \right)}^n& =-({\nabla }^n{u}_n+{\nabla }_n{u}^n)=-2\frac{\partial u^n}{\partial x_n}.\hfill \end{array}$$

(71)

Hence, we immediately obtain (68) by combining (26), (6), (70), and (71). □

In boundary normal coordinates, the operator $-\frac{\partial }{\partial x_n}{|}_{\partial M}$ can be represented as the pseudodifferential operator $Q(x,{\partial }_{x^{\prime }})$ (modulo a smoothing operator) of order one in $x^{\prime }$ depending smoothly on $x_n$. Hence, we have the following proposition.

Proposition 6. 

In boundary normal coordinates, the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$ can be represented as

$$\begin{array}{c}\hfill {\mathsf{\Lambda }}_g=(AQ-K){|}_{\partial M}\end{array}$$

(72)

modulo a smoothing operator, where A and K are given by (32) and (69), respectively.

Proof. 

We use the boundary normal coordinates $(x^{\prime },x_n)$ with $x_n\in [0,T]$. Since the principal symbol of the thermoelastic operator $T_g$ is negative definite, the hyperplane $x_n=0$ is non-characteristic. Hence, $T_g$ is partially hypoelliptic with respect to this boundary (see [58], p. 107). Therefore, the solution to the equation $T_g\mathit{U}=0$ is smooth in normal variable, that is, $\mathit{U}\in {\left[C^{\infty }([0,T];{\mathfrak{D}}^{\prime }\left({\mathbb{R}}^{n-1}\right))\right]}^{n+1}$ locally. From Proposition 4, we see that (5) is locally equivalent to the following system of equations for $\mathit{U},\mathit{W}\in {\left[C^{\infty }([0,T];{\mathfrak{D}}^{\prime }\left({\mathbb{R}}^{n-1}\right))\right]}^{n+1}$:

$$\begin{array}{cc}\hfill \left(I_{n+1}\frac{\partial }{\partial x_n}+Q\right)\mathit{U}& {=\mathit{W},\mathit{U}|}_{x_n=0}=\mathit{V},\hfill \\ \hfill \left(I_{n+1}\frac{\partial }{\partial x_n}+B-Q\right)\mathit{W}& =\mathit{Y}\in {\left[C^{\infty }([0,T]\times {\mathbb{R}}^{n-1})\right]}^{n+1}.\hfill \end{array}$$

Inspired by [2] (cf. [21]), if we substitute $t=T-x_n$ into the second equation above, then, we get a backwards generalized heat equation

$$\begin{array}{c}\hfill \frac{\partial \mathit{W}}{\partial t}-(B-Q)\mathit{W}=-\mathit{Y}.\end{array}$$

Since $\mathit{U}$ is smooth in the interior of the manifold M by interior regularity for elliptic operator $T_g$, it follows that $\mathit{W}$ is also smooth in the interior of M, and so ${\mathit{W}|}_{x_n=T}$ is smooth. In view of the real part of $q_1$ (the principal symbol of Q) is positive definite (see (41)), we get that the solution operator for this heat equation is smooth for $t>0$ (see [57], p. 134). Therefore,

$$\begin{array}{c}\hfill \frac{\partial \mathit{U}}{\partial x_n}+Q\mathit{U}=\mathit{W}\in {\left[C^{\infty }([0,T]\times {\mathbb{R}}^{n-1})\right]}^{n+1}\end{array}$$

locally. If we set ${\mathcal{R}\mathit{V}=\mathit{W}|}_{\partial M}$, then, $\mathcal{R}$ is a smoothing operator and

$$\begin{array}{c}\hfill \frac{\partial \mathit{U}}{\partial x_n}{|}_{\partial M}{=-Q\mathit{U}|}_{\partial M}+\mathcal{R}\mathit{V}.\end{array}$$

(73)

Combining (73) and (68), we immediately obtain (72). □

3. Determining Coefficients on the Boundary

In this section we will prove the uniqueness results for the coefficients $\lambda ,\mu ,\alpha$, and $\beta$ on the boundary. We first prove Theorem 1.

Proof of Theorem 1. 

Let $\sigma \left({\mathsf{\Lambda }}_g\right)\sim {\sum }_{j⩽1}{p}_j(x,{\xi }^{\prime })$ be the full symbol of the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$. According to (72) and (69) we have

$$\begin{array}{cc}\hfill p_1(x,{\xi }^{\prime })& =A{q}_1(x,{\xi }^{\prime })-k_1,\hfill \end{array}$$

(74)

$$\begin{array}{cc}\hfill p_0(x,{\xi }^{\prime })& =A{q}_0(x,{\xi }^{\prime })-k_0,\hfill \end{array}$$

(75)

$$\begin{array}{cc}\hfill p_{-m}(x,{\xi }^{\prime })& =A{q}_{-m}(x,{\xi }^{\prime }),m⩾1,\hfill \end{array}$$

(76)

where A is given by (32), and

$$\begin{array}{c}\hfill k_1=\left[\begin{array}{ccc}0& i\mu \left[{\xi }^{\alpha }\right]& 0\\ i\lambda \left[{\xi }_{\beta }\right]& 0& 0\\ 0& 0& 0\end{array}\right],k_0=\left[\begin{array}{ccc}0& 0& 0\\ \lambda \left[{\mathsf{\Gamma }}_{\alpha \beta }^{\alpha }\right]& \lambda {\mathsf{\Gamma }}_{\alpha n}^{\alpha }& -\beta \\ 0& 0& 0\end{array}\right].\end{array}$$

(77)

Substituting (41), (32), and (77) into (74), we immediately obtain (8). Similarly, (9) and (10) can also be obtained. □

We then prove the uniqueness of the coefficients on the boundary.

Proof of Theorem 2. 

It follows from (8)–(10) that the Lamé coefficients $\lambda$ and $\mu$ only appear in the $n\times n$ submatrices. In the Lamé system, the uniqueness of $\frac{{\partial }^{\left|J\right|}\lambda }{\partial x^J}$ and $\frac{{\partial }^{\left|J\right|}\mu }{\partial x^J}$ on the boundary for all multi-indices J has been proved in [4]. Clearly, this particular result also holds in thermoelastic system, and the proof is the same as that of [4]. Thus, we only need to prove the uniqueness of the coefficients $\alpha$ and $\beta$ on the boundary.

From (8), we know that the $(n+1,n+1)$-entry of $p_1$ is

$$\begin{array}{c}\hfill {\left(p_1\right)}_{n+1}^{n+1}=\alpha \left|{\xi }^{\prime }\right|.\end{array}$$

This shows that $p_1$ uniquely determines $\alpha$ on the boundary. Furthermore, the tangential derivatives $\frac{\partial \alpha }{\partial x_{\gamma }}$ for $1⩽\gamma ⩽n-1$ can also be uniquely determined by $p_1$ on the boundary.

Let $b_0^{\prime }$ and $c_1^{\prime }$ be the terms that only involve the partial derivatives of $\lambda$ and $\mu$ in the expressions (34) and (36), respectively. That is,

$$\begin{array}{cc}\hfill b_0^{\prime }& =\left[\begin{array}{ccc}\frac{1}{\mu }\frac{\partial \mu }{\partial x_n}{I}_{n-1}& \frac{1}{\mu }\left[{\nabla }^{\alpha }\lambda \right]& 0\\ \frac{1}{\lambda +2\mu }\left[\frac{\partial \mu }{\partial x_{\beta }}\right]& \frac{1}{\lambda +2\mu }\frac{\partial (\lambda +2\mu )}{\partial x_n}& 0\\ 0& 0& 0\end{array}\right],\hfill \\ \hfill c_1^{\prime }& =i\left[\begin{array}{ccc}\frac{1}{\mu }\left({\xi }_{\alpha }{\nabla }^{\alpha }\mu \right)I_{n-1}+\frac{1}{\mu }\left[{\xi }_{\beta }{\nabla }^{\alpha }\lambda +{\xi }^{\alpha }\frac{\partial \mu }{\partial x_{\beta }}\right]& \frac{1}{\mu }\frac{\partial \mu }{\partial x_n}\left[{\xi }^{\alpha }\right]& 0\\ \frac{1}{\lambda +2\mu }\frac{\partial \lambda }{\partial x_n}\left[{\xi }_{\beta }\right]& \frac{1}{\lambda +2\mu }{\xi }_{\alpha }{\nabla }^{\alpha }\mu & 0\\ 0& 0& 0\end{array}\right].\hfill \end{array}$$

It follows from (42) that

$$\begin{array}{cc}\hfill q_0& =\widetilde{q_0}+\frac{1}{2|{\xi }^{\prime }|}{E}_1^{\prime }-\frac{\lambda +\mu }{4(\lambda +3\mu )|{\xi }^{\prime }{|}^2}(F_2{E}_1^{\prime }+E_1^{\prime }{F}_1)+\frac{{(\lambda +\mu )}^2}{4{(\lambda +3\mu )}^2{\left|{\xi }^{\prime }\right|}^3}{F}_2{E}_1^{\prime }{F}_1,\hfill \end{array}$$

where $E_1^{\prime }=b_0^{\prime }{q}_1-c_1^{\prime }$, and $\widetilde{q_0}$ is the solution of the corresponding equation with constant coefficients (see [3], p. 13). Hence, we see that $q_0$ has the form (see [3], p. 13)

$$\begin{array}{c}\hfill q_0=\left[\begin{array}{ccc}\ast & \ast & \frac{i\beta }{(\lambda +3\mu )|{\xi }^{\prime }|}\left[{\xi }_{\alpha }\right]\\ \ast & \ast & -\frac{\beta }{\lambda +3\mu }\\ \frac{\mu \omega \beta {\theta }_0}{\alpha (\lambda +3\mu )|{\xi }^{\prime }|}\left[{\xi }_{\beta }\right]& \frac{i\mu \omega \beta {\theta }_0}{\alpha (\lambda +3\mu )}& \ast \end{array}\right],\end{array}$$

(78)

where ∗ denotes the terms which we do not care (of course, they can be computed explicitly).

Therefore, combining (78), (75), and (77), we get the $(n,n+1)$-entry ${\left(p_0\right)}_{n+1}^n$, that is,

$$\begin{array}{c}\hfill {\left(p_0\right)}_{n+1}^n=\beta -\frac{\beta (\lambda +2\mu )}{\lambda +3\mu }=\frac{\beta \mu }{\lambda +3\mu }.\end{array}$$

This implies that $p_0$ uniquely determines $\beta$ on the boundary and the tangential derivatives $\frac{\partial \beta }{\partial x_{\gamma }}$ on the boundary for $1⩽\gamma ⩽n-1$, since $\lambda$ and $\mu$ have been determined on the boundary by the previous arguments.

According to the above discussion, we see from (75) that $q_0$ is uniquely determined by $p_0$ since the boundary values of $\lambda ,\mu ,\alpha$, and $\beta$ have been uniquely determined. By (54), we can determine $E_1$ from the knowledge of $q_0$. For $k⩾0$, we denote by ${\mathcal{T}}_{-k}={\mathcal{T}}_{-k}(\lambda ,\mu ,\alpha ,\beta )$ the terms that involve only the boundary values of $\lambda ,\mu ,\alpha ,\beta$, and their normal derivatives of order ar most k (which have been uniquely determined). Note that ${\mathcal{T}}_{-k}$ may be different in different expressions.

From (38), we have

$$\begin{array}{c}\hfill E_1=b_0{q}_1+\frac{\partial q_1}{\partial x_n}-c_1+{\mathcal{T}}_0.\end{array}$$

(79)

By (76) and (54), we know that $q_{-1}$ is uniquely determined by $p_{-1}$, and $E_0$ can be determined from the knowledge of $q_{-1}$. From (39), we see that

$$\begin{array}{c}\hfill E_0=\frac{\partial q_0}{\partial x_n}+{\mathcal{T}}_{-1}.\end{array}$$

(80)

From (78), we find that the $(n,n+1)$-entry ${\left(\frac{\partial q_0}{\partial x_n}\right)}_{n+1}^n$ and the $(n+1,n)$-entry ${\left(\frac{\partial q_0}{\partial x_n}\right)}_n^{n+1}$ of $\frac{\partial q_0}{\partial x_n}$ are, respectively,

$$\begin{array}{cc}\hfill {\left(\frac{\partial q_0}{\partial x_n}\right)}_{n+1}^n& =-\frac{\frac{\partial \beta }{\partial x_n}(\lambda +3\mu )-\beta (\frac{\partial \lambda }{\partial x_n}+3\frac{\partial \mu }{\partial x_n})}{{(\lambda +3\mu )}^2}\hfill \\ & =-\frac{1}{\lambda +3\mu }\frac{\partial \beta }{\partial x_n}+{\mathcal{T}}_{-1},\hfill \end{array}$$

(81)

$$\begin{array}{cc}{\left(\frac{\partial q_0}{\partial x_n}\right)}_n^{n+1}\hfill & =\frac{-\beta \mu (\lambda +3\mu )\frac{\partial \alpha }{\partial x_n}+\alpha \mu (\lambda +3\mu )\frac{\partial \beta }{\partial x_n}+\alpha \beta (\lambda \frac{\partial \mu }{\partial x_n}-\mu \frac{\partial \lambda }{\partial x_n})}{{\alpha }^2{(\lambda +3\mu )}^2}\hfill \\ & =-\frac{\beta \mu }{{\alpha }^2(\lambda +3\mu )}\frac{\partial \alpha }{\partial x_n}+\frac{\mu }{\alpha (\lambda +3\mu )}\frac{\partial \beta }{\partial x_n}+{\mathcal{T}}_{-1}.\hfill \end{array}$$

(82)

Since $\alpha ,\beta ,\lambda ,\mu ,\frac{\partial \lambda }{\partial x_n}$ and $\frac{\partial \mu }{\partial x_n}$ have been determined on the boundary, then, $\frac{\partial \beta }{\partial x_n}$ can be determined by ${\left(\frac{\partial q_0}{\partial x_n}\right)}_{n+1}^n$ on the boundary, and $\frac{\partial \alpha }{\partial x_n}$ can be determined by ${\left(\frac{\partial q_0}{\partial x_n}\right)}_n^{n+1}$ on the boundary. This implies that $p_{-1}$ uniquely determines $\frac{\partial \alpha }{\partial x_n}$ and $\frac{\partial \beta }{\partial x_n}$ on the boundary.

By (54), we have

$$\begin{array}{c}\hfill (q_1-b_1)\frac{\partial q_0}{\partial x_n}+\frac{\partial q_0}{\partial x_n}{q}_1=\frac{\partial E_1}{\partial x_n}+{\mathcal{T}}_{-1}.\end{array}$$

This implies that $\frac{\partial E_1}{\partial x_n}$ can be determined from the knowledge of $\frac{\partial q_0}{\partial x_n}$. By (76) and (54), we know that $q_{-2}$ is uniquely determined by $p_{-2}$, and $E_{-1}$ can be determined from the knowledge of $q_{-2}$. From (40), we see that

$$\begin{array}{c}\hfill E_{-1}=\frac{\partial q_{-1}}{\partial x_n}+{\mathcal{T}}_{-2}.\end{array}$$

By (54), we have

$$\begin{array}{c}\hfill (q_1-b_1)\frac{\partial q_{-1}}{\partial x_n}+\frac{\partial q_{-1}}{\partial x_n}{q}_1=\frac{\partial E_0}{\partial x_n}+{\mathcal{T}}_{-2}.\end{array}$$

This implies that $\frac{\partial E_0}{\partial x_n}$ can be determined from the knowledge of $\frac{\partial q_{-1}}{\partial x_n}$. From (80), we have

$$\begin{array}{c}\hfill \frac{\partial E_0}{\partial x_n}=\frac{{\partial }^2{q}_0}{\partial x_n^2}+{\mathcal{T}}_{-2}.\end{array}$$

Thus, it follows from (81) and (82) that

$$\begin{array}{cc}\hfill {\left(\frac{{\partial }^2{q}_0}{\partial x_n^2}\right)}_{n+1}^n& =-\frac{1}{\lambda +3\mu }\frac{{\partial }^2\beta }{\partial x_n^2}+{\mathcal{T}}_{-2},\hfill \\ \hfill {\left(\frac{{\partial }^2{q}_0}{\partial x_n^2}\right)}_n^{n+1}& =-\frac{\beta \mu }{{\alpha }^2(\lambda +3\mu )}\frac{{\partial }^2\alpha }{\partial x_n^2}+\frac{\mu }{\alpha (\lambda +3\mu )}\frac{{\partial }^2\beta }{\partial x_n^2}+{\mathcal{T}}_{-2}.\hfill \end{array}$$

Since $\lambda ,\mu ,\alpha ,\beta ,\frac{\partial \lambda }{\partial x_n},\frac{\partial \mu }{\partial x_n},\frac{{\partial }^2\lambda }{\partial x_n^2},\frac{{\partial }^2\mu }{\partial x_n^2},\frac{\partial \alpha }{\partial x_n}$, and $\frac{\partial \beta }{\partial x_n}$ have been determined on the boundary, then, $\frac{{\partial }^2\beta }{\partial x_n^2}$ can be determined by ${\left(\frac{{\partial }^2{q}_0}{\partial x_n^2}\right)}_{n+1}^n$ on the boundary, and $\frac{{\partial }^2\alpha }{\partial x_n^2}$ can be determined by ${\left(\frac{{\partial }^2{q}_0}{\partial x_n^2}\right)}_n^{n+1}$ on the boundary. This implies that $p_{-2}$ uniquely determines $\frac{\partial {\alpha }^2}{\partial x_n^2}$ and $\frac{{\partial }^2\beta }{\partial x_n^2}$ on the boundary.

Finally, we consider $p_{-m-1}$ for $m⩾1$. By (76) and (54), we have $p_{-m-1}$ uniquely determines $q_{-m-1}$, and $E_{-m}$ can be determined from the knowledge of $q_{-m-1}$. From (40), we obtain

$$\begin{array}{c}\hfill E_{-m}=\frac{\partial q_{-m}}{\partial x_n}+{\mathcal{T}}_{-m-1}.\end{array}$$

We see from (54) that

$$\begin{array}{c}\hfill (q_1-b_1)\frac{\partial q_{-m}}{\partial x_n}+\frac{\partial q_{-m}}{\partial x_n}{q}_1=\frac{\partial E_{-m+1}}{\partial x_n}+{\mathcal{T}}_{-m-1}.\end{array}$$

This implies that $\frac{\partial E_{-m+1}}{\partial x_n}$ can be determined from the knowledge of $\frac{\partial q_{-m}}{\partial x_n}$.

We end this proof by induction. Suppose we have shown that, by iteration, $E_{-m}$ uniquely determines

$$\begin{array}{c}\hfill \frac{{\partial }^m{E}_0}{\partial x_n^m}=\frac{{\partial }^{m+1}{q}_0}{\partial x_n^{m+1}}+{\mathcal{T}}_{-m-1},\end{array}$$

(83)

which further determines $\frac{{\partial }^{m+1}\alpha }{\partial x_n^{m+1}}$ and $\frac{{\partial }^{m+1}\beta }{\partial x_n^{m+1}}$ on the boundary since we have

$$\begin{array}{cc}\hfill {\left(\frac{{\partial }^{m+1}{q}_0}{\partial x_n^{m+1}}\right)}_{n+1}^n& =-\frac{1}{\lambda +3\mu }\frac{{\partial }^{m+1}\beta }{\partial x_n^{m+1}}+{\mathcal{T}}_{-m-1},\hfill \\ \hfill {\left(\frac{{\partial }^{m+1}{q}_0}{\partial x_n^{m+1}}\right)}_n^{n+1}& =-\frac{\beta \mu }{{\alpha }^2(\lambda +3\mu )}\frac{{\partial }^{m+1}\alpha }{\partial x_n^{m+1}}+\frac{\mu }{\alpha (\lambda +3\mu )}\frac{{\partial }^{m+1}\beta }{\partial x_n^{m+1}}+{\mathcal{T}}_{-m-1}.\hfill \end{array}$$

By (76) and (54), we know that $q_{-m-2}$ is uniquely determined by $p_{-m-2}$, and $E_{-m-1}$ can be determined from the knowledge of $q_{-m-2}$. Hence, $E_{-m-1}$ uniquely determines $\frac{{\partial }^{m+2}{q}_0}{\partial x_n^{m+2}}$ by iteration. It follows that

$$\begin{array}{cc}\hfill {\left(\frac{{\partial }^{m+2}{q}_0}{\partial x_n^{m+2}}\right)}_{n+1}^n& =-\frac{1}{\lambda +3\mu }\frac{{\partial }^{m+2}\beta }{\partial x_n^{m+2}}+{\mathcal{T}}_{-m-2},\hfill \\ \hfill {\left(\frac{{\partial }^{m+2}{q}_0}{\partial x_n^{m+2}}\right)}_n^{n+1}& =-\frac{\beta \mu }{{\alpha }^2(\lambda +3\mu )}\frac{{\partial }^{m+2}\alpha }{\partial x_n^{m+2}}+\frac{\mu }{\alpha (\lambda +3\mu )}\frac{{\partial }^{m+2}\beta }{\partial x_n^{m+2}}+{\mathcal{T}}_{-m-2}.\hfill \end{array}$$

This implies that $p_{-m-2}$ uniquely determines $\frac{{\partial }^{m+2}\alpha }{\partial x_n^{m+2}}$ and $\frac{{\partial }^{m+2}\beta }{\partial x_n^{m+2}}$ on the boundary.

Therefore, by combining the uniqueness result of $\frac{{\partial }^{\left|J\right|}\lambda }{\partial x^J}$, $\frac{{\partial }^{\left|J\right|}\mu }{\partial x^J}$ (see [4]), and the above arguments, we conclude that the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$ uniquely determines $\frac{{\partial }^{\left|J\right|}\lambda }{\partial x^J}$, $\frac{{\partial }^{\left|J\right|}\mu }{\partial x^J}$, $\frac{{\partial }^{\left|J\right|}\alpha }{\partial x^J}$, and $\frac{{\partial }^{\left|J\right|}\beta }{\partial x^J}$ on the boundary for all multi-indices J. □

4. Global Uniqueness of Real Analytic Coefficients

This section is devoted to proving the global uniqueness of real analytic coefficients $\lambda ,\mu ,\alpha$, and $\beta$ on a real analytic manifold. More precisely, we prove that the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$ uniquely determines the real analytic coefficients on the whole manifold $\overline{M}$.

We recall that the definitions of real analytic functions and real analytic hypersurfaces of a Riemannian manifold. Let $f\left(x\right)$ be a real-valued function defined on an open set $\mathsf{\Omega }\subset {\mathbb{R}}^n$. For $y\in \mathsf{\Omega }$, we call $f\left(x\right)$ real analytic at y if there exist $a_J\in \mathbb{R}$ and a neighborhood $N_y$ of y such that $f\left(x\right)={\sum }_J{a}_J{(x-y)}^J$ for all $x\in N_y$ and $J\in {\mathbb{N}}^n$. We say $f\left(x\right)$ is real analytic on an open set $\mathsf{\Omega }$ if $f\left(x\right)$ is real analytic at each $y\in \mathsf{\Omega }$.

Let $(M,g)$ be a Riemannian manifold. A subset U of M is said to be an $(n-1)$-dimensional real analytic hypersurface if U is nonempty and if for every point $x\in U$, there is a real analytic diffeomorphism of a unit open ball $B(0,1)\subset {\mathbb{R}}^n$ onto an open neighborhood $N_x$ of x such that $B(0,1)\cap \{x\in {\mathbb{R}}^n|x_n=0\}$ maps onto $N_x\cap U$.

In order to prove Theorem 3, we need the following lemma (see [59], p. 65).

Lemma 7. 

(Unique continuation of real analytic functions) Let $M\subset {\mathbb{R}}^n$ be a connected open set and $f\left(x\right)$ be a real analytic function defined on M. Let $y\in M$. Then, $f\left(x\right)$ is uniquely determined in M if we know $\frac{{\partial }^{\left|J\right|}f\left(y\right)}{\partial x^J}$ for all $J\in {\mathbb{N}}^n$. In particular, $f\left(x\right)$ is uniquely determined in M by its values in any nonempty open subset of M.

Note that Lemma 7 still holds for real analytic functions defined on real analytic manifolds. Finally, we prove Theorem 3.

Proof of Theorem 3. 

According to Theorem 2, it has been proved that the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$ uniquely determines $\frac{{\partial }^{\left|J\right|}\lambda }{\partial x^J}$, $\frac{{\partial }^{\left|J\right|}\mu }{\partial x^J}$, $\frac{{\partial }^{\left|J\right|}\alpha }{\partial x^J}$, and $\frac{{\partial }^{\left|J\right|}\beta }{\partial x^J}$ on the boundary for all multi-indices J. Hence, for any point $x_0\in \mathsf{\Gamma }$, the coefficients can be uniquely determined in some neighborhood of $x_0$ by the analyticity of the coefficients on $M\cup \mathsf{\Gamma }$. Furthermore, it follows from Lemma 7 that the coefficients can be uniquely determined in M. Therefore, by combining Theorem 2, we conclude that the coefficients $\lambda ,\mu ,\alpha$, and $\beta$ can be uniquely determined on $\overline{M}$ by the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$. □

Remark 8. 

By applying the method of Kohn and Vogelius [12], we can also prove that the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$ uniquely determines the coefficients $\lambda ,\mu ,\alpha$, and β on $\overline{M}$, provided the manifold and the coefficients are piecewise analytic.