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\begin{document}

\title[Unique continuation]{Unique continuation for the elasticity system with residual stress and its applications}

\author{Gen Nakamura}
\address{\hskip-\parindent
Gen Nakamura\\
Department of Mathematics\\
Hokkaido University\\
Sapporo 060-0810\\
Japan}
\email{gnaka@math.hokudai.ac.jp}

\author{Jenn-Nan Wang}
\address{\hskip-\parindent
        Jenn-Nan Wang\\
        Department of Mathematics\\
        National Cheng-Kung University\\
        Tainan 701, Taiwan}
\email{jnwang@msri.org}

\thanks{JW is currently a postdoc at MSRI. Research at MSRI is supported in part by NSF grant DMS-9701755. GN is partially supported by Grant in Aid for Scientific Research (C) (No.12640153) of Japan Society for the Promotion of Science.}


\begin{abstract}
In this paper we prove the unique continuation property for the elasticity system with small residual stress. The elasticity system becomes anisotropic due to the existence of residual stress. The main technique in the proof is Carleman estimates. Having proved the unique continuation property, we will study the inverse problem of identifying the inclusion or cavity. 
\end{abstract}

\maketitle

\section{Introduction}\label{sec1}
\setcounter{equation}{0}

Let ${\mathcal B}$ be an isotropic elastic body with residual stress and the reference
configuration of ${\mathcal B}$ be $\Omega$, a bounded open set in
$\R^n$ with smooth boundary. The residual stress is modeled by a
symmetric, smooth, second-rank tensor $T(x)=(t_{ij}(x))_{1\leq
i,j\leq n}$ satisfying
\begin{equation}\label{resid}
\partial_{x_j}t_{ij}=0\quad\mbox{in}\ \Omega,\quad
1\leq i\leq n
\end{equation}
and
\begin{equation}\label{residb}
t_{ij}\nu_j=0\quad\mbox{on}\
\partial\Omega,\quad 1\leq i\leq n,
\end{equation}
where $\nu=(\nu_1,\cdots,\nu_n)$ is the unit outer normal to
$\partial\Omega$. Hereafter, we adopt the summation convention. Let $u:\Omega\to\R^n$ be the displacement
vector, then the first Piola-Kirchhoff stress is written as $$
\begin{array}{ll}
\sigma=&T+(\nabla u)T+\lambda(\mbox{tr}\epsilon)
I+2\mu\epsilon+\beta_1(\mbox{tr}\epsilon)(\mbox{tr}T)I+\beta_2(\mbox{tr}T)\epsilon\\
{}&+\beta_3((\mbox{tr}\epsilon)T+\mbox{tr}(\epsilon T)I)+\beta_4(\epsilon
T+T\epsilon),
\end{array}
$$ where $\lambda,\mu$ are the Lam\'e moduli and
$\beta_1\cdots\beta_4$ are material parameters and
$$\epsilon=(\epsilon_{ij})=\frac{1}{2}(\nabla u+(\nabla u)^t)$$ is the
strain tensor \cite{ma}. Moreover, we assume that the Lam\'e moduli satisfy the strong ellipticity condition 
\begin{equation}\label{ellip}
\mu(x)>\delta>0,\  \lambda(x)+2\mu(x)>\delta>0\qquad \forall x\in\Omega
\end{equation}
and $$ \beta_3=\beta_4=0,$$ i.e., $$\sigma=T+(\nabla
u)T+\tilde{\lambda}(\mbox{tr}\epsilon)I+2\tilde{\mu}\epsilon,$$ where
$$\tilde{\lambda}=\lambda+\beta_1(\mbox{tr}
T),\quad\tilde{\mu}=\mu+\frac{1}{2}\beta_2(\mbox{tr} T).$$ The
stationary elasticity system is expressed as
\begin{equation}\label{elastic}
(Lu)_i=(\nabla\cdot
\sigma)_i+\omega^2\rho(x)u_i=\partial_{j}\sigma_{ij}+\omega^2\rho(x)u_i=0\quad\mbox{in}\
\Omega,\ 1\leq i\leq n,\ \omega\in\R,
\end{equation}
where $\rho(x)>0$ is the density of the medium. In another setting, if we define the elasticity tensor $\C$ with components
\begin{equation}\label{etensor}
C_{ijkl}=\tilde{\lambda}\delta_{ij}\delta_{kl}+(\tilde{\mu}\delta_{jl}+t_{jl})\delta_{ik}+\tilde{\mu}\delta_{il}\delta_{jk}
\end{equation}
and denote
$$(\C E)_{ij}=C_{ijkl}E_{kl}\quad\mbox{for any matrix}\quad E,$$
then \eqref{elastic} is equivalent to 
$$
(Lu)_i=(\nabla\cdot \C\nabla u)_i+\omega^2\rho u_i=\partial_j(C_{ijkl}\partial_lu_k)+\omega^2\rho u_i=0\quad\mbox{in}\ \Omega,\ 1\leq i\leq n.
$$
It is clear to see that \eqref{elastic} is an anisotropic elasticity system. In this paper, we will investigate the (weak) unique continuation property (UCP) for the system \eqref{elastic}, i.e. if $u\in H^2_{\text{loc}}(\Omega)$ is a solution to \eqref{elastic} in $\Omega$ and vanishes in a non-empty open subset of $\Omega$, then $u$ vanishes identically in $\Omega$.

The unique continuation property for differential equations has a
long history. Many deep results about scalar elliptic equations or
elliptic systems have been established. We refer the reader to
\cite{bjs} and references therein for details. Recently, few
attempts have been made at studying the unique continuation
property for systems of equations in mathematical physics such as
the Dirac equations and the Maxwell equations \cite{deok},
\cite{je}, \cite{ok}, \cite{vo1}, \cite{vo2}. Here we mention two
interesting articles \cite{vo2} and \cite{ok} in which Vogelsang
and Okaji, respectively, proved the strong unique continuation
property for the Maxwell system with anisotropic coefficients. In
this paper we pay our attention to the elasticity system. Several
results of weak continuation property for the inhomogeneous
isotropic elasticity have been obtained in \cite{aity}, \cite{dero},
(stationary) and \cite{eint}, \cite{is2}
(non-stationary). Moreover, a strong unique continuation property
were recently proven by Alessandrini and Morassi \cite{almo}.
Unlike the isotropic case, the unique continuation property for
the inhomogeneous anisotropic elasticity has not been fully
explored.

Our study of the unique continuation property for the
inhomogeneous anisotropic elasticity is motivated by its
application to inverse problems. It was first recognized by Lax
\cite{la} that the Runge approximation property is a consequence
of the weak unique continuation property. The Runge approximation
property is shown to be a useful technique in dealing with some
inverse problems, especially the inverse problem of recovering
inclusions or cavities (see \cite{is1}, \cite{ik1}, \cite{ik2},
\cite{int}, \cite{ikna}, \cite{kovo} and references therein). It should be
noted that the Runge approximation property with constraint for
the anisotropic elasticity were proved in \cite{int} and
\cite{ikna}. However, the elasticity tensor there is assumed to be
either homogeneous or real-analytic. The weak unique continuation
property is an obvious fact in these two situations.

To prove the unique continuation property for the general
inhomogeneous anisotropic elasticity is very challenging and
difficult. Here we want to consider the system \eqref{elastic} which has the simplest form of anisotropy. It turns out we are able to establish the UCP for \eqref{elastic} provided the residual stress is sufficiently small. Our main idea comes from Weck's recent article \cite{we1} where he proved the UCP for the isotropic elasticity system with zeroth or first order perturbations which contains the results previous obtained by \cite{aity}, \cite{dero}. Weck actually proved something more, namely, he established the UCP for a rather general system of second order differential inequalities with the Laplacian principal part. Like many literature on the UCP, the key step in \cite{we1} is to prove appropriate Carleman estimates. Here we will adopt Weck's approach to \eqref{elastic} with small residual stress, but we have to work a little harder to derive the desired Carleman estimates because we need to deal with variable coefficients second order principal parts due to the presence of residual stress. As indicated previously, having established the UCP, we can prove the Runge approximation property for \eqref{elastic} with constraints on Dirichlet data. With this tool at hand, we can solve the inverse problem of identifying inclusions or cavities inside an elastic body with small residual stress by the localized Dirichlet-to-Neumann map using the methods in \cite{int} and \cite{ikna}.

This paper is organized as follows. In Section~2, we state and prove the UCP for \eqref{elastic} with small residual stress based on suitable Carleman estimates. The derivation of these Carleman estimates is given in Section~3. In Section~4, we will discuss the application of UCP for \eqref{elastic} to the aforementioned inverse problem. In the paper, $C$ stands for a generic constant and its value may vary from line to line.

\section{Unique continuation}\label{sec2}
\setcounter{equation}{0}

To begin, let us denote $v_i=u_i$ for $1\leq i\leq n$ and $v_{n+1}=\partial_iu_i$. Then, it follows from \eqref{elastic} that
\begin{equation}\label{eq1}
\begin{array}{rl}
0 &=(Lu)_i\\
{}&=(\tilde{\mu}\Delta+t_{kj}\partial_j\partial_k)v_i+(\tilde{\lambda}+\tilde{\mu})\partial_iv_{n+1}+(\partial_jt_{kj})\partial_kv_i+(\partial_i\tilde{\lambda})v_{n+1}\\
{}&\quad +(\partial_j\tilde{\mu})(\partial_iv_j+\partial_jv_i)+\omega^2\rho v_i\\
{}&=(\tilde{\mu}\Delta+t_{kj}\partial_j\partial_k)v_i+R_i^{(1)}(v_1,\cdots,v_n,v_{n+1})\quad\mbox{in}\ \Omega,\ 1\leq i\leq n,
\end{array}
\end{equation}
where $R_i^{(1)}$'s are some first order differential operators. Next, by taking the divergence of \eqref{elastic}, we obtain that
\begin{equation}\label{eq2}
\begin{array}{rl}
0&=\partial_i(Lu)_i\\
{}&=((\tilde{\lambda}+2\tilde{\mu})\Delta+t_{kj}\partial_j\partial_k)v_{n+1}+2(\partial_i\tilde{\mu})\Delta v_i+(\partial_it_{kj})\partial_j\partial_kv_i+2\partial_i(\tilde{\lambda}+\tilde{\mu})\partial_iv_{n+1}\\
{}&\quad+(\partial_jt_{kj})\partial_kv_{n+1}+(\partial_i\partial_jt_{kj})\partial_kv_i+(\Delta\tilde{\lambda})v_{n+1}+(\partial_i\partial_j\tilde{\mu})(\partial_iv_j+\partial_jv_i)+\omega^2(\partial_i\rho)v_i+\omega^2\rho v_{n+1}\\
{}&=((\tilde{\lambda}+2\tilde{\mu})\Delta+t_{kj}\partial_j\partial_k)v_{n+1}+R^{(2)}(v_1,\cdots,v_n)+R_{n+1}^{(1)}(v_1,\cdots,v_{n+1}),
\end{array}
\end{equation}
where $R^{(2)}$ is a pure second order (containing only second derivatives) and $R^{(1)}_{n+1}$ is a first order differential operators, respectively. It should be mentioned that $R^{(2)}$ acts only on $v_1,\cdots,v_n$. In view of \eqref{ellip}, we can see that if 
\begin{equation}\label{srs}
\underset{kj}{\max}\|t_{kj}\|_{L^{\infty}(\Omega)}<\varepsilon
\end{equation}
with $\varepsilon\ll 1$, then 
$$\tilde{\mu}>\delta'>0\quad\mbox{and}\quad\tilde{\lambda}+2\tilde{\mu}>\delta'>0\quad\forall x\in\Omega.$$


With the equations \eqref{eq1} and \eqref{eq2} in mind, motivated by Weck's paper \cite{we1}, we will prove the UCP for the following system of differential inequalities
\begin{equation}\label{eq5}
\begin{array}{ll}
|A_1(x,\partial)u^1|\leq CQ(u^1,u^2)^{1/2},\\
|A_2(x,\partial)u^2|\leq C\{\underset{ijk}{\sum}|\partial_i\partial_ju_k^1|+Q(u^1,u^2)^{1/2}\},
\end{array}
\end{equation}
where $u^l:\Omega\to\R^{m_l}, m_l\in{\mathbb Z}_+ (\mbox{psitive integers})$ and $A_l(x,\partial)=a_{ij}^l\partial_i\partial_j$ with real symmetric matrix $(a^l_{ij})$, $l=1,2$ and $Q(u^1,u^2)=\underset{ikl}{\sum}(|\partial_iu_k^l|^2+|u_k^l|^2)$. 
\begin{theorem}\label{thm1}
Let $a_{ij}^l\in W^{1,\infty}(\Omega)$ and $(u^1,u^2)\in H^2_{\text{loc}}(\Omega)\times H^2_{\text{loc}}(\Omega)$ satisfy \eqref{eq5}. Then there exists an $\varepsilon>0$
such that if
\begin{equation}\label{s1}
\underset{ij}{\max}\|a_{ij}^l(x)-\delta_{ij}\|_{L^{\infty}(\Omega)}<\varepsilon,
\end{equation}
then $(u^1,u^2)$ vanishes identically in $\Omega$ if it vanishes in a non-empty open subset of $\Omega$.
\end{theorem}
Theorem~\ref{thm1} immediately implies the UCP for \eqref{elastic} with small residual stress.
\begin{corollary}\label{thm2}
Let coefficients $\lambda,\mu,\beta_1,\beta_2,t_{kj}$ belong to $W^{2,\infty}(\Omega)$ and $\rho$ be in $W^{1,\infty}(\Omega)$. Then there exists an ${\varepsilon}>0$ such that if \eqref{srs} is satisfied with this $\varepsilon$, then the system \eqref{elastic} possesses the UCP.
\end{corollary}


The proof of Theorem~\ref{thm1} relies on the following Carleman estimates.
\begin{pr}\label{care}
Assume that the differential operators $A_1$ and $A_2$ satisfy the assumptions in Theorem~\ref{thm1}. Let $r_0<1$ and $U_{r_0}=\{u\in C_0^{\infty}(\R^n\setminus\{0\}): \mbox{\rm supp}(u)\subset B_{r_0}\}$, where $B_{r_0}$ is the ball centered at the origin with radius $r_0$. Then there exist positive constants $\beta_0$ and $\varepsilon_0$ such that if \eqref{s1} is satisfied with $\varepsilon\leq \varepsilon_0$, then for all $\beta\geq\beta_0$ and $u\in U_{r_0}$ we have that
\begin{equation}\label{car1}
\int r^{-\sigma}\psi^2\underset{ij}{\sum}|\partial_i\partial_ju|^2dx\leq C\int r^{-\sigma}\psi^2(\beta^2 r^{-2\beta-2}|\nabla u|^2+|A_lu|^2)dx
\end{equation}
and
\begin{equation}\label{car2}
\beta^2\int r^{-\sigma-\beta-1}\psi^2(|\nabla u|^2+|u|^2)dx\leq C\int r^{-\sigma}\psi^2|A_lu|^2dx
\end{equation}
for $l=1,2$, where $r=|x|$, $\psi=\exp(r^{-\beta})$ and $\sigma=\sigma_0+c\beta$ with $\sigma_0,c\in\R$.
\end{pr}

The proof of Proposition~\ref{care} is postponed until the next section. Here we want to prove Theorem~\ref{thm1} based on this proposition.

\medskip
\noindent{\bf Proof of Theorem~\ref{thm1}}: It suffices to prove the theorem for $m_1=m_2=1$ case. Let $(u^1,u^2)$ vanish in a neighborhood of $x_0\in\Omega$. Without loss of generality, we assume $x_0=0$. We set $\tilde{r}=\min\{1/2,\text{dist}(0,\partial\Omega)\}$. Now let $\chi\in C_0^{\infty}(\R^n)$ be a cut-off function satisfying $0\leq \chi\leq 1$, $\chi |_{B_{\tilde{r}/2}}=1$ and $\text{\supp}(\chi)\subset B_{\tilde{r}}$. Denote $v_l=\chi u^l$, $l=1,2$. From \eqref{eq5} we have that
\begin{equation}\label{eq10}
\begin{array}{ll}
|A_1v_1|\leq C(e(v_1)+e(v_2))^{1/2}+f_1,\\
|A_2v_2|\leq C[\underset{ij}{\sum}|\partial_i\partial_j v_1|+(e(v_1)+e(v_2))^{1/2}]+f_2,
\end{array}
\end{equation}
where $e(v)=|\nabla v|^2+|v|^2$ and $f_l$ is supported in $B_{\tilde{r}}\setminus B_{\tilde{r}/2}$ for $l=1,2$. It follows from \eqref{eq10} that
\begin{equation}\label{eq15}
I:=\gamma\int r^{-\beta}\psi^2|A_1v_1|^2dx+\int r\psi^2|A_2v_2|^2dx\leq C(F+G+\int r\psi^2\underset{ij}{\sum}|\partial_i\partial_j v_1|^2dx),
\end{equation}
where
$$\begin{array}{ll}
F=\gamma\int r^{-\beta}\psi^2f_1^2dx+\int r\psi^2f_2^2dx,\\
G=\int (r+\gamma r^{-\beta})\psi^2(e(v_1)+e(v_2))dx.
\end{array}
$$ Here $\gamma$ is a large positive parameter which will be chosen later on. By the standard approximation argument, we can see that $v_1$ and $v_2$ satisfy estimates \eqref{car1} and \eqref{car2}. Taking $\sigma=-1$ in the estimate \eqref{car1} for $l=1$ and substituting it into \eqref{eq15} yield
\begin{equation}\label{eq20}
I\leq C(F+G+\int r\psi^2|A_1v_1|^2dx+\beta^2\int r^{-2\beta-1}\psi^2|\nabla v_1|^2dx).
\end{equation}
Replacing the last term of \eqref{eq20} with the help of \eqref{car2} for $\sigma=\beta$ and $l=1$, we obtain that
\begin{equation}\label{eq25}
I\leq C(F+G+\int r^{-\beta}\psi^2|A_1v_1|^2dx).
\end{equation}
Now taking $\gamma$ sufficiently large, we can absorb the last term of \eqref{eq25} and get
\begin{equation}\label{eq30}
I\leq C(F+G).
\end{equation}
From now on we fix the parameter $\gamma$.

Next using $\sigma=\beta$ in \eqref{car2} for $l=1$ and $\sigma=-1$ in \eqref{car2} for $l=2$, we find that
\begin{equation}\label{eq35}
\begin{array}{rl}
H:&=\beta^2\int r^{-2\beta-1}\psi^2e(v_1)dx+\beta^2\int r^{-\beta}\psi^2e(v_2)dx\\
{}&\leq C(\int r^{-\beta}\psi^2|A_1v_1|^2dx+\int r\psi^2|A_2v_2|^2dx).
\end{array}
\end{equation}
Combining \eqref{eq30} and \eqref{eq35} gives
\begin{equation}\label{eq40}
H\leq C(F+G)\leq C(F+\int (r+\gamma r^{-\beta})\psi^2(e(v_1)+e(v_2))dx).
\end{equation}
Now observing that $r<r^{-\beta}<\beta r^{-\beta}<\beta r^{-2\beta-1}$ when $r\leq \tilde{r}$ and $\beta>1$, we obtain from \eqref{eq40} that
\begin{equation}\label{eq45}
H\leq C(F+\beta\int r^{-2\beta-1}\psi^2e(v_1)dx+\beta\int r^{-\beta}\psi^2e(v_2)dx).
\end{equation}
Taking $\beta$ sufficiently large in \eqref{eq45}, we get that
$$H\leq CF,$$
i.e.
$$\beta^2\int r^{-2\beta-1}\psi^2e(v_1)dx+\beta^2\int r^{-\beta}\psi^2 e(v_2)dx\leq C(\int r^{-\beta}\psi^2f_1^2dx+\int r\psi^2f_2^2dx)$$ from which we immediately have
\begin{equation}\label{eq50}
\beta^2\int_{B_{\tilde{r}/2}} r^{-\beta}\psi^2(v_1^2+v_2^2)dx\leq C\int_{B_{\tilde{r}}\setminus B_{\tilde{r}/2}}r^{-\beta}\psi^2(f_1^2+f_2^2)dx.
\end{equation}
Since $r^{-\beta}\psi^2$ is a strictly decreasing function, \eqref{eq50} implies that
$$
\beta^2\int_{B_{\tilde{r}/2}}(v_1^2+v_2^2)dx\leq C\int_{B_{\tilde{r}}\setminus B_{\tilde{r}/2}}(f_1^2+f_2^2)dx
$$
and therefore $(v_1,v_2)=0$ on $B_{\tilde{r}/2}$ if we choose $\beta$ sufficiently large. Clearly, $(u^1,u^2)$ must be zero throughout $\Omega$.\eproof

 
\section{Proof of Carleman estimates}\label{sec3}
\setcounter{equation}{0}

This section is devoted to the proof of Proposition~\ref{care}. It suffices to prove \eqref{car1} and \eqref{car2} for $A_1$. Therefore, we denote $a^1_{ij}=a_{ij}$ and $A_1=A$. To prove \eqref{car1}, we first recall the following estimate in \cite{we1}
$$\int r^{-\sigma}\psi^2\underset{ij}{\sum}|\partial_i\partial_ju|^2dx\leq C\int r^{-\sigma}\psi^2(\beta^2r^{-2\beta-2}|\nabla u|^2+|\Delta u|^2)dx$$ (see \cite[Lemma~2]{we1}) from which we have that
$$
\begin{array}{rl}
\int r^{-\sigma}\psi^2\underset{ij}{\sum}|\partial_i\partial_ju|^2dx&\leq C\int r^{-\sigma}\psi^2(\beta^2r^{-2\beta-2}|\nabla u|^2+|Au|^2+|\Delta u-Au|^2)dx\\
{}&\leq C\int r^{-\sigma}\psi^2(\beta^2r^{-2\beta-2}|\nabla u|^2+|Au|^2+\varepsilon^2\underset{ij}{\sum}|\partial_i\partial_j u|^2)dx.
\end{array}
$$ 
Thus, choosing $\varepsilon$ small enough immediately implies the estimate \eqref{car1}. 

The proof of \eqref{car2} is lengthy. Here we will adopt some techniques from \cite{pr}, \cite{we1} and \cite{we2}. Let $\phi=\psi^{-1}$ and $u=r^{\tau/2}\phi z$, then
$$
\begin{array}{rl}
r^{-\sigma/2}\psi Au&=r^{-\sigma/2}\psi A(r^{\tau/2}\phi z)\\
{}&=r^{-\sigma/2}\psi[r^{\tau/2}\phi Az+2a_{ij}\partial_i z\partial_j(r^{\tau/2}\phi)+zA(r^{\tau/2}\phi)].
\end{array}
$$ 
By virtue of the inequality $(a+b+c)^2\geq 2ab+2bc$, we have that
\begin{equation}\label{eq60}
\begin{array}{rl}
\int r^{-\sigma}\psi^2|Au|^2dx&\geq 4\int r^{-\sigma}\psi^2a_{ij}\partial_i z\partial_j(r^{\tau/2}\phi)r^{\tau/2}\phi Az dx\\
{}&\quad+4\int r^{-\sigma}\psi^2a_{ij}\partial_i z\partial_j(r^{\tau/2}\phi)zA(r^{\tau/2}\phi)dx.
\end{array}
\end{equation}
With the choice of $\tau=\sigma+\beta+2$, we can compute
$$
\begin{array}{l}
I:=\int r^{-\sigma}\psi^2 a_{ij}\partial_i z\partial_j(r^{\tau/2}\phi)r^{\tau/2}\phi Azdx\\
\quad =\beta\int a_{ij}\partial_iz x_jAzdx+\tau/2\int r^{\beta}a_{ij}\partial_izx_jAzdx.
\end{array}
$$
It is readily seen that the leading term (for large $\beta$) of $I$ is
$\beta\int a_{ij}\partial_izx_jAzdx$.
Repeated integration by parts shows that
\begin{equation}\label{eq100}
\begin{array}{rl}
2\int a_{ij}\partial_izx_jAzdx&=2\int a_{ij}\partial_i zx_ja_{kl}\partial_k\partial_lzdx\\
{}&=-\int \partial_iz\partial_l(a_{kl}a_{ij}x_j)\partial_kzdx+\int\partial_kz\partial_i(a_{kl}a_{ij}x_j)\partial_lzdx\\
{}&\quad-\int\partial_lz\partial_k(a_{kl}a_{ij}x_j)\partial_lzdx.
\end{array}
\end{equation}
Using \eqref{eq100} we obtain that
\begin{equation}\label{eq110}
\begin{array}{l}
|I|\leq C\beta|\int \partial_iz\partial_l(a_{kl}a_{ij}x_j)\partial_kzdx|\\
\leq C\beta\|\nabla z\|^2\\
\leq C\beta(\|\nabla(r^{-\tau/2}\psi)u\|^2+\|r^{-\tau/2}\psi\nabla u\|^2)\\
\leq C(\beta^3\int r^{-\sigma-3\beta-4}\psi^2|u|^2dx+\beta\int r^{-\sigma-\beta-2}\psi^2|\nabla u|^2 dx).
\end{array}
\end{equation}
Next we observe that
$$
\begin{array}{l}
J:=\int r^{-\sigma}\psi^2a_{ij}\partial_iz\partial_j(r^{\tau/2}\phi)zA(r^{\tau/2}\phi)dx\\
\quad =\beta\int r^{-\sigma+\tau/2-\beta-2}\psi a_{ij}\partial_izx_jzA(r^{\tau/2}\phi)dx\\
\qquad +\tau/2\int r^{-\sigma+\tau/2-2}\psi a_{ij}\partial_izx_jzA(r^{\tau/2}\phi)dx.
\end{array}
$$
Straightforward calculations show that
$$\quad \partial_i\partial_j\phi=(\beta^2x_ix_jr^{-2\beta-4}+\beta\delta_{ij}r^{-\beta-2}-\beta(\beta+2)x_ix_jr^{-\beta-4})\phi$$ and
$$\quad\partial_i\partial_j r^{\tau/2}=(\tau/2)(\tau/2-2)r^{\tau/2-4}x_ix_j+(\tau/2)r^{\tau/2-2}\delta_{ij}.$$ So the leading term of $J$ is
$$\beta^3\int r^{-2\beta-4}a_{ij}\partial_izx_ja_{kl}x_kx_lzdx.$$ Note that we have chosen $\tau=\sigma+\beta+2$. Performing the integration by parts, we can see that
$$\begin{array}{l}
\beta^3\int r^{-2\beta-4}a_{ij}\partial_izx_ja_{kl}x_kx_lzdx\\
=-2\beta^3\int z\partial_i(r^{-2\beta-4}a_{ij}x_ja_{kl}x_kx_l)zdx\\
\geq (4-o(\beta))\beta^4\int r^{-2\beta-6}a_{ij}x_ix_ja_{kl}x_kx_l|z|^2dx\\
\geq (4-o(\beta))\beta^4(1-O(\varepsilon))\int r^{-2\beta-2}|z|^2dx\\
\geq (4-o(\beta))\beta^4(1-O(\varepsilon))\int r^{-\sigma-3\beta-4}\psi^2|u|^2dx,
\end{array}
$$
where $o(\beta)\to 0$ as $\beta\to\infty$ and $O(\varepsilon)$ is a constant bounded by $C\varepsilon$. In other words, we have that
\begin{equation}\label{eq120}
J\geq (4-o(\beta))\beta^4(1-O(\varepsilon))\int r^{-\sigma-3\beta-4}\psi^2|u|^2dx.
\end{equation}
Notice that we need to keep track the leading constant here in order to obtain the desired estimate. Combining \eqref{eq60}, \eqref{eq110} and \eqref{eq120} gives
$$\begin{array}{l}
\int r^{-\sigma}\psi^2|Au|^2dx+C(\beta^3\int r^{-\sigma-3\beta-4}\psi^2|u|^2dx+\beta\int r^{-\sigma-\beta-2}\psi^2|\nabla u|^2 dx)\\
\geq 4(4-o(\beta))\beta^4(1-O(\varepsilon))\int r^{-\sigma-3\beta-4}\psi^2|u|^2dx
\end{array}
$$ from which we can derive that
\begin{equation}\label{eq150}
\int r^{-\sigma}\psi^2|Au|^2dx+C\beta\int r^{-\sigma-\beta-2}\psi^2|\nabla u|^2 dx\geq 4(4-o(\beta))\beta^4(1-O(\varepsilon))\int r^{-\sigma-3\beta-4}\psi^2|u|^2dx.
\end{equation}


By the ellipticity condition and performing the integration by parts, we can get that
\begin{equation}\label{eq180}
\begin{array}{l}
(1-O(\varepsilon))\int r^{-\sigma-\beta-2}\psi^2|\nabla u|^2dx\\
\leq \int r^{-\sigma-\beta-2}\psi^2a_{ij}\partial_iu\partial_judx\\
\leq |\int u\partial_i(r^{-\sigma-\beta-2}\psi^2)a_{ij}\partial_judx|+|\int r^{-\sigma-\beta-2}u\partial_i(a_{ij})\partial_judx|\\
\quad +|\int r^{-\sigma-\beta-2} ua_{ij}\partial_i\partial_judx|\\
:=K_1+K_2+K_3.
\end{array}
\end{equation}
Using the relation $|ab|\leq (a^2+b^2)/2$, we can estimate
\begin{equation}\label{eq200}
\begin{array}{rl}
K_1&=|\int u\partial_i(r^{-\sigma-\beta-2}\psi^2)a_{ij}\partial_judx|\\
{}&\leq\int 2\beta r^{-\sigma-2\beta-4}\psi^2|ua_{ij}x_i\partial_ju|dx\\
{}&\leq (2+o(\beta))\beta^2(1+O(\varepsilon))\int r^{-\sigma-3\beta-4}\psi^2|u|^2dx+(1+O(\varepsilon))/2\int r^{-\sigma-\beta-2}\psi^2|\nabla u|^2dx.
\end{array}
\end{equation}
Likewise, for $K_1$ and $K_2$, we have that
\begin{equation}\label{eq250}
K_2\leq C(r_0^{\beta}\int r^{-\sigma-3\beta-4}\psi^2|u|^2dx+r_0^{\beta+2}\int r^{-\sigma-\beta-2}\psi^2|\nabla u|^2dx)
\end{equation}
and
\begin{equation}\label{eq300}
K_3\leq C(r_0^\beta\beta^2\int r^{-\sigma-3\beta-4}\psi^2|u|^2dx+\beta^{-2}\int r^{-\sigma}\psi^2|Au|^2dx).
\end{equation}
Plugging \eqref{eq200}, \eqref{eq250} and \eqref{eq300} into \eqref{eq180} and multiplying the new inequality by $\beta^2$, we obtain that
\begin{equation}\label{eq500}
\begin{array}{l}
\beta^2(1-O(\varepsilon))\int r^{-\sigma-\beta-2}\psi^2|\nabla u|^2dx\\
\leq (2+o(\beta))\beta^4(1+O(\varepsilon))\int r^{-\sigma-3\beta-4}\psi^2|u|^2dx+\beta^2(1+O(\varepsilon))/2\int r^{-\sigma-\beta-2}\psi^2|\nabla u|^2dx\\
\quad + C(r_0^{\beta}\beta^2\int r^{-\sigma-3\beta-4}\psi^2|u|^2dx+r_0^{\beta+2}\beta^2\int r^{-\sigma-\beta-2}\psi^2|\nabla u|^2dx)\\
\quad + C(r_0^\beta\beta^4\int r^{-\sigma-3\beta-4}\psi^2|u|^2dx+\int r^{-\sigma}\psi^2|Au|^2dx).
\end{array}
\end{equation}
Adding \eqref{eq500} to \eqref{eq150} and taking $\beta$ sufficiently large and $\varepsilon$ small enough, we conclude that
$$\beta^4\int r^{-\sigma-3\beta-4}\psi^2|u|^2dx+\beta^2\int r^{-\sigma-\beta-2}\psi^2|\nabla u|^2dx\leq C\int r^{-\sigma}\psi^2|Au|^2dx$$ which immediately implies \eqref{car2}.



\section{Applications to inverse problems}
\setcounter{equation}{0}

In this section we will discuss the application of the UCP for \eqref{elastic} to the inverse problem of identifying inclusions or cavities by boundary measurements. To begin, assume that $D$ is an open subset of $\Omega$ with Lipschitz boundary such that $\Omega\setminus\bar{D}$ is connected. The domain $D$ stands for region of the inclusion or cavity embedded in $\Omega$. Let the reference elasticity tensor $\C(x)$ with components $C_{ijkl}(x)$ be defined by \eqref{etensor}, i.e.
$$C_{ijkl}=\tilde{\lambda}\delta_{ij}\delta_{kl}+(\tilde{\mu}\delta_{jl}+t_{jl})\delta_{ik}+\tilde{\mu}\delta_{il}\delta_{jk},$$
where $\tilde{\lambda}=\lambda+\beta_1(\mbox{tr}T)$ and $\tilde{\mu}=\mu+(1/2)\beta_2(\mbox{tr}T)$. Here we require that the Lam\'e moduli satisfy the strong convexity condition
\begin{equation}\label{convex}
\mu(x)>\delta>0\quad\mbox{and}\quad n\lambda(x)+2\mu(x)>\delta>0\quad\forall\ x\in\Omega
\end{equation}
which is equivalent to
\begin{equation}\label{acon}
\C(x)E\cdot E\geq\kappa E_{ij}E_{ij}=\kappa|E|^2,\quad\kappa>0,\ \forall\ x\in\Omega
\end{equation}
for all matrix $E$ provided that $\varepsilon$ in \eqref{srs} is sufficiently small. It is obvious that \eqref{convex} implies \eqref{ellip}. Next we assume that $\tilde{\C}$ is some fourth-rank tensor such that $\C+\chi_D\tilde{\C}$ satisfies the strong convexity condition \eqref{acon}, where $\chi_D$ denotes the characteristic function of $D$. Moreover, suppose that $\tilde{\C}$ satisfies the following jump condition
\begin{equation}\label{jump}
\forall\ x\in\partial D,\ \exists\ C_x>0,\ \exists\ \delta_x>0\ \mbox{such that}\ \tilde{\C}(y)E\cdot E\geq C_x|E|^2\ \mbox{or}\ \tilde{\C}(y)E\cdot E\leq -C_x|E|^2
\end{equation}
for almost all $y\in B_{\delta_x}(x)\cap D$ and all real matrices $E$. Let all components of $\C(x)$ and $\tilde{\C}(x)$ be in $L^{\infty}(\Omega)$, then it is easy to show that there exists a unique solution $u\in H^1(\Omega)$ to
$$
\begin{cases}
\nabla\cdot ((\C+\chi_{D}\tilde{\C})\nabla u)=0\quad\mbox{in}\ \Omega,\\
u=f\quad\mbox{on}\ \partial\Omega
\end{cases}
$$
for any $f\in H^{1/2}(\partial\Omega)$. In this case, the domain $D$ is an inclusion. So we can define the Dirichlet-to-Neumann (displacement-to-traction) map $\Lambda_I:H^{1/2}(\partial\Omega)\to H^{-1/2}(\partial\Omega)$ by
$$\Lambda_I(f)=(\C\nabla u)\nu|_{\partial\Omega}.$$ We are interested in the following inverse problem


{\bf IP.A} Reconstruct the inclusion $D$ from the knowledge of $\Lambda_I(f)_{\Gamma_0}$ for infinitely many $f\in H^{1/2}(\partial\Omega)$ with $\supp(f)\subset\Gamma_0$, where $\Gamma_0$ is a non-empty subset of $\partial\Omega$.

Likewise, in the extreme case, if the tensor $\tilde{\C}$ becomes $-{\C}$, then the domain $D$ corresponds to a cavity. In the same way, we can prove that there exists a unique solution $u\in H^1(\Omega\setminus\bar{D})$ to the following boundary value problem 
$$
\begin{cases}
\nabla\cdot (\C\nabla u)=0\quad\mbox{in}\ \Omega\setminus\bar{D},\\
(\C\nabla u)\nu=0\quad\mbox{on}\ \partial D,\quad (\C\nabla u)\nu=g\quad\mbox{on}\ \partial \Omega
\end{cases}
$$ 
for any $g\in H^{1/2}(\partial\Omega)$. Therefore, we can define the Dirichlet-to-Neumann map $\Lambda_C:H^{1/2}(\partial\Omega)\to H^{-1/2}(\partial\Omega)$ by
$$\Lambda_C(g)=(\C\nabla u)\nu|_{\partial\Omega}.$$ Similarly, we will consider the inverse problem


{\bf IP.B} Reconstruct the cavity $D$ from the knowledge of $\Lambda_C(g)|_{\Gamma_0}$ for infinitely many $g$ with $\supp(g)\subset\Gamma_0$. 

Note that explicit reconstruction algorithms for recovering the inclusion or cavity embedded in an elastic body have been developed in \cite{int} and \cite{ikna} where the reference medium is assumed to be either inhomogeneous isotropic or anisotropic with homogeneous or analytic elasticity tensors. Here we want to extend their results to the elasticity system with residual stress \eqref{elastic}. To this end, we will need the Runge approximation property with constraint for
\eqref{elastic} which is a consequence of the UCP (see Corollary~\ref{thm2}). Its proof
can be found in \cite{ikna}.
\begin{pr}\label{runge}
Assume that all coefficients of $\C$ are in $W^{2,\infty}(\Omega)$ and the residual stress satisfies \eqref{srs} with $\varepsilon$ given in Corollary~\ref{thm2}. Let $U$ and $\Omega$ be two
open bounded domains with Lipschitz and $C^2$ boundaries, respectively, such
that $\bar{U}\subset\Omega$. Denote $\Gamma_0$ a subset of the
boundary $\partial\Omega$. Let $u\in H^1(U)$ satisfy $$\nabla\cdot(\C\nabla u)=0\quad\mbox{in}\quad U.$$ Then for any compact subset
$K\subset U$ such that $\Omega\setminus K$ is connected and any $\tilde{\varepsilon}>0$ there exists $w\in H^1(\Omega)$
satisfying $$\nabla\cdot(\C\nabla w)=0\quad\mbox{in}\quad\Omega$$ with
$\supp(w|_{\partial\Omega})\subset\Gamma_0$ such that
$$\|w-u\|_{H^1(K)}<\tilde{\varepsilon}.$$
\end{pr}
{\bf Remark}: The reason for using $C^2$ boundary on $\Omega$ is that we want to extend all coefficients of $\C$ into a larger domain $\tilde{\Omega}$ and the newly extended coefficients have the same regularity $W^{2,\infty}$ in $\tilde{\Omega}$.


Having the Runge approximation property Proposition~\ref{runge} at hand, we now can apply the methods in \cite{int} and \cite{ikna} to solve {\bf IP.A} and {\bf IP.B}. It should be pointed out that the reference elasticity tensor in \cite{int} and \cite{ikna} satisfies the full symmetry properties, i.e.
$$C_{ijkl}=C_{klij}=C_{jikl}.$$ Nevertheless, it is not hard to check that the proofs in \cite{int} and \cite{ikna} are still valid if we only assume $C_{ijkl}=C_{klij}$ which is the case for the elasticity system with residual stress \eqref{elastic}. Precisely, we can prove that
\begin{theorem}[Identification of inclusion]\label{inc}
Let the domain $\Omega$ have $C^2$ boundary. Assume that the elasticity tensor $\C$ given by \eqref{etensor} possesses $W^{2,\infty}(\Omega)$ coefficients satisfying \eqref{convex}. Furthermore, the residual stress tensor $T$ in $\C$ satisfies the smallness condition described in Corollary~\ref{thm2}. Let $(D_1,\tilde{\C}_1)$ and $(D_2,\tilde{\C}_2)$ be two inclusions such that $\C+\chi_D\tilde{\C}_i$ and $\tilde{\C}_i$ satisfy \eqref{acon} and \eqref{jump}, respectively, and $\Omega\setminus\bar{D}_i$ is connected, $i=1,2$. If 
$$\Lambda_{I_1}(f)=\Lambda_{I_2}(f)\quad\mbox{on}\quad\Gamma_0$$ for all $f\in H^{1/2}(\partial\Omega)$ with $\supp(f)\subset\Gamma_0$, then 
$$D_1=D_2.$$
\end{theorem}

\begin{theorem}[Identification of cavity]\label{cav}
Let the assumptions in Theorem~\ref{inc} on $\Omega$ and $\C$ hold. Assume that $D_1$ and $D_2$ are two cavities and $\Omega\setminus\bar{D}_1$ and $\Omega\setminus\bar{D}_2$ are connected. Let 
$$\Lambda_{C_1}(f)=\Lambda_{C_2}(f)\quad\mbox{on}\quad\Gamma_0$$
for all $f\in H^{1/2}(\partial\Omega)$ with $\supp(f)\subset\Gamma_0$, then $D_1=D_2$.
\end{theorem}

\noindent {\bf Remarks}:\\
{\bf 1}. We do not assume $\tilde{\C}_1=\tilde{\C}_2$ in Theorem~\ref{inc}. Also, the regularity of the medium inside of the inclusions is only assumed to be essentially bounded. \\
{\bf 2}. In \cite{ikna}, a reconstruction algorithm for recovering the cavity $D$ is given. The same algorithm can be applied to Theorem~\ref{cav} as well.






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\end{document}

