---
abstract: 'We describe the main results and the ideas of the proofs in the papers \[E\] and \[ER2\] (see References). In addition, we simplify the construction of asymptotic solutions in \[E\], using the results of \[ER2\], and we simplify the proof of estimate (3.9) that was given in \[ER2\].'
author:
- |
    Gregory Eskin and James Ralston,\
    Department of Mathematics, UCLA,\
    Los Angeles, CA 90095-1555, USA.  E-mail: eskin@math.ucla.edu
title: '**Inverse Boundary Value Problems for Systems of Partial Differential Equations.**'
---

The Schrödinger Equation with an External Yang-Mills Potential. {#section 1}
===============================================================

Let $\Omega$ be a smooth bounded domain in ${{\bf R}}^n,\ n\geq 3$. Consider the Dirichlet problem, $u=f$ on $\partial \Omega$, for a system of differential equations of the form $$\label{eq:1.1}
-\Delta u-2i\Sigma_{k=1}^n A_k(x)\frac{\partial u}{\partial x_k} + Bu=0,
\ x\in  \Omega,$$ where $u=(u_1,\dots,u_m)$, and $A_k,\ k=1,\dots,n,$ and $B$ are smooth $m\times m$ matrix functions. We assume that $\Omega$ is such that the Dirichlet problem has a unique solution $u\in    H_1(\Omega)$ for every $f\in H_{\frac{1}{2}}(\partial \Omega)$. Let $\Lambda$ be the Dirichlet-to-Neumann operator, $$\Lambda f=\frac{\partial u}{\partial\hat{n}}+iA\cdot\hat{n} u\mbox{ on }\partial\Omega,$$ where $\hat{n}$ is the exterior unit normal to $\partial\Omega$ and $u$ is the solution of (\[eq:1.1\]) with $u=f$ on $\partial\Omega$. The inverse boundary value problem is to find the coefficients in (\[eq:1.1\]) given $\Lambda$. When one rewrites (\[eq:1.1\]) in the form $$\label{eq:1.2}
(-i\frac{\partial}{\partial x}+A(x))^2u +V(x)u=0$$ with $A=(A_1,..,A_n)$ and $V=B-\Sigma_{k=1}^n A_k^2+i\Sigma_{k=1}^n \frac{\partial  A_k}{\partial x_k}$, it becomes the time-independent Schrödinger equation for a particle under the influence of the Yang-Mills potential $(A,V)$. We say that two Yang-Mills potentials $(A^{(1)},V^{(1)})$ and $(A^{(2)},V^{(2)})$ are gauge equivalent if there exists a smooth, invertible matrix function $g(x)$ on $\overline{\Omega}$ such that $$\label{eq:1.3}
A^{(2)}=g^{-1}A^{(1)}g-ig^{-1}\frac{\partial g}{\partial x}
\mbox{ and }  V^{(2)}=g^{-1}V^{(1)}g.$$ The following theorem was proven in \[E\].

\[theo:1.1\] Let $L^{(j)}u=0,\ j=1,2,$ be two Schrödinger equations of the form (\[eq:1.2\]) in $\Omega\subset {{\bf R}}^n,\ n\geq 3$, with Yang-Mills potentials $(A^{(j)},V^{(j)}),
\ j=1,2,$ and let $\Lambda^{(j)},\ j=1,2,$ be their Dirichlet-to-Neumann operators. Assume that $\Omega$ is convex. Then $\Lambda^{(1)}=\Lambda^{(2)}$ if and only if $(A^{(1)},V^{(1)})$ and $(A^{(2)},V^{(2)})$ are gauge equivalent.

The proof of Theorem \[theo:1.1\] is based on the method of complex exponential solutions with a large parameter that was introduced in \[SU\]. Let $\mu$,$\nu$ and $l$ be pairwise orthogonal vectors in ${{{\bf R}}}^n$ with $|\mu|=|\nu|=1$. We set $\theta=\mu+i\nu$, and for $\tau >>0$ we set $\zeta =l/2 + (\tau^2-|l|^2/4)^{1/2}\mu$ and $\delta=\zeta+i\tau\nu$. We look for solutions of (\[eq:1.2\]) of the form $u=v\exp(ix\cdot \delta)$. Hence $v$ must satisfy $$\label{eq:1.4}
 L_\delta v=_{\mbox{def.}}(-i\frac{\partial}{\partial x} 
+\delta)^2v -2iA(x)\cdot  (\frac{\partial}{\partial x} 
+i\delta)v +B(x)v=
0.$$ In order to solve systems of the form (\[eq:1.4\]) or, more generally, inhomogeneous systems $$\label{eq:1.5}
L_\delta v =f\mbox{ in }\Omega_0,$$ where $\overline{\Omega}\subset \Omega_0$, we will need solutions of matrix equations of the form $$\label{eq:1.6}
i\theta\cdot \frac{\partial C(x,\theta)}{\partial  x}
=\theta\cdot A(x)C(x,\theta),\ x\in\Omega_0,$$ where $C$ is an invertible $m\times m$ matrix function. One can show that, when $A(x)$ is extended to a function of compact support in ${{{\bf R}}}^n$, there may be no solution of (\[eq:1.6\]) which tends to the identity matrix as $|x|\rightarrow\infty$. However, there are always solutions which grow polynomially. The following lemma was proven in \[E\].

\[lma:1.1\] There exists an invertible matrix function $C(x,\theta)$, solving (\[eq:1.6\]) and depending smoothly on $(x,\theta)$ on the domain $\overline{\Omega_0}
  \times\{ \theta=\mu+i\nu: \mu\cdot\nu=0,\ |\mu|=|\nu|=1\}$. This solution is not unique, but it can be chosen to satisfy $C(x,e^{i\omega}\theta)=C(x,\theta)$.

Using Lemma \[lma:1.1\], we can prove (see \[E\]):

\[lma:1.2\] For any $f\in L^2(\Omega_0)$ there is a $v(\tau)\in H^2(\Omega_0)$ such that for $\tau$ sufficiently large $v(\tau)$ solves (\[eq:1.5\]) in $\Omega$, and $$\|v\|_{H^l(\Omega_0)}\leq \frac{C}{ (1+\tau)^{1-l}}\|f\|_{L^2(\Omega_0)},\ l=0,1,$$ where $C$ is independent of $\tau$ and $f$.

To prove Lemma \[lma:1.2\] we proceed as follows. Let $C(x,\mu+i\nu)$ be as in Lemma \[lma:1.1\], and let $c(x,D,\tau)$ be the pseudo-differential operator with symbol $C(x,\frac{\xi^\prime +\zeta}{  |\xi^\prime +\zeta|} +i\nu)\chi$, where $\xi^\prime =\xi-(\xi\cdot\nu)\nu$ and $\chi$ is a suitable cutoff function in $(\zeta,\xi,\tau)$. We look for $v$ in the form $$\label{eq:1.7}
v=c(x,D,\tau)E(\tau)g,\mbox{ where }E(\tau)=
(-i\frac{\partial}{\partial x}  +\delta)^{-2}\mbox{ and }g\in L^2(\Omega_0).$$ Substituting (\[eq:1.7\]) into (\[eq:1.5\]) one gets $g+T(\tau)g=f$, where the norm of $T(\tau)$ goes to zero as $\tau\rightarrow\infty$. Therefore $I+T(\tau)$ is invertible for $\tau$ large. Note that the proof of Lemma \[lma:1.2\] in \[E\] is a generalization of the method in \[ER1\] which treated the (scalar) case of electro-magnetic potentials. Lemma \[lma:1.2\] is used in \[E\] to prove the following:

\[lma:1.3\] For every vector of polynomials, $p(z)$, in the complex variable $z=\theta\cdot x$ there is a solution $v(\tau)$ of (\[eq:1.4\]) satisfying $$\label{eq:1.8}
\lim_{\tau
\rightarrow \infty}v(\tau)=C(x,\theta)\Pi^+(C^{-1}p),$$ where $C$ is the matrix function from Lemma \[lma:1.1\] and $\Pi^+$ is a Toeplitz projection.

Using (\[eq:1.7\]) to construct the leading order term in $v(\tau)$ made both the proof of Lemma \[lma:1.3\] and its applications somewhat complicated in \[E\]. In \[ER2\] we found a way to construct such solutions more simply with explicit higher order asymptotics as $\tau\to \infty$, and this lead to simpler proofs. We will give this construction in §2.

Now we can complete the proof of Theorem \[theo:1.1\]. Following the strategy of \[SU\], we can use the assumption $\Lambda^{(1)}=\Lambda^{(2)}$, Green’s formula and Lemma \[lma:1.3\] to derive integral identities involving $C^{(j)}$ and $(A^{(j)},V^{(j)}),\ j=1,2$. Then arguments involving $\overline{\partial}$-equations in the parameters in these identities (see §5 and §6 in \[E\]) lead to the proof of Theorem \[theo:1.1\].

Construction of Complex Exponential Solutions. {#section 2}
==============================================

To construct solutions of (\[eq:1.1\]) of the form $u=v\exp(ix\cdot \delta)$ we will proceed as follows. Substituting $u=v\exp(ix\cdot \delta)$ into (\[eq:1.1\]), one sees that $v$ must satisfy $$\label{eq:2.1}
L_\delta v=_{\mbox{def.}}-\Delta v-2i\delta\cdot\frac{\partial v}{\partial x}
-2i A\cdot(i\delta v +\frac{\partial v}{\partial x})+Bv=0.$$ We will construct $v$ in the form $v=\Sigma_{k=0}^nv_k +\tilde{v}_n$, where $v_k=O(\tau^{-k})$ and is explicit modulo solutions of (\[eq:1.6\]), and $\tilde v_n$ is $O(\tau^{-n-1})$. We have $$\delta =\tau\theta +\frac{l}{2} +O(\tau^{-1})
=_{\mbox{def.}}\tau\theta +\delta^\prime  \mbox{ and }$$ $$L_\delta
=_{\mbox{def.}}-2i\tau\theta\cdot\frac{\partial}{\partial x} 
+2\tau \theta\cdot A  +
M_{\delta^\prime}.$$ To solve (\[eq:2.1\]) modulo terms of order $\tau^{-n}$ we require $$2\tau(i\theta\cdot\partial_x -\theta\cdot A)v_k=M_{\delta^\prime}v_{k-1}$$ for $k=0,\dots,n$ with $v_{-1}=0$. We set $v_0=C_0(x,\theta)p(x\cdot \theta)$, where $C_0$ is a solution of (\[eq:1.6\]) with the properties described in Lemma \[lma:1.1\] and $p(z)$ is a vector of polynomials in the complex variable $z$. Since we only require that $v$ satisfy (\[eq:2.1\]) on $\Omega$, we can introduce a cut-off function $\psi\in C_0^\infty(\Omega_0)$, $\psi =1$ on a neighborhood of $\Omega$, and set $$\label{eq:2.2}
v_k=-i(2\tau)^{-1}C_k(x,\theta)(\theta\cdot\partial_x)^{-1}
(C_k^{-1}(\cdot,\theta)\psi  M_{\delta\prime}v_{k-1}),\ k=1,\dots,n.$$ In (\[eq:2.2\]) the operator $(\theta\cdot\partial_x)^{-1}$ multiplies the Fourier transform by $(i\theta\cdot \xi)^{-1}$, and $C_k$ is again a solution of (\[eq:1.6\]) as in Lemma \[lma:1.1\] (in applications so far we have taken $C_k=C_0$, but this is not necessary). Since $M_{\delta^\prime}$ does not increase the order of a term in $\tau$ and $i(2\tau)^{-1}(\theta\cdot\partial_x)^{-1}$ adds a factor of $\tau^{-1}$, we have $v_k=O(\tau^{-k})$ and $$L_\delta(v_0+v_1+\cdots +v_n)=M_{\delta^\prime }v_n=O(\tau^{-n})$$ on the neighborhood of $\Omega$ where $\psi=1$. Hence, taking $\psi_0\in C^\infty(\Omega_0)$ supported in the set where $\psi=1$ such that $\psi_0=1$ on $\Omega$, $v=v_0+\cdots +v_n+\tilde v_n$ will be a solution of (\[eq:2.2\]) in $\Omega$ if $$\label{eq:2.3}
L_\delta\tilde v_n=-\psi_0 M_{\delta^\prime} v_n$$ in $\Omega_0$.

To solve (\[eq:2.3\]) apply Lemma \[lma:1.2\]. Lemma \[lma:1.2\] holds in a more general form (see \[ER2\]): if $f\in  H^k(\Omega_0),\ k\geq 0$, then $v\in H^{k+2}(\Omega_0)$, and one has the estimate $$\|v\|_{H^{k+l}(\Omega_0)}\leq \frac{C}{(1+\tau)^{1-l}}\|f\|_{H^k(\Omega_0)},\ l=0,1.$$ Since $v_n$ is bounded by $C\tau^{-n}$, the solutions given by Lemma \[lma:1.2\] will be bounded by $C\tau^{-n-1}$ in $H^k(\Omega_0)$ for $k\geq 0$. Thus the $v(\tau)$ that we have constructed is a solution of (\[eq:2.1\]) whose asymptotics in $H^k(\Omega)$ up to order $\tau^{-n}$ are given by the asymptotics of $v_0+\cdots +v_n$. For the leading term in the asymptotics we have $$\label{eq:2.4}
v(\tau)=C_0(x,\theta)p(\theta\cdot x)+O(\tau^{-1}),$$ where $O(\tau^{-1})$ means bounded by $C\tau^{-1}$ in $H^k(\Omega), k\geq 0$. Since the limit of $v(\tau)$ no longer involves a Toeplitz projection, one no longer needs one of the arguments (Lemma 5.1) in \[E\].

The Equations of Isotropic Elasticity. {#section 3}
======================================

The construction presented in §2 can be used to construct solutions of the system of isotropic elasticity as well. Using subscripts for derivatives, this system is given by $$\label{eq:3.1}
 \Sigma_{j=1}^3(\lambda w^j_{x_j})_{x_k} + \Sigma_{j=1}^3(\mu(w^k_{x_j}
+w^j_{x_k}))_{x_j}=0,\ k=1,2,3,$$ where $w=(w^1,w^2,w^3)$ is the deformation of an elastic body with Lamé parameters" $\lambda(x)$ and $\mu(x)$. Let $w$ be the solution of the Dirichlet problem for (\[eq:3.1\]) with $w=h$ on $\partial \Omega$. Then the inverse boundary value problem for this system is to recover $\lambda$ and $\mu$ from the Dirichlet-to-Neumann map $$\Lambda(h)^k=\Sigma_{j=1}^3(\lambda   w^j_{x_j})\hat{n}^k+\Sigma_{j=1}^3\mu(w^k_{x_j}+w^j_{x_k})\hat{n}^j,
\  k=1,2,3.$$ There is as yet no proof that $\Lambda$ determines $\lambda$ and $\mu$. Partial results are given in \[NU2\] and \[ER2\]. The system (\[eq:3.1\]) is not in the form (\[eq:1.1\]). However, Ang, Ikehata, Trong and Yamamoto in \[AITY\] show that $w=\mu^{-1/2}u+\mu^{-1}\nabla f-f\nabla \mu^{-1}$ will satisfy (\[eq:1.1\]) when the 4-vector $(u,f)$ satisfies the system $$\label{eq:3.2}
\Delta\left(\begin{array}{c} u\\ f\end{array}\right) 
+V_1(x)\left(\begin{array}{c} \nabla f\\ \nabla\cdot  u\end{array}\right)
+V_0(x)\left(\begin{array}{c} u\\ f \end{array}\right)
=\left(\begin{array}{c} 0\\  0\end{array}\right).$$ Here $$V_1(x) =\left(\begin{array}{ll} -2\mu^{1/2}\nabla^2\mu^{-1}
& -\mu^{-1}\nabla \mu\\
0
& \frac{\lambda +\mu}{\lambda +2\mu}\mu^{1/2}\end{array} \right)$$ and $\nabla^2f$ denotes the Hessian matrix $\partial^2f/\partial x_j\partial x_k$. The matrix $V_0$ is a complicated expression in $\lambda$, $\mu$ and their derivatives, but it vanishes when $\mu$ is constant. Since (\[eq:3.2\]) does have the form (\[eq:1.1\]), the method above can be used to construct solutions $(u,f)=\exp(i\delta\cdot x)(r,s)$ of (\[eq:3.1\]) with prescribed asymptotics as $\tau\rightarrow\infty$.

Suppose that one has two elastic bodies occupying the region $\Omega$ with Lamé parameters $(\lambda^{(j)},\mu^{(j)}).\ j=1,2$. Let $\Lambda^{(j)},\ j=1,2,$ be the corresponding Dirichlet-to-Neumann maps. Using Green’s formula, one can verify that $\Lambda^{(1)}=\Lambda^{(2)}$ is equivalent to $$\begin{aligned}
                             \label{eq:3.3}
  H(w^{(2)},w^{(1)})
=_{\mbox{def.}}\int_\Omega[(\lambda^{(2)}-\lambda^{(1)})
(\nabla \cdot\overline{w^{(2)}})(\nabla \cdot w^{(1)})
\\
+ \frac{1}{2}(\mu^{(2)}-\mu^{(1)})
\Sigma_{1\leq j,k\leq 3}
(\overline{w^{(2),j}_k+w^{(2),k}_j})(w^{(1),j}_k+w^{(1),k}_j)]dx=0
\nonumber\end{aligned}$$ for all solutions to $L^{(1)}w^{(1)}=0,\ L^{(2)}w^{(2)}=0$, where now $L^{(j)}w^{(j)}=0$ is the system (\[eq:3.1\]) with $\lambda=\lambda_j$ and $\mu=\mu_j$. We can use (\[eq:3.3\]) in the following way. The method of §2 yields solutions of (\[eq:3.2\]) with prescribed asymptotics. For the problem corresponding to $L^{(1)}$ we use these with $\delta^{(1)}=\delta$ as defined earlier, but for the problem corresponding to $L^{(2)}$ we set $\delta^{(2)}=\overline{\delta} -l$. Taking $w^{(j)}=
\mu_j^{-1/2}u^{(j)}+\mu_j^{-1}\nabla f^{(j)}-f^{(j)}\nabla \mu_j^{-1}$, we substitute $w^{(1)}$ and $w^{(2)}$ into (\[eq:3.3\]). When we collect the terms of each order in $\tau$, this gives $$0=H(w^{(2)},w^{(1)})=\tau^2H_2+\tau^1H_1+H_0+ \cdots + \tau^{-N}H_{-N}+O(\tau^{-N-1}),$$ where each $H_j$ is independent of $\tau$. How far one can continue this expansion depends on the choice of $n$ in §2. Note that each $H_j$ must vanish when $\Lambda^{(1)}=\Lambda^{(2)}$. In particular, one has $$\label{eq:3.4}
H_2=\int_\Omega e^{il\cdot x}(\theta\cdot\overline{r}_0^{(2)},
\overline{s}_0^{(2)})V(x,\theta)(\theta\cdot r_0^{(1)},s_0^{(1)})^tdx=0,$$ where $$V(x,\theta)=\left(\begin{array}{ll}
(\lambda_1+\mu_1-\lambda_2-\mu_2)
\frac{(\mu_1\mu_2)^{1/2}}{(\lambda_1+2\mu_1)(\lambda_2+2\mu_2)}
&   2(\mu_2^{-1}-\mu_1^{-1})\mu_2^{-1/2}\theta\cdot\partial_xb_2
\\ 
2(\mu_2^{-1}-\mu_1^{-1})\mu_1^{-1/2}\theta\cdot\partial_xb_1
&  2(\mu_2^{-1}-\mu_1^{-1})(b_1a_1+b_2a_2)\end{array}\right).$$ The functions $a_j$ and $b_j,\ j=1,2$ are given by $$a_j=(\theta\cdot\partial_x)^2\mu_j^{-1}\ \mbox{ and }\ b_j=\frac{\mu_j}
{ 2}\frac{\lambda_j+\mu_j}{\lambda_j+2\mu_j}.$$ The functions $(r_0^{(j)},s_0^{(j)})$ are vector solutions of the following version of (\[eq:1.6\]) $$\label{eq:3.5}
-2\theta\cdot \partial_x\left(
\begin{array}{c}r_0\\s_0\end{array}\right)=V_1\left(\begin{array}{c} s_0\theta^t\\
\theta\cdot r_0\end{array}\right).$$ This is (\[eq:1.6\]) with $$A=-\frac{i}{2}V_1\left(\begin{array}{ll}0_{3\times 3}&\theta^t\\
\theta&0 \end{array}\right).$$ One can analyze $H_2$ by the techniques that were used for the Yang-Mills case in \[E\]. However, here that does not lead to the conclusion that $\lambda_1=\lambda_2$ and $\mu_1=\mu_2$. Instead one arrives at the identity (Theorem 1.3 of \[ER2\]) $$b_1^{1/2}(\theta\cdot\partial_x)^2(b_1^{-1/2})-b_1(\theta\cdot\partial_x)^2
 (\mu_1^{-1})=b_2^{1/2}(\theta\cdot\partial_x)^2(b_2^{-1/2})
-b_2(\theta\cdot\partial_x)^2
 (\mu_2^{-1}).$$ Since this holds identically in $\theta$, it is equivalent to a system of five partial differential equations satisfied by $(\lambda_1,\mu_1,\lambda_2,\mu_2)$. However, even in the case that $\lambda_1$ and $\mu_1$ are constant, these equations have solutions with $\lambda_2$ and $\mu_2$ nonconstant.

A more readily useful result is the following (Theorem 2 of \[ER2\]):

\[theo:3.1\] Let $$\|f\|_\alpha^2=\int_\Omega |f|^2e^{2\alpha
|x|^2}dx.$$ Then, if $\Lambda^{(1)}=\Lambda^{(2)}$, for $\alpha>\alpha_0$, one has $$\|\lambda_1+\mu_1-\lambda_2-\mu_2\|_\alpha\leq 
\frac{C}{\alpha}\|\mu_1-\mu_2\|_\alpha$$ with $C$ independent of $\alpha$.

This result comes from $\overline{\partial}$-equations that arise when one follows the method of §6 in \[E\]. Theorem \[theo:3.1\] can be used to deduce uniqueness for constrained forms of the inverse problem. For instance, if one is given either that $\lambda_1=\lambda_2$ or that $\mu_1=\mu_2$, and that the Dirichlet-to-Neumann maps are equal, then both Lamé parameters must be equal.

The second way that one can use (\[eq:3.4\]) is to choose $\theta$ as a function of $l/|l|$ keeping $\theta(l/|l|)\cdot l=0$. This makes (\[eq:3.4\]) (and each of the other equations in the sequence $\{H_j=0\}_{j=2}^{-\infty}$) equivalent to a pseudo-differential equation of the form $$\label{eq:3.6}
P(x,D)(\lambda_2-\lambda_1)+Q(x,D)(\mu_2-\mu_1)=0$$ To prove uniqueness for the inverse boundary value problem we would like to use (\[eq:3.6\]) to bound $\lambda_2-\lambda_1$ in terms of $\mu_2-\mu_1$ or to bound $\mu_2-\mu_1$ in terms of $\lambda_2-\lambda_1$. Boundary determination for this inverse problem (see \[NU1\]) implies that $\lambda_2-\lambda_1$ and $\mu_2-\mu_1$ vanish to infinite order on $\partial \Omega$. Nonetheless, (\[eq:3.6\]) does not imply any estimates of this kind without more information on the operators $P$ and $Q$. Note that the construction in §2 makes $(r,s)$ a function of $\theta$, so that it contributes to the symbols of $P$ and $Q$,and, since the solutions of (\[eq:3.5\]) are not explicit in general, one usually does not know what $P$ and $Q$ are. However, (\[eq:3.6\]) becomes useful when one assumes that $\nabla \mu_j$ is small in $C^\infty$. In that case all entries in $V_1$ except $(V_1)_{44}$ become small and one can solve (\[eq:3.5\]) explicitly modulo small terms. In fact (\[eq:3.5\]) has a unique matrix solution tending to the identity as $|x|\rightarrow \infty$ and given by $$\label{eq:3.7}
C_{00}(x,\theta)=
\left(\begin{array}{ll} I_{3\times 3}
&  0  \\
\varphi\theta
&   1     \end{array} \right),$$ modulo small terms, where $\varphi=(\theta\cdot \partial_x)^{-1}(-\psi b\mu^{-1})$. Choosing $$p=(\hbox{Re}\{\theta(l/|l|)\},0)$$ in (\[eq:2.4\]), one gets $$\label{eq:3.8}
(r^{(j)},s^{(j)})=(\hbox{Re}\{\theta(l/|l|)\},0)+
(\tilde{r}_0^{(j)},\tilde{s}_0^{(j)})+ O(\tau^{-1}),$$ where $\tilde{r}_0^{(j)},\tilde{s}_0^{(j)}$ are symbols of order zero and $\tilde{r}_0^{(j)}$ is small. Since $\theta\cdot\hbox{Re}\{\theta\}=1$, (\[eq:3.8\]) makes $P(x,D)$ in (\[eq:3.6\]) simply multiplication by $(\mu_1\mu_2)^{1/2}(\lambda_1+2\mu_1)^{-1}(\lambda_2+2\mu_2)^{-1}$ plus an operator of order zero whose norm can be made arbitrarily small by taking $\nabla \mu_j,\ j=1,2,$ sufficiently small in $C^k(\Omega)$. Thus (\[eq:3.6\]) implies $$\label{eq:3.9}
\|\lambda_2-\lambda_1\|_{H^k(\Omega)}\leq C_k\|\mu_2-\mu_1\|_{H^k(\Omega)}$$ for all $k$, when $\nabla \mu_j,\ j=1,2,$ is sufficiently small.

Uniqueness for the inverse boundary value problem when $\nabla \mu_j$ is small will follow from (\[eq:3.9\]) if we can find another estimate bounding a norm of $\mu_2-\mu_1$ by a small constant times a norm of $\lambda_2-\lambda_1$. One can get this estimate by using $H_0=0$ in the following way. The contributions to $H_0$ come from the expansion of $(r,s)$ in $\tau$ up to order $\tau^{-2}$, and are therefore quite complicated in general. Nonetheless, using (\[eq:3.7\]) and taking $p=(0,0,0,1)$ in (\[eq:2.4\]), we find that (\[eq:3.5\]) has the solution $(r_0,s_0)=(0,0,0,1) +(\tilde{r}_0,\tilde{s}_0)$, where $(\tilde{r}_0,\tilde{s}_0)$ and its derivatives can be made arbitrarily small by taking $\nabla \mu_j,\ j=1,2,$ sufficiently small in $C^k$-norm. Moreover, checking further, one sees that the higher order terms in the expansion of $(r,s)$ that one constructs using (\[eq:2.2\]) are also small under these hypotheses. When one uses this choice of $(r,s)$ in the construction of $w^{(1)}$ and $w^{(2)}$, one gets $$\begin{aligned}
  \nonumber                       
H_0=
\int_\Omega e^{il\cdot x}[(\lambda_1-\lambda_2)A(x,\theta(l/|l|),l)\\+
(\mu_1-\mu_2)[(2\mu_1\mu_2)^{-1}|l|^4
+ B(x,\theta(l/|l|),l)]]dx,\nonumber\end{aligned}$$ where $A$ and $B$ are small symbols in $l$ of orders two and four respectively. Thus we have (\[eq:3.6\]) with $\|P(x,D)(\lambda_2-\lambda_1)\|_{L^2(\Omega)}$ bounded by a small constant times $\|\lambda_2-\lambda_1\|_{H^2(\Omega)}$ and $$Q(x,D)(\mu_2-\mu_1)=(2\mu_1\mu_2)^{-1}(\Delta)^2(\mu_2-\mu_1)
+R(x,D)(\mu_2-\mu_1),$$ where $\|R(x,D)(\mu_2-\mu_1)\|_{L^2(\Omega)}$ is bounded by a small constant times $\|\mu_2-\mu_1\|_{H^4(\Omega)}$. Thus we have $$\label{eq:3.10}
\|\mu_2-\mu_1\|_{H^4(\Omega)}\leq\varepsilon (\|\mu_2-\mu_1\|_{H^4(\Omega}
+\|\lambda_1-\lambda_2\|_{H^2(\Omega)}),$$ where $\varepsilon$ can be taken arbitrarily small when $\nabla \mu_j,\ j=1,2$ is sufficiently small in $C^k(\Omega)$. Combining (\[eq:3.9\]) and (\[eq:3.10\]), we have the following result.

\[theo:3.2\] Given that $\lambda_j,\mu_j$ and $\mu_j^{-1},\ j=1,2$, belong to a bounded set, $B$, in $C^k(\Omega)$ for $k$ sufficiently large, there is an $\varepsilon(B)>0$ such that $||\nabla \mu_j||_{C^{k-1}(\Omega)}<\varepsilon(B),\ j=1,2,$ implies $(\lambda_1,\mu_1)=(\lambda_2,\mu_2)$, if $\Lambda^{(1)}=\Lambda^{(2)}$.

This is the main result of \[NU2\] and it is also Theorem 1 of \[ER2\]. The first version of \[ER2\] (July 2001) contains Theorem \[theo:3.2\] with additional hypothesis $\|\nabla \lambda_j\|_{C^{k-1}(\Omega)}<\varepsilon(B),\ j=1,2.$ We did not notice that our proof did not use this hypothesis until we received a preprint of \[NU2\] (November 2001). We are grateful to G.Nakamura and G.Uhlmann for sending this to us.

[9999]{} Ang, D.D., Ikehata, M., Trong, D.D., and Yamamoto, M.: Unique continuation for a stationary isotropic Lamé system with variable coefficients, CPDE [**23**]{} (1998), 371-376. Eskin, G.: Global uniqueness in the inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials, Commun. Math. Phys. [**222**]{}(2001), 503-531. Eskin, G. and Ralston, J.: Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy, Commun. Math. Phys. [**173**]{}(1995), 199-224. Eskin, G. and Ralston, J.: On the inverse boundary value problem for linear isotropic elasticity, to appear Inverse Problems. Nakamura, G., Uhlmann, G.: Inverse problems at the boundary for an elastic medium, SIAM J. Math. Anal. [**26**]{}(1995), 263-279. Nakamura, G. and Uhlmann, G.: Correction to: Global uniqueness for an inverse boundary problem arising in elasticity(Inventiones Mathematica 118, 457-474(1994)), preprint November 2001. Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math.[**125**]{}(1987), 153-169.
