---
abstract: |
    We study the inverse problem for the second order self-adjoint hyperbolic equation with the boundary data given on a part of the boundary. This paper is the continuation of the author’s paper \[E\].

    In \[E\] we presented the crucial local step of the proof. In this paper we prove the global step. Our method is a modification of the BC-method with some new ideas. In particular, the way of the determination of the metric is new.
author:
- |
    G.Eskin,     Department of Mathematics, UCLA,\
    Los Angeles, CA 90095-1555, USA.  E-mail: eskin@math.ucla.edu
title: 'A new approach to hyperbolic inverse problems II (Global step)'
---

Introduction. {#section 1}
=============

Let $\Omega$ be a bounded domain in ${{\bf R}}^n,\ n\geq 2,$ with smooth boundary $\partial\Omega$. Consider the hyperbolic equation of the form: $$\begin{aligned}
                               \label{eq:1.1}
Lu\stackrel{def}{=}
\frac{\partial^2 u}{\partial t^2}
+\sum_{j,k=1}^n\frac{1}{\sqrt{g(x)}}
\left(-i\frac{\partial}{\partial x_j}+A_j(x)
\right)
 \sqrt{g(x)}g^{jk}(x)
\left(-i\frac{\partial}{\partial x_k}+A_k(x)\right)u
\nonumber
\\
+V(x)u=0\end{aligned}$$ in $\Omega\times(0,T_0)$ with $C^\infty(\overline{\Omega})$ coefficients. Here $ \|g^{jk}(x)\|^{-1}$ is the metric tensor, $g(x)=\det\|g^{jk}\|^{-1}$. We assume that $$\label{eq:1.2}
u(x,0)=u_t(x,0)=0 \ \ \mbox{in}\ \ \ \Omega,\ \ $$ $$u\left|_{\partial\Omega\times(0,T_0)}\right. = f(x,t).$$ Denote by $\Lambda$ the Dirichlet-to-Neumann (D-to-N) operator, i.e. $$\label{eq:1.4}
\Lambda f=\sum_{j,k=1}^n g^{jk}(x)\left(\frac{\partial u}{\partial x_j}+
iA_j(x)u\right)\nu_k
\left(\sum_{p,r=1}^n g^{pr}(x)\nu_p\nu_r\right)^{-\frac{1}{2}}
{\Huge |}_{\partial\Omega\times(0,T_0)},$$ where $\nu=(\nu_1,...,\nu_n)$ is the unit exterior normal to $\partial\Omega$ with respect to the Euclidean metric. Let $\Gamma_0$ be an open subset of $\partial\Omega$. We say that the D-to-N operator is given on $\Gamma_0\times(0,T_0)$ if $\Lambda f{\Huge |}_{\Gamma_0\times(0,T_0)}$ is known for all smooth $f(x,t)$ with supports in $\Gamma_0\times (0,T_0]$.

Denote by $G_0(\overline{\Omega})$ the group of all complex-valued functions $c(x)$ such that $c(x)\neq 0$ in $\overline{\Omega}$ and $c(x)=1$ on $\overline{\Gamma_0}$. We say that potentials $A(x)=(A_1(x),...,A_n(x))$ and $A'(x)=(A_1'(x),...,A_n'(x))$ are gauge equivalent if there exists $c(x)\in G_0(\overline{\Omega})$ such that $$A_j'(x)=A_j(x)-ic^{-1}(x)\frac{\partial c}{\partial x_j},\ \ \ \
1\leq j\leq n.$$ Note that if $Lu=0$ then $$\label{eq:1.5}
u'=c^{-1}(x)u$$ satisfies the equation $L'u'=0$ where $L'$ has the form (\[eq:1.1\]) with $A_j(x)$ replaced by $A_j'(x), \  1\leq j\leq n$. We shall call (\[eq:1.5\]) the gauge transformation.

We shall prove the following theorem:

\[theo:1.1\] Let $L^{(p)},p=1,2$, be two operators of the form (\[eq:1.1\]) in domains $\Omega^{(p)},p=1,2,$ respectively. Let $\Gamma_0\subset\partial\Omega^{(1)}\cap\partial\Omega^{(2)}$ and let $\Lambda^{(p)},p=1,2,$ be the D-to-N operators corresponding to $L^{(p)}, p=1,2.$ Assume that $L^{(p)}$ are self-adjoint, i.e. coefficients $A_1^{(1)}(x),...,A_n^{(1)}(x),V^{(1)}(x)$ and $A_1^{(2)}(x),...,A_n^{(2)}(x),$\
$V^{(2)}(x)$ are real-valued. Suppose $T_0>2\max_{x\in \bar{\Omega}^{(1)}}d_1(x,\Gamma_0)$, where $d_1(x,\Gamma_0)$ is the distance in $\overline{\Omega_1}$ with respect to the metric $\|g_1^{jk}(x)\|^{-1}$ from $x\in \overline{\Omega^{(1)}}$ to $\Gamma_0$. Suppose that the D-to-N operators $\Lambda^{(1)}$ and $\Lambda^{(2)}$ are equal on $\Gamma_0\times(0,T_0)$ for all $f$ with $\mbox{supp\ } f\subset
\Gamma_0\times (0,T_0]$. Then there exists a diffeomorphism $\varphi$ of $\overline{\Omega_2}$ onto $\overline{\Omega_1}$, $\varphi=I$ on $\Gamma_0$, and there exists a gauge transformation $c(x)\in G_0(\overline{\Omega^{(1)}}$ such that $c\circ\varphi\circ L^{(2)}=L^{(1)}$ in $\Omega^{(1)}$.

An important case of the inverse problems with boundary data on a part of the boundary are the inverse problems in domains with obstacles. In this case $\Omega=\Omega_0\setminus\cup_{r=1}^m \overline{\Omega_r}$, where $\Omega_0$ is diffeomorphic to a ball, $\overline{\Omega_1},...,\overline{\Omega_m}$ are smooth nonintersecting domains in $\Omega_0$ called obstacles, $\Gamma_0=\partial\Omega_0$ and zero Dirichlet boundary conditions hold on $\partial\Omega_r,\ 1\leq r\leq m$ (c.f. \[E1\]).

The first result on the inverse problems with the data on a part of the boundary was obtained in \[I\]. The general self-adjoint case was studied by the BC-method (see \[B1\], \[B2\], \[K\], \[KK\], \[KKL\], \[KL1\]). The present paper is a continuation of the paper \[E\] (see also \[E2\]). In \[E\] the crucial local step was considered, i.e. the unique determination of the coefficients of (\[eq:1.1\]) modulo a diffeomorphism and a gauge transformation near $\Gamma_0$.

In this paper we complete the proof of Theorem \[theo:1.1\]. In §2 we state the main results proven in \[E\] and prove the extension lemma. In §3 we refine the results of §2, and in §4 we complete the proof of Theorem \[theo:1.1\].

The summary of the local step and the extension lemma. {#section 2}
======================================================

Let $L^{(p)}, p=1,2,$ be two operators of the form (\[eq:1.1\]) in $\Omega^{(p)}\times(0,T_0),\ L^{(p)}u_p=0$ in $\Omega^{(p)}\times (0,T_0),\ u_p(x,0)=u_{pt}(x,0)=0,\ 
x\in \Omega^{(p)},\ u_p|_{\Gamma_0\times(0,T_0)}=f,\ ,p=1,2.$

Let $\Gamma$ be an open connected subset of $\Gamma_0$ and let $x=(x',x_n)$ be a system of coordinates in a neighborhood $V\subset {{\bf R}}^n$ of $\Gamma$ such that $x_n=0$ is the equation of $\Gamma$ and $ x'=(x_1,...,x_{n-1})$ are local coordinates on $\Gamma$. Introduce semigeodesic coordinates in $V$ corresponding to $\Gamma$ and to the metric $\|g_p^{jk}\|^{-1},p=1,2$: $$\label{eq:2.1}
y=\varphi_p(x).$$ Note that $\varphi_p(x)=(\varphi_{p1}(x),...,\varphi_{pn}(x)),p=1,2,$ satisfy the following differential equations (see \[E\], page 817) $$\begin{aligned}
                                    \label{eq:2.2}
\sum_{j,k=1}^n g_p^{jk}(x)\frac{\partial\varphi_{pn}}{\partial x_j}
\frac{\partial\varphi_{pn}}{\partial x_k}=1,\ \ 0\leq x_n<\delta,
\\
\varphi_{pn}(x',0)=0,
\nonumber\end{aligned}$$

$$\begin{aligned}
                                    \label{eq:2.3}
\sum_{j,k=1}^n g_p^{jk}(x)\frac{\partial\varphi_{pn}}{\partial x_j}
\frac{\partial\varphi_{pr}}{\partial x_k}=0,
\\
\varphi_{pr}(x',0)=x_r, \ \ 1\leq r\leq n-1.
\nonumber\end{aligned}$$

Denote by $\hat{L}^{(p})$ the operator $L^{(p)}$ in semigeodesic coordinates: $$\begin{aligned}
\hat{L}^{(p)}v_p\stackrel{def}{=}
\frac{\partial^2 v_p(y,t)}{\partial t^2}
+\frac{1}{\sqrt{\hat{g}_p(y)}}\left(-i\frac{\partial}{\partial y_n}+\hat{A}_{pn}(y)\right)
\sqrt{\hat{g}_p(y)}
\left(-i\frac{\partial }{\partial y_n}+\hat{A}_{pn}\right)v_p
\nonumber
\\
+\sum_{j,k=1}^{n-1}\frac{1}{\sqrt{\hat{g}_p(y)}}
\left(-i\frac{\partial}{\partial y_j}+\hat{A}_{pj}(y)\right)
\sqrt{\hat{g}_p}\hat{g}_p^{jk}(y)
\left(-i\frac{\partial}{\partial y_k}+\hat{A}_{pk}(y)\right)v_p(y,t)
\nonumber
\\ 
+\hat{V}_p(y)v_p(y,t)=0,
\nonumber\end{aligned}$$ where $v_p(\varphi_p(x),t)=u_p(x,t),\ L^{(p)}u_p=0,\ p=1,2$. Denote $$\label{eq:2.4}
u_p^{(1)}(y,t)=(\hat{g}_p(y',y_n))^{\frac{1}{4}}e^{-i\psi_p(y',y_n)}
u_p(\varphi_p^{-1}(y),t),\ \ \ p=1,2,$$ where the gauge transformation $c_p(y)= e^{-i\psi_p(y)}\in G_0(\overline{U})$ is such that $$\label{eq:2.5}
A_{pn}^{(1)}(y)=\hat{A}_{pn}(y)+\frac{\partial \psi_p}{\partial y_n}=0,\ \ \ \
\psi_p(y',0)=0.$$ Then $u_p^{(1)}$ satisfies the equation (c.f. \[E\], page 819): $$\begin{aligned}
                               \label{eq:2.6}
L_1^{(p)}u_p^{(1)}\stackrel{def}{=}\frac{\partial^2 u_p^{(1)}}{\partial t^2}-
\frac{\partial^2 u_p^{(1)}}{\partial y_n^2}
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  
\\
 +
\sum_{j,k=1}^{n}\left(-i\frac{\partial}{\partial y_j}+A_{pj}^{(1)}(y)\right)
\hat{g}_p^{jk}(y) \left(-i\frac{\partial}{\partial y_k}+A_{pk}^{(1)}(y)\right)u_p^{(1)}
\nonumber
\\
+V_{p1}u_p^{(1)}=0,\ \ \ \ p=1,2.
\nonumber\end{aligned}$$ Let $T\in (0,T_0)$ be small. As in \[E\] denote by $\mathcal{D}_p(\Gamma\times(0,T))$ the forward domain of influence of $\overline{\Gamma}\times[0,T]$ for $y_n\geq 0,p=1,2,$ and let $\overline{\mathcal{G}_p}
=\{y':(y',y_n,t)\in \mathcal{D}_p(\Gamma\times(0,T)),y_n=0,t=T\}$. It follows from the proof of Lemma 2.4 in \[E\] that $\mathcal{G}_1=
\mathcal{G}_2\stackrel{def}{=}\mathcal{G}$ since $\Lambda^{(1)}=\Lambda^{(2)}$ on $\Gamma_0\times(0,T)$.

Analogously, let $\mathcal{D}_p(\mathcal{G}\times(0,T))$ be the forward domain of influence of $\overline{\mathcal{G}}\times[0,T]$ for $y_n\geq 0,p=1,2$. Let $Y_{20}^{(p)}$ be the intersection of $\mathcal{D}_p(\mathcal{G}\times(0,T))$ with the plane $T-y_n-t=0$ and let $X_{20}^{(p)}$ be the part of $\mathcal{D}_p(\mathcal{G}\times(0,T))$ below $Y_{20}^{(p)},p=1,2$ (c.f. \[E\], page 819). Note that $\Lambda^{(1)}=\Lambda^{(2)}$ on $\Gamma_0\times(0,T_0)$ implies that $\hat{\Lambda}^{(1)}=\hat{\Lambda}^{(2)}$ on $\mathcal{G}\times(0,T)$, where $\hat{\Lambda}^{(p)}f'=\frac{\partial u_{p}^{(1)}}{\partial y_n}{\Huge |}_{y_n=0}$ is the D-to-N operator corresponding to $L_1^{(p)}$ (see \[E\], page 819). Here $f'=u_p^{(1)}{\Huge |}_{y_n=0},u_p^{(1)}=u_{pt}^{(1)}=0$ for $t=0,y_n>0$.

The main result of \[E\] is the following lemma:

\[lma:2.1\] Let $T_p,p=1,2,$ be such that the semigeodesic coordinates hold in $X_{20}^{(p)},\ \overline{\mathcal{G}}
\subset\Gamma_0$ and $X_{20}^{(p)}$ does not intersect $\partial\Omega^{(p)}\times[0,T_p]$ when $y_n>0$. Suppose $\hat{\Lambda}^{(1)}=\hat{\Lambda}^{(2)}$ on $\mathcal{G}\times(0,T)$, where $T=\min(T_1,T_2)$. Then $u_1^{(1)}=u_2^{(1)}$ for $y'\in \overline{\Gamma},
0\leq y_n\leq \frac{T}{2},y_n\leq t\leq T-y_n$ and $L_1^{(1)}=L_1^{(2)}$ in $\overline{\Gamma}\times[0,\frac{T}{2}]$.

[**Remark 2.1**]{} In \[E\] we assumed that $\mathcal{D}_p(\mathcal{G}\times(0,T_p))$ does not intersect $\partial\Omega^{(p)}\times[0,T_p]$ when $y_n>0$. It is easy to see that it is enough to assume that $X_{20}^{(p)}$ does not intersect $\partial\Omega^{(p)}\times[0,T_p]$ for $y_n>0$ since by the domain of dependence argument the solution of $L_1^{(p)}u_p^{(1)}=0,\ u_p^{(1)}=u_{pt}^{(1)}=0$ for $t=0,\ u_p^{(1)}=f'$ for $y_n=0,\ u_p^{(1)}=0$ for $(\partial\Omega^{(p)}\times[0,T_p])\cap\{y_n>0\}$ restricted to $X_{20}^{(p)}$ does not change whether $(\partial\Omega^{(p)}\times
[0,T_p])\cap\{y_n>0\}$ intersects $\mathcal{D}(\mathcal{G}\times(0,T_p))\setminus X_{20}^{(p)}$ or not.

Denote $D_p=\varphi_p^{-1}(\Gamma\times[0,\frac{T}{2}])\subset\Omega^{(p)}, p=1,2.$ Then $\varphi=\varphi_1^{-1}\circ\varphi_2$ is a diffeomorphism of $\overline{D}_2$ onto $\overline{D}_1$. Note that $\hat{g}_1(y',0)=\hat{g}_2(y',0)$ (c.f. Remark 2.2 in \[E\]). It follows from (\[eq:2.4\]) that $u_1(\varphi_1^{-1}(y),t)=c_1(y)u_2(\varphi_2^{-1}(y),t)$, where $c_1(y)\in G_0(\overline{\Gamma}\times[0,\frac{T}{2}])$. Therefore $$\label{eq:2.7}
c_2\circ \varphi\circ  L^{(2)}=L^{(1)}\  \mbox{in}\  \overline{D}_1,$$ where $c_2(x)=c_1(\varphi_1(x))\in G_0(\overline{D}_1)$.

[**Remark 2.2**]{} The subset $\Gamma\subset\Gamma_0$ does not have to be small. There may be no global coordinates $x=(x_1,x_2,...,x_n)$ near $\Gamma$ such that $x_n=0$ is the equation of $\Gamma$. In this case we take a finite cover $\{V_j\}, j=1,...,N,$ of $\overline{\Gamma}$ and apply Lemma \[lma:2.1\] to each $\Gamma_j=\Gamma\cap V_j,\ j=1,...,N$. Let $T_{pj},
\psi_{pj},\ p=1,2,\ j=1,...,N$, be the same as $T_p,\varphi_p$ in Lemma \[lma:2.1\].

Let $T_p=\min_{1\leq j\leq N}T_{pj},\ T=\min(T_1,T_2)$. Maps $\psi_{pj},1\leq j\leq N,$ define a diffeomorphism $\varphi_p$ of $D_p$ onto the manifold $\overline{\Gamma}\times[0,\frac{T}{2}]$ where $\overline{D}_p\stackrel{def}{=}\varphi_p^{-1}(\overline{\Gamma}\times[0,\frac{T}{2}])
\subset \overline{\Omega}^{(p)},\ p=1,2,
\ \varphi_p=I$ on $\overline{\Gamma}$.

Note that sets $\mathcal{G},\ X_{20}^{(p)}$ corresponding to $\Gamma_j,\ 1\leq j\leq N$ determine sets $\mathcal{G},\ X_{20}^{(p)}$ corresponding to the manifold $\Gamma$. Note also that $u_p^{(1)}(y',y_n)$ (see (\[eq:2.4\]) ) is not a scalar function on $\Gamma\times[0,T]$, but a half-density as $(g_p(y',y_n))^{\frac{1}{4}}$.

Therefore Lemma \[lma:2.1\] implies:

\[lma:2.2\] Let $\Gamma$ and $T$ be such that $\overline{\mathcal{G}}\subset\Gamma_0$, the semigeodesic coordinates hold in $X_{20}^{(p)},p=1,2,$ and $X_{20}^{(p)}\cap(\partial\Omega^{(p)}\setminus
(\Gamma_0\cap\cup_{j=1}^NV_j))=\emptyset$. Suppose $\hat{\Lambda}^{(1)}=\hat{\Lambda}^{(2)}$ on $\mathcal{G}
\times (0,T)$. Denote $\varphi=\varphi_1^{-1}\circ\varphi_2$. There exists a gauge transformation $c_2\in  G_0(\overline{D}_1)$ such that (\[eq:2.7\]) holds. or equivalently $$\label{eq:2.8}
L_{1j}^{(1)}=L_{1j}^{(2)}\ \ \mbox{in}\ \ 
\overline{\Gamma}_j\times [0,\frac{T}{2}],\ \ \ 1\leq j\leq N,$$ where $L_{1j}^{(p)},p=1,2,$ have the form (\[eq:2.6\]) in local semigeodesic coordinates in $\overline{\Gamma}_j\times[0,\frac{T}{2}]$. For the brevity of notations we shall write $L_1^{(1)}=L_1^{(2)}$ in $\overline{\Gamma}
\times[0,\frac{T}{2}]$ instead of (\[eq:2.8\]).

Note that $\overline{D}_p$ is the union of all geodesics in $\overline{\Omega}^{(p)},p=1,2,$ starting at $\overline{\Gamma}$, orthogonal to $\overline{\Gamma}$ and having the length $\frac{T}{2}$.

[**Remark 2.3**]{} In §4 we shall deal often with the following situation: Let $V$ be the same neighborhood as in the beginning of this section. Suppose $L^{(p)}$ are defined also in $\overline{V},\ p=1,2,$ and $L^{(1)}=L^{(2)}$ when $x_n<0$, i.e. in $V_-\stackrel{def}{=}V\setminus
\overline{\Omega}^{(p)}$. Let $y=\varphi_p(x)$ be the same as in (\[eq:2.2\]), (\[eq:2.3\]). Therefore $\varphi=\varphi_1^{-1}\circ\varphi_2$ is the diffeomorphism of $\overline{D}_2$ onto $\overline{D}_1$. We shall show that defining $\varphi=I$ on $V_-$ we get a diffeomorphism of $\overline{D}_2\cup\overline{V}_-$ onto $\overline{D}_1\cup\overline{V}_-$.

It follows from (\[eq:2.2\]), (\[eq:2.3\]) and (\[eq:2.7\]) that $\varphi(x)$ satisfies the equations: $$\|g_1^{jk}\|=\frac{D\varphi}{D x}\ \|g_2^{jk}\|\left(\frac{D\varphi}{D x}\right)^T
\ \ \ \mbox{in}\ \ \ V,$$ where $\frac{D\varphi}{D x}$ is the Jacobi matrix of $\varphi$. Since $\varphi=I$ when $x_n=0$ and since $\|g_1^{jk}\|=\|g_2^{jk}\|$ when $x_n\leq 0$ we have that $\varphi=I$ for $x_n\leq 0.$

Let $B\subset\Gamma\times[0,\frac{T}{2}]$ be a domain such that $\partial B\cap\Gamma$ is open and connected. Denote $B_p=\varphi_p^{-1}(B)$. We assume that $\Omega^{(p)}\setminus \overline{B}_p$ is smooth. Note that the restriction of $\varphi=
\varphi_1^{-1}\circ\varphi_2$ to $\overline{B}_2$ maps $\overline{B}_2$ onto $\overline{B}_1$. The following extension lemma holds (see \[Hi\], Chapter 8):

\[lma:2.3\] There exists a diffeomorphism $\varphi_3$ of $\overline{\Omega}^{(2)}$ onto $\overline{\Omega}^{(3)}
\stackrel{def}{=}\varphi_3(\overline{\Omega}^{(2)})$ such that $\varphi_3|_{B_2}=\varphi$ and $\varphi_3|_{\Gamma_0}=I.$

[**Proof:**]{} Since $\overline{D}_2
=\varphi_2^{-1}(\overline{\Gamma}\times[0,\frac{T}{2}])$ and $\varphi=\varphi_1=\varphi_2=I$ on $\overline{\Gamma}$ there exists a smooth family $\psi_t,0\leq t\leq 1,$ of embedding of $\overline{D}_2$ in ${{\bf R}}^n$ such that $\psi_0=\varphi$ and $\psi_1=I$ on $\overline{D}_2$. The proof of this fact is similar to the proof of Theorem 8.3.1. in \[Hi\]. Denote $A=\overline{B}_2\cup\overline{\Gamma}_0$. Define $\varphi=I$ on $\overline{\Gamma}_0$. Denote by $B_2'$ the union of a small neighborhood of $\overline{D}_2$ and a small neighborhood of $\overline{\Gamma}_0$. We can extend $\psi_t,0\leq t\leq 1$ from $\overline{D}_2$ to $B_2'$ preserving the properties that $\psi_0=\varphi$ on $A,\ \psi_t=I$ on $\overline{\Gamma}_0,
\  \psi_1=I$ on $B_2',\ \psi_t$ is an embedding of $B_2'$ in ${{\bf R}}^n$. Now applying Theorem 8.1.4 in \[Hi\] we get that there exists a diffeomorphism $\varphi_3$ of ${{\bf R}}^n$ onto ${{\bf R}}^n,\ \varphi_3=I$ for $|x|>N,\ N$ is large, such that $\varphi_3|_A=\varphi$. Taking the restriction of $\varphi_3$ to $\overline{\Omega}^{(2)}$ we prove Lemma \[lma:2.3\].

Denote by $c_3\in G_0(\overline{\Omega}^{(3)})$ the extension of $c_2$ from $\overline{B}_1\cup\overline{\Gamma}_0$ to $\overline{\Omega}^{(3)}$, where $c_2$ is the same as in (\[eq:2.7\]), $c_2=1$ on $\Gamma_0\ $, $\Omega^{(3)}\stackrel{def}{=}\varphi_3(\Omega^{(2)})$. Let $L^{(3)}=c_3\circ\varphi_3\circ L^{(2)}$ be the differential operator in $\Omega^{(3)}$. We have that $L^{(3)}=L^{(1)}$ in $\overline{B}_1$. Note that $B_1\subset \Omega^{(1)}\cap\Omega^{(3)}$ and $\Gamma_0\subset\partial\Omega^{(1)}\cap\partial\Omega^{(3)}$.

Let $\Gamma_1=(\Gamma_0\setminus(\partial B_1\cap\Gamma_0))\cup
(\partial B_1\setminus\overline{\Gamma}_0)$ and let $\delta=\max_{x\in\overline{B}_1}d_1(x,\Gamma_0)$, where $d_1(x,\Gamma_0)$ is the distance in $\overline{B}_1$ from $x\in \overline{B}_1$ to $\Gamma_0$. Note that $\Gamma_1\subset\partial(\Omega^{(1)}\setminus
\overline{B}_1)\cap\partial(\Omega^{(3)}\setminus\overline{B}_1)$.

Let $\Lambda^{(1)}$ and $\Lambda^{(3)}$ be the D-to-N operators corresponding to $L^{(1)}$ and $L^{(3)}$ in domains $\Omega^{(1)}\times(0,T_0)$ and $\Omega^{(3)}\times(0,T_0)$, respectively. For the simplicity of notations we continue to denote by $\Lambda^{(1)},\ \Lambda^{(3)}$ the D-to-N operators corresponding to $L^{(1)}$ and $\L^{(3)}$ in smaller domains $(\Omega^{(1)}\setminus\overline{B}_1)\times
(\delta,T_0-\delta),\ 
(\Omega^{(3)}\setminus\overline{B}_1)
\times(\delta,T_0-\delta)$, respectively.

The following lemma was proven in \[E\], Lemma 3.3.

\[lma:2.4\] If $\Lambda^{(1)}=\Lambda^{(3)}$ on $\Gamma_0\times(0,T_0)$ then the D-to-N operators corresponding to $L^{(1)},\ L^{(3)}$ on domains $(\Omega^{(1)}\setminus\overline{B}_1)\times(\delta,T_0-\delta)\ $, $(\Omega^{(3)}\setminus\overline{B}_1)\times(\delta,T_0-\delta)$ respectively are equal on $\Gamma_1\times(\delta,T_0-\delta)$.

Note that the inverse of Lemma \[lma:2.4\] is also true:

\[lma:2.5\] Suppose the D-to-N operators corresponding to $L^{(1)},\ L^{(3)}$ on domains $(\Omega^{(1)}\setminus\overline{B}_1)\times(0,T_0)$, $(\Omega^{(1)}\setminus\overline{B}_1/\times(0,T_0)$, respectively, are equal on $\Gamma_1\times(0,T_0)$. Then $\Lambda^{(1)}=\Lambda^{(3)}$ on $\Gamma_0\times(0,T_0)$.

[**Proof:**]{} Let $f$ be a smooth function on $\partial\Omega^{(1)}\times(0,T_0),\ 
\mbox{supp\ }f\subset\Gamma_0\times(0,T_0]$. Let $u_1$ be the solution of the initial-boundary value problem $$\begin{aligned}
L^{(1)}u_1=0 \ \ \ \mbox{in}\ \ \ \Omega^{(1)}\times(0,T_0),
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\nonumber
\\
u_1(x,0)=u_{1t}(x,0)=0,\ \ x\in\Omega^{(1)},
\ \ \ \ \ \ u_1|_{\partial\Omega^{(1)}\times(0,T_0)}=f.
\nonumber\end{aligned}$$ Denote by $f_1$ the restriction of $u_1$ to $\Gamma_1\times(0,T_0)$. Let $u_3$ be the solution of $$\begin{aligned}
L^{(3)}u_3=0 \ \ \ \mbox{in}\ \ \ (\Omega^{(3)}
\setminus\overline{B}_1)\times(0,T_0),
\nonumber
\\
u_3(x,0)=u_{3t}(x,0)=0,\ \ x\in\Omega^{(3)}\setminus\overline{B}_1,
\nonumber
\\
u_3|_{\Gamma_1\times(0,T_0)}=f_1,\ \ u_3=0\ \ \mbox{on}\ \ \
((\partial(\Omega^{(3)}\setminus\overline{B}_1))
\setminus\overline{\Gamma}_1)\times(0,T_0).
\nonumber\end{aligned}$$ Note that $u_1$ is also zero on $((\partial(\Omega^{(1)}
\setminus\overline{B_1}))\setminus\overline{\Gamma_1})\times(0,T_0)$. We have that $\Lambda^{(1)}f_1=\Lambda^{(3)}f_1$ on $\Gamma_1\times(0,T_0)$ by the assumption.

Let $\tilde{u}_3=u_3$ in $(\Omega^{(3)}\setminus\overline{B}_1)\times(0,T_0)$ and $\tilde{u}_3=u_1$ in $\overline{B}_1\times(0,T_0)$. Since $\Lambda^{(1)}f_1=\Lambda^{(3)}f_1$ and $L^{(3)}=L^{(1)}$ in $\overline{B_1}$ we get that $\tilde{u}_3$ is the solution of $L^{(3)}\tilde{u}_3=0$ in $\Omega^{(3)}\times(0,T_0)$. Note that $\tilde{u}_3(x,0)=\tilde{u}_{3t}(x,0)=0$ for $x\in \Omega^{(3)}$. Also $\tilde{u}_3|_{\Gamma_0\times(0,T_0)}=u_1|_{\Gamma_0\times(0,T_0)}=f,\ 
\ \tilde{u}_3=0$ on $(\partial\Omega^{(3)}\setminus\Gamma_0)\times
(0,T_0)$. Moreover, $\Lambda^{(3)}f=
\Lambda^{(1)}f$ on $\Gamma_0\times(0,T_0)$ since $u_1=u_3$ in $\overline{B_1}\times(0,T_0)$ and $\Lambda^{(1)}=\Lambda^{(3)}$ on $\Gamma_1\times(0,T_0)$.

Refinements of Lemma \[lma:2.2\]. {#section 3}
=================================

In this and the next sections we shall show how to use repeatedly Lemmas \[lma:2.1\] - \[lma:2.5\] to prove Theorem \[theo:1.1\]. We shall prove the following lemma considering, for the simplicity, first the case when $\Gamma=\Gamma_0$ and $\partial\Gamma_0=\emptyset$.

\[lma:3.1\] Let $\partial\Gamma_0=\emptyset$ and let $T_1$ be such that the semigeodesic coordinates for $L^{(1)}$ hold in $D_1=\varphi_1^{-1}(\Gamma_0\times[0,\frac{T_1}{2}])$ and $D_1\cap(\partial\Omega^{(1)}\setminus\Gamma_0)=\emptyset$. Let $T_2^*$ be such that the semigeodesic coordinates for $L^{(2)}$ hold in $\overline{D}_2=\varphi_2^{-1}(\Gamma_0\times[0,\frac{T_2}{2}])$ for any $T_2<T_2^*$ and $D_2\cap(\partial\Omega^{(2)}\setminus\Gamma_0)=\emptyset$ for any $T_2<T_2^*$. Suppose that there exists a focal point of a geodesics $\gamma_0$ in $\overline{\Omega}^{(2)}$, starting at $\Gamma_0$, orthogonal to $\Gamma_0$ and such that $y_{n0}=\frac{T_2^*}{2}$, where $(y_0',y_{n0})$ are semigeodesic coordinates of the focal point. Then, assuming that $\Lambda^{(1)}=\Lambda^{(2)}$ on $\Gamma_0\times(0,T_0)$ and $T_0>T_1$, we have that $T_2^*>T_1$ and $L_1^{(1)}=L_1^{(2)}$ in $\Gamma_0\times[0,\frac{T_1}{2}]$.

[**Proof:**]{} Consider the bicharacteristics system: $$\begin{aligned}
                         \label{eq:3.1}
\frac{dx_{rj}}{dt}=\frac{\partial H_r(x_r,p_r)}{\partial p_j},\ \ x_{rj}(0)=y_j,
\ 1\leq j\leq n-1,\ r=1,2,
\nonumber
\\
 \frac{dx_{rn}}{dt}=\frac{\partial H_r(x_r,p_r)}{\partial p_n},\ \ x_{rn}(0)=0,
\ \ r=1,2,
\\
\frac{dp_{rj}}{dt}=-\frac{\partial H_r(x_r,p_r)}{\partial x_j},\ \ p_{rj}(0)=0,
\ 1\leq j\leq n-1,\ r=1,2,
\nonumber
\\
\frac{dp_{rn}}{dt}=-\frac{\partial H_r(x_r,p_r)}{\partial x_{n}},\ \ p_n(0)=
\frac{1}{\sqrt{g_r^{nn}(y',0)}},
\ \ r=1,2.
\nonumber\end{aligned}$$ Here $H_r(x,p)=\sqrt{\sum_{j,k=1}^n g_r^{jk}(x)p_jp_k}$. Let $\varphi_r(x)=(\varphi_{r1}(x),...,\varphi_{rn}(x))$ be the same as in (\[eq:2.2\]), (\[eq:2.3\]). Then $p_{rj}(t)=\frac{\partial \varphi_{rn}(x_r(t))}{\partial x_j},
\ 1\leq j\leq n,
\ r=1,2,$ and (\[eq:2.2\]) implies that $\frac{d\varphi_{rn}(x(t))}{dt}
=\sum_{j=1}^n p_{rj}(t)\frac{\partial H_r(x_r,p_r)}{\partial p_j}
=H(x_r(t),p_r(t))=1$.

Since $\varphi_{rn}(x(0))=0$ we get that $$\label{eq:3.2}
\varphi_{rn}(x_r(t))=t.$$ Note that (\[eq:2.3\]) implies: $$\begin{aligned}
\frac{d\varphi_{rj}(x_r(t))}{dt}
=\sum_{k=1}^n\frac{\partial\varphi_{rj}(x_r(t))}{\partial x_k}
\frac{dx_{rk}}{dt}
=\sum_{k=1}^n\frac{\partial\varphi_{rj}(x_r(t))}{\partial x_k}
\frac{\partial H_r(x_r,\frac{\partial \varphi_{rn}}{\partial x})}{\partial p_k}=0,
\nonumber
\\
\ \ \ 1\leq j\leq n-1.
\nonumber\end{aligned}$$ Therefore $$\label{eq:3.3}
\varphi_{rj}(x_r(t))=\varphi_{rj}(x_r(0))
=\varphi_{rj}(y',0)=y_j,\ 1\leq j\leq n-1,\ r=1,2.$$ Denote $y_n=t$. The coordinates $(y_1,...,y_{n-1},y_n)$ are the semigeodesic coordinates. Consider the change of variables $$\label{eq:3.4}
x_{j}=x_{rj}(y_1,...,y_n),\ \ \ \ 1\leq j\leq n,\ \ \ r=1,2.$$ Assume that the Jacobian $$\det\frac{D(x_{r1},...,x_{rn})}{D(y_1,...,y_n)}\neq 0.$$ It follows from (\[eq:3.2\]), (\[eq:3.3\]) that (\[eq:3.4\]) is the inverse to the change of variables (\[eq:2.1\]), since we have : $$\begin{aligned}
y_j=\varphi_{rj}(x_r(y_1,...,y_{n-1},t)),
\nonumber
\\
y_n=\varphi_{rn}(x_r(y_1,...,y_{n-1},t))=t.
\nonumber\end{aligned}$$ Therefore $\frac{D\varphi_r(x)}{Dx}=(\frac{Dx_r}{Dy})^{-1}$. Note that the solution of the system (\[eq:3.1\]) exists for all $t\in {{\bf R}}$. Therefore $\frac{Dx_r}{Dy}$ is smooth for all $y$. However the Jacobian $\det\frac{Dx_r}{Dy}$ may vanish for some $y$. Such $y$ are called the focal points (caustics) for the Hamiltonian system (\[eq:3.1\]). Then $\det\frac{D\varphi_r}{Dx}\rightarrow\infty$ when $y=\varphi_r(x)$ approaches a focal point. Suppose $T_2^*\leq T_1$. Then there exists a focal point $y^{(0)}=(y_{0}',y_{n0})\in \Gamma_0\times [0,\frac{T_1}{2}]$ for the system (\[eq:3.1\]) when $r=2$ and $y_{n0}\leq \frac{T_1}{2}$. We have near $y^{(0)}$(see (1.8) in \[E\]): $$\label{eq:3.5}
\det\|\hat{g}_r^{jk}(\varphi_r(x))\|=\det\|g_r^{jk}(x)\|
\left(\det\frac{D\varphi_r}{Dx}\right)^2,$$ where $\|\hat{g}_r^{jk}(y)\|^{-1}$ is the metric tensor in semigeodesic coordinates. If $y\in\Gamma_0\times[0,\frac{T_2}{2}]$ and $\frac{T_2}{2}$ is close to $y_{n0},\ \frac{T_2}{2}<y_{n0}\leq \frac{T_1}{2}$, we have, by Lemma \[lma:2.2\] that $\hat{g}_1^{jk}=\hat{g}_2^{jk}$. Consider (\[eq:3.5\]) when $r=1$. Since $y^{(0)}$ is not a focal point for $r=1$ we have that $\det\frac{D\varphi_1}{Dx}$ is bounded. Therefore $\det\|\hat{g}_1^{jk}\|$ is bounded near $y^{(0)}$. Now consider (\[eq:3.5\]) for $r=2$. Then $\left(\det\frac{D\varphi_2}{Dx}\right)^2
=\frac{\det\|\hat{g}_1^{jk}\|}{\det\|g_2^{jk}\|}$ is also bounded when $y\rightarrow y^{(0)}$. Therefore $y^{(0)}$ is not a focal point for $r=2$ and this is a contradiction. Therefore we can take $T_2=T_1$ and we have $L_1^{(1)}=L_1^{(2)}$ in $\Gamma_0\times[0,\frac{T_1}{2}]$ (c.f. (\[eq:2.8\])).

\[lma:3.2\] Let $\partial\Gamma_0 =\emptyset$ and let $T_1$ be such that the semigeodesic coordinates for $L^{(1)}$ hold in $D_1=\varphi_1^{-1}(\Gamma_0\times[0,\frac{T_1}{2}])$ and $D_1\cap(\partial\Omega^{(1)}\setminus\Gamma_0)=\emptyset$. Then semigeodesic coordinates for $L^{(2)}$ hold also in $D_2=\varphi_2^{-1}(\Gamma_0\times[0,\frac{T_1}{2}]),\ 
D_2\cap(\partial\Omega^{(2)}\setminus\Gamma_0)=\emptyset$ and $L_1^{(1)}=L_1^{(2)}$ in $\Gamma_0\times[0,\frac{T_1}{2}]$.

[**Proof:**]{} Suppose $D_2$ intersects $\partial\Omega^{(2)}\setminus \Gamma_0$. Let $x^{(0)}$ be the point in $D_2\cap
(\partial\Omega^{(2)}\setminus\Gamma_0)$ closest to $\Gamma_0$ and let $y^{(0)}=(y_0',y_{n0})$ be the semigeodesic coordinates of $x^{(0)}$. We have by assumption that $y_{n0}\leq\frac{T_1}{2}$. For any $\frac{T_2}{2}
<y_{n0}$ we have by Lemma \[lma:3.1\] that $L_1^{(1)}=L_1^{(2)}$ in $\Gamma_0\times[0,\frac{T_2}{2}]$. Therefore by the continuity $L_1^{(1)}=L_1^{(2)}$ in $\Gamma_0\times[0,y_{n0}]$.

Denote by $\gamma_0$ the geodesics in $\overline{\Omega}^{(2)}$ starting at $x^{(1)}\in \Gamma_0,$ orthogonal to $\Gamma_0$ and reflecting at $\partial\Omega^{(2)}\setminus\Gamma_0$ at point $x^{(0)}$. Since $x^{(0)}$ is the closest point to $\Gamma_0$ in $D_1$, the angle of reflection at $x^{(0)}$ is $\frac{\pi}{2}$. Denote by $u_2$ the geometric optics solution depending on large parameter associated with $\gamma_0$ and its successive reflections (c.f. \[E\], page 824, and \[E2\], pages 28-29). We have $L^{(2)}u_2=0$ in $\Omega^{(2)}\times(0,T_0),\ u_2=u_{2t}=0$ for $t=0$, and $u_2|_{\partial\Omega^{(2)}\times(0,T_0)}=f$ where $\mbox{supp\ }f$ is contained in a small neighborhood of $(x^{(1)},t_1)\in 
\Gamma_0\times(0,T_0),t_1>0,$ where $(x^{(1)},t_1)$ is the starting point of the broken ray $\gamma_0$. In semigeodesic coordinates $u_2$ is concentrated (modulo lower order terms) in a small neighborhood of a broken ray $y'=y_0',
\ y_n-t=-t_1$ for $t_1<t<y_{n0}+t_1,\ y'=y_0',\ y_n+t=2y_{n0}+t_1$ for $y_{n0}+t_1<t<t_1+2y_{n0},\ y'=y_0',\ y_n-t=-t_1-2y_{n0}$ for $t>t_1+2y_{n0}$, etc.

Denote by $u_1$ the solution of $L^{(1)}u_1=0$ in $\Omega^{(1)}\times(0,T_0),\ 
u_1=u_{1t}=0$ for $t=0,\ u_1|_{\partial\Omega^{(1)}\times(0,T_0)}=f$. Let $u_p^{(1)}$ be related to $u_p,p=1,2,$ by formulas (\[eq:2.4\]). Since $\Lambda^{(1)}=\Lambda^{(2)}$ on $\Gamma_0\times(0,T_0)$ we have that $u_1^{(1)}$ and $u_2^{(1)}$ have tha same Cauchy data when $y_n=0$. Since $L_1^{(1)}=L_1^{(2)}$ in $\Gamma_0\times[0,y_{n0}]$ we get, by the unique continuation theorem (see \[T\]), that $u_1^{(1)}=u_2^{(1)}$ for $y\in\Gamma_0\times[0,y_{n0}],\ 0<t<T_0- y_{n0}$. However, $y^{(0)}$ is not a point of reflection for $u_1^{(1)}$. Therefore $u_1^{(1)}\neq u_2^{(1)}$ for $y_{n0}+t_1<t<y_{n0}+t_1+{{\varepsilon}},\ y_n<y_{n0}$, where ${{\varepsilon}}>0$ is small. This contradiction proves that $(\partial\Omega^{(2)}\setminus\Gamma_0)\cap
\bar{D}_2=\emptyset$.

Now we shall consider the case when $\Gamma\subset\Gamma_0$ may have a boundary, i.e. when $\partial\Gamma\neq\emptyset$. Let $T_1$ be such that all conditions of Lemma \[lma:2.2\] are satisfied. Let $X_{j0}^{(p)},j=1,2,$ be the same as in Lemma \[lma:2.2\]. Denote by $\Delta_j^{(p)}$ the projection of $X_{j0}^{(p)}$ on the plane $t=0,j=1,2,p=1,2.$ Note that $\Delta_j^{(p)}$ are contained in the strip $0\leq y_n\leq \frac{T_1}{2}$. Let $\tilde{\Gamma}\subset\Gamma$ be such that $\overline{\tilde{\Delta}}_{2}^{(1)}\subset \Gamma\times[0,\frac{T_1}{2}]$ where $\tilde{\Delta}_{2}^{(p)}$ is the same as $\Delta_{2}^{(p)}$ when $\Gamma$ is replaced by $\tilde{\Gamma}$.

We will need the following proposition:

\[lma:3.3\] Let $L^{(p)},p=1,2,$ be two operators such that $L_1^{(1)}=L_1^{(2)}$ in $(\overline{\Gamma}\setminus\tilde{\Gamma})\times[0,\frac{T_1}{2}]$. Suppose $\overline{\tilde{\Delta}}_{2}^{(1)}\subset \Gamma\times[0,\frac{T_1}{2}]$. Then $\overline{\tilde{\Delta}}_{2}^{(2)}\subset
\overline{\tilde{\Delta}}_{2}^{(1)}$.

[**Proof:**]{} Suppose there exists $(y_0',y_{n0})\in\overline{\tilde{\Delta}}_{2}^{(2)}$, where $y_{n0}\leq \frac{T_1}{2},\ (y_0',y_{n0})\notin \overline{\tilde{\Delta}}_{2}^{(1)}$. Note that $(y_0',y_{n0})\in\overline{\tilde{\Delta}}_{2}^{(2)}$ means that there exists $y_1'\in \overline{\tilde{\mathcal{G}}}$ such that the shortest path $\gamma$ connecting $(y_1',0)$ and $(y_0',y_{n0})$ has the length $|\gamma|\leq T_1-y_{n0}$. Here $\tilde{\mathcal{G}}$ is the same for $\tilde{\Gamma}$ as $\mathcal{G}$ for $\Gamma$ (see §2). If $\gamma$ is contained completely in $(\overline{\Gamma}\setminus\tilde{\mathcal{G}})\times[0,\frac{T_1}{2}]$ then we must have $(y_0',y_{n0})\in\overline{\tilde{\Delta}}_{2}^{(1)}$ since $L_1^{(1)}=L_1^{(2)}$ in $(\overline{\Gamma}\setminus\tilde{\Gamma)}
\times[0,\frac{T_1}{2}]$. Suppose there is a part of $\gamma$ that does not belong to $(\overline{\Gamma}\setminus\tilde{\mathcal{G}})\times[0,\frac{T_1}{2}]$. Let $(y_2',y_{n2})\in\partial\tilde{\mathcal{G}}\times[0,\frac{T_1}{2}]$ be such that the part $\gamma_1$ of $\gamma$ connecting $(y_2',y_{n2})$ and $(y_0',y_{n0})$ is in $(\overline{\Gamma}\setminus\tilde{\mathcal{G}})\times[0,\frac{T_1}{2}]$. Denote by $\gamma_2$ the remaining part of $\gamma$. Let $\gamma_2'$ be the straight line connecting $(y_{2}',y_{n2})$ and $(y_{2}',0)$. Since the metric for $L_1^{(2)}$ has the form $(dy_n)^2+\sum_{j,k=1}^{n-1}\hat{g}_{2jk}dy_jdy_k$ we have that the length $|\gamma_2'|$ of $\gamma_2'$ is less or equal than $|\gamma_2|$. Therefore $T-y_{n0}\geq |\gamma|=|\gamma_1|+|\gamma_2|\geq |\gamma_1|+|\gamma_2'|$. The path $\gamma_2'\cup\gamma_1$ is contained in $(\bar{\Gamma}\setminus\tilde{\mathcal{G}})
\times[0,\frac{T_1}{2}]$. Since $L_1^{(1)}=L_1^{(2)}$ in $(\bar{\Gamma}\setminus\tilde{\Gamma})\times[0,\frac{T_1}{2}]$ we have that $\gamma_2'\cup\gamma_1$ is contained in $\overline{\tilde{\Delta}}_{2}^{(1)}$. Therefore $(y_0',y_{n0})\in \overline{\tilde{\Delta}}_{2}^{(1)}$. This contradiction proves Lemma \[lma:3.3\].

The following lemma generalizes Lemmas \[lma:3.1\] and \[lma:3.2\] to the case when $\partial\Gamma\neq \emptyset$.

\[lma:3.4\] Consider $\Gamma\subset \Gamma_0$ such that $\partial\Gamma\neq \emptyset$. Let $0<T_1<T_0$ be such that the semigeodesic coordinates for $L^{(1)}$ hold in $\varphi_1^{-1}(\bar{\Gamma}\times[0,\frac{T_1}{2}])$ and $(\partial\Omega^{(1)}\setminus\Gamma)\cap \varphi_1^{-1}(\bar{\Gamma}
\times[0,\frac{T_1}{2}])=\emptyset$. Let $\tilde{\Gamma} \subset\Gamma$ be such that $\overline{\tilde{\Delta}_{2}^{(1)}}\subset \Gamma\times[0,\frac{T_1}{2}]$. Suppose that semigeodesic coordinates for $L^{(2)}$ hold in $(\bar{\Gamma}\setminus\tilde{\Gamma})\times[0,\frac{T_1}{2}],
\ (\partial\Omega^{(2)}\setminus\Gamma)\cap \varphi_2^{-1}((\bar{\Gamma}
\setminus\tilde{\Gamma})
\times[0,\frac{T_1}{2}])=\emptyset$ and $L_1^{(1)}=L_1^{(2)}$ in $(\bar{\Gamma}\setminus\tilde{\Gamma})
\times[0,\frac{T_1}{2}]$. Then the semigeodesic coordinates for $L^{(2)}$ hold in $\bar{\Gamma}
\times[0,\frac{T_1}{2}],\  
(\partial\Omega^{(2)}\setminus\Gamma)\cap \varphi_2^{-1}(\bar{\Gamma}
\times[0,\frac{T_1}{2}])=\emptyset$ and $L_1^{(1)}=L_1^{(2)}$ in $\bar{\Gamma}\times[0,\frac{T_1}{2}]$.

[**Proof:**]{} Since $\overline{\tilde{\Delta}}_{2}^{(2)}\subset
\overline{\tilde{\Delta}}_{2}^{(1)}\subset
\Gamma\times[0,\frac{T_1}{2}]$ (see Lemma \[lma:3.3\]) we have that there is no focal points for $L_1^{(2)}$ in $\tilde{\Delta}_{2}^{(2)}\setminus(\tilde{\Gamma}\times[0,\frac{T_1}{2}])$. Repeating the proof of Lemma \[lma:3.1\] with $\Gamma_0$ replaced by $\tilde{\Gamma}$ we get that there is no focal points for $L_1^{(2)}$ in $\overline{\tilde{\Gamma}}\times[0,\frac{T_1}{2}]$ and $L_1^{(1)}=L_1^{(2)}$ in $\overline{\tilde{\Gamma}}\times[0,\frac{T_1}{2}]$ assuming that $Z\stackrel{def}{=}(\partial\Omega^{(2)}\setminus\Gamma)\cap\varphi_2^{-1}
(\overline{\tilde{\Gamma}}\times[0,\frac{T_1}{2}])=\emptyset$. Now we shall show that the set $Z$ is empty. Suppose $Z\neq \emptyset$. Since $(\partial\Omega^{(2)}\setminus\Gamma)\cap\varphi_2^{-1}
(\partial\tilde{\Gamma}\times[0,\frac{T_1}{2}])=\emptyset$ the point $(y_{n0}',y_{n0})\in Z$ closest to $\Gamma$ belongs to $\tilde{\Gamma}\times[0,\frac{T_1}{2}]$, i.e. $y_0'$ is an interior point of $\tilde{\Gamma}$.

From this point we can repeat the proof of Lemma \[lma:3.2\] to get a contradiction.

The global step. {#section 4}
================

We start this section with a lemma (Lemma \[lma:4.1\]) that will play a key role in the global step of the proof of Theorem \[theo:1.1\]

Let $O(\delta_1)$ be a ball in $\Gamma_0:O(\delta_1)=\{x'\in\Gamma_0,\ 
d_1(x',x_0')<\delta_1\}$, where $x_0'\in \Gamma_0,\ d_1(x',x_0')$ is the distance on $\Gamma_0$ induced by the metric $\| g_1^{jk}\|^{-1}$. Let $R$ be the union of geodesics in $\Omega^{(1)}$ with respect to the metric $\|g_1^{jk}\|^{-1}$, starting on $\overline{O}(\delta_1)$, orthogonal to $\Gamma_0$ and having the lengths $T,\ 2T<T_0$. We assume that these geodesics have no focal points in $\overline{R}$ and do not intersect $\partial\Omega^{(1)}\setminus \overline{O}(\delta_1)$. Therefore we can introduce semigeodesic coordinates $y=\psi_1(x)$ in $\overline{R}$ using these geodesics. By the continuity the semigeodesic coordinates $y=\psi_1(x)$ hold in a larger domain $R_0\supset R$, i.e. there exists a small $\delta_2>0$ such that $\overline{O}(\delta_1+\delta_2)\subset \Gamma_0$ and $y=\psi_1(x)$ is a diffeomorphism of $\overline{R}_0$ onto $\overline{O}(\delta_1+\delta_2)\times[0,T]$.

\[fig:figure1\]

 \
Figure . Domains $R_k$ and boundaries $\hat{\Sigma}_{k-1}$, $k \ge
  2$. $\hat{\Sigma}_{k-1}$ is drawn in bold.

Denote by $R_1$ the union of all geodesics in $R_{0}$ with the lengths $\delta_3$. We will choose $\delta_3<\delta_2$. Let $R_1'$ be the union of geodesics in $\Omega^{(2)}$ with respect to the metric $\|g_2^{jk}\|^{-1}$ starting on $\overline{O}(\delta_1+\delta_2)$ orthogonal to $\Gamma_0$ and having the lenghts $\delta_3$. Let $y=\psi_2(x)$ be the semigeodesic coordinates on $R'_1$, i.e. $\psi_2$ is a diffeomorphism of $\overline{R}_1'$ onto $\overline{O}(\delta_1+\delta_2)
\times[0,\delta_3]$.

Note that the metrics on $\Gamma_0$ induced by $\|g_1^{jk}\|^{-1}$ and $\|g_2^{jk}\|^{-1}$ are the same since $\Lambda^{(1)}=\Lambda^{(2)}$ on $\Gamma_0\times(0,T_0)$ (c.f., for example, Remark 2.2 in \[E\]). Note that $\psi_3=\psi_1^{-1}\circ\psi_2$ is a diffeomorphism of $\overline{R}_1'$ onto $\overline{R}_1$. Since $\delta_3>0$ is small we can apply Lemma \[lma:2.1\] or Lemma \[lma:2.2\] to get $L^{(1)}=L^{(3)}$ in $\overline{R}_1$ where $L^{(3)}=c_3\circ \psi_3\circ L^{(2)},\ c_3\in G_0(\overline{R}_1),\ 
\psi_3=I$ and $c_3=1$ on $O(\delta_1+\delta_2),
 \ L^{(3)}$ is an operator on $\overline{R}_1$.

Let $\hat{X}_{20}^{(p)}$ be the same as in §2 when $\Gamma$ is replaced by $O(\delta_1+\frac{\delta_2}{2})$ and let $\hat{\Delta}_{20}^{(p)}$ be the projection of $\hat{X}_{20}^{(p)}$ on the plane $t=0,p=1,3$. If $\delta_3$ is much smaller than $\delta_2$, we have that $\overline{\hat{\Delta}}_{20}^{(1)}\subset O(\delta_1+\delta_2)\times[0,\delta_3]$ and therefore $\overline{\tilde{\Delta}}_{20}^{(3)}\subset
\overline{\tilde{\Delta}}_{20}^{(1)}$ (c.f. Lemma \[lma:3.3\]).

Let $\Sigma_1=\psi_1^{-1}(\sigma_1)$ where $\sigma_1=\overline{O}(\delta_1+\delta_2)\setminus O(\delta_1)$ when $y_n=0,\ \sigma_1=\partial O(\delta_1)$ when $0\leq y_n\leq \delta_3,\ 
\sigma_1=O(\delta_1)$ when $y_n=\delta_3$ (see Fig. 1). Here $y=\psi_1(x)$ are the semigeodesic coordinates in $\overline{R}_{0}$. Note that $\sigma_1$ has edges when $y_n=0,\ y'\in \partial O(\delta_1)$ and when $y_n=\delta_3,\ y'\in\partial O(\delta_1)$. We will smooth $\Sigma_1=\psi_1^{-1}(\sigma_1)$ near these edges to obtain a smooth surface $\Sigma_1$. We can arrange the smoothing in such a way that $\Sigma_1$ and $\hat{\Sigma}_1$ differ only in a small neighborhood of edges of the size $O(\varepsilon)$ where $0< {{\varepsilon}}\ll \delta_3$. Denote by $R_2$ the domain bounded by $\hat{\Sigma}_1$ and $O(\delta_1+\delta_2)$. Note that $R_2\subset R_1$. Using Lemma \[lma:2.3\] we can extend $\psi_3$ from $\overline{R}_2'\stackrel{def}{=}\psi_3^{-1}(\overline{R}_2)
\subset\overline{\Omega}^{(2)}$ to $\overline{\Omega}^{(2)}$ as a diffeomorphism of $\overline{\Omega}^{(2)}$ onto $\overline{\Omega}^{(3)}\stackrel{def}{=}\psi_3(\Omega^{(2)}),\ \psi_3=I$ on $\Gamma_0$ and extend $c_3$ from $\overline{R}_2$ to $\Omega^{(3)}$ as an element of $G_0(\overline{\Omega}^{(3)})$. It follows from Lemma \[lma:2.4\] that $\Lambda^{(1)}=
\Lambda^{(3)}$ on $\Sigma_1\times(\delta_3,T_0-\delta_3)$.

Denote by $S_1$ the union of all geodesics in $\overline{\Omega}^{(1)}$ with respect to the metric $\|g_1^{jk}\|^{-1}$, starting at $\hat{\Sigma}_1$, orthogonal to $\hat{\Sigma}_1$ and having lengths ${{\varepsilon}}>0$. Since ${{\varepsilon}}>0$ is small (we assume that $0<{{\varepsilon}}\ll \delta_3\ll \delta_2$) there is no focal points in $\overline{S}_1$ and the interior of $S_1$ does not intersect $\partial\Omega^{(1)}$. Since ${{\varepsilon}}\ll\delta_3$ we can apply Lemma \[lma:3.4\] to $\overline{S}_1$ and $L^{(1)},L^{(3)}$. We get that there is a diffeomorphism $\psi_{41}$ of $\overline{S}_1$ onto $\overline{S}_1'\stackrel{def}{=}\psi_{41}(\overline{S}_1)
\subset
\overline{\Omega}^{(3)}$ and a gauge transformation $c_{41}\in G_0(\overline{S}_1)$, such that $\varphi_{41}=I$ on $\hat{\Sigma}_1,\ c_{41}=1$ on $\hat{\Sigma}_1$ and $L^{(1)}=L^{(4)}$ on $S_1$, where $L^{(4)}\stackrel{def}{=}c_{41}\circ\psi_{41}^{-1}\circ L^{(3)}$.

Define $\psi_4'=\psi_{41}^{-1}$ on $\overline{S}_1',\ \psi_4'=I$ on $\overline{R}_2,\ c_4'=c_{41}$ on $\overline{S}_1,\ c_4'=1$ on $\overline{R}_2$. Since $L^{(1)}=L^{(3)}$ in $\overline{R}_2$ we have that $\psi_4'$ is a $C^\infty$ diffeomorphism on $\overline{S}_1'\cup\overline{R}_2$ and $c_4\in G_0(\overline{S}_1\cup\overline{R}_2)$ ( c.f. Remark 2.3 ). Using Lemma \[lma:2.3\] we can extend the diffeomorphism $\psi_4'$ from $\overline{S}_1'\cup \overline{R}_2\subset\overline{\Omega}^{(3)}$ to $\overline{\Omega}^{(3)}$. We can also extend $c_4'$ from $\overline{S}_1\cup
\overline{R}_2$ to $\Omega^{(4)}\stackrel{def}{=}\psi_4'(\overline{\Omega}^{(3)})$.

Define $\psi_4=\psi_4'\circ\psi_3,\ c_4=c_4'(x)c_3(\psi_4^{-1}(x))$. Then $L^{(1)}=L^{(4)}$ in $\overline{S}_1\cup\overline{R}_2$ and $L^{(4)}=c_4\circ
\psi_4\circ L^{(2)}$ is an operator in $\Omega^{(4)}$.

Let $\sigma_2$ be the same as $\sigma_1$ with $\delta_3$ replaced by $\delta_3+{{\varepsilon}}$, let $\Sigma_2=\psi_1^{-1}(\sigma_2)$ and let $\hat{\Sigma}_2$ be the smoothing of $\Sigma_2$ (see Fig.1). Denote by $R_3$ the domain bounded by $\hat{\Sigma}_2$ and $O(\delta_1+\delta_2)$. Note that $R_2\subset R_3\subset \overline{S}_1\cup\overline{R}_2$. Therefore $L^{(1)}=L^{(4)}$ in $\overline{R}_3$. It follows from Lemma \[lma:2.4\] that $\Lambda^{(1)}=\Lambda^{(4)}$ on $\hat{\Sigma}_2\times(\delta_2+{{\varepsilon}},T_0-\delta_2-{{\varepsilon}})$.

Now repeat the same construction with $\hat{\Sigma}_1$ replaced by $\hat{\Sigma_2}$.

We will get a domain $S_2\subset\overline{\Omega}^{(1)}$ consisting of all geodesics with respect to the metric $\|g_1^{jk}\|^{-1}$ starting on $\hat{\Sigma}_2$, orthogonal to $\hat{\Sigma}_2$ and having the length ${{\varepsilon}}$, where ${{\varepsilon}}\ll \delta_3$ is the same as above. Applying Lemma \[lma:3.4\] and Remark 2.3 we get a diffeomorphism $\psi_5'$ of $\overline{S}_2'\cup\overline{R}_3$ onto $\overline{S}_2\cup\overline{R}_3$ and a gauge transformation $c_5'\in G_0(\overline{S}_2\cup\overline{R}_3)$ such that $L^{(1)}=L^{(5)}$ on $\overline{S}_2\cup\overline{R}_3$ where $L^{(5)}=c_5'\circ\psi_5'\circ L^{(4)},\ S_2'\subset\overline{\Omega}^{(4)}$ is the same as $S_2$ with respect to the metric $\|g_4^{jk}\|^{-1}$. Using Lemma \[lma:2.3\] we extend $\psi_5'$ from $\overline{S}_2'\cup\overline{R}_3\subset \overline{\Omega}^{(4)}$ to $\Omega^{(4)}$ as a diffeomorphism of $\overline{\Omega}^{(4)}$ onto $\overline{\Omega}^{(5)}\stackrel{def}{=}\psi_5'(\overline{\Omega}^{(4)})$. Define $\psi_5=\psi_5'\circ\psi_4,\ c_5(x)=c_5'(x)c_4(\psi_5^{-1}(x))$. Then $\psi_5(x)$ is a diffeomorphism of $\overline{\Omega}^{(2)}$ onto $\overline{\Omega}^{(5)},\ c_5(x)\in G_0(\overline{\Omega}^{(5)},\ 
L^{(5)}=c_5\circ\psi_5\circ L^{(2)}$.

Analogously, let $\sigma_3$ be the same as $\sigma_1$ with $\delta_2$ replaced by $\delta+2{{\varepsilon}},\ \Sigma_3=\psi_1^{-1}(\sigma_3)$ and let $\hat{\Sigma}_3$ be a smoothing of $\Sigma_3$. Denote by $R_4$ the domain bounded by $\hat{\Sigma}_3$ and $O(\delta_1+\delta_2)$. Then $\overline{R}_3\subset\overline{R}_4\subset(\overline{S}_2\cup\overline{R}_3)$ and $L^{(1)}=c_6\circ\psi_6\circ L^{(2)}$ on $\overline{R}_4$, where $\psi_6$ is a diffeomorphism of $\overline{\Omega}^{(2)}$ onto $\overline{\Omega}^{(6)}
\stackrel{def}{=}\psi_6(\overline{\Omega}^{(2)})$ and $c_6\in G_0(\overline{\Omega}^{(6)})$. After $\frac{T-\delta_3}{{{\varepsilon}}}=N$ steps we get a domain $\overline{R}_N
\supset\overline{R}$, a diffeomorphism $\psi_{N+2}$ of $\overline{\Omega}^{(2)}$ onto $\overline{\Omega}^{(N+2)}$ and a gauge transformation $c_{N+2}\in G_0(\overline{\Omega}^{(N+2)})$ such that $L^{(1)}=c_{N+2}\circ\psi_{N+2}\circ L^{(2)}$ on $\overline{R}_N,\ \psi_{N+2}=I$ and $c_{N+2}=1$ on $\Gamma_0$.

We proved the following lemma:

\[lma:4.1\] Let $O'\subset \Gamma_0$ be a small neighborhood of $O(\delta_1+\delta_2)$. Suppose $\Lambda^{(1)}=\Lambda^{(2)}$ on $O'\times(0,T_0)$. Let $R\subset\Omega^{(1)}$ be the same as above and $T<\frac{T_0}{2}$. Then there exists a diffeomorphism $\varphi$ of $R$ onto $\overline{R'}\stackrel{def}{=}\varphi(\overline{R})\subset\overline{\Omega}^{(2)}$ and a gauge tranformation $c\in G_0(\overline{R})$ such that $L^{(1)}=
c\circ\varphi^{-1}\circ L^{(2)}$ on $\overline{R},\ \varphi=I$ and $c=1$ on $O(\delta_1)$.

[**Remark 4.1**]{} In order to prove Lemma \[lma:4.1\] we used $L^{(p)},p=1,2$ only in a small neighborhoods of $\overline{R}$ and $\overline{R'}$. The properties of $L^{(p)},p=1,2,$ outside of these neighborhoods play no role. For example, $L^{(p)},p=1,2,$ are not required to be self-adjoint outside of neighborhoods of $R$ and $R'$.

[**Remark 4.2**]{} The following generalization of Lemma \[lma:4.1\] holds:

Let $x^{(0)}\in\Gamma_0$ and let $x^{(1)}$ be an arbitrary point in $\Omega^{(1)}$. Let $\gamma$ be an arbitrary curve in $\Omega^{(1)}$ connecting points $x^{(0)}$ and $x^{(1)}$ and having the length less than $\frac{T_0}{2}$. There exists a neighborhood $V_0\subset \Omega^{(1)}$ of $\gamma$, a diffeomofphism $\psi_0$ of $\overline{V}_0$ onto $\overline{V}_0'\stackrel{def}{=}
\psi_0(\overline{V}_0)\subset\Omega^{(2)}$ and there exists a gauge $c_0(x)\in G_0(\overline{V}_0)$ such that $L^{(1)}=c_0\circ\psi_0^{-1}\circ L^{(2)}$ in $\overline{V}_0,\ \psi_0=I$ and $c_0=1$ on $\overline{V}_0\cap\Gamma_0$ assuming that $\Lambda^{(1)}=\Lambda^{(2)}$ on $\tilde{O}\times(0,T_0)$ where $\tilde{O}\subset\Gamma_0$ is a small neighborhood of $\overline{V}_0\cap\Gamma_0$.

To prove this result we approximate $\gamma$ by a piece-wise smooth curve consisting of geodesic segments and prove Lemma \[lma:4.1\] successively for each geodesic segment (see similsr arguments below).

Now we actually start the global construction (c.f. \[KKL\] and \[KKL1\]). Since $\overline{\Omega}^{(1)}$ is compact there exists $\delta_0>0$ such that for any point $x^{(0)}\in \overline{\Omega}^{(1)}$ the geodesics starting at $x^{(0)}$ form a local system of coordinates in $B(x^{(0)},\delta_0)\setminus B(x^{(0)},\delta)$ for any $0<\delta<\delta_0$. Here $B(x^{(0)},r)=\{x:d(x,x^{(0)})<r\}$ is a ball of radius $r$. We assume that the metric $\|g_1^{jk}\|^{-1}$ is extended to a $\delta_0$-neighborhood of $\overline{\Omega}^{(1)}$. Therefore $B(x^{(0)},\delta_0)$ make sense when $B(x^{(0)},\delta_0)\cap\partial\Omega^{(1)}
\neq\emptyset$. Let ${{\varepsilon}}_1>0$ be such that the semigeodesic coordinates with respect to $\partial\Omega^{(1)}$ hold in $2{{\varepsilon}}_1$-neighborhood of $\partial\Omega^{(1)}$ and the interior of this neighborhood does not intersect $\partial\Omega^{(1)}$. For each $x^{(0)}\in\partial\Omega^{(1)}$ consider a ball $B(x^{(0)},{{\varepsilon}}_1)$. For each $x^{(0)}\in\Omega^{(1)}$ consuder a ball $B(x^{(0)},r)$ such that $r<\frac{\delta_0}{2}$ and $\overline{B}(x^{(0)},r)\cap\partial\Omega^{(1)}
=\emptyset$. Such balls form an open cover of $\overline{\Omega}^{(1)}$ and since $\overline{\Omega}^{(1)}$ is compact there exists a finite subcover $\{B(x^{(k)},r_k)\},k=1,...,N$. Denote by $\Omega_{{{\varepsilon}}_1}$ the union of all balls $B(x^{(k)},r_k)$ such that $\overline{B}(x^{(k)},r_k)\cap\partial\Omega^{(1)}=\emptyset$.

Let $\Gamma$ and $T$ be the same as in Lemma \[lma:2.2\]. Repeating the proof of Lemmas \[lma:2.2\] and \[lma:2.3\] and using th same notations we get a domain $B_1\subset\Omega^{(1)},\ 
\partial B_1\cap \Gamma\neq\emptyset$ and connected, a diffeomorphism $\varphi_3$ of $\overline{\Omega}^{(2)}$ onto $\overline{\Omega}^{(3)}\stackrel{def}{=}
\varphi_3(\overline{\Omega}^{(2)})$ and a gauge transformation $c_3\in G_0(\overline{\Omega}^{(3)})$ such that $L^{(3)}=L^{(1)}$ in $B_1\subset \Omega^{(1)}\cap\Omega^{(3)}$ where $L^{(3)}=c_3\circ\varphi_3\circ L^{(2)}$ is an operator in $\Omega^{(3)},\ \varphi_3=I$ on $\Gamma_0$. Since $\varphi_3=I$ on $\Gamma_0$ we have that $\Lambda^{(3)}=\Lambda^{(1)}$ on $\Gamma_0\times(0,T_0)$ where $\Lambda^{(3)}$ is the D-to-N for $L^{(3)}$. Let $\Gamma_1=
(\Gamma_0\setminus(\Gamma_0\cap\partial B_1))\cup(\partial B_1\setminus \Gamma_0)$. Note that $\Gamma_1\subset\partial(\Omega^{(1)}\setminus \overline{B}_1)\cap
\partial(\Omega^{(3)}\setminus \overline{B}_1)$.

\[fig:figure2\]

 \
Figure . The domain $U_0$. The boundary $S_0$ is drawn bold.

Take arbitrary point $x^{(0)}\in B_1$ such that $d(x^{(0)},\Gamma_1)=2{{\varepsilon}}_2$ where ${{\varepsilon}}_2$ is much smaller than $\delta_0$. We have that $B(x^{(0)},{{\varepsilon}}_2)
\subset B_1$. Pick some geodesics $\gamma_0$ starting at $x^{(0)}$ and denote by $U_0$ the part of $B(x^{(0)},\delta)\setminus\overline{B}(x^{(0)},{{\varepsilon}}_2)$ consisting of all geodesics $\gamma$ starting at $x^{(0)}$ and having an angle less than $\pi-{{\varepsilon}}_3$ with $\gamma_0$ at $x^{(0)}$. We choose $\delta\leq\delta_0,\gamma_0$ and ${{\varepsilon}}_3$ such that $\overline{U}_0
\cap\partial\Omega^{(1)}=\emptyset$. Denote by $S_0$ a smooth surface in $B_1$ that contains $\overline{U}_0
\cap\partial B(x^{(0)},{{\varepsilon}}_2)$ and divides $B_1$ in two domains, $B_1^+$ and $B_1^-$, where $B_1^+$ is bounded by $\Gamma_1$ and $S_0$, and $B_1^-=B_1\setminus\overline{B}_1^+$ (see Fig.2). We assume that $\Omega^{(1)}\setminus \overline{B}_1^-$ has a smooth boundary that includes $S_0$. Denote $\Gamma_2=S_0\cup (\Gamma_1\setminus\partial B_1^+)$. Since $L^{(3)}=L^{(1)}$ in $\overline{B}_1^-$ and $\Lambda^{(3)}=\Lambda^{(1)}$ on $\Gamma_0\times(0,T) $ we get from the Lemma \[lma:2.4\] that $\Lambda^{(3)}=\Lambda^{(1)}$ on $\Gamma_2\times(\delta',T-\delta')$, where $\delta'=\max_{x\in\bar{B}_1^-}d_1(x,\Gamma_0)$ and we consider the D-to-N operators corresponding to $L^{(1)},L^{(3)}$ in domains $(\Omega^{(1)}\setminus \overline{B}_1^-)\times(\delta',T_0-\delta'),\ 
(\Omega^{(3)}\setminus \overline{B}_1^-)\times(\delta',T_0-\delta')$.

Now apply Lemma \[lma:4.1\] to $U_0\subset(\Omega^{(1)}\setminus B_1^-)$ instead of $R$. Let $U_0'$ be the union of all geodesics in $\Omega^{(3)}$ with respect to the metric $\|g_3^{jk}\|^{-1}$ starting at $\overline{U}_0
\cap\partial B(x^{(0)},{{\varepsilon}}_2)$, orthogonal to $\partial B(x^{(0)},{{\varepsilon}}_2)$ and having the length $\delta-{{\varepsilon}}_2$. Note that $B(x^{(0)},{{\varepsilon}}_2)\subset B_1\subset
\Omega^{(1)}\cap\Omega^{(3)}$. Since $\Lambda^{(1)}=\Lambda^{(3)}$ on $\Gamma_2\times(\delta',T_0-\delta')$, the Lemma \[lma:4.1\] implies that there exists a diffeomorphism $\varphi$ of $\overline{U}_0'
\subset\overline{\Omega}^{(2)}$ onto $\overline{U}_0$ and there exists $c(x)\in C^\infty(\overline{U}_0),\ c(x)\neq 0$ in $\overline{U}_0$ such that $c\circ\varphi\circ L^{(3)}=L^{(1)}$ in $\overline{U}_0,\ \varphi=I$ and $c=1$ on $S_0\cap\overline{U}_0$. Define $\varphi=I$ on $\overline{B}_1\setminus U_0$ and $\Gamma_0$. Also define $c=1$ on $\overline{B}_1\setminus U_0$ and $\Gamma_0$. Since $L^{(3)}=L^{(1)}$ in $\overline{U}_0\cap\overline{B}_1$ we have that $\varphi=I$ and $c=1$ in $\overline{U}_0\cap\overline{B}_1$. Therefore $\varphi$ and $c$ are $C^\infty$ on $\overline{U}_0\cup\overline{B}_1\cup\Gamma_0$ (c.f. Remark 2.3). Applying Lemma \[lma:2.3\] we can extend $\varphi$ from $\overline{U}_0\cup\overline{B}_1\cup\Gamma_0$ to $\overline{\Omega}^{(3)}$ as a diffeomorphism of $\overline{\Omega}^{(3)}$ onto $\overline{\Omega}^{(4)}\stackrel{def}{=}\varphi(\overline{\Omega}^{(3)})$ and extend $c$ from $\overline{U}_0\cup\overline{B}_1\cup\Gamma_0$ to $\overline{\Omega}^{(4)}$ as an element of $G_0(\overline{\Omega}^{(4)})$.

Let $\varphi_4=\varphi\circ\varphi_3$. Then $\varphi_4$ is a diffeomorphism of $\overline{\Omega}^{(2)}$ onto $\overline{\Omega}^{(4)}$. Let $c_4(x)=
c(x)c_3(\varphi_4^{-1}(x))\in G_0(\Omega^{(4)}$. Then we have $L^{(4)}=c_4\circ\varphi_4\circ L^{(2)}$ is an operator in $\Omega^{(4)}$ equal to $L^{(1)}$ in $\overline{U}_0\cup\overline{B}_1 
 \subset\overline{\Omega}^{(1)}\cap\overline{\Omega}^{(4)}$.

Therefore applying Lemma \[lma:4.1\] to $U_0$ we gained that $\overline{B}_1$ is replaced by a larger domain $\overline{U}_0\cup\overline{B}_1$.

Taking a point $x^{(1)}\in U_0\cup B_1$ instead of $x^{(0)}$ we can construct a domain $U_1$ similar to $U_0$. For the brevity we shall call by $U$-type domains the domains similar to $U_0$. We shall show that adding a finite number of $U$-type domains we can cover $\overline{\Omega}^{(1)}$.

Take any ball $B(x^{(p)},r_p)\subset \Omega_{{{\varepsilon}}_1}$. Since $\overline{\Omega}^{(1)}$ is connected, the point $x^{(p)}$ can be connected with $x^{(0)}\in B_1$ by a broken geodesics (c.f. \[KKL\]), more exactly, there exist points $y_1=x^{(0)},
y_2,...,y_{N_1}=x^{(p)}$ such that $y_k$ and $y_{k+1},1\leq k\leq N_1-1$, can be connected by a geodesics of length $\leq \frac{\delta_0}{2}$. Using a sequence of $U$-type domains we can cover this broken geodesics including $x^{(p)}$. Adding more $U$-type domains if needed we can cover $B(x^{(p)},r_p)$ too. We can do this with any ball $B(x^{(p)},r_p)\subset \Omega_{{{\varepsilon}}_1}$. Therefore inserting $M$ of $U$-type domains $U_1,...,U_M$ in $\Omega^{(1)}$ we get a sequences of domains $\Omega^{(k)},4\leq k\leq M+4$, diffeomorphisms $\varphi_k$ of $\overline{\Omega}^{(2)}$ onto $\overline{\Omega}^{(k)}$ and gauge transformations $c_k\in G_0(\overline{\Omega}^{(k)})$ such that $L^{1)}=L^{(k)}$ in $\overline{B}_1\cup(\cup_{j=0}^{k-4}\overline{U}_j),\ 4\leq k\leq M+4$, where $L^{(k)}=c_k\circ\varphi_k\circ L^{(2)}$ are operators in $\Omega^{(k)},\ 
B_1\cup(\cup_{j=0}^{k-4}U_j)\subset\Omega^{(1)}\cap\Omega^{(k)},\ 
\Lambda^{(1)}=\Lambda^{(k)}$ on $\Gamma_0\times(0,T_0)$ $\ 4\leq k\leq M+4$.

Let $M$ be such that $\Omega_{{{\varepsilon}}_1}\subset B_1\cup(\cup_{j=0}^{M}U_j)\subset\Omega^{(1)}\cap\Omega^{(M+4)}$. We have $L^{(1)}=L^{(M+4)}$ in $\overline{\Omega}_{{{\varepsilon}}_1}$, and $\Lambda^{(1)}=\Lambda^{(M+4)}$ on $\Gamma_0\times
(0,T_0)$.

Note that we choose $U_j,0\leq j\leq M$, taking into account the geometry of $\Omega^{(1)}$ and the metric $\|g_1^{jk}\|^{-1}$, regardless of the geometry of $\Omega^{(p)}$ and the metric $\|g_p^{jk}\|^{-1}$, $\ 2\leq p$.

Now consider the cover of $\overline{\Omega}^{(1)}\setminus \Omega_{{{\varepsilon}}_1}$ by $U$-type domains. Let $\alpha=\partial\Omega^{(1)}\cap B(x^{(k)},{{\varepsilon}}_1)$, where $\{B(x^{(k)},{{\varepsilon}}_1)\}$ is a finite cover of $\partial\Omega^{(1)}$. Denote by $U_{M+1}$ the union of all geodesics with respect to the metric $\|g_1^{jk}\|^{-1}$ starting on $\overline{\alpha}$, orthogonal to $\overline{\alpha}$ and having the same lengths $2{{\varepsilon}}_1$. Let $W_{M+1}$ be the set of endpoints of these geodesics. It follows from the definition of $\Omega_{{{\varepsilon}}_1}$ that $W_{M+1}$ is located inside $\Omega_{{{\varepsilon}}_1}$.

Denote by $U_{M+1}'$ the union of all geodesics with respect to metric $\|g_{M+4}^{jk}\|^{-1}$ corresponding to $L^{(M+4)}$ that start on $W_{M+1}$, orthogonal to $W_{M+1}$ and having the lengths $2{{\varepsilon}}_1$. Note that $W_{M+1}\subset\Omega_{{{\varepsilon}}_1}\subset\Omega^{(M+4)}$.

Let $\tilde{W}_{M+1}$ bea surface in $\Omega_{{{\varepsilon}}_1}$ containing $W_{M+1}$ and similar to $S_0$ in the case of domain $U_0$. Since $\Lambda^{(1)}=\Lambda^{(M+4)}$ on $\Gamma_0\times(0,T_0)$ and $L^{(1)}=L^{(M+4)}$ on $\overline{\Omega}_{{{\varepsilon}}_1}$, Lemma \[lma:2.4\] implies that $\Lambda^{(1)}=\Lambda^{(M+4)}$ on $\tilde{W}_{M+4}\times(T',T_0-T')$, where $T'=\max_{x\in\overline{\Omega}_{{{\varepsilon}}_1}} d_1(x,\Gamma_0)$. Now we can repeat for $U_{M+1}$ the same arguments as for $U_0$. Applying Lemma \[lma:4.1\] we get that there exists a diffeomorphism $\varphi'_{M+5}$ of $\overline{U}_{M+1}'$ onto $\overline{U}_{M+1},\ 
\varphi_{M+5}'=I$ on $W_{M+1}$ and $c_{M+5}'(x)\in G_0(\overline{U}_{M+1}),\ 
c'=1$ on $W_{M+1}$ such that $$\label{eq:4.1}
c_{M+5}'\circ\varphi_{M+5}'\circ L^{(M+4)}=L^{(1)}
 \ \mbox{in\ } U_{M+1}.$$ Since $L^{(M+4)}=L^{(1)}$ in $\Omega_{{{\varepsilon}}_1}$ we have that $\varphi_{M+5}'=I$ on $\overline{U}_{M+5}\cap\overline{\Omega}_{{{\varepsilon}}_1}$. Therefore $\varphi_{M+5}'$ is a diffeomorphism on $\overline{U}_{M+5}\cup\overline{\Omega}_{{{\varepsilon}}_1}$ after extending $\varphi_{M+5}'$ as $I$ on $\overline{\Omega}_{{{\varepsilon}}_1}$ (c.f. Remark 2.3). Analoguously taking $c_{M+5}'=1$ on $\overline{\Omega}_{{{\varepsilon}}_1}$ we get a $C^\infty$-function on $\overline{\Omega}_{{{\varepsilon}}_1}\cup\overline{U}_{M+1}$. Note that Lemma \[lma:4.1\] implies that $U_{M+1}'\subset \Omega^{(M+4)}$.

We shall consider first the case when $\alpha\cap\Gamma_0\neq\emptyset$. We shall prove that $\varphi_{M+5}'=I,\ c_{M+5}'=1$ on $\alpha_{1}\stackrel{def}{=}
\alpha\cap\Gamma_0$.

Let $g$ be arbitrary smooth function with the support in $\overline{\alpha}_{1}
\times(T',T_0-T')$.

Let $u_1$ be the solution of $L_1u_1=0$ on $\Omega^{(1)}\times(0,T_0),\ 
u_1=u_{1t}=0$ for $t=0,\ u_1|_{\Gamma_0\times(0,T_0)}=g$. Also let $u_{M+4}(x)$ be the solution of $L^{(M+4)}u_{M+4}=0$ in $\Omega^{(M+4)}\times(0,T_0),\ 
u_{M+4}=\frac{\partial u_{M+4}}{\partial t}=0$ for $t=0$ and $u_{M+4}|_{\Gamma_{0}\times(0,T_0)}=g$. Since $\Lambda^{(1)}=\Lambda^{(M+4)}$ on $\Gamma_0\times(0,T)$ and since $L^{(M+4)}=L^{(1)}$ in $\overline{\Omega}_{{{\varepsilon}}_1}$ we get, by the unique continuation theorem (c.f \[T\]), that $u_1=u_{M+4}$ in $\overline{\Omega}_{{{\varepsilon}}_1}\times
(T',T_0-T')$.

It follows from (\[eq:4.1\]) that $u_1(x,t)$ and $u_{M+5}(x,t)=c_{M+5}'(x)u_{M+4}((\varphi_{M+5}')^{-1}(x),t)$ satisfy the same equation $L^{(1)}u=0$ in $U_{M+1}\times(T',T_0-T'))$. Since $u_1(x)=u_{M+4}(x),\ \varphi_{M+5}'=I$ and $c_{M+5}'=1$ on $\Omega_{{{\varepsilon}}_1}\times(T',T_0-T')$ we have that $u_1$ and $u_{M+5}$ have the same Cauchy data on $\tilde{W}_{M+1}\times(T',T_0-T')$.

We have, using again the unique continuation theorem that $$\label{eq:4.2}
c_{M+5}'(x)u_{M+4}((\varphi_{M+5}')^{-1}(x),t)=u_1(x,t)
\  \mbox{in  \ }\overline{U}_{M+1}\times(T_1+2{{\varepsilon}}_1,T_0-T_1-2{{\varepsilon}}_1).$$ Taking the restriction of (\[eq:4.2\]) to $\overline{\alpha}_1\times(T'+2{{\varepsilon}}_1,T_0-T'-2{{\varepsilon}}_1)$ we obtain $$\label{eq:4.3}
c_{M+5}'(x)g((\varphi_{M+5}')^{-1}(x),t)=g(x,t)  \ \mbox{in\  }
\overline{\alpha}_{1}\times(T_1+2{{\varepsilon}}_1,T_0-T_1-2{{\varepsilon}}_1).$$

Since $g$ is arbitrary we get that $\varphi_{M+5}'=I$ on $\overline{\alpha}_{1},\ 
c_{M+5}'(x)=1$ on $\overline{\alpha}_{1}$. Therefore, we take $\varphi_{M+5}'=I,\ 
c_{M+1}'(x)=1$ on $\Gamma_0$ and get a $C^\infty$-functions on $\Gamma_0\cup\overline{\Omega}_{{{\varepsilon}}_1}\cup\overline{U}_{M+1}$. There is no any obstruction to the extension $\varphi_{M+5}'=I,\ 
c_{M+5}'=1$ on $\Gamma_0$ in the case when $\alpha\cap\Gamma_0=\emptyset$. Thus in both cases applying Lemma \[lma:2.3\] we get a diffeomorphism $\varphi_{M+5}'$ of $\Omega^{(M+4)}$ onto $\overline{\Omega}^{(M+5)}\stackrel{def}{=}
\varphi_{M+5}'(\overline{\Omega}^{(M+4)})$ and $c_{M+5}'\in G_0(\Omega^{(M+5)})$ such that $\varphi_{M+5}'=I$ on $\Gamma_0$.

As in the case of the $U$-type domain $U_0$, denote $\varphi_{M+5}=
\varphi_{M+5}'\circ\varphi_{M+4},\ c_{M+5}=c_{M+5}'c_{M+4}(\varphi_{M+4}^{-1}(x))$. Then we get that $$L^{(M+5)}=L^{(1)}\ \ \ \mbox{in\ \ \ } \overline{\Omega}_{{{\varepsilon}}_1}\cup\overline{U}_{M+1}
\subset\overline{\Omega}^{(1)}\cap\overline{\Omega}^{(M+5)},$$ where $L^{(M+5)}=c_{M+5}\circ\varphi_{M+5}\circ L^{(2)}$ is an operator on $\overline{\Omega}^{(M+5)}$, $\ \varphi_{M+5}$ is a diffeomorphism of $\overline{\Omega}^{(2)}$ onto $\overline{\Omega}^{(M+5)}$, $\varphi_{M+5}=I$ on $\Gamma_0$, $c_{M+5}\in
G_0(\overline{\Omega}^{(M+5)})$. In particular, we have $\alpha\subset\partial\Omega^{(1)}\cap\partial\Omega^{(M+5)}$ and $\varphi_{M+5}^{-1}$ maps $\alpha\subset\partial\Omega^{(1)}$ onto $\alpha'\in\partial\Omega^{(2)}$. Repeating the same construction with each of $B(x^{(k)},{{\varepsilon}}_1)$ such that $B(x^{(k)},{{\varepsilon}}_1)\cap\partial\Omega^{(1)}\neq\emptyset$ we get $\overline{\Omega}_{{{\varepsilon}}_1}\cup
(\cup_{j=1}^{M_1}\overline{U}_{M+j})=\overline{\Omega}^{(1)},\ \
\overline{\Omega}_{{{\varepsilon}}_1}\cup
(\cup_{j=1}^{M_1}\overline{U}_{M+j})\subset\overline{\Omega}^{(M+M_1+4)}$ and $L^{(1)}=c_{M+M_1+4}\circ\varphi_{M+M_1+4}\circ L^{(2)}$ in $\overline{\Omega}_{{{\varepsilon}}_1}\cup
(\cup_{j=1}^{M_1}\overline{U}_j)$, where $\overline{\Omega}^{M+M_1+4)}\stackrel{def}{=}
\varphi_{M+M_1+4}(\overline{\Omega}^{(2)}),\
\varphi_{M+M_1+4}(x)$ is a diffeomorphism, $\varphi_{M+M_1+4}=I$ on $\Gamma_0,\ c_{M+M_1+4}\in G_0(\Omega^{M+M_1+4)})$.

We claim that $\Omega^{(M+M_1+4)}=\Omega^{(1)}$. If $\overline{\Omega}^{(M+M_1+4)}  \neq \overline{\Omega}^{(1)}$ then there exists an interior point of $\Omega^{(M+M_1+4)}$ that is a boundary point of $\Omega^{(1)}$. It follows from the arguments similar to the proof of Lemma \[lma:3.2\] that this is impossible.

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