Abstract
We study the inverse problems for the second order hyperbolic equations of general form with time-dependent coefficients assuming that the boundary data are given on a part of the boundary. The main result of this paper is the determination of the time-dependent Lorentzian metric by the boundary measurements. This is achieved by the adaptation of a variant of the boundary control method developed by Eskin (Inverse Probl 22(3):815–833, 2006; Inverse Probl 23:2343–2356, 2007).
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1 Introduction
Consider a second order hyperbolic equation in \(\mathbb {R}^{n+1}\) of the form
where \(x=(x_0,x_1,\ldots ,x_n)\in \mathbb {R}^{n+1},\ x_0\) is the time variable. In (1.1) \(g(x)=\det [g_{jk}(x)]_{j,k=0}^n\), where \([g_{jk}(x)]_{j,k=0}^n=\big ([g^{jk}]_{j,k=0}^n\big )^{-1}\) is the metric tensor, \(A(x)=(A_0(x),A_1(x),\ldots ,A_n(x))\) is the vector potential. We assume that all coefficients in (1.1) belong to \(C^\infty (\mathbb {R}^{n+1})\) and that
Let \((\xi _0,\xi _1,\ldots ,\xi _n)\) be dual variables to \((x_0,x_1,\ldots ,x_n)\). The strict hyperbolicity of (1.1) with respect to \(\xi _0\) means that the quadratic equation
has two real distinct roots \(\xi _0^-(\xi _1,\ldots ,\xi _n)<\xi _0^+(\xi _1,\ldots ,\xi _n)\) for all \((\xi _1,\ldots ,\xi _n)\ne (0,\ldots ,0)\) and all \(x\in \mathbb {R}^{n+1}\). We have
The strict hyperbolicity implies that
for all \((\xi _1,\ldots ,\xi _n)\ne 0, \ x\in \mathbb {R}^n\).
In this paper we assume a more restrictive condition that
i.e. we assume that the spatial part of the equation (1.1) is elliptic for any \(x\in \mathbb {R}^{n+1}\).
Note that the quadratic form (1.3) has the signature \((+1,-1,\ldots ,-1)\). Therefore \((-1)^ng(x)>0\). We assume also that \(A_j(x),0\le j\le n,\) are real-valued. Thus the operator in (1.1) is formally self-adjoint.
We consider the following class of domains \(D\subset \mathbb {R}^{n+1}\). Let \(D_t=D\cap \{x_0=t\}\) be the intersection of D with the plane \(\{t=x_0\}, t\in \mathbb {R}\). We assume that \( D_t\) is a smooth closed bounded domain in \(\mathbb {R}^n\) smoothly dependent and uniformly bounded in t and such that \(D_t\) is diffeomorphic to \(D_0\) for all \(t\in \mathbb {R}\). More precisely we assume that there exists a diffeomorphism
that maps \(D_{x_0}\) onto \(D_0\) and smoothly depends on \(x_0\). We shall call such domains D admissible.
Let \(S(x_0,x_1,\ldots ,x_n)=0\) be the equation of \(\partial D=\bigcup _{t\in \mathbb {R}}\partial D_t\). Sometimes we denote \(\partial D\) by S. We assume that S is a time-like smooth surface in \(\mathbb {R}^{n+1}\), i.e.
where \(x\in S\) and \((\nu _0,\nu _1,\ldots ,\nu _n)\) is a normal vector to S. The vector \((\nu _0,\ldots ,\nu _n)\) satisfying (1.8) is called a space-like vector. Also, the surface \(\Sigma \) in \(\mathbb {R}^{n+1}\) is called space-like if \(\sum _{j,k=0}^ng^{jk}(x)\nu _j(x)\nu _k(x)>0\), where \(x\in \Sigma \) and \((\nu _0(x),\ldots ,\nu _n(x))\) is the normal vector to \(\Sigma \).
Consider the initial–boundary value problem
where f is a smooth function on \(S=0\) with compact support, \(Lu=0\) is the same as in (1.1).
It is well known that the initial-value problem (1.9), (1.10), (1.11) is well-posed (cf. [15]), assuming that (1.2), (1.5) and (1.8) are satisfied.
Let \(S_0\subset S\) be a part of S such that (see Fig. 1) \(S_{0t}=\partial D_t\cap S_0\) has a nonempty interior for all \(t\in \mathbb {R}\). We assume also that for any \(x^{(0)}\in \partial S_0\) a vector \(\tau ^{(1)}\), tangent to S and normal to \(S_0\), is not parallel to \((1,0,\ldots ,0)\).
We include in the definition of the admissibility of D (see (1.7)) that the map \(y={\hat{y}}(x_0,x)\) is such that
\(S_0\) is part of the boundary \(S, \ S_0\cap \partial D_t\ne \emptyset \) for \(\forall t\)
Note that (1.7) maps the admissible domain D onto \(D=D_0\times \mathbb {R},\ S_0\) onto \({\hat{S}}_0=\Gamma _0\times \mathbb {R}\) (cf. (1.12)), where \(\Gamma _0=S_0\cap \partial D_0\), i.e. \({\hat{D}},{\hat{S}}_0\) are cylindrical domains (see Fig. 2).
\({\hat{D}}={\hat{D}}_0\times \mathbb {R}, \ \ {\hat{S}}_0=\Gamma _0\times \mathbb {R}\) are cylindrical domains
The Dirichlet-to-Neumann operator \(\Lambda \) that maps the Dirichlet data to the Neumann data on \(\partial D\) is defined as
where u(x) is the solution of (1.9), (1.10), (1.11) and \((\nu _0(x),\ldots ,\nu _n(x))\) is the unit outward normal to S.
Denote by
any proper diffeomorphism of \({\overline{D}}\) onto some domain \(\overline{{\hat{D}}}\) such that
We call a diffeomorphism of \({\overline{D}}\) onto \(\overline{{\hat{D}}}\) proper if for any \([t_1,t_2]\subset \mathbb {R}\) the image of \({\overline{D}}\cap \{t_1\le x_0\le t_2\}\) is a domain \(\overline{{\hat{D}}}\cap \{S_-^{(t_1)}(y_1,\ldots ,y_n)\le y_0 \le S_+^{(t_2)}(y_1,\ldots ,y_n)\}\), where \(y_0=S_+^{(t_1)},y_0=S_-^{(t_0)}\) are space-like surfaces.
Let \({\hat{L}}{\hat{u}}=0\) be the Eq. (1.1) in y-coordinates, \(y\in {\hat{D}}\). We have
where
Here \({\hat{u}}(y)=u(x)\), \(y=y(x),\ {\hat{g}}(y)=\det [{\hat{g}}_{jk}(y)]_{j,k=0}^n,\ [{\hat{g}}_{jk}(y)]_{j,k=0}^n=\big ([{\hat{g}}^{jk}(y)]_{j,k=0}^n\big )^{-1}\).
Note that (1.17), (1.18) are equivalent to the equalities
where y and x are related by (1.14). Metric tensors \([g_{jk}(x)]_{j,k=0}^n\) and \([{\hat{g}}_{jk}(x)]_{j,k=0}^n\), related by (1.20), are called isometric.
We assume that conditions (1.2), (1.6) hold also in y-coordinates, i.e.
Let \(c(x)\in C^\infty ({\overline{D}})\) be such that
The group \(G_0({\overline{D}})\) of such c(x) is called the gauge group.
If \(Lu=0\) then \(u'=c^{-1}(x)u(x)\) satisfies the equation of the form (1.1) with \(A_j(x)\) replaced by
We shall call potentials \((A_0',\ldots ,A_n'(x))\) and \((A_0(x),\ldots ,A_n(x)))\) related by (1.23) gauge equivalent. Note that when D is simply connected then \(c(x)=\exp i\varphi \) where \(\varphi (x)\in C^\infty ({\overline{D}}),\ \varphi (x)\) is real-valued and \(\varphi (x)=0\) on \(S_0\).
Let \(y=y(x)\) be the change of variables, such that \(y(x)=x,\ x\in S_0\), transforming the equation \(Lu=0\) in D to the equation of the form (1.16) in \({\hat{D}}\). We consider the initial–boundary value problem
Note that \({\hat{S}}_0=S_0\), since \({\hat{y}}(x)=x\) on \(S_0\).
Since (1.21) holds, the initial–boundary value problem (1.24), (1.25), (1.26) is also well-posed. Let \({\hat{c}}(y)\in G_0(\overline{{\hat{D}}})\). Make the gauge transformation \(u'(y)={\hat{c}}^{-1}(y){\hat{u}}(y)\) and let \(L^{\prime }\) be such that \(L'u'=0\). We have
Note that \(u'={\hat{u}}\) on \({\hat{S}}_0\) since \({\hat{c}}(y)=1\) on \({\hat{S}}_0\) and \(L^{\prime }u^{\prime }\) has the form
\(A_j'(y),\ 0\le j\le n,\) are potentials gauge equivalent to \({\hat{A}}_j(y),\ 0\le j\le n\).
Let \(\Lambda ^{\prime }\) be the DN operator for (1.27), (1.28), (1.29)
where f is the same as in (1.10) and (1.29).
It can be shown that
if the operator \(L^{\prime }\) is obtained from L by the change of variables (1.14), (1.15) and the gauge transformation c(y) such that (1.22) holds.
Therefore the inverse problem of the determination of the coefficient of (1.1) can be solved only up to the changes of variables (1.14), (1.15) and the gauge transformations (1.22).
We shall formulate now some conditions which will be required to solve the inverse problem.
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(1)
Real analyticity in the time variable
One of the crucial steps in solving the inverse problem will be the use of the following unique continuation theorem of Tataru and Robbiano and Zuily (cf. [25, 29]) that requires the analyticity in \(x_0\):
Theorem 1.1
Let the coefficients of (1.1) be analytic in \(x_0\). Consider the equation \(Lu=0\) in a neighborhood \(U_0\) of a point \(P_0\). Let \(\Sigma =0\) be a noncharacteristic surface with respect to L passing through \(P_0\). If \(u=0\) in \(U_0\cap \{\Sigma <0\}\) then \(u=0\) in \(U_0\cap \{\Sigma >0\}\) near \(\Sigma =0\).
We assume also that the gauge c(x) and the map (1.7) are analytic in \(x_0\).
Let \(y=\varphi (x)\) be a diffeomorphism of neighborhood \(U_0\) onto the neighborhood \({\overline{V}}_0=\varphi ({\overline{U}}_0)\). Here \(\varphi (x)\) is smooth but not analytic in any variable. It is clear that if the unique continuation property for the operator L holds in \(U_0\) then it holds in \(V_0\) for the operator \({\tilde{L}}=\varphi \circ L\), though the coefficients of \({\tilde{L}}\) are not analytic. Here \(\varphi \circ L\) is the operator L in y-coordinates (cf. (1.16)). Therefore the following more general class of operators L with non-analytic coefficients has the unique continuation property: For each point \(x^{(0)}\) on D there is a neighborhood \(U_0\) and the diffeomorphism \(\psi (x)\) of \(U_0\) onto \(V_0=\psi (U_0)\) such that the coefficients of the operators \(\psi \circ L\) in \(V_0\) are analytic in \(x_0\). Thus, the unique continuation property holds for L in \(U_0\).
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(2)
The Bardos–Lebeau–Rauch condition
Consider the initial–boundary value problem
in the cylindrical domain \(D_0\times \mathbb {R},\ f\) has a compact support in \(\Gamma _0\times \mathbb {R},\ \Gamma _0\subset \partial D_0\). We say that BLR condition holds on \([t_0,T_{t_0}]\) if the bounded map from \(f\in H_1(\Gamma _0\times (t_0,T_{t_0}))\) to \(\big (u\big |_{x_0=T_{t_0}},\frac{\partial u}{\partial x_0}\big |_{x_0=T_{t_0}}\big )\in H_1(D_0)\times L_2(D_0)\), is onto in \(H_1(D_0)\times L_2(D_0)\), where \(u=0\) for \(x_0<t_0,\ f=0\) for \(x_0< t_0\).
Note that BLR condition obviously holds on \([t_0,T]\) for any \(T>T_{t_0}\) if it holds on \([t_0,T_{t_0}]\).
Let \(\{x=x(s),\xi =\xi (s)\}\in T_0^*({\overline{D}}_0\times [t_0,T_{t_0}])\), where
be the equations of null-bicharacteristics. Here \(L_0(x,\xi )=\sum _{j,k=0}^ng^{jk}(x)\xi _j\xi _k, L_0(y,\eta )=0\).
We assume that for any \(t_0\) there exists \(T_{t_0}\) depending continuously on \(t_0\) such that the BLR condition holds on \([t_0,T_{t_0}]\). It follows from [1] that BLR condition holds if any null bicharacteristic in \(T_0^*({\overline{D}}_0\times [t_0,T])\) intersects \(T_0^*({\overline{\Gamma }}_0\times [t_0,T])\) when \(T\ge T_{t_0}\).
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(3)
Domains of dependence
Let \(G(x,\xi )=\sum _{j,k=0}^ng_{jk}(x)\xi _j\xi _k,\ [g_{jk}]_{j,k=0}^n=\big ([g^{jk}]_{j,k=0}^n\big )^{-1}\). We say that \(x=x(\tau )\) is a forward time-like ray in \(D_0\times \mathbb {R}\) if \(x=x(\tau )\) is piece-wise smooth, \(G(x(\tau ),\frac{d x(\tau )}{d\tau })>0\) and \(\frac{d x_0}{d\tau }>0,\ 0\le \tau \). If \(G(x(\tau ),\frac{d x(\tau )}{d\tau })>0\) and \(\frac{d x_0}{d\tau }<0\) the ray \(x=x(\tau )\) is called the backward time-like ray.
One can show (cf [7]) that the forward domain of influence \(D_+(F)\) of a closed set \(F\subset D_0\times \mathbb {R}\) is the closure of the union of all piece-wise smooth forward time-like rays in \(D_0\times \mathbb {R}\) starting on F.
Analogously, the backward domain of influence \(D_-(F)\) of the closed set \(F\subset D_0\times \mathbb {R}\) is the closure of the union of all backward time-like piece-wise smooth rays in \(D_0\times \mathbb {R}\) starting at F. The domain of dependence of F is the intersection \(D_+(F)\cap D_-(F)\).
Let \(\Gamma \subset \partial D_0\) and let \(Lu=0\) in \(D_0\times \mathbb {R}\). A consequence of the unique continuation property is that \(u\big |_{\Gamma \times (t_1,t_2)} =\frac{\partial u}{\partial \nu }\Big |_{\Gamma \times (t_1,t_2)}=0\) implies \(u=0\) in the domain of dependence of \(\Gamma \times [t_1,t_2]\). Here \(\frac{\partial }{\partial \nu }\) is the normal derivative to \(\Gamma \). This fact follows from [19] in the case of time-independent coefficients. The proof in the time-dependent case is similar.
The following fact follows from the BLR condition:
Consider \(\Gamma \times [t_1,t_2],\Gamma \subset \partial D_0\). Suppose \([t_1,t_2]\) is arbitrary large. Then the domain of dependence of \({\overline{\Gamma }}\times [t_1,t_2]\) contains \({\overline{D}}_0\times [t_1+\delta ,t_2-\delta ]\) for some \(\delta >0\) dependent of the metric and the domain.
In this paper we will not attempt to estimate \(\delta >0\) since \([t_0+\delta ,t_2-\delta ]\) is also arbitrary large if \([t_1,t_2]\) is arbitrary large. \(\square \)
Now we shall state the main result of this paper.
Consider an admissible domain D in \(\mathbb {R}^{n+1}\) and an initial–boundary value problem in D.
Using the map of the form (1.7) defining the admissibility of the domain D we get a cylindrical domain \(D_0\times \mathbb {R}\) with \(S_0=\Gamma _0\times \mathbb {R}\) (cf. Fig. 2) and the initial–boundary value problem
where L has the form (1.1) and f has a compact support in \({\overline{\Gamma }}_0\times \mathbb {R}\). Consider another admissible domain \({\hat{D}}\). Making again the change of variables (1.7) we get a cylindrical domain \({\hat{D}}_0\times \mathbb {R}\) and another initial–boundary value problem
where \(L^{\prime }u^{\prime }\) has the form (1.30), \(f^{\prime }\) has a compact support in \({\overline{\Gamma }}_0\times \mathbb {R}\). Therefore the inverse problems for the admissible domains are reduced to the inverse problems in cylindrical domains.
We shall prove the following theorem:
Theorem 1.2
Consider two initial–boundary value problems (1.34), (1.35), (1.36) and (1.37), (1.38), (1.39) in domains \(D_0\times \mathbb {R}\) and \({\hat{D}}_0\times \mathbb {R}\), respectively. Suppose \(A_j(x), A_j'(y),0\le j\le n,\) are real-valued. Assume that \(\Gamma _0\subset \partial D_0\cap \partial {\hat{D}}_0\) is nonempty and open. Let \(\Lambda \) and \(\Lambda ^{\prime }\) be the corresponding DN operators for L and \(L^{\prime }\). Assume that \(\Lambda f\big |_{\Gamma _0\times \mathbb {R}}=\Lambda ' f\big |_{\Gamma _0\times \mathbb {R}}\) for all smooth f with compact support in \({\overline{\Gamma }}_0\times \mathbb {R}\). Suppose the conditions (1.2), (1.6) hold for L and \(L^{\prime }\). Assume that the coefficients of L and \(L^{\prime }\) are analytic in \(x_0\) and \(y_0\), respectively. Suppose also that BLR condition holds for (1.34), (1.35), (1.36) on \([t_0,T_{t_0}]\) for each \(t_0\in \mathbb {R}\). Then there exists a proper map \(y=y(x)\) of \({\overline{D}}_0\times \mathbb {R}\) onto \(\overline{{\hat{D}}_0}\times \mathbb {R},\ y=x\) on \(\Gamma _0\times \mathbb {R}\), and there exists a gauge transformation with the gauge \(c'(y)\in G_0(\overline{{\hat{D}}_0}\times \mathbb {R}),\ c'(y)=1\) on \({\overline{\Gamma }}_0\times \mathbb {R}\) such that \(L'= c'\circ y^* L\). Here \(y^*\circ L\) is the operator with \([{\hat{g}}^{jk}(y)]_{j,k=0}^n\) and \({\hat{A}}_k(y),\ 0\le k\le n\) as in (1.17), (1.18), \(c'\circ y^*\circ L\) is the operator with potentials \(A_j'(y),\ 0 \le j\le n\), gauge equivalent to \({\hat{A}}_k(y),\ 0\le k\le n\).
We end the introduction with the outline of the previous work and a short description of the content of the paper.
The first result on inverse hyperbolic problems with the data on the part of the boundary was obtained by Isakov [17]. The powerful boundary control (BC) method was discovered by Belishev [2] and was further developed by Belishev [3,4,5], Belishev and Kurylev [6], Kurylev and Lassas [21, 22] and others (see [19, 20]). In [8, 9] the author proposed a new approach to hyperbolic inverse problems that uses substantially the idea of BC method. This approach was extended in [10] to some class of time-independent metrics with time-dependent vector potentials and in [11] to the case of hyperbolic equations of general form with time-independent coefficients without vector potentials. The generalization to the case of Yang–Mills potentials was considered in [14]. The inverse problems for the D’Alambert equation with the time-dependent scalar potentials were considered earlier by Stefanov [S] and Ramm and Sjostrand [24] (see also Isakov [18]). The case of the D’Alambert equation with time-dependent vector potentials was studied by Salazar [26, 27]:
As it was mentioned in [1] the study of hyperbolic equations with time-dependent coefficients is very important because the linearization of basic nonlinear hyperbolic equations of mathematical physics leads to time-dependent linear hyperbolic equations.
The main result of the present paper is the determination of the time-dependent Lorentzian metric by the boundary measurements given on the part of the boundary. We consider the second order hyperbolic equations of general form (1.1) with time-dependent coefficients and vector potentials. The method is the extension of the approach in [8, 9, 11] to the case of time-dependent metrics. We adapt some lemmas of [8,9,10,11] to the time-dependent situation and simultaneously give sharper and simpler proofs.
The main step in the proof is the local step of solving the inverse problem in a small neighborhood near the boundary. This is done in Sects. 2–6.
In Sect. 2 we make a change of variables in a neighborhood \(U_0\subset \mathbb {R}^{n+1}\) of a point \(x^{(0)}\in S_0\). We called the new coordinates the Goursat coordinates since they are similar to coordinates arising in a solution of the Goursat problem in the case of hyperbolic equations with one space variable. The Goursat coordinates allow to simplify the equation (1.1). Another advantage of the Goursat coordinates is that the characteristic surface is a plane in these coordinates. Also it is proven in Sect. 2 that the original DN operator determines the new DN operator corresponding to the equation in Goursat coordinates.
In Sect. 3 we derive the Green formula in Goursat coordinates and prove the crucial density lemma (Lemma 3.1).
In Sect. 4 we establish the main formula that states that some integrals of solutions of the initial–boundary value problems are determined by the DN operator [see Theorem 4.5, formula (4.29)]. To establish this formula one needs to compare Sobolev norms on the characteristic surfaces corresponding to different operators having the same DN data. This is done by using the BLR condition. Note that in the case of time-independent coefficients there is an additional energy identity that allows to avoid the use of the BLR condition (see Remark 4.1). Also note that proofs in Sect. 4 require that the hyperbolic operators are formally self-adjoint. Thus the vector potentials \(A=(A_0,A_1,\ldots ,A_n)\) are required to be real-valued.
In Sect. 5 we construct geometric optics type solutions depending on a large parameter k. Substituting the geometric optics solutions in the main formula, we prove in Sects. 5 and 6 the local version (Theorem 6.2) of the main theorem (Theorem 1.2). In the last Sect. 7 we consider the global case. At first we study the case of a finite time interval (Theorem 7.7) and then finally prove Theorem 1.2.
2 The Goursat coordinates
We shall prove first the Theorem 1.2 in the small neighborhood of the boundary \(\partial D\).
Let \(x^{(0)}\in S_0\) and let \(U_0\subset \mathbb {R}^{n+1}\) be a small neighborhood of \(x^{(0)}\).
Suppose that we already did the change of variables (1.7) to make \(\partial D\) and \(S_0\) cylindrical, i.e. \(\partial D=\partial D_0\times \mathbb {R}\) and \(S_0=\Gamma _0\times \mathbb {R}\). We assume that we have chosen the coordinates \((x_0,x',x_n),\ x'=(x_1,\ldots ,x_{n-1})\) in \(U_0\) such that \(x_n=0\) is the equation of \(U_0\cap \partial D\) and \(U_0\cap D\) is contained in the half-space \(x_n>0\). Let \((x_0^{(0)},x_1^{(0)},\ldots ,x_{n-1}^{(0)},0)\) be the coordinates of the point \(x^{(0)}\). Let \(T_1<x_0^{(0)}<T_2, \ T_2-T_1\) is small.
Consider the initial–boundary value problem in \(U_0\cap D\):
We assume that L has the form (1.1). For the simplicity, we shall not change the notations when choosing the local coordinates such that the equation of \(U_0\cap S_0\) is \(x_n=0\). Assume that \(\hbox {supp}\,g \subset U_0\cap (\Gamma _0\times [T_1,T_2]),\ g=0\) for \(x_0<T_1\). Note that \(\hbox {supp}\,u(x_0,x',x_n)\cap [T_1,T_2]\subset U_0\cap [T_1,T_2]\) for \(x_n>0\) if \(T_2-T_1\) is small.
We introduce new coordinates to simplify the operator L (cf. [11, pp. 327–329]) that we called the Goursat coordinates.
Denote by \(\psi ^\pm (x),x=(x_0,x',x_n)\) the solutions of the eikonal equations
with initial conditions
It is well known that the solution \(\psi ^\pm (x)\) of (2.4), (2.5) exists for \(0\le x_n\le {\varepsilon }\), where \({\varepsilon }>0\) is small (see, for example, [12], §64).
Since (2.4) is a quadratic equation in \(\psi _{x_n}^\pm \) one has to specify the sign of the square root. We have
We will need below that \(\psi _{x_n}^++\psi _{x_n}^-<0\) for \(x_n>0\). So we choose the plus sign of the square root:
Note that \(g^{nn}(x)<0,\ \psi _{x_0}^\pm \big |_{x_n=0}=\pm 1\). Therefore \(\psi _{x_n}^\pm \big |_{x_n=0}= \frac{\mp g^{n0}+\sqrt{(g^{n0})^2-g^{nn}g^{00}}}{g^{nn}}\). The solutions \(\psi ^\pm (x)\) exists for \(0<x_n<\delta ,\ \delta \) is small. For given \(T_1, T_2\) we assume that \(\delta \) is such that surfaces \(\psi ^+=0\) and \(\psi ^-=0\) intersect when \(x_n<\delta \) and are inside \(U_0\) when \(x_n<\delta \).
Let \(\varphi _j(x_0,x',x_n),1\le j\le n-1,\) be solutions of the linear equation
with initial condition
Make the following change of variables in \(U_0\cap [T_1,T_2]\):
Equation (1.1) has the following form in \((s,\tau ,y')\) coordinates where \(y'=(y_1,\ldots ,y_{n-1})\)
where
Note that terms containing \(\frac{\partial ^2}{\partial s^2},\frac{\partial ^2}{\partial \tau ^2},\frac{\partial ^2}{\partial y_j\partial \tau }\) vanished because of (2.4), (2.7), and
It follows from (2.6) for \(x_n=0\) that \(g^{+,-}>0\).
In (2.10) \({\hat{u}}(s,\tau ,y')=u(x_0,x',x_{n}),\)
Now we shall introduce a new system of coordinates (cf. [11])
where \(\psi ^+,\psi ^-,\varphi _j,\ 1\le j\le n-1\), are the same as in (2.4), (2.7).
Note that
Therefore \(y=\varphi (x)=(\varphi _0(x_1),\varphi _1(x),\ldots ,\varphi _n(x))\) is the identity on \(x_n=0\):
Here
Note that \(y_n=\varphi _n(x)>0\) when \(x_n>0\) since \(\psi _{x_n}^++\psi _{x_n}^-<0\) (cf. (2.6)),
Thus one can easily rewrite (2.10) in \((y_0,y',y_n)\) coordinates.
We shall further simplify (2.10) by making a gauge transformation
Then \(u^{\prime }\) satisfies the equation
where \(L^{\prime }\) is the same as \({\hat{L}}\) with \({\hat{A}}_j, {\hat{A}}_+, {\hat{A}}_-\) replaced by \(A_j', A_+', A_-',\ 1\le j\le n-1\), where
We choose \(d(s,\tau ,y')\) such that
Let
Note that
Since \(\sqrt{|{\hat{g}}|}=\frac{\sqrt{g_1}}{2{\hat{g}}^{+,-}}\) (cf (2.11)) we can rewrite \(L'u'=0\) in the form (cf. [8]):
where
Make the change of unknown function
where \( g_1=|\det [{\hat{g}}^{jk}]_{j,k=1}^{n-1}|^{-1}\) (cf. (2.11)). Then dividing \(L'u'=0\) by \({\hat{g}}^{+,-}\) we get (cf. [11])
where \(L_1u_1=0\) has the form (cf. (2.24))
where
and \(V_1\) is the same as in (2.25). Using that \(\frac{\partial }{\partial \tau }+iA_-'=\frac{1}{2}\big (-\frac{\partial }{\partial y_0}-\frac{\partial }{\partial y_n}\big )+iA_-' = -\frac{1}{2}\big [\big (\frac{\partial }{\partial y_0}-iA_-'\big )+\big (\frac{\partial }{\partial y_n}-iA_-'\big )\big ]\) and \(\frac{\partial }{\partial s}=\frac{1}{2}\big (\frac{\partial }{\partial y_0}-\frac{\partial }{\partial y_n}\big ) = \frac{1}{2}\big [\big (\frac{\partial }{\partial y_0}-iA_-'\big )-\big (\frac{\partial }{\partial y_n}-iA_-'\big )\big ]\) we can rewrite \(L_1u_1\) in \((y_0,y',y_n)\) coordinates:
Note that we transformed the equation \(Lu=0\) to the equation \(L_1u_1=0\) in two steps. First, we transformed \(Lu=0\) to \(L'u'=0\) by making the change of variables \(y=\varphi (x)\) of the form (2.15) and gauge transformation with the gauge \(e^{-id(s,\tau ,y')}\) belonging to the group \(G_0\) (cf. (1.22). Then we transform \(L'u'=0\) to \(L_1u_1=0\) by using the change of variables (2.26), i.e. by using a gauge \(e^A\), where \(A=\ln g_1^{\frac{1}{4}}\) and then dividing \(L'u'=0\) by \({\hat{g}}^{+,-}\). \(\square \)
The DN operator for L has the form
since the outward normal to \(x_n=0\) is \((0,0,\ldots ,-1)\).
Rewrite \(L'u'=0\) in \((y_0,y',y_n)\) coordinates using (2.17).
Denote \(-{\hat{g}}^{00}={\hat{g}}^{nn}={\hat{g}}^{+,-},\ {\hat{g}}^{nj}={\hat{g}}^{jn}={\hat{g}}^{+,j},\ 1\le j\le n-1\). Note that \({\hat{g}}^{+,-}>0.\)
The DN operator for \(L'u'=0\) has the following form in \((y_0,y',y_n)\) coordinates:
where (cf. (2.18))
\(y=y(x)\) is the same as in (2.14).
Since \(L^{\prime }\) is obtained from (1.1) by the change of variables (2.14)and the gauge transformation (2.18) and since (2.15), (2.21) hold, we have \(\Lambda g=\Lambda ^{\prime }g \) on \(\{y_n=0\}\cap U_0\) for all g with \(\hbox {supp}\,g\) in \((\Gamma _0\times [T_1,T_2])\cap U_0\). Using the expression of \(L_1u_1=0\) in \((y_0,y',y_n)\) coordinates (see (2.28)) we get that DN operator \(\Lambda _1g\) has the form
where \(g_0^{+,j}=\frac{{\hat{g}}^{nj}}{{\hat{g}}^{+,-}}\).
We shall show that the DN operators \(\Lambda ^{\prime }\) determines the DN operator \(\Lambda _1\) in \(U_0\cap \Gamma _0\).
The following lemma is well known, especially in the elliptic case (cf. [12, 23, § 57]). For the hyperbolic case see [8, Remark 2.2].
Lemma 2.1
The DN operator \(\Lambda ^{\prime }\) determines
and the derivatives of (2.31) in \(y_n\) at \(y_n=0\).
Proof
The principal symbol of operator \(L^{\prime }\) has the form \({\hat{g}}^{+,-}p(y,\eta )\), where (cf. (2.10) in y-coordinates)
where
Since (1.6) holds the quadratic form \(\sum _{j,k=1}^{n-1}g_0^{jk}\eta _j\eta _k\) is negative definite. Therefore for \({\varepsilon }>0\) in the region \(\Sigma =\{\eta _0^2 +\big (\sum _{j=1}^{n-1}g_0^{+,j}\eta _j\big )^2 -{\varepsilon }\sum _{j=1}^{n-1}\eta _j^2<0\}\) of the cotangent space \(T^*=U_0\times (\mathbb {R}^{n+1}\setminus \{0\})\) the operator \(p(y,\eta )\) is elliptic. We shall call \(\Sigma \) the elliptic region.
There is a parametrix of the Dirichlet problem in the elliptic region and DN operator microlocally in \(\Sigma \) is a pseudodifferential operator on \(y_n=0\). We shall find the principal symbol of this operator in \(\Sigma \). Let \(\lambda _\pm \) be the roots in \(\eta _n\) of \(p(y,\eta _0,\eta ',\eta _n)=0\):
where \(\mathfrak {I}\lambda _+>0\) in \(\Sigma \). Therefore the symbol of DN in \(\Sigma \) is (cf. [12, § 57]):
Knowing \(\Lambda ^{\prime }\) we know the symbol (2.35) for all \(\eta _0,\eta ^{\prime }\). In particular, we can find (2.31). Computing the next term of the parametrix (cf. [12, § 57]) we can find the normal derivatives of (2.31). \(\square \)
We have
where \(u_1\big |_{y_n=0}=f',\ u_1= g_1^{\frac{1}{4}}u',\ f'=g_1^{\frac{1}{4}}\big |_{y_n=0}f\). Note that \(\frac{\partial }{\partial y_k}u_1 = g_1^{\frac{1}{4}}\frac{\partial }{\partial y_k}u'+\big (\frac{\partial }{\partial y_k} g_1^{\frac{1}{4}}\big )u^{\prime }\). Therefore
It follows from the Lemma 2.1 that \( g_1, \frac{\partial g_1}{\partial y_n}, {\hat{g}}^{+,-}, g_0^{+,j}\) are known on \(y_n=0\) if \(\Lambda ^{\prime }\) is known. Therefore knowing \(\Lambda 'f\) we can determine \(\Lambda _1f^{\prime }\). Note that
3 The Green’s formula
First, we introduce some notations.
Let \(\Gamma _1\subset U_0\cap \Gamma _0\). Denote by \(D_{1T_1}\) the forward domain of influence of \({\overline{\Gamma }}_1\times [T_1,T_2]\) in the half-space \(y_n\ge 0\). We shall define \(\Gamma _2\) as the intersection \(D_{1T_1}\cap \{y_n=0\}\cap \{y_0=T_2\}\). Analogously, let \(\Gamma _3=D_{2T_1}\cap \{y_n=0\}\cap \{y_0=T_2\}\), where \(D_{2T_1}\) is the forward domain of influence of \({\overline{\Gamma }}_2\times [T_1,T_2]\). We assume that \(\Gamma _1\subset \Gamma _2\subset \Gamma _3\subset (\Gamma _0\cap U_0)\).
\(Y_{jT_1}\) is the intersection of the plane \(\tau =0\) with \(D_{jT_1}\), \(\Gamma _{j+1}\) is the intersection of \(Y_{jT_1}\) with the plane \(y_n=0\)
Let \(D_{js_0}\) be the forward domain of influence of \({\overline{\Gamma }}_j\times [s_0,T_2],1\le j\le 3\), where \(T_1\le s_0\le T_2\). Denote by \(Y_{js_0}\) the intersection of \(D_{js_0}\) with the plane \(\tau =T_2-y_n-y_0=0\) (cf. Fig. 3). Let \(X_{js_0}\) be the part of \(D_{js_0}\) below \(Y_{js_0}\) and let \(Z_{js_0}=\partial X_{js_0}\setminus (Y_{js_0}\cup \{y_n=0\})\).
We assume also that \(D_{3,T_1}\cap \{y_n=0\}\subset \Gamma _0\cap U_0\) and that \(D_{3T_1}\) does not intersect \(\Gamma _0\times [T_1,T_2]\) outside of \(y_n=0\).
Consider the following initial–boundary value problem:
where \(\hbox {supp}\,f\subset {\overline{\Gamma }}_3\times [T_1,T_2]\). Also let \(v^g\) be such that
where \(\hbox {supp}\,g\subset {\overline{\Gamma }}_3\times [T_1,T_2]\).
Note that \(L_1^*=L_1,\) i.e. \(L_1\) is formally self-adjoint.
Let (u, v) be the \(L_2\) inner product \(\int _{X_{3T_1}}u(y){\overline{v}}(y)dy\). We have
since \(L_1u^f=0,\ L_1v^g=0\). The Jacobian \(\frac{\partial (y_n,y_0)}{\partial (s,\tau )}\) is equal to \(\frac{1}{2}\). Thus \(dy_0dy_n=\frac{1}{2}dsd\tau \). Integrating by parts in s we get
Integrating by parts in \(\tau \) we get
We used in (3.4), (3.5) that \(u^f, v^g\) are equal to zero on \(Z_{3T_1}\). Note that \(s=y_0-T_1,\tau =T_2-y_0\) on \(y_n=0\), and \(\frac{\partial }{\partial s}=\frac{1}{2}\big (\frac{\partial }{\partial y_0} -\frac{\partial }{\partial y_n}\big ), \frac{\partial }{\partial \tau }=-\frac{1}{2}\big (\frac{\partial }{\partial y_0}+\frac{\partial }{\partial y_n}\big )\). Therefore, making changes of variable \(\tau =T_2-y_0\) in the first integral and \(s=y_0-T_1\) in the second, we get
Analogously, integrating by parts other terms of \(\int _{X_{3T_1}}(L_1u^f)\overline{v^g}dsd\tau \) we get (cf. [10], p. 316)
where \(\Lambda _1\) has the form (2.30)
It follows from (3.7) that
is determined by the boundary data, i.e. by the DN operator on \(\Gamma ^{(3)}\times (T_1,T_2)\).
We shall denote the \(L_2\) inner product in \(Y_{3T_1}\) by \((u,v)_{Y_{3T_1}}\), or simply (u, v) when it is clear what is the domain of integration.
Let \(D_j^-\) be the backward domain of influence of \({\overline{\Gamma }}_j\times [T_1,T_2]\). Thus \( D_{jT1}\cap D_j^-\) is the domain of dependence of \({\overline{\Gamma }}_j\times [T_1,T_2]\). Denote by \(Q_j\) the intersection of \(D_j^-\) with \(\tau =0\). Let \(R_{js_0}=Y_{js_0}\cap Q_j\) be the rectangle \(\{s_0-T_1\le s\le T_2-T_1,\tau =0,y'\in {\overline{\Gamma }}_j\}\). Note that \(R_{js_0}\) belongs to the domain of dependence of \({\overline{\Gamma }}_j\times [s_0,T_2]\). Let \(H_0^1(R_{js_0})\) be the subspace of the Sobolev space \(H^1(R_{js_0})\) consisting of \(w\in H^1(R_{js_0})\) such that \(w=0\) on \(\partial R_{j0}\setminus \{y_n=0\}\). Analogously, let \(H_0^1(Y_{js_0})\) be the subspace of \(H^1(Y_{js_0})\) consisting of \(v\in H^1(Y_{js_0})\) such that \(v=0\) on \(\partial Y_{js_0}\setminus \{y_n=0\}\). Note that \(R_{js_0}\subset Y_{js_0}\subset R_{j+1,s_0}\) (cf. Fig. 4).
The rectangle \(R_{js_0} =\{s_0-T_1\le s\le T_2-T_1,\tau =0,y'\in \Gamma _j\}\), \(Y_{js_0}\) is the intersection of the domain of influence of \([s_0,T_2]\times {\overline{\Gamma }}_j\) with the plane \(\tau =0\). Note that \(R_{js_0}\subset Y_{js_0}\subset R_{j+1.s_0}\)
Note that \(H_0^1(R_{js_0})\) is a subspace of \(H_0^1(Y_{js_0})\).
Lemma 3.1
(Density lemma) For any \(w\in H_0^1(R_{js_0})\) there exists a sequence \(\{u^{f_n}\}\) where \(u^{f_n}\) are solutions of the initial–boundary value problem (3.1), \(f_n(y_0,y')\in H_0^1(\Gamma _j\times [s_0,T_2])\), such that \(\Vert w-u^{f_n}\Vert _{1,Y_{js_0}}\rightarrow 0\) when \(n\rightarrow \infty , \ j=1,2,3.\)
Here \(\Vert w\Vert _{1,Y_{js_0}}\) is the norm in \(H_0^1(Y_{js_0})\) and \(f\in H_0^1(\Gamma _j\times [s_0,T_2])\), i.e. \(f=0\) on \(\partial (\Gamma _j\times [s_0,T_2])\setminus (\Gamma _j\times \{y_0=T_2\})\).
The domain \(\Delta _2^{\prime }\) is bounded by \(\Gamma _1^{\prime }\) and \(\Gamma _2^{\prime }\)
Proof
The proof of Lemma 3.1 is a simplification of the proof of Lemmas 2.2 and 3.2 in [10]. We shall prove Lemma 3.1 for the case \(s_0=T_1\). The proof for the case \(T_1<s_0<T_2\) is identical.
Denote by \(\Delta _2^{\prime }\) the domain bounded by the half-plaines \(\Gamma _1'=\{y_n=0,y_0<T_2,y'\in \mathbb {R}^{n-1}\}\) and \(\Gamma _2'=\{\tau =T_2-y_n-y_0=0,s<T_2,y'\in \mathbb {R}^{n-1}\}\) (cf. Fig. 5). Let \(\Gamma _\infty ^{\prime }\) be the plane \(\tau =0\). Denote by \(H_0^{-1}(\Gamma _2')\) the Sobolev space of \(h\in H^{-1}(\Gamma _\infty ')\) such that \(\hbox {supp}\,h\subset {\overline{\Gamma }}_2^{\prime }\), i.e. \(h(s,y')=0\) when \(s>T_2\). Note that \(H_0^{-1}(\Gamma _2')\) is dual to \(H^1(\Gamma _2')\) with respect to the extension of the \(L_2\) inner product on \(\Gamma _\infty ^{\prime }\) (cf. [12]). \(\square \)
Lemma 3.2
For any \(h(s,y')\in H_0^{-1}(\Gamma _2')\) there exists a distribution \(u(s,\tau ,y')\) such that
Proof
Since \(h(s,y')=0\) for \(s>T_2\), there exists \(v(s,y')=0\) for \(s>T_2, v(s,y')\) belongs to \(L_2\) in s and to \(H^{-1}\) in \(y^{\prime }\) and such that \(\frac{\partial v}{\partial s}=h\) in \(\Gamma _\infty ^{\prime }\). We can define \(v(s,y')\) by the formula
where \({\tilde{h}}(z_0,\xi _1,\ldots ,\xi _{n-1})\) is the Fourier transform of \(h(s,y')\) and \(F^{-1}\) is the inverse Fourier transform, \(z_0\) is the dual variable to s.
The distribution \(\theta (-\tau )u\) satisfies the equation
in the half-space \(y_n>0\) with the boundary condition
where \(\theta (s)=1\) for \(s>0\) and \(\theta (s)=0\) for \(s<0\).
We look for \(\theta (-\tau )u\) in the form
where w satisfies
where \(L_{11}=L_1+\frac{4\partial ^2}{\partial s\partial \tau }\). Note that \(L_{11}\) is a differential operator in \(\frac{\partial }{\partial s},\frac{\partial }{\partial y_k},1\le k\le n-1\).
We impose the zero initial conditions on w requiring that
Therefore w is the solution of the hyperbolic equation \(L_1w=\varphi \) in the half-space \(y_n>0\) with the boundary condition (3.17) and the zero initial conditions (3.19). It follows from ([15] and [13]) that initial–boundary value problem has a unique solution in appropriate Sobolev space of negative order.
Since \(\varphi \) belongs to \(L_2\) in \(\tau \) and to Sobolev spaces of negative order in s and \(y'\), we get that w belongs to \(H^1\) in \(\tau \). Therefore \(w{\big |}_{\tau =\tau _0}\) is continuous function of \(\tau _0\) with the values in Sobolev’s spaces of negative order in \((s,y')\). Since \(\varphi =0\) for \(\tau <0\) we have that \(w=0\) for \(\tau <0\) by the domain of influence argument. Therefore by the continuity \(w{\big |}_{\tau =0}=0\) and \(\frac{\partial w}{\partial s}\big |_{\tau =0}=0\).
Therefore \(u=v(s)+w(s,\tau ,y')\) is the distribution solution of (3.10), (3.11), (3.12) in \(\Delta _2'\).
Note that the restrictions of any distribution solution of \(L_1u=0\) to \(y_n=0\) exists since \(y_n=0\) is not a characteristic surface for \(L_1\). This property is called the partial hypoellipticity (cf., for example, [12]). \(\square \)
Now using Lemma 3.2 we can prove Lemma 3.1. If \(\{u^f,f\in H_0^1(\Gamma _{jT_1}), \Gamma _{jT_1}=\Gamma _j\times [T_1,T_2]\}\) is not dense in \(H_0^1(R_{jT_1})\) then there exists nonzero \(h\in H_0^{-1}(R_{jT_1})\) such that \((u^f,h)=0,\forall u^f, f\in H_0^1(\Gamma _{jT_1})\). Let v be such that \(\frac{\partial v}{\partial s}=h\). Then \(\big (u^f,\frac{\partial v}{\partial s}\big )_{Y_{jT_1}}=0, \forall f\in H_0^1(\Gamma _{jT_1})\). Let u be the same as in Lemma 3.2, i.e. \(L_1u=0\) in \(\Delta _2'\), \(u\big |_{Y_{T_1}}=v,\ u\big |_{y_n=0}=0\). Then
Note that \((u^f,h)_{Y_{jT_1}}\) is understood as the extension of \(L_2\) inner product in \((u_1,h)\) in \(\Gamma _\infty '\), where \(u_1\) is an arbitrary extension of \(u^f\) for \(s>T_2\). Analogously for the right hand side of (3.20). (Note that \(u=0\) for \(y_0>T_2)\).
To justify (3.20) we take a sequence \(h_j\in C_0^\infty (\Gamma _2'), \ h_j\rightarrow h\) in \(H_0^{-1}(\Gamma _2')\). By Lemma 3.2 there exists smooth \(v_j\) such that \(L_1v_j=0\) in \(\Delta _2',v_j\big |_{y_n=0}=0, \frac{\partial v_j}{\partial s}\big |_{\tau =0}=h_j\). Applying the Green’s formula (3.7) to \(u^f\) and \(v_j\) we get
since \(v_j\big |_{y_n=0}=0.\) Integrating by parts we get
Since \(v_j=0\) for \(s>T_2\) we have that \(v_j\big |_{y_n=0,y_0=T_2}=0\). Therefore, taking the limit when \(j\rightarrow \infty \) we get (3.20).
Since f is arbitrary and \(\big (u^f,\frac{\partial v}{\partial s}\big )=0\) we get that \(\frac{\partial u}{\partial y_n}=0\) on \(\Gamma _{jT_1}\). Therefore \(L_1u=0\) in \(\Delta _2'\) and u has zero Cauchy data on \(\Gamma _{2T_1}\). Then \(u=0\) in the domain of dependence \(D_j^-\cap D_{jT_1}\), in particular, \(u=0\) on \(R_{jT_1}\). Thus \(v=0\) on \(R_{jT_1}\) and this contradicts the assumption that \(h\ne 0\). \(\square \)
We shall prove two more theorems in this section that will be used in Sect. 4.
We shall need some known results on the initial–boundary hyperbolic problem. The following theorem holds:
Lemma 3.3
Let \(L_1u=F\) in \(\mathbb {R}_+^n\times (-\infty ,T_2)\) where \(F\in H_+^s(\mathbb {R}^n\times (-\infty ,T_2),\mathbb {R}_+^n=\{y_n>0,y'\in \mathbb {R}^{n-1}\}\). Let \(u\big |_{y_n=0}=f\), where \(f\in H_+^{s+1}(\mathbb {R}^{n-1}\times (-\infty ,T_2))\). Then for any \(s\ge 0\) and any \(f\in H_+^{s+1}(\mathbb {R}^{n-1}\times (-\infty ,T_2))\) and \(F\in H_+^s(\mathbb {R}_+^n\times (-\infty ,T_2))\) there exists a unique \(u\in H_+^{s+1}(\mathbb {R}_+^n\times (-\infty ,T_2)) \) such that
Moreover,
Here \(H_+^s(\mathbb {R}_+^n\times (-\infty ,T_2))\) is the Sobolev’s space \(H^s(\mathbb {R}_+^n\times (-\infty ,T_2))\) with norm \(\Vert u\Vert _s\) consisting of u(y) with the support in \(y_0\ge T_1,\ [f]_s\) is the norm in \(H_+^s(\mathbb {R}^{n-1}\times (-\infty ,T_2))\).
We assume that F(y) and \(f(y_0,y')\) have compact supports in \(y'\). Note that \(f=0,\ F=0,\ u=0\) for \(y_0<T_1\).
Then \(u(y_0,y',y_n)\) has also a compact support in \(y'\).
The proof of Lemma 3.3 in the case of time-dependent coefficients is given in [15] and [13].
Note that Lemma 3.3 holds also in the case when \(\mathbb {R}_+^n\) is replaced by an arbitrary smooth domain \(\Omega \subset \mathbb {R}^n\).
The following lemma follows from Lemma 3.3.
Lemma 3.4
Let, for the simplicity, \(F=0\). For any \(f\in H_+^1(\mathbb {R}^{n-1}\times (-\infty ,T_2]),f=0\) for \(y_0\le T_1\), there exists \(u\in C(H^1(\mathbb {R}_+^n),[T_1,T_2])\cap C^1(L_2(\mathbb {R}_+^n), [T_1,T_2])\) such that \(L_1u=0\) in \(\mathbb {R}_+^n\times (-\infty ,T_2], u=0\) for \(y_0<T_1\),
Here \(C(H^1(\mathbb {R}_+^n),[T_1,T_2])\cap C^1(L_2(\mathbb {R}_+^n),(T_1,T_2))\) means that \(u(y),\frac{\partial u(y)}{\partial y_0}\) are continuous functions of \(y_0\) with values in \(H^1(\mathbb {R}_+^n), L_2(\mathbb {R}_+^n)\), respectively.
Proof
Take \(s>\frac{3}{2}\). Consider the equation \(L_1u=0\) in \(\mathbb {R}_+^n\times (-\infty ,T_2), u=0\) for \(y_0<T_1\), using \((y_0,y',y_n)\) coordinates (cf. (2.28)). We have
Let \((u,v)_{T'}\) be the \(L_2\)-inner product in \(\mathbb {R}_+^{n+1}\times (T_1,T'),T'\le T_2\). Integrating by parts the identity
we get (cf. [11])
where
\(\Lambda _1f\) is the same as in (2.36),
Note that \(I_1=0\) when the coefficients of \(L_1\) do not depend on \(y_0\).
Let \(\Vert u\Vert _{s,T'}\) be the norm in \(H^s(\mathbb {R}_+^n)\) when \(y_0=T'\). We have
where
We have
if \(T_2-T_1\) is small.
Since \(T_2-T_1\) is small, (3.25) implies
Note that
Therefore (3.22), (3.25), (3.31), (3.32), (3.33) imply (3.23).
Since \(H_+^s\) is dense in \(H_+^1\) when \(s>1\) we can approximate \(f\in H_+^1(\mathbb {R}^{n-1}\times (-\infty ,T_2))\) by functions from \(H_+^s(\mathbb {R}^{n-1}\times (-\infty ,T_2)),s>\frac{3}{2}\) and therefore the inequality (3.23) holds for \(f\in H_+^1\). \(\square \)
We shall study the Goursat problem (see Fig. 6).
Domain \(\Delta _1\) is bounded by the planes \(\Gamma _2,\Gamma _3,\Gamma _4\)
We use the same notations that we used in the proof of the Lemma 3.1 in [10]: Let \(\Gamma _2,\Gamma _3\) and \(\Gamma _4\) be the following planes:
Let \(\Delta _1\) be the domain bounded by \(\Gamma _2,\Gamma _3,\Gamma _4\). The following lemma is similar to Lemma 3.1 in [10].
Lemma 3.5
For any \(v_0\in H^1(\Gamma _4), v_1\in L_2(\Gamma _4)\) there exists \(u\in H^1(\Delta _1)\) such that \(L_1u=0\) in \(\Delta _1,\ u\big |_{\Gamma _4}=v_0,u_{y_0}\big |_{\Gamma _4}=v_1\). Moreover, the traces \(\varphi =u\big |_{\Gamma _2},\psi =u\big |_{\Gamma _3}\) exists and belongs to \(H^1(\Gamma _2),H^1(\Gamma _3)\), respectively. The following estimate holds:
Vice versa, for any \(\varphi \in H^1(\Gamma _2),\psi \in H^1(\Gamma _3),\varphi =\psi \) at \(y_0=\frac{T_2+T_1}{2}\) there exists \(u\in H^1(\Delta _1),\ L_1u=0\) in \(\Delta _1\) such that \(u\big |_{\Gamma _2}=\varphi ,\ u\big |_{\Gamma _3}=\psi \) and the following estimate holds:
Proof
Let \(\Delta _{1,T'}\) be the domain bounded by \(\Gamma _2,\Gamma _3\) and \(\Gamma _{4,T'}\), where \(\Gamma _{4,T'}\) is the plane \(y_0=T',\frac{T_1+T_2}{2}\le T'\le T_2\). Denote by \((u,v)_{\Delta _{1,T'}}\) the \(L_2\)-inner product in \(\Delta _{1,T'}\). Integrating by parts the identity
we get, as in [11]:
where \(E_{T'}(u,u)\) is the same as in (3.26),
(cf. (3.22) in [11]),
Here \(\Gamma _{2T'}, \Gamma _{3T'}\) are parts of \(\Gamma _2,\Gamma _3\) for \(\frac{T_1+T_2}{2}\le T'\). When \(T_2-T_1\) is small, \(Q_{T'}(u,u)\) is positive definite (cf. [11, 3.23]). Therefore
Analogously,
Having (3.31), (3.39), (3.40), (3.41) we can complete the proof of Lemma 3.5 exactly as the proof of Lemma 3.1 in [10]. \(\square \)
Combining Lemmas 3.4 and 3.5 we can prove the following lemma:
Lemma 3.6
The map \(f\rightarrow u^f\) is a bounded operator from \(H_0^1(\Gamma _j\times [s_0,T_2])\) to \(H_0^1(Y_{js_0})\):
Proof
It follows from Lemma 3.4 that
Then (3.34) gives
Combining (3.43) and (3.44) and taking into account that \(\hbox {supp}\, u^f\big |_{\Gamma _2}=Y_{js_0}\), we get (3.42). \(\square \)
4 The main formula
Let \(L_1^{(i)},i=1,2,\) be two operators of the form (2.27) such that the corresponding DN operators \(\Lambda _1^{(1)}\) and \(\Lambda _1^{(2)}\) are equal on \(U_0\cap \{y_n=0\}\). We choose \(\Gamma _1^{(1)}=\Gamma _1^{(2)}=\Gamma _1\) in a neighborhood of \(x^{(0)}\) in \(U_0\cap \{y_n=0\}\). Let \(\Gamma _j^{(i)}, j=2,3,i=1,2,\) be defined as before (see Fig. 3) for \(i=1,2,\) respectively.
Lemma 4.1
We have \(\Gamma _j^{(1)}=\Gamma _j^{(2)},j=2,3\) (cf. [8]).
Proof
Let \(\Delta _{2T_1}^{(i)}\) be the intersection of the domain of influence \(D_{2T_1}^{(i)},i=1,2,\) with the plane \(y_n=0\). Note that \(\Delta _{2T_1}^{(i)}\) is the intersection of \(y_n=0\) with the closure of the union \(\bigcup \hbox {supp}\, u_i^f\) where the union is taken over all \(f\in H_0^1(\Gamma _1\times [T_1,T_2]), L_1^{(i)}u_i^f=0\).
Let \({\tilde{\Delta }}_{2T_1}^{(i)}\) be the closure of the union \(\bigcup \hbox {supp}\,\Lambda _1^{(i)}f\), where the union is taken also over all \(f\in H_0^1(\Gamma _1\times [T_1,T_2])\). We shall show that \({\tilde{\Delta }}_{2T_1}^{(i)}=\Delta _{2T_1}^{(i)}\).
If \(x^{(0)}\not \in \Delta _{2T_1}^{(i)}\) then \(u_i^f=0,\ \forall f\), in some neighborhood of \(x^{(0)}\) in \(U_0\). Then \(\Lambda _1^{(i)} f=0\) in a neighborhood of \(x^{(0)},\ \forall f\). Thus \(x^{(0)}\not \in {\tilde{\Delta }}_{2T_1}^{(i)}\), i.e. \({\tilde{\Delta }}_{2T_1}^{(i)}\subset \Delta _{2T_1}^{(i)}\). Let now \(x_0'\not \in {\tilde{\Delta }}_{2T_1}^{(i)}\). Then \(\Lambda _1^{(i)}f=0\) in a neighborhood of \(x_0'\) for any \(f\in H_0^1(\Gamma _1\times [T_1,T_2])\) and also \(f=0\) in a neighborhood of \(x_0'\). Then by the uniqueness of the Cauchy problem (see [25, 29]) we have that all \(u^f=0\) in a neighborhood of \(x_0'\) in \(\mathbb {R}^{n+1}\). Therefore \(x_0'\not \in \Delta _{2T_1}^{(i)}\). Thus \(\Delta _{2T_1}^{(1)}={\tilde{\Delta }}_{2T_1}^{(1)}\). Since \(\Lambda _1^{(1)}=\Lambda _1^{(2)}\), we have \({\tilde{\Delta }}_{2T_1}^{(1)}={\tilde{\Delta }}_{2T_1}^{(2)}\). Therefore \(\Delta _{2T_1}^{(1)}=\Delta _{2T_1}^{(2)}\), i.e. \(\Gamma _2^{(1)}=\Gamma _2^{(2)}\). Analogously one shows that \(\Gamma _3^{(1)}=\Gamma _3^{(2)}\). \(\square \)
Since \(\Gamma _j^{(1)}=\Gamma _j^{(2)}\) we shall write \(\Gamma _j, 1\le j\le 3,\) instead of \(\Gamma _j^{(i)}\). It follows from (3.7) that (3.9) is determined by the boundary data. Integrating by part we have
Since \(u^f(T_2-T_1,0,y')=f(T_2,y'),\ v^g(T_2-T_1,0,y')=g(T_2,y')\), we have that
is also determined by the boundary data.
Lemma 4.2
Let \(f\in H_0^1(\Gamma _1\times [T_1,T_2])\). For any \(s_0\in [T_1,T_2)\) there exists \(u_0\in H_0^1(R_{2s_0})\) such that
for any \(v'\in H_0^1(Y_{3s_0})\). Note that \(R_{2s_0}=\{\tau =0,s_0-T_1\le s\le T_2-T_1,y'\in \Gamma _2\}\) (cf. Fig. 7).
Proof
Note that \(Y_{1T_1}\cap \{s_0-T_1\le s\le T_2-T_1\}\subset R_{2s_0}\). Let \(w_1\) be such that \(w_{1s}=0\) in \(R_{2s_0},\ w_1=u^f\) when \(s=s_0-T_1,y'\in \Gamma _2\). Then \(u_0=u^f-w_1\) for \(s\ge s_0-T_1,\ u_0=0\) for \(s\le s_0-T_1\), belongs to \(H_0^1(R_{2s_0})\) and solves (4.3). \(\square \)
\(R_{j+1,s_0}\) is the rectangle \(\{s_0-T_1\le s\le T_2-T_1,\tau =0,y'\in \Gamma _{j+1}\}, Y_{jT_1}\cap \{s_0-T_1\le s\le T_2-T_1\}\subset R_{j+1,s_0}\)
If \(v'=v^{g'}\) where \(g'\in H_0^1(\Gamma _2\times [s_0,T_2])\) then \((u_0,v^{g'})=(u^f,v^{g'})\) is determined by the DN operator. Let \(g\in H_0^1(\Gamma _1\times [T_1,T_2])\). We shall show that still \((u_0,v^g)\) is determined by the DN operator. The following theorem holds.
Theorem 4.3
Let \(L_1^{(i)},i=1,2\), be two operators of the form (2.27). Let f be in \(H_0^1(\Gamma _1\times [T_1,T_2])\) and let \(u_0^{(i)}\) be the same as in (4.3) for \(i=1,2\). Then
for all \(g\in H_0^1(\Gamma _2\times [T_1,T_2]).\)
Here \(u_i^f,v_i^g\) are the same as in (3.1), (3.2) for \(i=1,2\), respectively. Operators \(L^{(i)}\) and, consequently, \(L_1^{(i)}\) are formally self-adjoint, \((u_0^{(i)},v_i^g)_{Y_{2s_0}^{(i)}}\) is the \(L_2\)-inner product over \(Y_{2s_0}^{(i)}, i=1,2.\)
To prove Theorem 4.3 we will need the Density Lemma 3.1 and the following lemma that uses the BLR condition:
Lemma 4.4
Let \(L^{(1)}\) and \(L^{(2)}\) be two operators in \(D\cap [t_0,T_2]\) having the same DN operator on \(\Gamma _0\times [t_0,T_2]\). Suppose \(L^{(1)}\) satisfies the BLR condition on \([t_0,T_2]\).
Let \(L_1^{(i)},u_i^f,X_{2s_0}\) be the same as in (3.1), \(i=1,2,f\in H_0^1(\Gamma _{2s_0})\), where \(\Gamma _{2s_0}=\Gamma _2\times [s_0,T_2]\). Then
Proof of Lemma 4.4
(cf. Lemma 2.3 in [10]): Suppose that BLR condition (see [1]) is satisfied for \(L^{(1)}\) on \([t_0,T_{t_0}]\) and \(t_0<T_1,\ T_2\ge T_{t_0}\). The BLR condition implies that the map \(f\rightarrow \big (u_1^f(x)\big |_{D_{T_2}},\frac{\partial u_1^f(x)}{\partial x_0}\big |_{D_{T_2}}\big )\) of \(H_+^1(\Gamma _0\times (t_0,T_2))\) to \(H^1(D_{T_2})\times L_2(D_{T_2})\) is onto, where \(D_{T_2}=D_0^{(1)}\times \{x_0=T_2\}\). It follows from [15] (cf. also Lemma 3.6) that
By the closed graph theorem we have
where \({\mathcal {F}}\subset H_+^1(\Gamma _0\times (t_0,T_2))\) is the set of \(f'\) such that
It follows from (4.7) that there exists \(f_0\in {\mathcal {F}},f_0=0\) for \(x_0<t_0\) such that
Note that
Let \(\Lambda ^{(i)}\) be the DN operator corresponding to \(L^{(i)},i=1,2,\) and let \(u_{i0}^{f_0}\) be the solution of \(L^{(i)}u_{i0}^{f_0}=0\) in \(D\times [t_0,T_2),\ u_{i0}^{f_0}=f_0\) on \(\Gamma _0\times [t_0,T],\ i=1,2\). Let \(L_1^{(i)}u_i^{f_0}=0\) be the solutions in a neighborhood \(U_0\) obtained from \(L^{(i)}u_{i0}^{f_0}=0\) as in Sect. 2 (cf. (2.38)) for \(i=1,2.\)
Consider the identity
where \(v^g\) is the same as in (3.2). Since \(\hbox {supp}\,v^g \subset D(\Gamma _{js_0})\), where \(D(\Gamma _{js_0})\) is the domain of influence of \(\Gamma _{js_0}\) for \(y_0\le T_2\), we have that \(v_i^g=0\) on \(Z_{js_0}\). Therefore integrating by parts in (4.11) we get as in (3.7):
Analogously, we have for \(L^{(2)}u_2^{f_0}=0, \ L^{(2)}v_2^g=0\):
We have that \(\Lambda ^{(1)}f_0=\Lambda ^{(2)}f_0\) on \(\Gamma _0\times [t_0,T_2]\). Therefore (2.38) implies that \(\Lambda _1^{(1)}f_0=\Lambda _1^{(2)}f_0\) in \(\Gamma _{js_0}\). Also \(\Lambda _1^{(1)}g=\Lambda _1^{(2)}g\) in \(\Gamma _{js_0}\). Integrating by parts we get
Note that \(v_i^g\big |_{s=0}=0\) and \(u_i^{f_0}\overline{v_1^g}\big |_{s=T_2-T_1}=f_0(T_2,y')\overline{g(T_2,y')}\). Therefore
for all \(g\in H_0^1(\Gamma _{3s_0})\).
Let \(\Gamma _2,\Gamma _3,\Gamma _4\) be the same as in Lemma 3.5. It was proven there that
It follows from \(u_1^f\big |_{\Gamma _4}=u_1^{f_0}\big |_{\Gamma _4},\ u_{1y_0}^f\big |_{\Gamma _4}= u_{1y_0}^{f_0}\big |_{\Gamma _4}\) that
by the domain of dependence argument. Comparing (4.12) with \((u_{1s}^{f},v_1^g)=(u_{2s}^{f},v_2^g)\) and taking into account (4.15) we get
By Lemma 3.1 \(\{v_2^g\}\) are dense in \(H_0^1(R_{3s_0}^{(2)})\). Since \(Y_{2s_0}^{(2)}\subset R_{3s_0}^{(2)}\) we get that \(\{v_2^g\}\) are dense in \(H_0^1(Y_{2s_0}^{(2)})\) and therefore \(u_{2s}^{f_0}=u_{2s}^f\) in \(Y_{2s_0}^{(2)}\). Since \(u_2^f\big |_{s=T_2-T_1}=f(T_2,y')=u_1^f(T_2,y',0), \ u_2^{f_0}\big |_{s=T_2-T_1}=f_0(T_2,y')=u_1^{f_0}(T_2,y',0)\) and since \(u_1^{f_0}(T_2,y',0)=u_1^{f}(T_2,y',0)\) we get that \(u_2^f\big |_{s=T_2-T_1}=u_2^{f_0}\big |_{s=T_2-T_1}.\) Thus
It follows from (4.13) that
since that \(u_1^f=0\) on \(\Gamma _3\) by the domain of dependence argument.
Since \(Y_{2s_0}^{(2)}\) belongs to the domain of dependence of \(D_{T_2}\) we get, similarly to (4.14), that
where \(D_{T_2}^{(2)}=D^{(2)}\cap \{y_0=T_2\}\).
We also have (cf. Lemma 3.6)
Combining (4.18), (4.9) with (4.19), (4.20) and taking into account (4.17), we get
Now we shall prove Theorem 4.3. \(\square \)
Proof of Theorem 4.3
Since \(u_0^{(1)}\in H_0^1(R_{2s_0}^{(1)})\) we get, using the Density Lemma 3.1, that there exists \(u_1^{f_n},f_n\in H_0^1(\Gamma _2\times [s_0,T_2])\) such that \(\big \Vert u_0^{(1)}-u_1^{f_n}\big \Vert _{1,Y_{2s_0}^{(1)}}\rightarrow 0\). By Lemma 4.4 \(\{u_2^{f_n}\}\) also converges in \(H_0^1(Y_{2s_0}^{(2)})\) to some function \(w\in H_0^1(Y_{2s_0}^{(2)})\). Passing to the limit in
we get
where \(\Gamma _{3T_1}=\Gamma _3\times [T_1,T_2]\) Note that (4.22) and therefore (4.23) hold also for any \(g'\in H_0^1(\Gamma _{3s_0}),\) i.e.
For such \(g'\) the equality (4.3) holds, i.e.
Comparing (4.24) and (4.25) we get
Since \(v_2^{g'}\in H_0^1(Y_{3s_0}^{(2)})\) are dense in \(H_0^1(R_{3s_0}^{(2)})\) and \(w\in H_0^1(Y_{2s_0}^{(2)}) \subset H_0^1(R_{3s_0}^{(2)})\), we have that \(u_{0s}^{(2)}=w_s\). Since \(u_0^{(2)}\) and w are zero on \(\partial Y_{3s_0}^{(2)}\setminus \{y_n=0\}\) we get that
Therefore (4.23) and (4.27) gives
for all \(g\in H_0^1(\Gamma _{3T_1})\), i.e. (4.4) holds. \(\square \)
The following formula will be the main tool in solving the inverse problem.
Theorem 4.5
For any \(T_1\le s_0\le T_2\) the integral
is determined by the DN operator on \(\Gamma _{jT_1}=\Gamma _j\times [T_1,T_2]\).
Proof
Since \(u_0^{(i)} =\frac{\partial u^f}{\partial s}\) for \(s\ge s_0-T_1, \ u_{0s}=0\) for \(s\le s_0-T_1\), formula (4.28) gives that \(\int _{Y_{jT_1}\cap \{s> s_0-T_1\}}\frac{\partial u^f}{\partial s}\overline{v^g}dsdy'\) is determined by the DN operator on \(\Gamma _{jT_1}\). The integral (4.29) is the difference \((u_s^f,v^g)-(u_{0s},v^g)\) thus (4.29) is determined by DN operator. \(\square \)
Remark 4.1
When the coefficients of \(L_1^{(i)}, i=1,2\), do not depend on \(y_0\), we can obtain the estimate (4.5) without assuming the BLR condition. In this case we can derive, in addition to (3.3), another Green’s formula (cf. (3.18) in [10]):
Consider the identity
Integrating by parts as in [10] and using that \(\frac{\partial }{\partial y_0}\) and \(L_1\) are commute, we get (cf. (3.20) in [10])
where
and
As in [10] (cf. (3.23) in [10]) we have that \({\tilde{Q}}(u,u)\) is a positive definite form when \(T_2-T_1\) is small and
Let \(u_i^f, i=1,2,\) be such that \(L_1^{(i)}u_i^f=0\) in \(X_{2s_0}^{(i)},\ u_i^f\big |_{y_n=0}=f,\ u_i^f=0\) for \(y_0<T_1,\ \hbox {supp}\,f\) is contained in \(\Gamma _2\times (T_1,T_2]\). We assume that \(\Lambda _1^{(1)}=\Lambda _1^{(2)}\) on \(\Gamma _{2T_1}=\Gamma _2\times (T_1,T_2)\). It follows from (4.13), (4.31), (4.33) that
where \({\tilde{Q}}_i\) corresponds to \(L_1^{(i)},i=1,2\). Thus, (4.34) implies that
i.e. the estimate (4.5) is proven.
5 The geometric optics construction
It follows from Theorem 4.5 that the DN operator allows to determine \( \int _{Y_{2s_0}\cap \{s\le s_0 - T_1\}}u_{s}^f\overline{v^g}dsdy'\) for all \(f\in H_o^1(\Gamma _{jT_1}),g\in H_0^1(\Gamma _{jT_1}),j=1,2,\) i.e. if \(u_i^f,v_i^g\) satisfy (3.1), (3.2), \(i=1,2,\) then
Let \(u_i\) be the solution of \(L^{(i)}u_i=0\) such that
\(u_i^{(N+1)}\) will be chosen below, k is a large parameter. We have the following equations for \(a_p^{(i)},0\le p\le N\) (see [11] and [12], §64, for more details on the construction of geometric optics type solutions):
with the initial conditions
where \( \chi _1(s)=0\) for \(|s-s_0'|>2\delta ,\ \chi _1(s)=1\) for \(|s-s_0'|\le \delta \), \(\chi _2(y')\in C_0^\infty (\Gamma _2),\ \chi _2(y')\ne 0\) when \(|y'-y_0'|<\delta ,\ y_0'\in \Gamma _2\) is arbitrary, \(g_{i0}^{j0}\) corresponds to \(L_1^{(i)},i=1,2\). Note that \(y_n=\frac{T_2-T_1-s-\tau }{2}=0\) when \(\tau =\tau _0\).
Let \(u_i^{(N+1)}\) be such that
\(u_i^{(N+1)}=u_{iy_0}^{(N+1)}=0\) when \(y_0=T_1, y_n>0, i=1,2, \ u_i^{(N+1)}\big |_{y_n=0}=0,\ y_n\le T_2\). Such \(u_i^{(N+1)}\) exists (cf. [15]) and \(L_1^{(i)}(u_N^{(i)}+u_i^{(N+1})=0\).
Since \(\hbox {supp}\,u_N^{(i)}\) is contained in a small neighborhood of the line \(\{s=s_0-T_1,y'=y_0'\}\), we have that \(\hbox {supp}\,(u_N^{(i)}+u_i^{N+1})\subset D_+(\Gamma _2\times [T_1,T_2])\) when \(s_0-T_1>0\). Here, as in Sect. 1, \(D_+(\Gamma _2\times [T_1,T_2])\) is the forward domain of influence of \(\Gamma _2\times [T_1,T_2]).\)
Let \(\beta ^{(i)}(s,\tau ,{\hat{y}}')=(\beta _1^{(i)},\beta _2^{(i)},\ldots ,\beta _{n-1}^{(i)})\) be the solution of the system (cf. [11])
where \({\hat{y}}'=({\hat{y}}_1,\ldots ,{\hat{y}}_{n-1})\in \Gamma _2\), s is a parameter in (5.7).
Let
be the inverse to the map
i.e.
Note that \(\alpha _j^{(i)}(s,\tau ,y'),1\le j\le n-1,\) satisfy the equation
Let \({\hat{a}}_0^{(i)}(s,\tau ,{\hat{y}}')=a_0^{(i)}(s,\tau ,y')\), where \(y'=\beta ^{(i)}(s,\tau ,{\hat{y}}')\). Then using (5.7) and (5.2) we get
where \(B^{(i)}(s,\tau ,y')=-iA_-'-i\sum _{j=1}^{n-1}g_{i0}^{0j} A_j'+\frac{1}{2}\sum _{j=1}^{n-1}\frac{\partial g_{0i}^{0j}}{\partial y_j},\ {\hat{B}}^{(i)}(s,\tau ,{\hat{y}}')=B^{(i)}(s,\tau ,\beta ^{(i)}(s,\tau ,{\hat{y}}')),\ {\hat{a}}_0^{(i)}(s,\tau ,{\hat{y}}')\big |_{\tau =\tau _0} =\chi _1(s)\chi _2({\hat{y}}')\).
Therefore
where \( b^{(i)}(s,\tau ,{\hat{y}}')=\int _{\tau _0}^\tau {\hat{B}}^{(i)}(s,{\hat{\tau }},{\hat{y}}')d{\hat{\tau }}\). Substituting \(u=u_N^{(i)}+u_i^{(N+1)}\) into (4.29) instead of \(u_i^f\), integrating by parts in s and taking the limit when \(k\rightarrow \infty \), we get
Note that \(\tau =0\) on \(Y_{2T_1}^{(i)},i=1,2.\) In (5.15) \(s_0\in (T_1,T_2]\) is arbitrary, \(s_0'=s_0-T_1\).
Denote by \(Y_{2T_1}^{(i)}(\tau ')\) the intersection of the plane \(\tau =\tau '\) with \(X_{2T_1}^{(i)}\). Let \(R_{2T_1}^{(i)}(\tau ')\subset Y_{2T_1}^{(i)}(\tau ')\) be the rectangle \(\{\tau =\tau ',0\le s\le T_2-T_1-\tau ',y'\in \Gamma _2\}\). Note that \(Y_{2T_1}^{(i)}(0)=Y_{2T_1}^{(i)}\) and \(R_{2T_1}^{(i)}(0)=R_{2T_1}^{(i)}\).
Repeating the proof of Theorem 4.5 with \(Y_{2T_1}^{(2)},R_{2T_1}^{(2)}\) replaced by \(Y_{2T_1}^{(2)}(\tau '), R_{2T_1}^{(2)}(\tau '), 0\le \tau '\le T_2-T_1\), we get again, using the geometric optics construction (5.1), that (5.15) holds for any \((s,\tau )\in \Sigma \), where \(\Sigma =\{(s,\tau ),s\ge 0,\tau \ge 0, s+\tau \le T_2-T_1\}\). Thus, we have
for \((s,\tau ,y')\in X_{2T_1}^{(i)}\).
Let \(\beta ^{(i)}(\Sigma \times {\overline{\Gamma }}_2)\) be the image of \(\Sigma \times {\overline{\Gamma }}_2\) under the map (5.10). Note that the support of geometric optics solution \(u_N^{(i)}+u_i^{(N+1)}\) is contained in \(D(\Gamma _2\times [T_1,T_2])\). Also we have that the curve \(y'=\beta ^{(i)}(s,{\hat{\tau }},{\hat{y}}')\) for \(\tau _0\le {\hat{\tau \le \tau }}\), is contained in \(X_{2T_1}^{(i)}\). Therefore \(\beta ^{(i)}(\Sigma \times {\overline{\Gamma }}_2)\subset X_{2T_1}^{(i)}\). Denote by \(X_{\Gamma _2}^{(i)}\) the intersection of \(\beta ^{(i)}(\Sigma \times {\overline{\Gamma }}_2)\) with \(\Sigma \times {\overline{\Gamma }}_2\). Note that \(\Sigma \times {\overline{\Gamma }}_2= \bigcup _{0\le \tau '\le T_2-T_1} R_{2T_1}^{(i)}(\tau ')\).
Finally, denote by \({\tilde{X}}_{\Gamma _2}^{(i)}\) the image of \(X_{\Gamma _2}^{(i)}\) under the inverse map (5.9). Note that \({\tilde{X}}_{\Gamma _2}^{(i)}\subset \Sigma \times {\overline{\Gamma }}_2\).
Making the change of variables (5.10) in (5.16) we get
where \(J_i\) is the Jacobian of the map (5.10), \((s,\tau ,{\hat{y}}')\in \Sigma \times \Gamma _2\).
Let \(b^{(i)}=b_1^{(i)}+ib_2^{(i)}\), where \(b_1^{(i)}, b_2^{(i)}\) are real.
Since \(\chi _2(y')\in C_0^\infty (\Gamma _2)\) is arbitrary, we have
Let
Our strategy will be to show that \(w_1^g(s,\tau ,{\hat{y}}')=w_2^g(s,\tau ,{\hat{y}}')\) in \({\tilde{X}}_{\Gamma _2}^{(1)}\) and then to show that the equations \({\tilde{L}}_1^{(1)}w_1^g=0\) and \({\tilde{L}}_1^{(2)}w_1^g=0\) have the same coefficients in \({\tilde{X}}_{\Gamma _2}^{(1)}\). Here \({\tilde{L}}_1^{(i)}\) is obtained from \(L_1^{(i)}\) by the change of variables (5.10), \(i=1,2\).
We shall show first that \(e^{2b_1^{(1)}}J_1(s,\tau ,{\hat{y}}')=e^{2b_1^{(2)}}J_2(s,\tau ,{\hat{y}}')\). Consider the geometric optics solutions \(v_{i,k}^g\) of the form (5.1), where \(g=\chi _1(s)\chi _3(y'),\chi _3(y')\in C_0^\infty (\Gamma _2)\) is arbitrary. Substituting \(v_{i,k}^g\) into (5.16), integrating by parts and passing to the limit when \(k\rightarrow \infty \), we get
where \(s_0'=s_0-T_1\).
Note that \(e^{b^{(i)}}e^{\overline{b^{(i)}}}=e^{2b_1^{(i)}}\).
Making the change of variables \(y'=\beta ^{(i)}(s_0,\tau ,{\hat{y}}')\) and using that \(\chi _2\) and \(\chi _3\) are arbitrary we get
Therefore, (5.18) and (5.21) imply
\(\square \)
As in (4.12) the integration by parts gives
Therefore \(\int \limits _{Y_{3T_1}}u^f\overline{v_s^g}dsdy'\) is determined by the boundary data since \(u^f\big |_{y_n=0}=f(T_2,y'),\overline{v^g}\big |_{y_n=0}= {{\overline{g}}(T_2,y')}\), i.e. the roles of \(u^f\) and \(v^g\) are reversed in comparison with (4.12). Therefore we get, as in (4.28),
Substituting in (5.23) the geometric optics solution (5.1), integrating by parts in s, multiplying by ik and, finally, taking the limit when \(k\rightarrow \infty \), we get (5.16) with \(v_i^g\) replaced by \(v_{is}^g\). Note that we assumed that \(v_i^g\in H_0^2(Y_{2T_1})\) when integrating by parts in (5.23). This can be achieved by requiring that \(g\in H_0^2(\Gamma _{2T_1})\) and using the regularity results for hyperbolic initial–boundary value problems (cf. [13, 15]). Therefore we get (5.18), with \(v_i^g\) replaced by \(v_{is}^g\):
Using (5.21) we get
We shall need the following lemma:
Lemma 5.1
The equalities
hold on \({\tilde{X}}_{\Gamma _2}^{(1)}\).
Proof
Making the change of variables \({\hat{y}}'=\alpha ^{(i)}(s,\tau ,y')\) in (5.19), we get
Differentiating in s we have
Returning back in (5.28) to \(y'=\beta ^{(1)}(s,\tau ,{\hat{y}}')\) coordinates we get
Subtracting (5.29) for \(i=1\) from (5.29) for \(i=2\) and taking into account (5.24) and (5.22) we get
for all \({\tilde{w}}_1^g(s,\tau ,{\hat{y}}')\) where \((s,\tau ,{\hat{y}}')\in \Sigma \times \Gamma _2\).
Fix \(\tau =\tau ', 0\le \tau '<T_0-T_1\). By the Density Lemma 3.1 \(\{v_i^g(s,\tau ',y')\}\) are dense in \(H_0^1(R_{2T_1}^{(i)}(\tau '))\), where \(g\in H_0^1(\Gamma _2\times \{T_1\le y_0\le T_2-\tau '\})\).
Let \({\tilde{R}}_{2T_1}^{(i)}(\tau ')\) be the image of \(R_{2T_1}^{(i)}(\tau ')\cap \beta ^{(i)}(\Sigma \times {\overline{\Gamma }}_2)\) under the map (5.9). Since \({\tilde{w}}_i^g=e^{-b^{(i)}}v_i^g(s,\tau ,\beta ^{(i)}(s,\tau ,{\hat{y}}'))\) we have that \({\tilde{w}}_i^g(s,\tau ',{\hat{y}}')\) are dense in \(H_0^1({\tilde{R}}_{2T}^{(i)}(\tau '))\). \(\square \)
The following lemma is similar to arguments in [10, pp. 1749–1750].
Lemma 5.2
Since \(\{w_1^g(s,\tau ',{\hat{y}}'), g\in H_0^1({\overline{\Gamma }}_2\times \{T_1\le y_0\le T_2-\tau '\})\}\) are dense in \({\tilde{R}}_{2T_1}^{(1)}(\tau ')\) we have
Proof
Let \(\gamma (s,\tau ',{\hat{y}}')\in C_0^\infty ({\tilde{R}}_{2T_1}^{(1)}(\tau '))\). There exists a sequence \({\tilde{w}}_1^{g_k}(s,\tau ',{\hat{y}}')\) convergent to \(\gamma (s,\tau ',{\hat{y}}')\) in \(H_0^1({\tilde{R}}_{2T_1}^{(1)}(\tau '))\). Therefore \({\tilde{w}}_1^{g_k}\) converges weakly to \(\gamma (s,\tau ',{\tilde{y}}')\). Passing in (5.30) to the limit when \(k\rightarrow \infty \) we get
For any point \((s,{\hat{y}}')\in {\tilde{R}}_{2T_1}^{(1)}(\tau ')\) we can find n \(C_0^\infty ({\tilde{R}}_{2T_1}^{(1)})\) functions \(\gamma _1(s,{\hat{y}}'),\ldots ,\gamma _n(s,{\hat{y}}')\) such that the determinant of \(n\times n\) matrix
is not equal to zero at the point \((s,{\hat{y}}')\). Therefore (5.31), (5.32) hold. \(\square \)
Repeating the same arguments for any \(0\le \tau '\le T_2-T_1\) we get that (5.31), (5.32) hold for any \(\tau ',\) i.e. it hold on \({\tilde{X}}_{\Gamma _2}^{(1)} = \bigcup _{0\le \tau '\le T_2-T_1} {\tilde{R}}_{2T_1}^{(1)}(\tau ')\), since \(\bigcup _{0\le \tau '\le T_2-T_1}R_{2T_1}^{(1)}(\tau ')=\Sigma \times \Gamma _2\) and \({\tilde{X}}_{\Gamma _2}\) is the image of \((\Sigma \times {\overline{\Gamma }}_2)\cap \beta ^{(1)}(\Sigma \times {\overline{\Gamma }}_2)\) under the map (5.9). This proves (5.25). To prove (5.26) we note that
Since (5.31), (5.32) hold, we have
Equation (5.34) is a linear homogeneous equation for \(b^{(1)}(s,\tau ,{\hat{y}}')- b^{(2)}(s,\tau ,{\hat{y}}')\) on \({\hat{X}}_{\Gamma _2}^{(1)}\). Since \(b^{(1)}=b^{(2)}=0\) when \(y_n=0\), we get
It follows from (5.35) and (5.22) that
where \(w_i^g=v_i^g(s,\tau ,\beta ^{(i)}(s,\tau ,{\hat{y}}'))\).
6 The conclusion of the local step
We shall prove the following theorem.
Theorem 6.1
Let \(L_1^{(i)}v_i^g=0,i=1,2.\) Make change of variables
Let \({\tilde{L}}_1^{(i)}w^g=0\) be the operator \(L_1^{(i)}\) in the new coordinates. Then the coefficients of \({\tilde{L}}_1^{(1)}\) and \({\tilde{L}}_1^{(2)}\) are equal on \({\tilde{X}}_{\Gamma _2}^{(1)}\).
Proof
Equations \(L_1^{(i)}v_i^g=0\) have the following form in \((s,\tau ,y')\) coordinates (cf. (2.27)):
where \(i=1,2,\ g_{i0}^{+,j}=g_{i0}^{0j},\ V_1^{(i)}\) is the same as in (2.27).
Making the change of variables (6.1) in (6.2) we get:
where \(w_i^g(s,\tau ,{\hat{y}}')=v_i^g(s,\tau ,\beta ^{(i)}(s,\tau ,{\hat{y}}'))\),
We used in (6.3) that (see (5.12))
Also we have
since \(g_{i0}^{-,k}=0\) (cf. (6.2)).
Note that \(A_+^{(i)}, A_-^{(i)}, A_j^{(i)}\) and \({\tilde{A}}_+^{(i)},{\tilde{A}}_-^{(i)},{\tilde{A}}_j^{(i)}\) are related by the equality
where \(A_+^{(i)}=0,i=1,2,\ s={\hat{s}},\ \tau ={\hat{\tau }},\ y_j=\beta _j(s,\tau ,{\hat{y}}')\).
Note that (5.21), (5.26) imply
The first order term containing \(\frac{\partial }{\partial \tau }\) in (6.3) is equal to
It follows from (6.7) that
Since \( A_+^{(i)}=0\) we have that (6.10) implies that (6.9) is equal to zero.
Taking into account that \(\alpha _{js}^{(1)}(s,\tau ,\beta ^{(1)})=\alpha _{js}^{(2)}(s,\tau ,\beta ^{(2)}),\ 1\le j\le n-1\), \(J_1=J_2\) and \(w_1^g(s,\tau ,{\hat{y}}')=w_2^g(s,\tau ,{\hat{y}}')\) we get that \({\tilde{L}}_1^{(1)}-{\tilde{L}}_1^{(2)}\) is a differential operator in \(\frac{\partial }{\partial s}, \frac{\partial }{\partial y_1},\ldots ,\frac{\partial }{\partial y_n}\). We have
Since \(\{w_1^g,g\in C_0^\infty (\Gamma ^{(1)}\times [T_1,T_2-\tau ']\}\) are dense in \(H_0^1({\tilde{R}}_{1T_1}(\tau '))\) we get as in Lemma 5.2 (cf. [10]) that all coefficients of \({\tilde{L}}_1^{(1)}\) and \({\tilde{L}}_1^{(2)}\) are equal in \({\tilde{R}}_{1T_1}^{(1)}(\tau ')\). Since \(\tau '\in [0,T_2-T_1]\) is arbitrary, we get that on \({\hat{X}}_{\Gamma _2}^{(1)}\):
Therefore \({\tilde{L}}_1^{(1)}={\tilde{L}}_1^{(2)}\) in \({\tilde{X}}_{\Gamma _2}^{(1)}\). \(\square \)
This completes the proof of Theorem 6.1. Let \(L_i'v_i^g=0\) be the equation of the form (2.24). Making the change of variables (5.10) we get the equation \({\tilde{L}}_i'w_i^g=0,i =1,2,\) on \({\tilde{X}}_{\Gamma _2}^{(1)}\).
Note that \(w_1^g=w_2^g\) on \({\tilde{X}}_{\Gamma _2}^{(1)}\). We shall prove that \({\tilde{L}}_1'={\tilde{L}}_2'\) on \({\tilde{X}}_{\Gamma _2}^{(1)}\).
Let \({\hat{g}}_i^{+,-},{\hat{g}}_i^{+,j},{\hat{g}}_j^{jk},1\le j\le n-1,1\le k\le n-1\), be the inverse metric tensor of \(L_i'\). Note that for \(L_1^{(i)}\) we have (cf. (2.27))
Therefore the equation \({\tilde{L}}_i'w_i^g=0\) has the inverse metric tensor with elements (cf. (6.4), (6.5), (6.6))
Since \(\alpha _{ks}^{(1)}=\alpha _{ks}^{(2)}\) and \({\tilde{g}}_{10}^{jk}={\tilde{g}}_{20}^{jk}\) (see (6.12)), we get that the metric tensors of \({\tilde{L}}_1'\) and \({\tilde{L}}_2'\) are equal if we can prove that
We shall prove first that
where \( g_1^{(i)}=\big |\det [{\hat{g}}_i^{jk}]_{j,k=1}^{n-1}\big |^{-1}\) (see (2.22)).
Note that \(V_1^{(i)}(s,\tau ,y')\) has the form (2.25) for \(i=1,2\), where \(A^{(i)}=\ln (g_1^{(i)})^{\frac{1}{4}}\). Making the change of variables (6.1) we get (cf. (6.14))
Note that the metric tensors for \({\tilde{L}}_1'\) and \({\tilde{L}}_2'\) are equal on \({\tilde{X}}_{\Gamma _2}\). Let \({\tilde{A}}^{(i)}(s,\tau ,{\hat{y}}')=A^{(i)}(s,\tau ,\beta ^{(i)}(s,\tau ,{\hat{y}}'))\).
Using the equality
and similar equality involving derivatives in s and \(\tau \) we can represent (6.18) as a homogeneous second order hyperbolic equation in \({\tilde{A}}^{(1)}-{\tilde{A}}^{(2)}\) with the coefficients depending on \({\tilde{A}}^{(1)}\) and \({\tilde{A}}^{(2)}\). Since the Cauchy data for \({\tilde{A}}^{(1)}-{\tilde{A}}^{(2)}=0\) at \(y_n=0\) (cf. Lemma 2.1) we get, by the uniqueness of the Cauchy problem (cf. [25, 29]), that \({\tilde{A}}^{(1)}={\tilde{A}}^{(2)}\) in \({\tilde{X}}_{\Gamma _2}^{(1)}\). Therefore (6.17) holds.
Note that \({\tilde{g}}_i^{jk}={\tilde{g}}_i^{+,-}{\tilde{g}}_0^{jk}\). Therefore
Since \({\tilde{g}}_{10}^{jk}={\tilde{g}}_{20}^{jk}\) and (6.17) holds, we get
and this proves (6.16), since we assumed that \(n>1\). Therefore metric tensors of \({\tilde{L}}_1'\) and \({\tilde{L}}_2'\) are equal. Combining this with (6.13), (6.14) we get \({\tilde{L}}_1'={\tilde{L}}_2'\) on \({\tilde{X}}_{\Gamma _2}^{(1)}\supset \Sigma \times {\overline{\Gamma }}_1 \).
Remark 6.1
Change \(\Gamma _2\) to \(\Gamma _1\). We have \(\beta ^{(i)}(\Sigma \times {\overline{\Gamma }}_1)\subset \beta ^{(i)}(\Sigma \times {\overline{\Gamma }}_2)\). Since \(\beta ^{(i)}(\Sigma \times {\overline{\Gamma }}_1)\subset X_{1T_1}^{(i)}\) and \(X_{1T_1}^{(i)}\subset (\Sigma \times {\overline{\Gamma }}_2)\), we get \(\beta ^{(i)}(\Sigma \times {\overline{\Gamma }}_1)\subset (\Sigma \times {\overline{\Gamma }}_2)\). Therefore, \(\beta ^{(i)}(\Sigma \times {\overline{\Gamma }}_1)\subset (\Sigma \times {\overline{\Gamma }}_2)\cap \beta ^{(i)}(\Sigma \times {\overline{\Gamma }}_2)= X_{\Gamma _2}^{(i)}\). Applying the map (5.9) to \(\beta ^{(i)}(\Sigma \times {\overline{\Gamma }}_1)\subset X_{\Gamma _2}^{(i)}\) we get \(\Sigma \times {\overline{\Gamma }}_1\subset {\tilde{X}}_{\Gamma _2}^{(i)}\). Therefore, \({\tilde{L}}_1'={\tilde{L}}_2'\) on \(\Sigma \times {\overline{\Gamma }}_1\).
We shall summarize the results of Sects. 2–6.
Theorem 6.2
(Local step) Consider two initial boundary value problems
where \(L^{(i)}\) have the form (1.1). Suppose \(\Gamma _0\subset \partial D_0^{(1)}\cap \partial D_0^{(2)}\) and suppose the BLR condition holds for \(L^{(1)}\) on \([t_0,T_{t_0}]\). Suppose the corresponding DN operators \(\Lambda ^{(i)}\) are equal on \(\Gamma _0\times (t_0,T_2),\ T_2\ge T_{t_0}\), i.e. \(\Lambda ^{(1)}f=\Lambda ^{(2)}f\) on \(\Gamma _0\times (t_0,T_{2})\) for all f with support in \(\overline{\Gamma _0}\times [t_0,T_{2}]\). Let \(T_2-T_1\) be small. Suppose coefficients of \(L^{(1)}\) and \(L^{(2)}\) are analytic in \(x_0\).
Let \(\varphi ^{(i)}\) be the changes of variables (2.14) for \(i=1,2\) and let \(\beta ^{(i)},i=1,2,\) be the changes of variables (5.10). Let \(c_i\) be the gauge transformation (2.20), (2.21) for \(i=1,2.\) Then
where
\(\Sigma =\{(s,\tau ),s\ge 0,\tau \ge 0, s+\tau \le T_2-T_1\}= \{(y_0,y_n):0\le y_n\le \frac{T_2-T_1}{2}, T_1+y_n<y_0<T_2-y_n\}\).
7 The global step
Let \(L^{i}u_i=0\) in \(D_i=D_0^{(i)}\times \mathbb {R},i=1,2,\ u_i=0\) for \(x_0\ll 0,\ \partial D_0^{(1)}\cap \partial D_0^{(2)}\supset \Gamma _0\) and \(u_i\big |_{\partial D_0^{(i)}\times \mathbb {R}}=f,i=1,2,\ f\) has a compact support in \(\overline{\Gamma _0}\times \mathbb {R}\).
First we extend the Theorem 6.2 for a larger time interval.
Let \([t_1,t_2]\) be an arbitrary time interval. Let \([t_0,T_{t_0}]\) be such that \(T_{t_0}\le t_1\) and the BLR condition holds on \([t_0,T_{t_0}]\). Thus the BLR condition is satisfied on \([t_0,t]\) for any \(t\in [t_1,t_2]\). Let \(\Gamma _1\) be arbitrary connected part of \(\Gamma _0,\ \overline{\Gamma _1}\subset \Gamma _0\). Note that we do not require \(\overline{\Gamma _1}\) to be small.
Let \(\psi _{0i}^\pm (x_0,x',x_n),i=1,2,\) be the solution of the form (2.4) in \([t_0-1,t_2+1]\times \overline{\Gamma '}\times [0,{\varepsilon }_\pm ]\) where \(\overline{\Gamma _1}\subset \Gamma '\subset \Gamma _0\).
We impose the following initial conditions on \(\psi _{0i}^\pm , i=1,2\),
Such solutions exist in \([t_0-1,t_2+1]\times \overline{\Gamma '}\times [0,{\varepsilon }_0]\subset D_0^{(i)} \times \mathbb {R}\), when \({\varepsilon }_0\) is small. We choose \(\psi _{0i}^\pm \) such that (2.6) is satisfied and we choose \({\varepsilon }_1>0\) such that \({\varepsilon }_1\le {\varepsilon }_0\) and \(\{0<x_n<{\varepsilon }_1,x'\in \overline{\Gamma '},x_0\in [t_0-1,t_2+1]\}\) do not intersect \(\partial D_0^{(i)}\times \mathbb {R}\).
Let \(\varphi _{ji}(x_0',x',x_n),1\le j\le n-1,\) be the solutions of the linear equations (cf. (2.7))
with initial conditions
Similarly to (2.14) consider the map \((y_0^{(i)}(x),y_i'(x),y_n^{(i)}(x))=(\varphi _0^{(i)},\varphi _i',\varphi _n^{(1)}),\ x\in [t_0-1,t_2+1]\times \overline{\Gamma '}\times [0,{\varepsilon }_1]\), where
As in (2.15) we have that the map \((x_0,x',x_n)\rightarrow (y_0,y',y_n)\) is the identity when \(x_n=0\):
Let \(u_s=\frac{1}{2}(u_{y_0}-u_{y_n}),u_\tau =-\frac{1}{2}(u_{y_0}+u_{y_n})\). Making the change of variables (7.4) in \(L^{i}u_i=0\), the gauge transformation (2.18), (2.21) and the change of unknown function (2.26), we get in \(t_0\le y_0\le t_2,\ 0\le y_n\le T_0,\ y'\in \overline{\Gamma '},\ T_0\) is small, the equation of the form
where \(L_1^{(i)}\) has the form (2.28). Here
We assume that \(u_1^{(i)}\) satisfy the zero initial conditions
and
We also assume that DN operators for \(L^{i}\) and subsequently for \(L_1^{(i)}\) are equal on \([t_0,t_2)\times \overline{\Gamma '}\).
Note that the change of variables
where \( \alpha ^{(i)}\) are the same as in (5.9), (5.11), are also defined globally on \({\hat{\Omega }}_{0}\).
Let \([T_1,T_2]\subset [t_1,t_2]\) be arbitrary such that \(T_2-T_1=2T_0.\)
Applying Theorem 6.1 to the interval \([T_1,T_2]\) we get that the coefficients of \({\tilde{L}}_1^{(1)}\) and \({\tilde{L}}_1^{(2)}\) and the coefficients of \(L_1'\) and \(L_2'\) are equal on \(\Sigma _{T_1T_2}\times \overline{\Gamma _1}\) where \(\Sigma _{T_1T_2} =\{0\le y_n\le T_0,T_1+y_n\le y_0\le T_2-y_n\}\). We assume that \(\Gamma '\supset \overline{\Gamma _1}\) is such that \(\overline{\Gamma _2}\subset \overline{\Gamma _3}\subset \Gamma _0'\) for all \([T_1,T_2]\subset [t_1,t_2]\). Here \(\Gamma _2,\Gamma _3\) are defined as in Sect. 3. Note that \(\Gamma _2,\Gamma _3\) may depend on \([T_1,T_2]\).
If two intervals \([T_1,T_2]\) and \([T_1',T_2']\) intersect, then the coefficients of \({\tilde{L}}_1^{(1)}\) and \({\tilde{L}}_1^{(2)}\) coincide in \((\Sigma _{T_1T_2}\bigcup \Sigma _{T1'T_1'})\times \Gamma _1\).
Therefore coefficients of \({\tilde{L}}_1^{(1)}\) and \({\tilde{L}}_1^{(2)}\) and consequently the coefficients of \(L_1'\) and \(L_2'\) (cf. (6.2), (6.3)) coincide for \(0\le y_n\le T_0,\ y'\in \overline{\Gamma _1},\ t_1+T_0<y_0<t_2-T_0.\)
Therefore we proved
Lemma 7.1
Suppose \([t_1,t_2]\) is arbitrary large, \(T_0>0\) is small, \(t_0\) is such that the BLR condition is satisfied on \([t_0,t_1]\). Let \(\Omega _0=\{y_0\in [t_0+T_0,t_2-T_0],\ y'\in \overline{\Gamma _1}, y_n\in [0,\frac{T_0}{2}]\}\). Assume that the coefficients of \(L^{(i)}\) are analytic in \(x_0,i=1,2\). Then
\(\square \)
Let \(\Omega _i=(\beta ^{(i)}\varphi ^{(i)})^{-1}\Omega _0,\ i=1,2\). Note that \(\Omega _i\subset D_0^{(i)}\times [t_0-1,t_2+1]\) since \(T_0\) is small. We have that \(\Phi _2=(\beta ^{(1)}\varphi ^{(1)})^{-1}\beta _2\varphi ^{(2)}\) maps \(\Omega _2\) onto \(\Omega _1\). Note that \(\partial \Omega _1\cap \partial \Omega _2\supset \Gamma _1\times [t_0,t_2]\) and \(\Phi _2=I\) on \(\Gamma _1\times [t_0+T_0,t_2-T_0]\). Note also that \(\beta ^{(i)}\circ c_i\) can be represented as \(c_i'\circ \beta ^{(i)}\) where \(c_i'\) is the gauge transformation in \((y_0,{\hat{y}}',y_n)\) coordinates. Analogously, \((\beta ^{(1)}\circ c_1\circ \varphi ^{(1)})^{-1}\beta ^{(2)}\circ c_2\circ \varphi =c_3\circ \Phi _2\), where \(c_3\) is the gauge transformation. Therefore
Let B be a smooth domain in \(D_0^{(1)}\) such that \(\partial B\cap \partial D_0^{(1)}=\gamma _1\subset \Gamma _0\). Suppose B is small and such that \(B\times [t_1+1,t_2-1]\subset \Omega _1\).
Let \(S_2=\Phi _2^{-1}(B\times [t_1+1,t_2-1]\subset D_0^{(2)}\times \mathbb {R}\) and let \(S_2^+=\Phi _2^{-1}(B\times \{x_0=t_2-1\}), \ S_2^-=\Phi _2^{-1}(B\times \{x_0=t_1+1\})\). Let \({\tilde{S}}_2^\pm \) be space-like surfaces in \(D_0^{(2)}\times \mathbb {R}\) such that \({\tilde{S}}_2^+\) is the extension of \(S_2^+\) and \({\tilde{S}}_2^-\) is the extension of \(S_2^-\).
We assume that the projections of \({\tilde{S}}_2^\pm \) on \(D_0^{(2)}\) is \(D_0^{(2)}\). Let \(D_1^{(2)}\) be the domain in \(D_0^{(2)}\times \mathbb {R}\) bounded by \({\tilde{S}}_2^+\) and \({\tilde{S}}_2^-\) (cf. Fig. 8).
It follows from [16, Chapter 8], that there exists an extension \({\tilde{\Phi }}_2\) of \(\Phi _2\) from \(S_2\subset D_1^{(2)}\) to \(D_1^{(2)}\) such that \({\tilde{\Phi }}_2\big |_{\Gamma _0\times [t_1+1,t_2-1]}=I\).
Define \({\overline{D}}_1^{(3)}={\tilde{\Phi }}_2({\overline{D}}_1^{(2)})\). There exists also an extension \({\tilde{c}}_3\) of the gauge \(c_3\) from \(S_2\) to \(D_1^{(2)}\) such that \({\tilde{c}}_3=1\) on \(\Gamma _0\times [t_1+1,t_2-1]\). Let \(L^{(3)}={\tilde{c}}_3\circ {\tilde{\Phi }}_2\circ L^{(2)},\ L^{(3)}\) is defined on \(D_1^{(3)}\). Thus \(L^{(3)}=L^{(1)}\) on \(B\times [t_1+1,t_2-1]\). Note that \(D_1^{(3)}\cap (D_0^{(1)}\times [t_1+1,t_2-1])\supset B\times [t_1+1,t_2-1], \ \partial ' D_1^{(3)}\cap (\partial D_0^{(1)}\times [t_1+1,t_2-1])\supset \Gamma _0\times [t_1+1,t_2-1]. \) We denote by \(\partial 'D_1^{(3)}\) the lateral (time-like) part of \(\partial D_1^{(3)}\) and by \(\partial _\pm D_1^{(3)}\) the top and the bottom space-like parts of \(\partial D_1^{(3)}\), i.e. \(\partial D_1^{(3)}=\partial ' D_1^{(3)}\cup \partial _+D_1^{(3)}\cup \partial _- D_1^{(3)}\).
The almost cylindrical domain \(D_1^{(2)}\) is the part of \(D_0^{(2)}\times \mathbb {R}\) bounded from above and from below by space-like surfaces \({\tilde{S}}_2^+\) and \({\tilde{S}}_2^-\)
The following lemma is the key lemma of this section. It allows to reduce the solution of the inverse problem to an inverse problem in a smaller domain.
Lemma 7.2
Consider two initial–boundary value problem \(L^{(1)}u_1=0\) in \(D_0^{(1)}\times [t_1,t_2]\) and \(L^{(3)}u_3=0\) in \(D_1^{(3)}\),
We assume that \((\partial D_0^{(1)}\times [t_1,t_2]) \cap \partial D_1^{(3)} \supset \Gamma _0\times [t_1,t_2]\). Assume that \(L^{(1)}=L^{(3)}\) in a smooth domain \(B\times [t_1,t_2]\) where \(B\times [t_1,t_2]\subset (D_0^{(1)}\times [t_1,t_2])\cap D_1^{(3)},\ \gamma _1=\partial D_0^{(1)}\setminus \Gamma _0,\ {\tilde{\Gamma }}_3=\partial ' D_1^{(3)}\setminus (\Gamma _0\times (t_1,t_2)),\ \partial B=\gamma _0\cup \gamma _0'\), where \(\gamma _0,\gamma _0'\) are smooth, \(\gamma _0\subset \Gamma _0\), (cf. Fig. 9).
Suppose \(\Lambda _1=\Lambda _3\) on \(\Gamma _0\times [t_1,t_2]\).
Consider \(L^{(1)}u_1=0\) and \(L^{(3)}u_3=0\) in smaller domains \((D_0^{(1)}\setminus B)\times (t_1+\delta ,t_2-\delta )\) and \((D_1^{(3)}\cap (t_1+\delta ,t_2-\delta ))\setminus (B\times (t_1+\delta ,t_2-\delta ))\). Note that \(\partial (D_0^{(1)}\setminus B)\supset (\Gamma _0\setminus \gamma _0) \cup \gamma _0')\). Then \(\Lambda _1'=\Lambda _3'\) are equal on \(((\Gamma _0\setminus \gamma _0)\cup \gamma _0')\times (t_1+\delta ,t_2-\delta )\) for some \(\delta >0\). Here \(\Lambda _1',\Lambda _3'\) are DN operators for the initial–boundary value problem
To prove Lemma 7.2 we will need the following version of the Runge theorem about the approximation of solutions of the equation in a smaller domain by solutions of the same equation in a larger domain.
Lemma 7.3
Denote by \(D_{{\varepsilon }}\) the domain bounded by \(\Gamma _0\) and \(\gamma _{\varepsilon }\) such that \(\gamma _{\varepsilon }\cup \gamma _1\) is smooth. Let \(W_0\) be the space of \(v\in H_s((D_0^{(1)}\setminus B)\times (t_1,t_2)),\ s\ge 1\), such that
where \(\gamma _1=\partial D_0^{(1)}\setminus \Gamma _0\).
Denote by K the closure of \(W_0\) in \(L_2((D_0^{(1)}\setminus B)\times (t_1,t_2))\). Consider the space W of \(u(x)\in H_s((D_0^{(1)}\cup D_{\varepsilon })\times (t_1,t_2)),s\ge 1\) such that
Then the closure of the restrictions of W to \(L_2((D_0^{(1)}\setminus B)\times (t_1,t_2))\) is also equal to K. Thus any function \(v\in W_0\) in (\(D_0^{(1)}\setminus B)\times (t_1,t_2)\) can be approximated in \(L_2((D_0^{(1)}\setminus B)\times (t_1,t_2))\) norm by the functions in W.
Proof
Let \(K^\perp \) be the orthogonal complement of K in \(L_2((D_0^{(1)}\setminus B)\times (t_1,t_2))\). Take any \(g\in K^\perp \) and denote by \(g_0\) the extension of g by zero outside \((D_0^{(1)}\setminus B)\times (t_1,t_2)\). Let w be the solution of the initial–boundary value problem
where \(L_1^*\) is the formally adjoint to \(L^{(1)}\). Note that \(\partial (D_0^{(1)}\cup D_{{\varepsilon }0})=\gamma _1\cup \gamma _{\varepsilon }\) (see Fig. 9).
The boundary of B is \(\gamma _0\cup \gamma _0'\). The boundary of \(D_{\varepsilon }\) is \(\gamma _{\varepsilon }\cup \Gamma _0,\ \partial D_0^{(1)}=\Gamma _0\cup \gamma _1\)
By Hormander [15] and Eskin [13] (see also Lemma 3.3) such w(x) exists and belongs to \(H_1((D_0^{(1)}\cup D_{\varepsilon }))\times (t_1,t_2))\). We shall show that \(w=0\) in \((B\cup D_{\varepsilon })\times (t_1,t_2)\). Let \(\varphi \in C_0^\infty ((B\cup D_{{\varepsilon }})\times (t_1,t_2))\) and let u(x) be the solution of
(cf. [13, 15] and Lemma 3.3), i.e. \(u\in W_0\) since \(\varphi =0\) in \((D_0^{(1)}\setminus B)\times (t_1,t_2)\).
Consider the \(L_2\) inner product \((\varphi ,w)\) in \((D_0^{1)}\cup D_{{\varepsilon }})\times (t_1,t_2)\). Since \(\varphi =L^{(1)}u\) we get \((\varphi ,w)=(L^{(1)}u,w)\). Integrating by parts we have \((L_1u,w)= (u,L_1^*w)=(u,g_0)=0\) since \(u\in W_0, g_0\in K^\perp \). Therefore \((\varphi ,w)=0,\ \forall \varphi \). Thus \(w=0\) in \((B\cup D_{{\varepsilon }})\times (t_1,t_2)\).
Let now \({\tilde{w}}\) be any function in W. We have \(({\tilde{w}},g_0)_0=({\tilde{w}},L_1^*w)_0\), where \((\,\,)_0\) means that we integrate over \((D_0^{(1)}\setminus B)\times (t_1,t_2)\). Since \(w=0\) in \((B\cup D_{{\varepsilon }})\times (t_1,t_2)\), we have that
where \(\frac{\partial }{\partial \nu }\) is the normal derivative.
Note that \((\Gamma _0\setminus \gamma _0)\cup \gamma _0'=\partial (D_{{\varepsilon }}\cup B)\setminus \gamma _{\varepsilon }\). Since w satisfies the homogenous equation \(L_1^*w=0\) in \(D_{\varepsilon }\cup B\) the restrictions of w and all derivatives on \(\partial (D_{{\varepsilon }}\cup B)\) exists by the partial hypoellipticity (see, for example, [12]). Note that \({\tilde{w}}\) and w have zero values on \(\gamma _1\). Therefore, integrating by parts, we have
since \(L^{(1)}{\tilde{w}}=0\) in \((D_0^{(1)}\setminus B)\times (t_1,t_2)\). Therefore \(({\tilde{w}},g_0)_0=0,\ \ \forall g_0\in K^\perp ,\) i.e. \({\tilde{w}}\in {\overline{K}}\).
To make the integration by parts rigorous we approximate \(\gamma _0'\cup (\Gamma _0\setminus \gamma _0)\) by \(\gamma _{{\varepsilon }_1}'\), similar to \(\gamma _{\varepsilon },\ \gamma _{{\varepsilon }_1}' \subset D_{\varepsilon }\cup B\). Note that \(w=0\) in \(D_{\varepsilon }\cup B\). Therefore integrating by parts over domain bounded by \(\gamma _1\cup \gamma _{{\varepsilon }_1}'\), and taking the limit when \(\gamma _{e_1}'\rightarrow \gamma _0'\cup (\Gamma _0\setminus \gamma _0)\) we get \(({\tilde{w}},g_0)=0\). \(\square \)
Now we shall proof Lemma 7.2.
Let \(\hbox {supp}\,f\subset \Gamma _0'\times (t_1,t_2),\ \Gamma _0'=(\Gamma _0\setminus \gamma _0)\cup \gamma _0'\). Let \(v_1\) be the solutions of
where \(\partial (D_0^{(1)}\setminus B)=\Gamma _0'\cup \gamma _1,\ f_1=0\) on \(\gamma _1\times (t_1,t_2), \ f_1=f\) on \(\Gamma _0'\times (t_1,t_2)\).
Let \(v_3\) be solution of \(L^{(3)}v_3=0\) in \(D_1^{(3)}\setminus (B\times (t_1,t_2))\)
Let \(\Lambda _1'\) be the DN operator for (7.14) and \(\Lambda _3'\) be the DN operator for (7.15). Assuming that \(\Lambda _1=\Lambda _2\) on \(\Gamma _0\times (t_1,t_2)\) we shall prove that
for all f with supports in \(\Gamma _0'\times (t_1+\delta ,t_2-\delta )\). By Lemma 7.3 there exists a sequence of smooth solutions \(w_{n1}\in W_0\) such that
where \(\Vert v_1\Vert _0\) is the norm in \(L_2((D_0^{(1)}\setminus B)\times (t_1,t_2))\). Note that \( L^{(1)}w_{n1}=0\) in \(D_0^{(1)}\times (t_1,t_2),\ w_{n1}\big |_{\gamma _1\times (t_1,t_2)}=0,\ w_{n1}\big |_{x_0=t_1}=\frac{\partial w_{n1}}{\partial x_0}\big |_{x_0=t_1}=0\), where \(\gamma _1=\partial D_0^{(1)}\setminus \Gamma _0\). Let \(f_n=w_{n1}\big |_{\Gamma _0\times (t_1,t_2)}\). Denote by \(w_{n3}\) the solution of
Since \(\Lambda _1=\Lambda _2\) on \(\Gamma _0\times (t_1,t_2)\), we have
Since \(\gamma _o\subset \Gamma _0\), the equality (7.17) implies
We have \(L^{(1)}=L^{(3)}\) on \(B\times (t_1,t_2)\). Using the uniqueness theorem of [25] and [29], we get
where \(\delta >0\) is determined by the metric and by the domain B (cf. Fig. 9). In particular,
Therefore
where \(\Gamma _0'=(\Gamma _0\setminus \gamma _0)\cup \gamma _0'\), i.e. \(\Lambda _1'f_n'=\Lambda _2'f_n'\) on \(\Gamma _0'\times (t_1+\delta ,t_2-\delta )\), where \(f_n'=w_{n1}\big |_{\Gamma _0'\times (t_1+\delta ,t_2-\delta )} =w_{n3}\big |_{\Gamma _0'\times (t_1+\delta ,t_2-\delta )}\). We have
since \(\partial (D_0^{(1)}\setminus B)=\Gamma _0'\cup \gamma _1\) and \(f=f_n'=0\) on \(\gamma _1\times (t_1+\delta ,t_2-\delta )\).
In (7.21) we again use the partial hypoellipticity property that restrictions of solutions of \(L^{(1)}u=0\) to the noncharacteristic boundary exists for any Sobolev’s space \(H_s\) (cf. [12]). The same is true for all normal derivatives of \(u_1\), and the same estimates hold as in the case of positive \(s>0\) (cf. [8] and [12]).
By Lemma 3.3 (see [13, 15]) we have
Analogously we have
Note that
\( \big \Vert f-f_n'\big \Vert _{-\frac{1}{2},\partial (D_0^{(1)}\setminus B)\times (t_1+\delta ,t_2-\delta )}= \big \Vert f-f_n'\big \Vert _{-\frac{1}{2},\partial '(D_1^{(3)}\cap (t_1+\delta ,t_2-\delta )\setminus (B\times (t_1+\delta ,t_2-\delta ))}\).
Therefore, taking the limit as \(n\rightarrow \infty \), we get, using (7.20), that
Thus we proved that
for any f with \(\hbox {supp}\,f\subset \Gamma _0'\times (t_1+\delta ,t_2-\delta )\). \(\square \)
Using Lemma 7.2 we reduce the inverse problem in \(D_0^{(1)}\times (t_1,t_2)\) to the inverse problem in smaller domains \((D_0^{(1)}\setminus B)\times (t_1+\delta ,t_2-\delta )\) and we can continue this process starting from \((D_0^{(1)}\setminus B)\times (t_1+\delta ,t_2-\delta )\) instead of \(D_0^{(1)}\times (t_1,t_2)\).
In all lemmas below we assume that DN operators for \(L^{(1)}\) and \(L^{(2)}\) are equal on \(\Gamma _0\times [t_1,t_2]\) and that the time interval \([t_1,t_2]\) is large enough. We shall continue to call the coordinates \((y_0,{\hat{y}}',y_n)\), given by the map \(\beta ^{(i)}\varphi ^{(i)}\), the Goursat coordinates for \({\tilde{L}}_1^{(i)},i=1,2\).
Lemma 7.4
Let \(\Gamma _1\subset \Gamma _0\) and let \({\tilde{\Gamma }}_1\subset \Gamma _0\) be such that \({\overline{\Gamma }}_1 \subset {\tilde{\Gamma }}_1\). We assume that \({\tilde{\Gamma }}_1\subset {\tilde{\Gamma }}_2\subset {\tilde{\Gamma }}_3 \subset \Gamma _0\) where \({\tilde{\Gamma }}_j,1\le j\le 3,\) are the same as \(\Gamma _j,1\le j\le 3,\) in Sect. 3. Suppose the Goursat coordinates for \(L^{(1)}\) exists in \(\Omega _1=(t_1,t_2)\times {\tilde{\Gamma }}_3\times [0,{\varepsilon }_0]\), i.e. \(L^{(1)}\) has the form \({\tilde{L}}_1^{(1)}\) in these coordinates (we include the gauge transformation (2.21) in \({\tilde{L}}_1^{(i)})\). Suppose the Goursat coordinates for \(L^{(2)}\) exist in \((t_1,t_2)\times (\overline{{\tilde{\Gamma }}}_3\setminus \Gamma _1)\times [0,{\varepsilon }_0]\). Let \({\tilde{\Omega }}_2= (t_1,t_2)\times ({\tilde{\Gamma }}_1\setminus \Gamma _1)\times [0,{\varepsilon }_0]\) and suppose \({\tilde{L}}_1^{(2)}={\tilde{L}}_1^{(1)}\) in \({\tilde{\Omega }}_2\). Then \(L^{(2)}\) has also Goursat coordinates in \(\Omega _3=(t_1,t_2)\times \Gamma _1\times [0,{\varepsilon }_0]\), and \({\tilde{L}}_1^{(1)}={\tilde{L}}_1^{(2)} \) in \(\Omega _3\).
Proof
Let \(y^{(i)}=\psi ^{(i)}(x)\) be the transformation to the Goursat coordinates, and let \(\frac{D\psi ^{(i)}(x)}{Dx}\) be the Jacobi matrix of this transformation. We have
where \([{\hat{g}}_i^{jk}]^{-1}\) is the metric tensor in the Goursat coordinates. The Goursat coordinates degenerate at point \(y_i^{(0)}=\psi ^{(i)}(x^{(0)})\), when \(\det \frac{D\psi ^{(i)}(x)}{Dx}\rightarrow \infty \) for \(y\rightarrow y^{(0)}\) (or \(x\rightarrow x^{(0)}\)) (cf. (2.11)). We call such point a focal point. We shall prove that there is no focal points for \(L^{(2)}\) in \(\Omega _3\).
We have
Suppose there exists the focal point \(y^{(0)}=(y_0^{(0)},y_0',y_n^{(0)}), y_n^{(0)}<{\varepsilon }_0,y_0'\in \Gamma _1\) such that there is no focal points for \(L^{(2)}\) when \(y_n<y_n^{(0)}\) for all \(y_0\in [t_1,t_2], y'\in {\overline{\Gamma }}_1.\)
Since \(L^{(1)}\) and \(L^{(2)}\) have Goursat coordinates for \(y_n<y_n^{(0)}\) we get, by Lemma 7.1, that \({\tilde{L}}_1^{(1)}={\tilde{L}}_1^{(2)}\) in \((t_1,t_2)\times {\overline{\Gamma }}_1\times [0,y_n^{(0)}-{\varepsilon }], \forall {\varepsilon }>0\), and hence \([{\hat{g}}_1^{jk}]=[{\hat{g}}_2^{jk}]\) for \({\varepsilon }>0\). Since \(\det [{\hat{g}}_2^{jk}]=\det [{\hat{g}}_1^{jk}]\) for \(y_n<y_n^{(0)}-{\varepsilon }\), we have that \(\big (\det \frac{D\psi ^{(2)}}{Dx}\big )^2=\frac{\det [{\hat{g}}_1^{jk}]}{\det [ g_2^{jk}]}\) is bounded when \({\varepsilon }\rightarrow 0\). Therefore \(y^{(0)}=(y_0^{(0)},y_0',y_n^{(0)})\) is not a focal point for \(L^{(2)}\). Thus \(L^{(2)}\) has no focal points in \(\Omega _3\) and then, by Lemma 7.1, we have \({\tilde{L}}_1^{(1)}={\tilde{L}}_1^{(2)}\) in \(\Omega _3\) (cf. [9]). \(\square \)
Lemma 7.5
Assume that DN operators for \(L^{(1)}\) and \(L^{(2)}\) are equal on \(\Gamma _0\times [t_1,t_2]\). Let \({\overline{\Gamma }}_1\subset \Gamma _0\). Assume that the Goursat coordinates for \(L^{(1)}\) exists in \((t_1,t_2)\times {\overline{\Gamma }}_1\times [0,\frac{T_0}{2}]\). Then the Goursat coordinates for \(L^{(2)}\) also exists in \(\Omega _1=(t_1+\delta ,t_2-\delta )\times {\overline{\Gamma }}_1\times [0,\frac{T_0}{2}]\) for some \(\delta >0\) and \({\tilde{L}}_1^{(1)}={\tilde{L}}_1^{(2)}\) in \({\overline{\Omega }}_1\), where \({\tilde{L}}_1^{(i)}\) are the operators \(L^{(i)}\) in the Goursat coordinates.
Proof
Let \({\overline{\Gamma }}_1\subset {\tilde{\Gamma }}_1,\ \overline{{\tilde{\Gamma }}}_1\subset \Gamma _0\). If \(0\le y_n\le {\varepsilon }\), where \({\varepsilon }>0\) is small enough, then \({\tilde{\Gamma }}_1\subset {\tilde{\Gamma }}_2\subset {\tilde{\Gamma }}_3\subset \Gamma _0\), where \({\tilde{\Gamma }}_j,j=1,2,3,\) are the same as in Lemma 7.4. Applying Lemma 7.1 we get that the Goursat coordinates for \({\tilde{L}}_1^{(1)}\) and \({\tilde{L}}_1^{(2)}\) exist in \(\Omega _{1{\varepsilon }}=[t_1,t_2]\times {\tilde{\Gamma }}_1\times [0,{\varepsilon }]\) and
Let \(\Sigma _1\) be the surface in \((y',y_n)\) space such that \(y_n=0\) on \({\tilde{\Gamma }}_1\setminus \Gamma _1,0\le y_n\le {\varepsilon }\) on \(\partial \Gamma _1,y_n={\varepsilon }\) on \(\Gamma _1\). Note that \(\Sigma _1\) is not smooth since it has edges when \(y_n=0,y'\in \partial \Gamma _1\) and when \(y_n={\varepsilon },y'\in \partial \Gamma _1\). We shall smooth \(\Sigma _1\) by replacing it by smooth surface \({\tilde{\Sigma }}_1\), where \({\tilde{\Sigma }}_1\) differs from \(\Sigma _1\) in a neighborhood of edges having the size \(O({\varepsilon })\). Let \(\Sigma _2\) be the surface, where \(y_n={\varepsilon }\) when \(y'\in {\tilde{\Gamma }}_1\setminus \Gamma _1({\varepsilon }),\ \Gamma _1({\varepsilon })\) is the \({\varepsilon }\)-neighborhood of \(\Gamma _1,{\varepsilon }\le y_n\le 2{\varepsilon }, \) when \(y'\in \partial \Gamma _1({\varepsilon }),\ y_n=2{\varepsilon }\) when \(y'\in \Gamma _1({\varepsilon })\) (cf. Fig. 10). \(\square \)
\(\Sigma _2'\) is the surface \(\{y_n=0\ \hbox {when}\ y'\in {\tilde{\Gamma }}'\setminus \Gamma _1,\ 0\le y_n\le 2{\varepsilon }\ \hbox {when}\ y'\in \partial \Gamma _1, y_n=2{\varepsilon }\ \hbox {when}\ y'\in \Gamma _1\}\), \(\Sigma _3\) is the surface \(\{y_n={\varepsilon }\ \hbox {when}\ y'\in {\tilde{\Gamma }}_1\setminus \Gamma _1({\varepsilon }), \ {\varepsilon }\le y_n\le 3{\varepsilon }\ \hbox {when}\ y'\in \partial \Gamma _1({\varepsilon }), y_n=3{\varepsilon }\ \hbox {when}\ y'\in \Gamma _1({\varepsilon })\}\), \(S_2\) is the region between \(\Sigma _3\) and \(\Sigma _2'\)
Let \({\tilde{\Sigma }}_2\) be the smoothing of \(\Sigma _2\). Denote by \({\tilde{S}}_1\) the domain between \({\tilde{\Sigma }}_1\) and \({\tilde{\Sigma }}_2\) when \(y'\in {\tilde{\Gamma }}_1\). Since \({\tilde{L}}_1^{(1)}={\tilde{L}}_1^{(2)}\) for \(0\le y_n\le {\varepsilon }\), we have, by Lemma 7.2, that DN operators for \(L^{(1)}\) and \(L^{(2)}\) are equal on \({\tilde{\Sigma }}_1\times (t_1+\delta _1,t_2-\delta _1)\) for some \(\delta _1>0\).
Suppose \({\varepsilon }>0\) is small and such that we can introduce the Goursat coordinates for \(L^{(1)}\) in \({\tilde{S}}_1\times [t_1+\delta _1,t_2-\delta _1]\).
Note that \({\varepsilon }\) and \(\delta _1\) are determined by \(L^{(1)}\) only and are independent of \(L^{(2)}\). It follows from Lemma 7.4 that Goursat coordinates for \(L^{(2)}\) also hold on \({\tilde{S}}_1\times (t_1+\delta _1,t_2-\delta _1)\) and \({\tilde{L}}_1^{(1)}={\tilde{L}}_1^{(2)}\) in \({\tilde{S}}_1\times (t_1+\delta _1,t_2-\delta _1)\).
By Lemma 7.2 DN operators for \(L^{(1)}\) and \(L^{(2)}\) are equal on \({\tilde{\Sigma }}_2\times (t_2+\delta _1,t_2-\delta _1)\).
Let \(\Sigma _2'\) be the surface in \((y',y_n)\) space such that \(y_n=0\) on \({\tilde{\Gamma }}_1\setminus \Gamma _1,\ 0\le y_n\le 2{\varepsilon }\) on \(\partial \Gamma _1,\ y_n=2{\varepsilon }\) on \(\Gamma _1\) and let \(\Sigma _3\) be the surface where \(y_n={\varepsilon }\) on \({\tilde{\Gamma }}_1\setminus \Gamma _1({\varepsilon }),\ {\varepsilon }\le y_n\le 3{\varepsilon }\) on \(\partial \Gamma _1({\varepsilon }),y_n=3{\varepsilon }\) on \(\Gamma _1({\varepsilon })\). Let \({\tilde{\Sigma }}_2'\) and \({\tilde{\Sigma }}_3\) be the smoothing of \(\Sigma _2',\Sigma _3\) and let \({\tilde{S}}_2\) be the domain between \({\tilde{\Sigma }}_2'\) and \({\tilde{\Sigma }}_3\) when \(y'\in {\tilde{\Gamma }}_1\). Since DN operators for \(L^{(1)}\) and \(L^{(2)}\) are equal on \({\tilde{\Sigma }}_2'\times (t_1+\delta _1,t_2-\delta _1)\) and since the Goursat coordinates for \(L^{(1)}\) hold on \({\tilde{S}}_2\times (t_1+\delta _1,t_2-\delta _1)\), Lemma 7.4 implies that the Goursat coordinates hold for \(L^{(2)}\) in \(S_2\times (t_1+\delta _1,t_2-\delta _1)\), \(L_1^{(1)}=L_1^{(2)}\) in \(S_2\times (t_1+\delta _1+\delta _2,t_2-\delta _1-\delta _2)\) for some \(\delta _2>0\), and DN operators for \(L^{(1)}\) and \(L^{(2)}\) are equal on \({\tilde{\Sigma }}_3\times (t_1+\delta _1+\delta _2,t_2-\delta _1-\delta _2)\).
Analogously, for \(k>2\) denote by \(\Sigma _k'\) the surface such that \(y_n=0\) on \({\tilde{\Gamma }}_1\setminus \Gamma _1, 0\le y_n\le k{\varepsilon }\) on \(\partial \Gamma _1\) and \(y_n=k{\varepsilon }\) on \(\Gamma _1\). Let \(\Sigma _{k+1}\) be the surface, where \(y_n={\varepsilon }\) for \({\tilde{\Gamma }}_1\setminus \Gamma _1({\varepsilon }),\ {\varepsilon }\le y_n\le (k+1){\varepsilon }\) on \(\partial \Gamma _1({\varepsilon })\) and \(y_n=(k+1){\varepsilon }\) on \(\Gamma _1({\varepsilon })\).
Denote by \({\tilde{\Sigma }}_k'\) and \({\tilde{\Sigma }}_{k+1}\) the smoothing of \(\Sigma _k',\Sigma _{k+1}\). Let \({\tilde{S}}_k\) be the domain between \({\tilde{\Sigma }}_k'\) and \({\tilde{\Sigma }}_{k+1}\). Applying successively the same arguments for \(k=3,\ldots ,m\), we prove as above that \({\tilde{L}}_1^{(2)}={\tilde{L}}_1^{(1)}\) in \({\tilde{S}}_k\times [t_1+\sum _{j=1}^k\delta _j,t_2-\sum _{j=1}^k\delta _j],k=3,\ldots ,m\).
Let m be such that \((m+1){\varepsilon }\ge \frac{T_0}{2}\). Then we get that \({\tilde{L}}_1^{(2)}={\tilde{L}}_1^{(2)}\) in \((t_1-\delta ,t_2+\delta )\times {\overline{\Gamma }}_1\times [0,\frac{T_0}{2}]\), where \(\delta =\sum _{j=1}^m\delta _j\). Note that we assume that \([t_1,t_2]\) is large. Thus \(t_2-t_1\gg \delta \). \(\square \)
Suppose that after several applications of Lemma 7.2 we have
where we are considering the interval \([t_1,t_2]\) instead of \([t_1+\delta ,t_2-\delta ]\) for the simplicity of notations. We assume that
We also assume that \(\Gamma _0\times (t_1,t_2)\subset \partial ' D_1^{(m)}\cap (\partial D_0^{(1)}\times (t_1,t_2))\) and \(\Omega _1\times (t_1,t_2)\subset (D_0^{(1)}\times (t_1,t_2))\cap D_1^{(m)}\) (cf. Fig. 11).
We assume that \(L^{(1)}=L^{(m)}\) in \(\Omega _1\times (t_1,t_2)\) and that DN operators \(\Lambda _1\) and \(\Lambda _2\) are equal on \(\partial \Omega _1\times (t_1,t_2)\) and \(\Gamma _0\times (t_1,t_2)\). Here, as above, \(\partial 'D_1^{(m)}\) means the time-like part of the boundary of \(D_1^{(m)}\). It follows from Lemmas 7.4, 7.5 that the enlargement of the domain \(\Omega _1\) depends only on \(L^{(1)}\) and does not depend on \(L^{(2)}\). Therefore, as in [9], we arrive to the situation when \(\Omega _1\) and \(D_1^{(0)}\) are close. To apply Lemma 7.2 to the domain \((D_1^{(0)}\setminus \Omega _1)\times \mathbb {R}\) we need new tools.
When \(\partial \Omega _1\) and \(\partial D_0^{(1)}\) are close, there is a narrow domain \(\sigma _1\subset D_0^{(1)}\setminus \Omega _1\) such that \(\gamma _1\subset \partial D_0^{(1)},\gamma _0\subset \partial \Omega _1\) and the distance between \(\gamma _0\) and \(\gamma _1\) is small (cf. Fig. 11)
\(\gamma _1\subset \partial D_0^{(1)},\gamma _0\subset \Omega _1,\ \gamma _1\) and \(\gamma _0\) are close
Introduce Goursat coordinates for \(L^{(1)}\) and \(L^{(m)}\) near \(\gamma _0\times (t_1,t_2)\). We assume that operators \(L^{(1)}\) and \(L^{(m)}\) are defined in domains slightly larger than \(D_0^{(1)}\times (t_1,t_2)\) and \(D_1^{(m)}\). Let \(L_1^{(1)}\) and \(L_1^{(m)}\) be the operators \(L^{(1)}\) and \(L^{(m)}\) in corresponding Goursat coordinates. Let \(y=\varphi _1(x)\) be the transformation to the Goursat coordinates for \(L^{(1)}\). Let \(\sigma _0=(t_1,t_2)\times \gamma _0\times [0,{\varepsilon }_0]\) be the domain where the Goursat coordinates for \(L^{(1)}\) hold. We assume that \(\sigma _1\) is so small that \(\varphi _1(\sigma _1\times (t_1,t_2))\subset \sigma _0\). Let \(\tau _0=\varphi _1(\gamma _1\times (t_1,t_2))\), i.e. \(\tau _0\) is the image of part of the boundary \(\partial D_0^{(1)}\times (t_1,t_2)\) in Goursat coordinates. Denote by \(\sigma _0^+\) the part of \(\sigma _0\) between \(\gamma _0\times (t_1,t_2)\) and \(\tau _0\), i.e. \(\sigma _0^+\) is the image of \(\sigma _1\times (t_1,t_2)\) in Goursat coordinates.
We assume that the Goursat coordinates for \(L^{(m)}\) also hold in \((t_1,t_2)\times \gamma _0\times [0,{\varepsilon }_0]\). Moreover, applying Lemmas 7.4, 7.5 repeatedly we get that \({\tilde{L}}_1^{(m)}={\tilde{L}}_1^{(1)}\) in \({\hat{\sigma }}_0^+\), where \({\hat{\sigma }}_0^+=\sigma _0^+\cap (t_1+\delta ,t_2-\delta )\). Here \({\tilde{L}}_1^{(1)},{\tilde{L}}_1^{(m)}\) are operators \(L^{(1)},L^{(m)}\) in Goursat coordinates.
Consider the initial–boundary value problem for \({\tilde{L}}_1^{(1)}\) in Goursat coordinates
where \(\hbox {supp}\,f\subset (t_1+\delta ,t_2-\delta )\times \gamma _0\).
Consider now the equation \({\tilde{L}}_1^{(m)}{\tilde{u}}_2=0\) in \({\hat{\sigma }}_0^+.\)
where f is the same as in (7.25).
Since \({\tilde{L}}_1^{(1)}={\tilde{L}}_1^{(m)}\) in \({\hat{\sigma }}_0^+\) and since \(\Lambda _{1}^{(1)}f =\Lambda _1^{(2)}f\) on \((t_1+\delta ,t_2-\delta )\gamma _{0}\)
we have, by the unique continuation theorem of [25] and [29] that \({\tilde{u}}_1={\tilde{u}}_2\) in \({\hat{\sigma }}_1^+\cap (t_1+2\delta ,t_2-2\delta )\). Therefore by the continuity \({\tilde{u}}_2\big |_{\partial {\hat{\sigma }}_1^+\cap (t_1+2\delta ,t_2-2\delta )}=0\).
Let \(\sigma _2^+=\varphi _2^{-1}({\hat{\sigma }}_0^+), \tau _2=\varphi _2^{-1}(\partial {\hat{\sigma }}_{0}^+)\), where \(y=\varphi _2(x)\) is the transformation to the Goursat coordinates for \(L^{(m)}\).
We shall show that \(\tau _2\) is a part of the boundary of \(D_1^{(m)}\).
Construct the geometric optic solution \(v_1(y)\) for \({\tilde{L}}_1^{(1)}u_1=0\) in Goursat coordinates as in (5.1). Since \(\tau _0\) is the boundary of the domain \(\sigma _{0}^{+}\) and since the zero Dirichlet boundary condition holds on \({\hat{\tau }}_{0}=\tau _0\cap (t_1+\delta ,t_2-\delta )\) this solution must reflect at \({\hat{\tau }}_{0}\) (cf. [14]).
Consider now the geometric optics solution \(v_2(y)\) for \({\tilde{L}}_1^{(m)}u_2=0\) with the same initial condition. Since \({\tilde{L}}_1^{(1)}v_1={\tilde{L}}_1^{(m)}v_2\) in \({\hat{\sigma }}_{0}^+\) we have that \(v_1(y)=v_2(y)\) before the reflection at \({\hat{\tau }}_{0}\). If \(\tau _2=\varphi _2^{-1}({\hat{\tau }}_{0})\) is not a part of the boundary of \(\partial D_1^{(m)}\) there will be no reflection for \(v_2(y)\) at \({\hat{\tau }}_{0}\). Thus, the solutions \(v_1(y)\) and \(v_2(y)\) will be different in \({\hat{\sigma }}_{0}^+\) near \({\hat{\tau }}_{0}\). This contradicts the fact that \(v_1(y)=v_2(y)\) in \({\hat{\sigma }}_{0}^+\).
Therefore \(\varphi _m=\varphi _2^{-1}\varphi _1\) maps the boundary \(\gamma _1\times (t_1,t_2)\) of \(\partial D_0^{(1)}\times (t_1,t_2)\) on the part of boundary of \(\partial D_1^{(m)}\). Let \(\sigma _2\subset D_1^{(m)}\) be the image of \(\varphi _m=\varphi _2^{-1}\varphi _1(\sigma _1\times (t_1+2\delta ,t_2-2\delta ))\). Let \(\partial _\pm (({\overline{\Omega }}_1\times [t_1+2\delta ,t_2-2\delta ])\cup {\overline{\sigma }}_2)\) be the space-like parts of the boundary of \(({\overline{\Omega }}_1\times [t_1+2\delta ,t_2-2\delta ])\cup {\overline{\sigma }}_2\). Extend \(\partial _+(({\overline{\Omega }}_1\times [t_1+2\delta ,t_2-2\delta ])\cup {\overline{\sigma }}_2)\) and \(\partial _-(({\overline{\Omega }}_1\times [t_1+2\delta ,t_2-2\delta ])\cup {\overline{\sigma }}_2)\) as space-like surfaces \(S_m^+\) and \(S_m^-\) to the whole domain \(D_1^{(m)}\). Let \({\tilde{D}}_1^{(m)}\) be part of \(D_1^{(m)}\) bounded from below and above by \(S_m^-\) and \(S_m^+\), respectively. Note that \(\partial ' D_1^{(m)}\supset \Gamma _0\times (t_1+2\delta ,t_2-2\delta )\). Define \({\tilde{\varphi }}_m=I\) on \(\Omega _1\times (t_1+2\delta ,t_2-2\delta ), \ {\tilde{\varphi }}_m=I\) on \(\Gamma _0\times (t_1+2\delta ,t_2-2\delta ),\ {\tilde{\varphi }}_m=\varphi _m\) on \(\sigma _1\times (t_1+2\delta ,t_2-2\delta )\). Let \(\Phi _{m+1}\) be the extension of \({\tilde{\varphi }}_m\) (cf. [16]) to the whole domain \({\tilde{D}}_1^{(m)}\). Denote by \(D_1^{(m+1)}\) the image of \({\tilde{D}}_1^{(m+1)}\) under the map \(\Phi _{m+1}\). If \(c_m\) is a gauge transformation on \({\overline{\sigma }}_1\times [t_1+2\delta ,t_2-2\delta ]\) we denote by \({\tilde{c}}_{m+1}\) the extension of \(c_m\) to \(D_1^{(m+1)}\) such that \({\tilde{c}}_{m+1}=1\) on \(\Omega _1\times [t_1+2\delta ,t_2-2\delta ],\ {\tilde{c}}_{m+1}=1\) on \(\Gamma _0\times (t_1+2\delta ,t_2-2\delta )\).
We just proved the following lemma:
Lemma 7.6
Let \(L^{(1)}\) and \(L^{(m+1)}={\tilde{c}}_{m+1}\circ \Phi _{m+1}\circ L^{(m)}\) be operators in \(D_0^{(1)}\times [t_1+2\delta ,t_2-2\delta ]\) and \(D_1^{(m+1)}\), respectively. Then \(\Omega _2\times (t_1+2\delta ,t_2-2\delta )\subset D_1^{(m+1)}\cap (D_0^{(1)}\times (t_1+2\delta ,t_2-2\delta ))\) where \({\overline{\Omega }}_2={\overline{\Omega }}_1\cup {\overline{\sigma }}_1\) and \(L^{(1)}=L^{(m+1)}\) in \(\Omega _2\times (t_1+2\delta ,t_2-2\delta )\).
We shall proceed with the enlargement of the domain \(\Omega _2\) using Lemmas 7.2 and 7.6. Therefore after finite number of steps (cf. [9]) we get a domain \(D_1^{(N)}\), operator \(L^{(N)}\) on \(D_1^{(N)}\) and the map \(\Phi _N\) of \(D_1^{(N)}\) onto \(D_0^{(1)}\times (t_1+\delta _N,t_2-\delta _N)\) such that \(c_N\circ \Phi _N\circ L^{(N)}=L^{(1)}\) in \(D_0^{(1)}\times (t_1+\delta _N,t_2-\delta _N)\) for some \(\delta _N>0\). Here \(c_N\) is the gauge transformation. Remind that \({\tilde{\Phi }}_2\) is the diffeomorphism of \( D_1^{(2)}\) onto \(D_1^{(3)},\ \Phi _3\) is the diffeomorphism of \({\tilde{D}}_1^{(3)}\subset D_1^{(3)}\) onto \(D_1^{(4)}\), etc.... \({\tilde{\Phi }}_{N-1}\) is the map of \({\tilde{D}}_1^{(N-1})\subset D_1^{(N-1)}\) onto \(D_1^{(N)}\) and \(\Phi _N\) is the map of \(D_1^{(N)}\) onto \(D_0^{(1)}\times (t_1+\delta _N,t_2-\delta _N)\).
Therefore, the diffeomorphism \(\Phi ^{-1}=\Phi _1^{-1}\Phi _3^{-1}...\Phi _N^{-1}\) maps \(D_0^{(1)}\times [t_0+\delta _N,t_2-\delta _N]\) onto \(D_1^{(2)}\). Thus \(\Phi \) maps \(D_1^{(2)}\) onto \(D_0^{(1)}\times [t_1+\delta _N,t_2-\delta _N]\).
Note that \(D_1^{(2)}\) is an almost cylindrical domain in \(D_0^{(2)}\times \mathbb {R}\), i.e. \(D_1^{(2)}= D_0^{(2)}\times \{S^-(x_1,\ldots ,x_n)\le x_0\le S^+(x_1,\ldots ,x_n)\}\), where \(x_0=S^\pm (x_1,\ldots ,x_n)\) are space-like surfaces, \((x_1,\ldots ,x_n)\in D_0^{(2)}\).
Note that \([t_1,t_2]\) is arbitrary large and therefore \([t_1',t_2']=[t_1+\delta ,t_2-\delta ]\) is also arbitrary large. Therefore we obtained the following theorem:
Theorem 7.7
Let \(L^{(1)}\) and \(L^{(2)}\) be two operators in \(D_0^{(1)}\times \mathbb {R}\) and \(D_0^{(2)}\times \mathbb {R}\), respectively. Suppose \(\Gamma _0\subset \partial D_0^{(1)}\cap \partial D_0^{(2)}\) and the DN operators, corresponding to \(L^{(i)}\), are equal on \(\Gamma _0\times \mathbb {R}\) for all f that have a compact support in \({\overline{\Gamma }}_0\times \mathbb {R}\). Suppose that the conditions (1.2), (1.6) hold for \(L^{(i)}, i=1,2\), and the coefficients of \(L^{(1)}\) and \(L^{(2)}\) are analytic in \(x_0\) in \(D_0^{(i)}\times \mathbb {R},\ i=1,2\). Suppose for each \(t_0\in R\) there exists \(T_{t_0}\) such that the BLR condition is satisfied for \(L^{(1)}\) on \([t_0,T_{t_0}]\). Let \([t_1',t_2']\) be an arbitrary sufficiently large time interval. Then there exists a diffeomorphism \(\Phi ^{-1}\) of \({\overline{D}}_0^{(1)}\times [t_1',t_2']\) on an almost cylindrical domain \({\overline{D}}_1^{(2)}\subset {\overline{D}}_0^{(2)}\times \mathbb {R},\ \Phi =I\) on \(\Gamma _0\times [t_1',t_2']\) and there exists a gauge transformation c(y) on \(D_1^{(2)}, \ |c(y)|=1\) on \( D_1^{(2)},\ c(y)=1\) on \(\Gamma _0\times [t_1',t_2']\) such that
Now we shall use Theorem 7.7 to prove Theorem 1.2.
Proof of Theorem 1.2
Let \(L^{(i)}\) be two operators in \(D_0^{(i)}\times \mathbb {R},i=1,2,\ \Gamma _0\subset \partial D_0^{(1)}\cap \partial D_0^{(2)}\) and all conditions of Theorem 7.7 are satisfied.
Let \((t_{j1},t_{j2})\) be an interval as in Theorem 7.7 and \(\bigcup _{j=-\infty }^\infty (t_{j1},t_{j2})=\mathbb {R}\). We have \({\overline{D}}_0^{(1)}\times \mathbb {R}\subset \bigcup _{j=-\infty }^\infty {\overline{D}}_0^{(1)}\times [t_{j1},t_{j2}]\). It follows from Theorem 7.7 that for each \(j\in \mathbb {Z}\) there exists a diffeomorphism \(\Psi _j\) on \(D_j^{(1)}\times [t_{j1},t_{j2}]\) and a gauge transformation \(c_j\) such that \(\Psi _j=I\) and \(c_j=1\) on \({\overline{\Gamma }}_0\times [t_{j1},t_{j2}]\), and
In (7.26) \(\Psi _j\) is a diffeomorphism of \({\overline{D}}_0^{(1)}\times [t_{j1},t_{j2}]\) onto an almost cylindrical domain \({\overline{D}}_0^{(2)}\times \{S_j^-(x_1,\ldots ,x_n)\le x_0\le S_j^+(x_1,\ldots ,x_n)\}\), where \(x_0= S_j^\pm (x_1,\ldots ,x_n)\) are space-like surfaces, \(\Psi _j=I\) on \({\overline{\Gamma }}_0\times [t_{j1},t_{j2}],\ |c_j(x)|=1\) for all \(x\in {\overline{D}}_0^{(1)}\times [t_{j1},t_{j2}],\ c_j=1\) on \(\Gamma _0\times [t_{j1},t_{j2}]\).
We shall show that
on \({\overline{D}}_0^{(1)}\times [t_{j+1,1},t_{j2}]\) where \([t_{j+1,1},t_{j2}]\) is the intersection of \([t_{j1},t_{j2}]\) and \([t_{j+1,1},t_{j+1,2}]\).
Let \([t_{j1},t_{j+1,2}]=[t_{j1},t_{j2}]\cup [t_{j+1,1},t_{j+1,2}]\). Note that the proof of Theorem 7.7 consists of several steps, each step is dealing with the arbitrary large time interval. Consider, for example, Lemma 7.1.
The changes of variables (7.4), (7.5) and (7.7) are defined on arbitrary time interval \([t',t'']\). Thus they are defined on \([t_{j1},t_{j+1,2}]\). Therefore the maps (7.4), (7.5), (7.7) are defined on \([t_{j1},t_{j2}]\) and \([t_{j+1,1},t_{j+1,2}]\) and they coincide on \([t_{j+1,1},t_{j2}]=[t_{j1},t_{j2}]\cap [t_{j+1,1},t_{j+1,2}]\).
Also using the extension theorems of [16, §8], we can find the extension \({\tilde{\Phi }}_2\cap [t_{j1},t_{j+1,2}]\) of \(\Phi _2\cap [t_{j1},t_{j+1,2}]\) (cf. (7.8)) in two steps: First extend \({\tilde{\Phi }}_2\cap [t_{j1},t_{j+1,1}]\) and then extend continuously \(\Phi _2\cap [t_{j+1,1},t_{j+1,2}]\) to get a continuous extension of \(\Phi _2\cap [t_{j1},t_{j+1,2}]\). This way we show that (7.27) holds for some \(j\in \mathbb {Z}\). Therefore starting with \(\Psi _0\) on \([t_{01},t_{02}]\) we can construct \(\Psi _1\) on \([t_{11},t_{12}]\) such that \(\Psi _0=\Psi _1\) on \([t_{01},t_{02}]\cap [t_{11},t_{12}]\). Continuing this construction we get (7.27) for any \(j\in \mathbb {Z}\).
Let \(\Psi =\Psi _j, c=c_j\) on \([t_{j1},t_{j2}],\forall j\in \mathbb {Z}\). Then \(\Psi \) is a proper diffeomorphism of \({\overline{D}}_0^{(1)}\times \mathbb {R}\) onto \({\overline{D}}_0^{(2)}\times \mathbb {R}, \Psi =I\) on \(\Gamma \times \mathbb {R}\) and
\(\square \)
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Communicated by Ari Laptev.
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Eskin, G. Inverse problems for general second order hyperbolic equations with time-dependent coefficients. Bull. Math. Sci. 7, 247–307 (2017). https://doi.org/10.1007/s13373-017-0100-2
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DOI: https://doi.org/10.1007/s13373-017-0100-2