M. M. LAVRENT'EV

ON AN INVERSE PROBLEM FOR THE WAVE EQUATION

(Presented by Academician N. N. Bogolyubov, 3 III 1964)

In the present note a theorem is proved on the uniqueness of the solution of an inverse problem for the wave equation.

Consider the equation

\[ n^2 \frac{\partial^2 u}{\partial t^2}=\Delta u, \tag{1} \]

where \(u\) is a function of the three variables \(x,y,t\); \(n\) is a function of the variables \(x,y\).

We shall consider the following formulation:

In the plane \(x,y\) a domain \(D_0\) is given. The function \(n(x,y)>0\) is continuous and is identically equal to one outside \(D_0\). In addition, in some domain \(D_1\), \(D_1\cap D_0\) empty, a family \(G\) of solutions of (1) is given for all \(t>0\). It is required to determine the function \(n(x,y)\) inside \(D\).

We note that an analogous formulation in the case of one variable was considered in [1]; close in character to the formulation considered here is the widely known inverse Sturm—Liouville problem.

Let us formulate the uniqueness theorem for the posed problem for one family \(G\).

Theorem. Let \(D_0\) and \(D_1\) be bounded simply connected domains, let \(D_2\) also be some bounded simply connected domain not intersecting \(D_0,D_1\), and let \(G\) be the set of solutions of (1) satisfying the following initial conditions:

\[ u(x,y,0)=0, \]

\[ \frac{\partial u(x,y,0)}{\partial t}=\delta(x-x_0,\;y-y_0), \tag{2} \]

where \(Q(x_0,y_0)\) is an arbitrary point of \(D_2\).

Then the solution of the formulated inverse problem is unique, i.e., the function \(n(x,y)\) is determined uniquely inside \(D_0\).

We shall present the brief content of the proof of the theorem. Denote the solution of the Cauchy problem (2) for equation (1) by \(u(x,y,x_0,y_0,t)\) and consider the function

\[ v(x,y,x_0,y_0,\lambda)=\int_0^\infty u(x,y,x_0,y_0,t)\cos \lambda t\,dt. \]

It is easy to see:

\[ \Delta v=-\delta(x-x_0,\;y-y_0)-\lambda^2 n^2 v. \tag{3} \]

The function \(v\) is the fundamental solution of the Helmholtz equation

\[ \Delta v=-\lambda^2 n^2 v \]

with a singularity at the point \(Q(x_0,y_0)\). As is known, the fundamental solutions of elliptic equations with analytic coefficients are analytic functions both of the independent variables \(x,y\) and of the coordinates of the singularity \(x_0,y_0\) everywhere outside the singularity.

By the conditions of the theorem the function \(v\) is given in the domain \(D_{12}\) of the four-dimensional space \(R(x,y,x_0,y_0)\), which is the direct product of the domains \(D_1,D_2\) of the spaces \(P(x,y),Q(x_0,y_0)\) (\(R\in D_{12}\) if \(P\in D_1,\ Q\in D_2\)). The function \(n(x,y)\) is identically equal to one outside \(D_0\). Consequently, by virtue of the uniqueness of analytic continuation, the function \(v\) may be considered given everywhere outside the domain \(D_{00}\)—the product of \(D_0\) by itself (\(R\in D_{00}\), if \(P\in D_0;\ Q\in D_0\)).

Let \(D_3\) be some bounded domain containing the domains \(D_0,D_1,D_2\), and let

\[ r^2=(x-\xi)^2+(y-\eta)^2,\qquad r_0^2=(x_0-\xi)^2+(y_0-\eta)^2. \]

It follows from (3) that for \(P \in D_3\):

\[ v(x,y,x_0,y_0,\lambda)=\frac{1}{2\pi}\ln\bigl[(x-x_0)^2+(y-y_0)^2\bigr] \]
\[ -\frac{1}{2\pi}\lambda n^2(x,y)\int_{D_0}v(\xi,\eta,x_0,y_0,\lambda)\ln r\,d\xi\,d\eta +\widetilde v(x,y,x_0,y_0,\lambda), \tag{4} \]
\[ \widetilde v=\frac{1}{2\pi}\int_{\Gamma_3} \left(v\frac{\partial}{\partial n}\ln r-\frac{\partial v}{\partial n}\ln r\right)\,ds, \]

where \(\Gamma_3\) is the boundary of \(D_3\).

Denote by \(v_1(x,y,x_0,y_0)\) the function

\[ v_1=\frac{\partial}{\partial\lambda} \left[ \frac{\partial^2 v(x,y,x_0,y_0,0)}{\partial x\,\partial x_0} + \frac{\partial^2 v(x,y,x_0,y_0,0)}{\partial y\,\partial y_0} \right]. \]

It is not difficult to show that, for \(R\in D_{33}\) (\(D_{33}\) is the product of \(D_3\) by itself), the function \(v_1\) is equal to

\[ v_1=\int_{D_0} n(\xi,\eta) \frac{(x-\xi)(x_0-\xi)+(y-\eta)(y_0-\eta)}{r^2 r_0^2} \,d\xi\,d\eta+\widetilde v_1, \tag{5} \]

where

\[ \widetilde v_1(x,y,x_0,y_0)=\frac{\partial}{\partial\lambda} \left[ \frac{\partial^2 \widetilde v}{\partial x\,\partial x_0} + \frac{\partial^2 \widetilde v}{\partial y\,\partial y_0} \right]. \]

In view of (5), the functions \(v_1,\widetilde v_1\) for \(R\in D_{33}\), \(R\notin D_{00}\) (\(D_{33}\) is the product of \(D_3\) by itself) are analytic functions of the variables \(x,y,x_0,y_0\).

Putting in (5)

\[ x=x_0,\qquad y=y_0;\qquad P(x,y)\in D_1, \]

we obtain

\[ v_2=v_1(x,y,x,y)-\widetilde v_1(x,y,x,y) =\int_{D_0} n(\xi,\eta)\frac{1}{r^2}\,d\xi\,d\eta. \tag{6} \]

Let us now consider the function

\[ w(x,y,x_1,y_1)=\int_{D_0} n(\xi,\eta)\frac{1}{\rho^2}\,d\xi\,d\eta, \]

where

\[ \rho^2=(x-\xi)^2+(y-\eta)^2+x_1^2+y_1^2. \]

The function \(w\) is the potential of a simple layer in the four-dimensional space \(R_1(x,y,x_1,y_1)\) with density \(n(x,y)\), distributed on the two-dimensional manifold:

\[ P(x,y)\in D_0;\qquad x_1=y_1=0 \qquad (D_0'). \]

In view of the foregoing, the function \(w\) may be regarded as given on the two-dimensional manifold:

\[ P(x,y)\in D_1;\qquad x_1=y_1=0 \qquad (D_1^1) \]

It can be shown that the function \(w\) is uniquely determined in the entire space \(R_1\) by its values in \(D_1^1\), whence, by the known theorems of potential theory, the assertion of the theorem follows.

We note that the theorem generalizes to the case of the wave equation in a space of any number of dimensions, to the heat equation, and also to certain equations of hyperbolic and elliptic type of higher orders.

Computing Center
of the Siberian Branch of the Academy of Sciences of the USSR

Received
12 II 1964

CITED LITERATURE

1. M. G. Krein, Dokl. Akad. Nauk SSSR, 82, 669 (1959).