UDC 517.946

MATHEMATICS

M. M. LAVRENT'EV, V. G. ROMANOV

ON THREE LINEARIZED INVERSE PROBLEMS FOR HYPERBOLIC EQUATIONS

(Presented by Academician S. L. Sobolev on 19 III 1966)

1. Consider the equation

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

where (n) is a function of the variables (x,y); (\Delta) is the Laplace operator with respect to these same variables. Let (u(x,x_0,y,t)) be a generalized solution of equation (1) in the half-plane (y>0), satisfying the following boundary and initial conditions:

[
u(x,x_0,y,0)=\frac{\partial}{\partial t}u(x,x_0,y,0)=0,
]

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

where (\delta(t)) is the Dirac delta function. Further, let (\tau(x_1,x_2)) be the largest of the numbers (\tau) such that the support of the function (u(x_1,x_0,0,t)), for given (x_1,x_0), is contained in the half-line*

[
\tau \leqslant t < \infty .
]

For equation (1) we pose the following inverse problem:

Problem 1. It is required, from the function (\tau(x_1,x_0)), to determine the function (n(x,y)).

In the case where the function (n) does not depend on the variable (x), the formulated inverse problem has been studied in the works ((^{1,2})).

The function (\tau(x_1,x_0)) is the minimum of the functional

[
I(\Gamma)=\int_{\Gamma} n(x,y)\,ds,
]

where (\Gamma) is a curve joining the points ((x_1,0)), ((x_0,0)) and lying in the upper half-plane.

Let now

[
n(x,y)=n_0(x,y)+n_1(x,y),
]

where the function (n_0(x,y)) is given, and the function (n_1(x,y)) is small. Correspondingly, the function (\tau(x_1,x_0)) is represented in the form of the sum

[
\tau(x_1,x_0)=\tau_0(x_1,x_0)+\tau_1(x_1,x_0).
]

Here the function (\tau_0(x_1,x_0)) corresponds to the function (n_0(x,y)).

Neglecting quantities of order (n_1^2(x,y)), we obtain that the function (\tau_1(x_1,x_0)) is represented in the form

[
\tau_1(x_1,x_0)=\int_{\Gamma(x_1,x_0)} n_1(x,y)\,ds,
\tag{3}
]

[
\text{* In the process described by equation (1), } \tau(x_1,x_0) \text{ is the time in which a disturbance produced at the point } (x_0,0) \text{ reaches the point } (x_1,0).
]

where (\Gamma(x_1,x_0)) is the curve joining the points (x_1, x_0), for which the minimum of the functional is attained

[
I(\Gamma)=\int_{\Gamma} n_0(x,y)\,ds.
]

We investigate the posed linearized problem for one special case of the function (n_0(x,y)). Let

[
n_0(x,y)=(ay+b)^{-1},
]

where (a>0,\ b\geqslant 0).

In this case (see (3)) the curve (\Gamma(x_1,x_0)) is an arc of a circle with center at the point (P((x_1+x_0)/2,-b/a)). Let us extend the functions (n_0(x,y)) and (n_1(x,y)) into the lower half-plane in an even manner. In this case the posed problem reduces to the problem of determining the function (n_1(x,y)) from the equation

[
\tau_1(x-r,x+r)=\frac{r}{2}\int_{0}^{2\pi} n_1(x+r\cos\varphi,r\sin\varphi)\,d\varphi .
\tag{4}
]

2. Denote by (u(M,M_0,t)) the generalized solution of the equation

[
\partial^2 u/\partial t^2=\Delta u+a(M)u+\delta(M-M_0)\delta(t)
\tag{5}
]

in the domain (z>0) under the following initial and boundary conditions:

[
u(M,M_0,0)=\frac{\partial}{\partial t}u(M,M_0,0)=0,
\tag{6}
]

[
\frac{\partial}{\partial z}u(M_1,M_0,t)=0.
\tag{7}
]

Here (M(x,y,z)), (M_0(x_0,y_0,0)), (M_1(x,y,0)) are points of three-dimensional space; (\Delta u) is the Laplace operator with respect to the variables (x,y,z).

Assuming that the function (a(M)) is small, consider the problem of reconstructing it from the solution of the problem (5), (6), (7) known in the plane (z=0).

It is convenient to reduce the posed problem to an equivalent problem for the whole space, extending the function (a(M)) into the half-space (z<0) in an even manner. Represent the solution (u(M,M_0,t)) in the form

[
u(M,M_0,t)=u_0(M,M_0,t)+u_1(M,M_0,t),
]

where the function (u_0(M,M_0,t)) satisfies the wave equation

[
\partial^2 u/\partial t^2=\Delta u+\delta(M-M_0)\delta t
]

and the conditions (6), (7). As is known, the function (u_0(M,M_0,t)) has the form

[
u_0(M,M_0,t)=\delta(t-r(M,M_0))/2\pi r(M,M_0).
\tag{8}
]

In this formula (r(M,M_0)) is the distance between the points (M) and (M_0).

For the function (u_1(M,M_0,t)) we obtain, neglecting the term of order (a(M)u_1(M,M_0,t)), the linearized equation

[
\partial^2 u_1/\partial t^2=\Delta u_1+a(M)u_0(M,M_0,t)
\tag{9}
]

and zero initial data. Taking formula (8) into account, we find that the function (u_1(M,M_0,t)) has the form

[
u_1(M,M_0,t)=
-\frac{1}{8\pi^2\sqrt{t^2-r^2(M,M_0)}}
\iint_{S}
\frac{a(P)\,ds}{\sqrt{r(P,M_0)\,r(P,M)}} ,
\tag{10}
]

where (P(\xi,\eta,\zeta)) is the variable point of integration, (S) is the surface of the ellipsoid

[
r(P,M_0)+r(P,M)=t.
]

From formula (10) follow two possible formulations of the problem of reconstructing the function (a(M)).

Problem II. The point (M_0) is fixed, and the function
(u_1(M_1, M_0, t)=\varphi(M_1, M_0, t)) is known. From formula (10) it follows in this case that the integral equation of the first kind is

[
\frac{1}{8\pi^2}\iint_S \frac{a(P)\,ds}{\sqrt{r(P,M_1)\,r(P,M_0)}}=
-\sqrt{t^2-r^2(M_1,M_0)}\,\varphi(M_1,M_0,t).
\tag{11}
]

Problem III. The point (M_0) is a variable point, and the function
(u_1(M_0,M_0,t)=\psi(M_0,t)) is known. In this case, letting the point (M) tend to the point (M_0) in formula (10), we obtain the equation

[
\frac{1}{4\pi^2 t^2}\iint_S a(P)\,ds=-\psi(M_0,t),
\tag{12}
]

in which the surface of integration is a sphere of radius (t/2) with center at the point (M_0). Consequently, we arrive at the problem of reconstructing a function even in (z) from the known values of its integrals over arbitrary spheres whose centers lie in the plane (z=0).

1. The uniqueness of the solution of Problems I and III follows directly from the results contained in the book of R. Courant ((^4)).

The uniqueness of the solution of Problem II can be proved by a method analogous to that contained in this book.

Problems I, II, and III are ill-posed in the sense of Hadamard, but well-posed in the sense of A. N. Tikhonov.

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

Received
12 III 1966

CITED LITERATURE

(^1) G. Herglotz, Zs. Math. u. Phys., 52, 275 (1905).
(^2) E. Wiechert, Nachr. Gesellschaft Wiss. Göttingen, 415 (1907).
(^3) V. I. Smirnov, A Course of Higher Mathematics, 4, Moscow, 1957.
(^4) R. Courant, Partial Differential Equations, “Mir,” 1964.