International Conference «Inverse and Ill-Posed Problems of Mathematical Physics»,
dedicated to Professor M.M. Lavrent’ev in occasion of his 75-th birthday, August 20-25, 2007, Novosibirsk,
Russia
An inverse problem for elliptic equation with an unknown coefficient in a main term
A. Sh. Lyubanova
Siberian Federal University
Krasnoyarskii Rabochii str., 95, Krasnoyarsk, Russia, 660025
E-mail: lubanova@mail.ru
This work is devoted to the identification of the unknown coefficient k in equation
kMu + g(x)u = f(x),
x  ,
under the boundary condition
u   (x)
and the condition of overdetermination
k
u
 N u ds  .
(1)
(2)
(3)

Here   R is a bounded domain with the boundary   C2; M is a linear selfajoint elliptic
differential operator of the second order:
M  div (M ( x))  m( x) I ,
M(x) is a matrix of functions mij(x), i,j = 1,2, …, n, m(x) is a scalar function, I is the identical
operator;  N is the conormal derivative. The functions f(x), g(x), (x) and constant  are given.
Various physical phenomena lead to a study of problem (1)-(3). Such problems occur in
studies of the inverse problems for evolution equations.
By the solution of problem (1)-(3) is meant a pair involving a function u( x) W22 () and
positive real constant k which satisfy (1)-(3). The existence of a unique solution to problem (1)-(3)
is established by the following theorem.
THEOREM 1. Let
(i)
f ( x)  L2 (), g ( x)  C (), ( x)  W23 2 ();
(ii)
f(x), (x) are nonnegative in ;
0  g(x)  g1 = const < + in  ;
a  (M( x)a, a) R dx  (m( x)a, a) 0  0,
n


where a = a(x) is the solution to problem
Ma  0, x  ,
a   ( x);
F ( x)  g ( x)a  f  0;
    ( f , a)  g1 a
2
0
 0.
Then problem (1)-(3) has a unique solution (u(x), k). Moreover,
0  u ( x)  a ( x)
for almost all x   .