Volume 14, Issue 8

The system under consideration is governed by the equation u tt = ∇ div u – rot µ rot u in Ω × (0, T ); its response operator ("input output" map) R T plays the role of the inverse data. As in the case of the proper Lamé system, such a model describes a dynamical system with two wave modes ( p -waves and s -waves) propagating with different velocities c p = and c s = correspondingly. We show that R 2T determines and , where and are the subdomains of Ω filled (at the moment T ) with p - and s -waves propagating from ∂Ω. Due to the wave splitting u = ∇ p + rot s the problem is reduced to the inverse problems for the acoustical and Maxwell subsystems governed by the equations p tt = Δ p and s tt = − µ rot rot s with the response operators and determined by R 2T . The first problem can be solved by the BC method (Belishev, 1986), the second one is solved by a version of the method based on a blow up effect. This version is the main subject of the paper. In addition, we derive the inequality, which can be used for approximate determination of the shape of .

The problem of determining source term in a semilinear wave equation is considered. The source term is represented as the product ƒ( u ( x, t )) p ( x ), where ƒ( s ) is a given function, u ( x, t ) is a solution to Cauchy problem for wave equation, p ( x ) is an unknown function. To determine p ( x ) the additional information on the solution of the Cauchy problem u ( α ( t ), t ) = g ( t ), u ( β ( t ), t ) = h ( t ) is used. Theorems of existence and uniqueness of solution to an inverse problem in the class of continuous functions p ( x ) and in the class of functions p ( x ) = p o + xq ( x ), where q ( x ) is continuous, are proved.

Akduman, Haddar and Kress [1, 12, 15] proposed a conformal mapping approach for solving inverse boundary value problems for the two-dimensional Laplace equation in a doubly connected domain D with interior boundary curve Γ 0 and exterior boundary curve Γ 1 . The inverse problem consists of reconstructing the interior boundary Γ 0 from the Cauchy data on Γ 1 of a harmonic function satisfying a homogeneous Dirichlet or Neumann boundary condition on the unknown interior boundary curve Γ 0 . The reconstruction method consists of two parts: In the first step, by successive approximations a nonlocal and nonlinear ordinary differential equation is solved to determine the boundary values of a holomorphic function Ψ on the outer boundary circle C 1 of an annulus B . Then in the second step via regularizing a Laurent expansion in the sense of Tikhonov an ill-posed Cauchy problem is solved to determine Ψ in the annulus and the unknown Γ 0 as the image Ψ( C 0 ) of the interior boundary circle C 0 of B . The present paper extends this approach to the case of a homogeneous impedance boundary condition. The analysis and the numerical implementation of the method differ from the limiting cases of the Dirichlet and Neumann conditions since the impedance problem in the annulus B that is associated with the impedance problem in the original domain D depends on the conformal map Ψ.