Some inverse problems with a ‘partial’ point source

Abstract

Let x ∊ Rk and y ∊ Rn. Let L(x, Dx) and L(y, Dy) be the elliptic operators of the second order in Rk and Rn respectively, and L(x, y, Dx, Dy) = L(x, Dx) + L(y, Dy). The forward problem is the Cauchy problem for either the parabolic equation ut = Lu with the initial data u(x, y, 0) = δ(x)f(y), i.e. with the ‘partial’ δ-function or for the hyperbolic equation utt = Lu. The inverse problem consists in the recovery of an unknown y-dependent coefficient given the data depending on n variables. An essentially new element here is the presence of the δ(x)-function in the initial condition ‘at the expense’ of the assumption that the unknown coefficient is independent of x. It is shown that the global uniqueness can be proven using a combination of an incomplete separation of variables in the parabolic case, an analogue of the Laplace transform connecting solutions of hyperbolic and parabolic Cauchy problems and the method of Carleman estimates. A convolution theorem is obtained for that analogue of the Laplace transform. It enables one to prove both global uniqueness theorems and stability estimates for the hyperbolic case with a finite time interval.