The Carleman formula for the Helmholtz equation on the plane

We consider the Cauchy problem for the Helmholtz equation in an arbitrary bounded planar domain with Cauchy data only on part of the boundary of the domain. We derive a Carleman-type formula for a solution to this problem and give a conditional stability estimate.     

Ill-posed problems of number theory and tomography

In honor of the 60th Birthday of Academician M. M. Lavrent'ev.     

Uniqueness in One Inverse Problem for the Elasticity System

We consider an inverse problem for the stationary elasticity system with constant Lame coefficients and a variable matrix coefficient depending on the spatial variables and frequency. The right-hand side contains a delta-function whose support (source) varies in some domain disjoint from the support of the variable coefficient. The inverse problem is to find the coefficient from the scattered wave measured at the same point at which the perturbation originates. A uniqueness theorem is proven. The proof bases on reduction of the inverse problem to a family of equations with the M. Riesz potential.