The Carleman formula and recovering the refractive index of the two-dimensional Helmholtz equation

Abstract

We consider the Cauchy problem for the Helmholtz equation with given bounded refrac- tive index in an arbitrary 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. From another side we obtain the inversion formula for smooth unknown refractive index using the Cauchy date set on the boundary. Based of this inversion formula we numerically reconstruct the refractive index on the unit disk.