On the mappings of plane domains by solutions to second order elliptic equations

In this paper we investigate the sufficient conditions, under which the solution to second order partial differential equation will be one-to-one in a plane Jordan domain. It is proved that if a function maps a boundary of Jordan domain to rectifiable boundary of the other Jordan domain continuously, one-to-one and keeping an orientation and the Cauchy integral with this measure function is bounded by defined constant in the exterior domain, then the solution to Dirichlet problem with boundary function in this domain maps these domains one-to-one. In the proof of the main result we use integral representations of solutions to equation, particularly, properties of Fredholm type integral equations on the boundary of domain.