On mappings of plane domains by solutions of second-order elliptic equations (Q1792508)

In the interesting paper under review, the author obtains sufficient conditions for one-to-one solvability of the second-order partial differential equation \[ \overline{\partial}\partial_\beta u=0 \] in a plane Jordan domain. Here \(\overline{\partial}\) is the Cauchy-Riemann operator, while \(\partial_\beta=\frac{\partial}{\partial x}+i\beta\frac{\partial}{\partial y}\) with \(\beta\in (-1,0).\) The main result, regarding a continuous one-to-one and orientation-keeping map of the boundary of a Jordan domain to the rectifiable boundary of some other Jordan domain, asserts the following: If the Cauchy integral with a measure generated by this map is bounded by some constant in the exterior domain, then the solution to the corresponding Dirichlet problem in the domain with this boundary function maps these domains in an one-to-one way. The proof relies on integral representations of the solutions and on Fredholm properties of the integral equations on the boundary.