A Representation Formula for the Robin Function
Marcus Stiemer · Complex Variables Theory and Application An International Journal · 2003
The Robin function R is the fundamental solution of a mixed boundary value problem in the complex plane, where Dirichlet conditions are posed on one part of the boundary and Neumann conditions on the remaining part. We prove a representation formula for R in a multiply connected domain under rather weak assumptions on the structure of the Dirichlet and the Neumann boundary. The key to this formula is to relate the Robin function to the Green function of a certain Riemann surface. This surface is embedded in a compact surface that arises from the action of a classical Schottky group in its domain of discontinuity.