Some harmonic green’s functions with mixed Dirichlet and Neumann boundary conditions for a ring, a half ring and a quarter ring in the plane

In this work we show the construction of a mixed harmonic Green Dirichlet-Neumann function for a ring, half-ring and quarter-ring in the complex plane. The construction is carried out with the parqueting-reflection method. With these mixed Green’s functions it is possible to obtain integral representation formulas for the solution of Poisson equation with mixed boundary conditions of the type Dirichlet-Neumann on different parts of the boundary of the domain, which is of particular interest for many related problems that arise in mathematical physics and engineering.