Neumann and Dirichlet Problems for the Cauchy–Riemann and the Poisson Equations in the Partial Eclipse Domain

In this paper, we consider the Neumann boundary value problem and the Dirichlet boundary value problem for complex partial differential equations in the partial eclipse domain. First, By the parqueting–reflection principle and the Cauchy–Pompeiu formula, a modified integral representation formula in the partial eclipse domain is constructed. Then, we explicitly solve the Neumann problem for the homogeneous equation and discuss the solvability conditions. Moreover, we investigate the Dirichlet problem for the Poisson equation in the partial eclipse domain. In other words, with the help of the Green’s function, we provide a unique solution for the Dirichlet boundary value problem for the Poisson equation and consider boundary behavior.