On a boundary-value problem for the Poisson equation and the Cauchy–Riemann equation in a lens

In this paper, we consider the Dirichlet boundary-value problem for complex partial differential equations in a lens. With the help of the harmonic Green function, the Dirichlet boundary-value problem is solved explicitly for the Poisson equation in a lens. In particular, the boundary behaviors at corner points are considered. In addition, we study the explicit solvability of the Dirichlet boundary-value problem for the homogeneous Cauchy–Riemann equation in the lens.