Three boundary value problems for complex partial differential equations in the lens domain

In this paper, we investigate some boundary value problems for the Cauchy–Riemann equations in the lens domain $M$. We apply the parqueting-reflection method for the domain to achieve the points of the complex plane. Then the Schwarz representation formula is constructed by the Cauchy–Pompeiu formula and an explicit solution for the Schwarz boundary value problem for the inhomogeneous Cauchy–Riemann equation on the domain is presented. We also discuss about the condition of solvability and by using the Schwarz boundary value problem, the homogeneous Neumann and the inhomogeneous Dirichlet boundary value problems are investigated.