Schwarz and Dirichlet problems for $ \bar{\partial} $-equation in a triangular domain

In this paper, we investigate some boundary value problems for the Cauchy–Riemann equations in the triangular domain $ M $. By the parqueting-reflection method, we obtain a covering of the complex plane through reflections of the domain. Then by the Cauchy–Pompeiu formula, we derive the Cauchy–Schwarz representation formula in $ M $. In addition, we consider the Schwarz boundary value problem for the inhomogeneous Cauchy–Riemann equation and the explicit expressions of solution and condition of solvability. Finally, by using the Schwarz boundary value problem, the solution of the Dirichlet boundary value problem is explicitly solved.