SCHWARZ AND DIRICHLET PROBLEMS FOR COMPLEX PARTIAL DIFFERENTIAL EQUATIONS IN THE PARTIAL ECLIPSE DOMAIN

In this article, we consider Schwarz boundary value problem and Dirichlet boundary value problem for the Cauchy–Riemann equations in the partial eclipse domain M. We apply the parqueting–reflection method for the domain to achieve the points of the complex plane. Then the Cauchy–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 and the inhomogeneous Dirichlet boundary value problems are investigated.