Complex boundary value problems on a circular triangle (Q6627087)

Let \(D\) be the triangle domain which is displayed as: \N\[\ND=\left\{z\in\mathbb{C}||z+a|<\sqrt2a,|z-\sqrt3ai|<\sqrt2a,|z-a|<\sqrt2a\right\}\,,\N\]\Nwhere \(a\) is the real number.\N\NThe boundary of \(D\) is denoted by \(\partial D\) and defined as: \N\[\N\partial D=\left\{z\in\overline{D}|z\in C_1,z\in C_2,z\in C_3\right\}\,,\N\]\Nwhere \(C_1=\{z\in\mathbb{C}||z+a|=\sqrt2a\}\), \(C_2=\{z\in\mathbb{C}||z-a|=\sqrt2a\}\), \(C_3=\{z\in\mathbb{C}||z-\sqrt3ai|=\sqrt2a\}\). Points \(ai\), \(\frac{\sqrt3}2a+\frac12ai\) and \(\frac{-\sqrt3}2a+\frac12ai\) are the intersection of \(C_1\), \(C_2\) and \(C_3\) that are the vertices of triangle.\N\NIn this paper, the explicit solutions and the solvability conditions for some basic boundary value problems in \(D\) are obtained. It is constructed the Cauchy-Schwarz representation formula. This formula is constructed by modifying the Cauchy-Pompeiu formula. This modification is achieved by placing the points gained from the parquetingreflection principle in the Cauchy-Pompeiu formula. By the Schwarz-Poisson formula the solution of the Schwarz BVP for the inhomogeneous Cauchy-Riemann equation is provided. The solution and the solvability condition of the Dirichlet and Neumann BVP are presented.