Schwarz-type problem of nonhomogeneous Cauchy–Riemann equation on a triangle

We consider the Schwarz-type boundary-value problem (BVP) of the nonhomogeneous Cauchy–Riemann equation on an isosceles orthogonal triangle. By the technique of plane parqueting and the Cauchy–Pompeiu formula on the triangle, the Schwarz–Poisson formula is obtained. We also investigate boundary behaviors of the Schwarz-type operator and the Pompeiu-type operator. Especially, boundary-values at the corners are proved to exist. Finally, the solution of the Schwarz-type BVP is explicitly obtained.