Half Dirichlet problem for the Hölder continuous matrix functions in Hermitian Clifford analysis

Abstract

The simultaneous null solutions of the two complex Hermitian Dirac operators are focused on Hermitian Clifford analysis, where the Hermitian Cauchy integral was constructed in the framework of circulant (2 × 2) matrix functions. Under this setting, we will present the half Dirichlet problem with boundary spaces of Hölder continuous circulant (2 × 2) matrix functions on the sphere of even-dimensional Euclidean space. We will give the unique solution to it merely by using the Hermitian Cauchy transformation, get the solutions to the Dirichlet problem on the unit ball for Hölder continuous circulant (2 × 2) matrix functions as the boundaries and the solutions to the classical Dirichlet problem as the special case, and derive a decomposition of the Poisson kernel for matrix Laplace operator.