Dirichlet Boundary Value Problems for Cauchy–Riemann and Polyanalytic Equations in a Half-Ring Domain

Abstract

In this work, we study the Dirichlet boundary value problem for several classes of
complex partial dierential equations in a half-ring domain. We begin with the homogeneous
Cauchy-Riemann equation and derive an explicit integral representation that satises the given
boundary conditions. The approach is then extended to the inhomogeneous Cauchy-Riemann
equation, where the presence of a nonhomogeneous term necessitates modications in the
solution technique. Finally, we investigate the Dirichlet problem for a polyanalytic equation of
nth order. By employing advanced tools from complex analysis, we construct solution formulas
that account for the higher-order structure of the equation and ensure the boundary conditions
are met.