Boundary Value Problems for Inhomogeneous Polyanalytic Equations in a Triangle

In this paper, we investigate Dirichlet and Schwarz type boundary value problems for both the inhomogeneous Cauchy-Riemann equation and higher-order polyanalytic equations in a nonstandard domain, specifically a triangular region formed by the intersection of three circular disks in the complex plane. Such domains introduce additional geometric complexity, which requires careful analytical treatment. By constructing appropriate kernel functions tailored to the geometry of the domain, we develop integral operator techniques that allow us to derive explicit solution formulas for the given boundary conditions. In addition, we establish necessary and sufficient conditions for the solvability of these problems, depending on the compatibility of boundary data and the properties of the inhomogeneous terms. Our approach generalizes classical methods used for standard domains, extending their applicability to more intricate geometric settings. The results presented in this work contribute to the broader theory of boundary value problems for complex partial differential equations and offer new tools for addressing similar problems in applied mathematical physics and complex analysis