Schwarz boundary value problems for polyanalytic equation in a sector ring

TL;DR

This paper develops explicit solutions for Schwarz boundary value problems in sector rings for polyanalytic equations using modified Schwarz-Pompeiu formulas and integral operators.

Contribution

It introduces a modified Schwarz-Pompeiu formula in sector rings and applies it to solve boundary problems for polyanalytic equations.

Findings

Explicit solutions for Schwarz problems in sector rings.

Introduction of new integral operators with specific properties.

Application to inhomogeneous and generalized polyanalytic equations.

Abstract

In this article,we first give a modified Schwarz-Pompeiu formula in a general sector ring by proper conformal mappings, and obtain the solution of the Schwarz problem for the Cauchy-Riemann equation in explicit forms. Furthermore, a class of integral operators is introduced together with their properties. Finally, by virtue of these operators, Schwarz problems for a inhomogeneous polyanalytic equation and for a generalized polyanalytic equation are investigated, respectively.