The Generalized Riemann-Hilbert Boundary Value Problem for Non-Homogeneous Polyanalytic Differential Equation of Order $n$ in the Sobolev Space $W_{n,p}(D)$

Abstract

Given is a nonlinear non-homogeneous polyanalytic differential equation of order $n$ in a simply-connected domain $D$ in the complex plane. Initially we prove (under certain conditions) the existence of its general solution in $W_{n,p}(D)$ by first transforming it into a system of integro-differential equations. Next we prove the solvability of a generalized Riemann-Hilbert problem for the differential equation. This is effected by first reducing the boundary value problem posed to a corresponding one for a polyanalytic function. The latter is then transformed into $n$ classical Riemann-Hilbert problems for holomorphic functions, whose solutions are known in the literature.