The Generalized Riemann Problem of Linear Conjugation for Non-Homogeneous Polyanalytic Equations of Order $n$ in $W_{n,p}(D)$

Abstract

We consider a non-homogeneous polyanalytic partial differential equation of order $n$ in a simply-connected domain $D$ with smooth boundary $\partial D$ in the complex plane $\mathbb{C}$. Initially we transform the given equation into an equivalent system of integro-differential equations and then find the general solution of the former in $W_{n,p}(D)$. Next we pose and prove the solvability of a generalized Riemann problem of linear conjugation to the differential equation. This is effected by first reducing the Riemann problem to a corresponding one for a polyanalytic function. The latter is solved by first transforming it into $n$ classical Riemann problems of linear conjugation for $n$ holomorphic functions expressed in terms of the analytic functions which define the polyanalytic function. The solution of the classical Riemann problem is available in the literature.