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Zeitschrift für Analysis und ihre Anwendungen


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Volume 18, Issue 3, 1999, pp. 611–624
DOI: 10.4171/ZAA/901

Published online: 1999-09-30

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

Ali Seif Mshimba[1]

(1) University of Dar es Salaam, Tanzania

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.

Keywords: Polyanalytic functions, generalized Cauchy-Pompeiu integral operators of higher order, Riemann-Hilbert problem

Mshimba Ali Seif: The Generalized Riemann-Hilbert Boundary Value Problem for Non-Homogeneous Polyanalytic Differential Equation of Order $n$ in the Sobolev Space $W_{n,p}(D)$. Z. Anal. Anwend. 18 (1999), 611-624. doi: 10.4171/ZAA/901