Zeitschrift für Analysis und ihre Anwendungen Vol. 18, No. 3, pp. 611624 (1999) 

The Generalized RiemannHilbert Boundary Value Problem for NonHomogeneous Polyanalytic Differential Equation of Order $n$ in the Sobolev Space $W_{n,p}(D)$Ali Seif MshimbaUniversity of Dar es Salaam, Department of Mathematics, P.O. Box 35062, Dar es Salaam, Tanzania; \ email amshimba@cs.udsm.ac.tzAbstract: Given is a nonlinear nonhomogeneous polyanalytic differential equation of order $n$ in a simplyconnected 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 integrodifferential equations. Next we prove the solvability of a generalized RiemannHilbert 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 RiemannHilbert problems for holomorphic functions, whose solutions are known in the literature. Keywords: polyanalytic functions, generalized CauchyPompeiu integral operators of higher order, RiemannHilbert problem Classification (MSC2000): 30G30, 35J40, 47G10 Full text of the article:
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