Zeitschrift für Analysis und ihre Anwendungen 24 (2005), No. 2, 341--351
Copyright Heldermann Verlag 2005
The Schwarz Problem for Polyanalytic Functions
Heinrich Begehr
I. Mathematisches Institut, Freie Universität, Arnimallee 3, 14195 Berlin, Germany
begehr@math.fu-berlin.de
Dieter Schmersau
I. Mathematisches Institut, Freie Universität, Arnimallee 3, 14195 Berlin, Germany
The analytic Schwarz kernel function the real part of which is the harmonic Poisson kernel was generalized to a polyanalytic Schwarz kernel in a joint work with G. N. Hile [Rocky Mountain J. Math. 27 (1997) 669--706], see also H. Begehr [Complex Analytic Methods for Partial Differential Equations. An Introductory Text, Singapore: World Scientific 1994]. It is here used to give some higher order Cauchy-Schwarz-Pompeiu representations and to solve the Schwarz problem for the Poisson, the inhomogeneous Bitsadze and the inhomogeneous polyanalytic equation.
Keywords: Schwarz kernel, Poisson equation, Bitsadze equation, polyanalytic equation, Schwarz boundary value problem.
MSC: 30E25, 30G20, 31A30, 35J40
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