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The Dirichlet Problem for the Slab with Entire Data and a Difference Equation for Harmonic Functions

Published online by Cambridge University Press:  20 November 2018

Dmitry Khavinson ,
Erik Lundberg  and
Hermann Render
Dmitry Khavinson
Affiliation:
Dept. of Mathematics and Statistics, University of South Florida, Tampa, FL, USA e-mail: dkhavins@usf.edu
Erik Lundberg
Affiliation:
Dept. of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL, USA e-mail: elundber@fau.edu
Hermann Render
Affiliation:
School of Mathematics and Statistics, University College Dublin, Dublin, Ireland e-mail: hermann.render@ucd.ie
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Abstract

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It is shown that the Dirichlet problem for the slab $\left( a,\,b \right)\,\times \,{{\mathbb{R}}^{d}}$ with entire boundary data has an entire solution. The proof is based on a generalized Schwarz reflection principle. Moreover, it is shown that for a given entire harmonic function $g$, the inhomogeneous difference equation $h\left( t\,+\,1,\,y \right)\,-\,h\left( t,\,y \right)\,=\,g\left( t,\,y \right)$ has an entire harmonic solution $h$.

Keywords

31B2031B05reflection principleentire harmonic functionanalytic continuation
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 1 , 01 March 2017 , pp. 146 - 153
Copyright
Copyright © Canadian Mathematical Society 2017

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