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

  Published:2016-04-19
 Printed: Mar 2017
  • Dmitry Khavinson,
    Dept. of Mathematics and Statistics , University of South Florida, Tampa, FL, USA
  • Erik Lundberg,
    Dept. of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL, USA
  • Hermann Render,
    School of Mathematics and Statistics, University College Dublin, Dublin, Ireland
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Abstract

It is shown that the Dirichlet problem for the slab $(a,b) \times \mathbb{R}^{d}$ 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 $h ( t+1,y) -h (t,y) =g ( t,y)$ has an entire harmonic solution $h$.
Keywords: reflection principle, entire harmonic function, analytic continuation reflection principle, entire harmonic function, analytic continuation
MSC Classifications: 31B20, 31B05 show english descriptions Boundary value and inverse problems
Harmonic, subharmonic, superharmonic functions
31B20 - Boundary value and inverse problems
31B05 - Harmonic, subharmonic, superharmonic functions
 

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