The Dirichlet problem for the slab with entire data and a difference equation for harmonic functions

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 $h\left(t+1,y\right)-h\left(t,y\right)=g\left(t,y\right)$ has an entire harmonic solution $h$.

8 pages