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

TL;DR

This paper proves that the Dirichlet problem for a slab with entire boundary data admits an entire harmonic solution and establishes the existence of entire harmonic solutions to certain difference equations, using a generalized Schwarz reflection principle.

Contribution

It introduces a method to find entire harmonic solutions for the Dirichlet problem in slabs and solves inhomogeneous difference equations with entire harmonic solutions, extending classical results.

Findings

Existence of entire harmonic solutions for the Dirichlet problem in slabs.

Solution of inhomogeneous difference equations with entire harmonic functions.

Application of a generalized Schwarz reflection principle.

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 $h$.