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

Khavinson, Dmitry ; Lundberg, Erik ; Render, Hermann

Canadian mathematical bulletin, Tome 60 (2017) no. 1, pp. 146-153

Résumé

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 $h$ .

Détail

DOI : 10.4153/CMB-2016-018-x

Mots-clés : 31B20, 31B05, reflection principle, entire harmonic function, analytic continuation

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