Uniqueness for the n-dimensional half space Dirichlet problem

David Siegel and Erik O. Talvila

Vol. 175 (1996), No. 2, 571–587

DOI: 10.2140/pjm.1996.175.571

Abstract

In ℝⁿ, we prove uniqueness for the Dirichlet problem in the half space x_n > 0, with continuous data, under the growth condition u = o(|x|sec^γ𝜃) as |x|→∞ (x_n = |x|cos𝜃, γ ∈ ℝ). Under the natural integral condition for convergence of the Poisson integral with Dirichlet data, the Poisson integral will satisfy this growth condition with γ = n − 1. A Phragmén-Lindelöf principle is established under this same growth condition. We also consider the Dirichlet problem with data of higher order growth, including polynomial growth. In this case, if u = o(|x|^N+1 sec^γ𝜃) (γ ∈ ℝ, N ≥ 1), we prove solutions are unique up to the addition of a harmonic polynomial of degree N that vanishes when x_n = 0.

Mathematical Subject Classification 2000

Primary: 35J05

Secondary: 31B35

Milestones

Received: 9 May 1994

Published: 1 October 1996