Uniqueness for the n-dimensional half space Dirichlet problem

Abstract

In ℝn, we prove uniqueness for the Dirichlet problem in the half space cursive Greek chin > 0, with continuous data, under the growth condition u = o(|cursive Greek chi| secγ θ) as |cursive Greek chi| → ∞ (cursive Greek chin = |cursive Greek chi| 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 PhragménLindelöf 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(|cursive Greek chi|N+1secγθ) (γ ∈ ℝ, N ≥ 1), we prove solutions are unique up to the addition of a harmonic polynomial of degree N that vanishes when cursive Greek chin = 0.