Uniqueness for the \(n\)-dimensional half space Dirichlet problem
type: Item
instance of: scholarly article
title: Uniqueness for the \(n\)-dimensional half space Dirichlet problem
zbMATH Open document ID: 0865.35038
author: David Siegel
author: E. O. Talvila
published in: Pacific Journal of Mathematics
publication date: 1997-06-15T00:00:00Z
review text: In \(\mathbb{R}^n\) we prove uniqueness for the Dirichlet problem in the half-space \(x_n> 0\), with continuous data, under the growth condition \(u= o(|x|\text{sec}^\gamma \theta)\) as \(|x|\to \infty\) \((x_n= |x|\cos \theta\), \(\gamma\in \mathbb{R})\). Under the natural integral condition for convergence of the Poisson integral with Dirichlet data, the Poisson integral will satisfy this growth condition with \(\gamma= n-1\). 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 \(u= o(|x|^{N+1} \text{ sec}^\gamma \theta)\) \((\gamma\in \mathbb{R}\), \(N\geq 1)\), we prove that solutions are unique up to the addition of a harmonic polynomial of degree \(N\) that vanishes when \(x_n =0\).
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 31B10
zbMATH DE Number: 950227
zbMATH Keywords: uniqueness
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: continuous data
zbMATH Keywords: Poisson integral
zbMATH Keywords: growth condition
zbMATH Keywords: Phragmén-Lindelöf principle
zbMATH Keywords: higher order growth
zbMATH Keywords: polynomial growth
MaRDI profile type: Publication
full work available at URL: pjm.1996.175.571
OpenAlex ID: W1506218469
DOI: 10.2140/PJM.1996.175.571
Recommended article: A type of uniqueness for the Dirichlet problem on a half-space with continuous data
Recommended article: Uniqueness of the solution of the Dirichlet problem for an \(n\)-dimensional pseudoparabolic equation
Recommended article: Q3726589
Recommended article: Uniqueness of solutions for some nonlinear Dirichlet problems
Recommended article: Uniqueness of solutions of nonlinear Dirichlet problems
Recommended article: Q4282719
Recommended article: Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems
Recommended article: Q4850816
Recommended article: Q3520089
Recommended article: Q4548129
description: scientific article; zbMATH DE number 950227