A type of uniqueness for the Dirichlet problem on a half-space with continuous data
type: Item
instance of: scholarly article
title: A type of uniqueness for the Dirichlet problem on a half-space with continuous data
zbMATH Open document ID: 0853.31004
author: Hidenobu Yoshida
published in: Pacific Journal of Mathematics
publication date: 1997-01-09T00:00:00Z
review text: Let \(T= \{(X, y): X\in \mathbb{R}^{n-1}\) and \(y>0\}\), where \(n\geq 2\). If \(f: \mathbb{R}^{n-1}\to \mathbb{R}\) is continuous and satisfies \(\int|f(X) |(1+|X|)^{-n} dX< +\infty\), then the Poisson integral of \(f\) yields a harmonic function \(h\) on \(T\) with boundary values \(f\) on \(\partial T\equiv \mathbb{R}^{n-1}\). It is known that, for functions \(f\) which fail to satisfy this integrability condition, the corresponding Dirichlet problem can be solved by suitably modifying the Poisson kernel. Although these solutions are certainly not unique, the paper under review presents a type of uniqueness theorem which answers a question posed by D. Siegel. To be more precise, suppose that \(\int|f(X) |(1+ |X|)^{- n-l} dX< +\infty\) for some \(l\) in \(\mathbb{N}\). Then there exists a harmonic function \(h\) on \(T\) with boundary values \(f\) such that \(r^{- l-2} {\mathcal D} (|h|, r)\to 0\) as \(r\to +\infty\), where \({\mathcal D} (|h|,r)\) denotes the mean value of \(y|h(x, y)|\) with respect to surface area measure on the hemisphere \(\{(X, y)\in T:|(X, y)|=r\}\). Further, all other solutions satisfying this growth condition are of the form \(h(X, y)+ y\Pi (X, y)\), where \(\Pi\) is a polynomial in \(\mathbb{R}^n\) of degree at most \(l-1\) and even with respect to the variable \(y\).
reviewed by: Stephen J. Gardiner
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 31B20
zbMATH DE Number: 910273
zbMATH Keywords: harmonic polynomial
zbMATH Keywords: harmonic function
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: uniqueness theorem
MaRDI profile type: Publication
full work available at URL: pjm.1996.172.591
OpenAlex ID: W1529810514
DOI: 10.2140/PJM.1996.172.591
description: scientific article; zbMATH DE number 910273