The Neumann problem on Lipschitz domains in Hardy spaces of order less than one
type: Item
instance of: scholarly article
title: The Neumann problem on Lipschitz domains in Hardy spaces of order less than one
zbMATH Open document ID: 0855.35021
author: R. M. Brown
published in: Pacific Journal of Mathematics
publication date: 1996-07-31T00:00:00Z
review text: Recently, B. E. J. Dahlberg and C. E. Kenig considered the Neumann problem, \(\Delta u= 0\) in \(D\), \(\partial u/\partial\nu= f\) on \(\partial D\), for Laplace's equation in a Lipschitz domain \(D\). One of their main results considers this problem when the data lies in the atomic Hardy space \(H^1(\partial D)\) and they show that the solution has gradient in \(L^1(\partial D)\). The aim of this paper is to establish an extension of their theorem for data in the Hardy space \(H^p(\partial D)\), \(1- \varepsilon< p< 1\), where \(0< \varepsilon< 1/n\) is a positive constant which depends only on \(m\), the maximum of the Lipschitz constants of the functions which define the boundary of the domain. We also extend G. Verchota's and Dahlberg and Kenig's theorem on the potential representation of solutions of the Neumann problem to the range \(1- \varepsilon< p< 1\). This has the interesting consequence that the double-layer potential is invertible on Hölder spaces \(C^\alpha(\partial D)\) for \(\alpha\) close to zero.
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 912151
zbMATH Keywords: nonsmooth domains
zbMATH Keywords: Neumann problem
zbMATH Keywords: atomic Hardy space
MaRDI profile type: Publication
full work available at URL: pjm.1995.171.389
OpenAlex ID: W1553366520
DOI: 10.2140/PJM.1995.171.389
description: scientific article; zbMATH DE number 912151