Global regularity of the \(\overline{\partial}\)-Neumann problem on circular domains
type: Item
instance of: scholarly article
title: Global regularity of the \(\overline{\partial}\)-Neumann problem on circular domains
zbMATH Open document ID: 0662.32016
DOI: 10.1007/BF01442668
author: So-Chin Chen
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164581
review text: We show that if \(D\subseteq {\mathbb{C}}^ n\), \(n\geq 2\), is a smooth bounded pseudoconvex circular domain with \(\sum^{n}_{i=1}z_ i\frac{\partial r}{\partial z_ i}\neq 0\) near the boundary, where r(z) is the defining function for D, then global regularity of the \({\bar \partial}\)-Neumann problem holds on D. More precisely, if given \(f\in W^ k_{p,q}(D)\), let \(u\in L^ 2_{q,p}(D)\) be the solution to the \({\bar \partial}\)- Neumann problem, \[ Q(u,v)=({\bar \partial}u,{\bar \partial}v)+({\vec \partial}u,{\vec \partial}v)=(f,v), \] for all \(v\in \tilde {\mathcal D}_{p,q}(D)\), then \(u\in W^ k_{p,q}(D)\) and \(\| u\|_ k\leq C_ k\| f\|_ k\), where \(W^ k_{p,q}(D)\) is the Sobolev space of order k on D and \(\tilde {\mathcal D}_{p,q}(D)\) is the completion of smooth (p,q)-forms with Neumann boundary conditions under Q. We also prove that same conclusions hold on any smooth bounded Reinhardt pseudoconvex domain. As an application we show that same results hold on Sibony's domain that is a smooth bounded weakly pseudoconvex domain in \({\mathbb{C}}^ 3\), but fails to have sup-norm estimate for the \({\bar \partial}\)-equation.
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 32A07
Mathematics Subject Classification ID: 32T99
Mathematics Subject Classification ID: 35N15
zbMATH DE Number: 4081958
zbMATH Keywords: pseudoconvex circular domain
zbMATH Keywords: global regularity of the \(\overline{\partial }\)-Neumann problem
zbMATH Keywords: Reinhardt pseudoconvex domain
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4081958