Necessary conditions for subellipticity of the \({\bar\partial}\)-Neumann problem
type: Item
instance of: scholarly article
title: Necessary conditions for subellipticity of the \({\bar\partial}\)-Neumann problem
zbMATH Open document ID: 0552.32017
DOI: 10.2307/2006974
published in: Annals of Mathematics. Second Series
publication date: 1983-01-01T00:00:00Z
review text: Let \(\alpha\) be a \({\bar \partial}\)-closed form of type (p,q) with \(L^ 2\)-coefficient on a smoothly bounded domain \(\Omega\) in \({\mathbb{C}}^ n\). One of the principal methods used in the investigation of the \({\bar \partial}\)-Neumann problem (existence and regularity properties of the solution w of \({\bar \partial}u=\alpha\)) is the proof of certain a priori subelliptic estimates i.e., if U is a neighborhood of point \(z_ 0\in \partial \Omega\), a subelliptic estimate holds in U if \((1)\quad \|| u\||^ 2_{\epsilon}\leq {\mathbb{C}}(\| {\bar \partial}u\|^ 2+\| {\bar \partial}^*u\|^ 2+\| u\|^ 2)\) is valid for all \(u\in {\mathcal D}^{p,q}(U)\) where \(\|| \cdot \||^ 2_{\epsilon}\) denotes the tangential Sobolev norm of order \(\epsilon\) and \({\mathcal D}^{p,q}(U)\) the space of smooth (p,q-1) forms u. The author presents geometric conditions that must hold if a subelliptic estimate of order \(\epsilon\) is valid. One of the main results is the following theorem: Suppose that \(\Omega\) is a domain in \({\mathbb{C}}^ n\) and that \(\partial \Omega\) is smooth and pseudoconvex in a neighborhood U of \(z_ 0\). Suppose further that there is a q-dimensional complex analytic variety V passing through \(z_ 0\) such that for all \(z\in V\), z sufficently close to \(z_ 0\), \(| r(z)| \leq c| z-z_ 0|^{\eta},\) where \(\eta >0\) and r(z) is the boundary-defining function of \(\Omega\). If a subelliptic estimate of order \(\epsilon\) of the form (1) holds in U then \(\epsilon\leq 1/\eta\).
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 32T99
Mathematics Subject Classification ID: 35N15
zbMATH DE Number: 3879278
zbMATH Keywords: order of contact
zbMATH Keywords: tangential operator
zbMATH Keywords: Neumann problem
zbMATH Keywords: subelliptic estimate
zbMATH Keywords: pseudoconvex
Wikidata QID: Q56701610
author: David W. Catlin
reviewed by: Rodolfo Salvi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2006974
OpenAlex ID: W2313785636
description: scientific article; zbMATH DE number 3879278