The Dirichlet problem for sublaplacians on nilpotent Lie groups - geometric criteria for regularity
type: Item
instance of: scholarly article
title: The Dirichlet problem for sublaplacians on nilpotent Lie groups - geometric criteria for regularity
zbMATH Open document ID: 0601.31007
DOI: 10.1007/BF01456983
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164218
review text: Given a sublaplacian L on a stratified Lie algebra \({\mathcal N}\) of type I, geometric criteria for the regularity (irregularity resp.) of open subsets are given. Assuming that \({\mathcal N}\) is of step r and that \({\mathcal N}\) is identified with the underlying space \({\mathbb{R}}^ m\) such an operator is of the form \(L=X^ 2_ 1+...+X^ 2_ n\) with smooth vectorfields \(X_ i\) which together with their commutators up to order r span the tangent space at each point of \({\mathbb{R}}^ m\). In terms of Euclidean geometry the following results are obtained: If \(n=2\) and \(m>2\) then an exterior (r/2-\(\epsilon)\)-Hölder condition is not sufficient for U to be regular. Whenever \(n\geq 3\) and \(m>3\) even an exterior (r- \(\epsilon)\)-Hölder condition is not sufficient. The only cases where all bounded domains with smooth boundary are regular are the cases \(r\leq 2\) and the few cases where \(n=2\) and \(r\leq 4\). The results generalize the well known cone conditions for the classical Laplacian.
Mathematics Subject Classification ID: 31D05
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 3969038
zbMATH Keywords: sublaplacian
zbMATH Keywords: stratified Lie algebra
zbMATH Keywords: regularity
Wikidata QID: Q115393464
author: Wolfhard Hansen
author: Hermann Hueber
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3969038