Partial regularity for sub-elliptic systems with Dini continuous coefficients involving natural growth terms in the Heisenberg group
type: Item
instance of: scholarly article
title: Partial regularity for sub-elliptic systems with Dini continuous coefficients involving natural growth terms in the Heisenberg group
zbMATH Open document ID: 1320.35128
author: Dongni Liao
author: Zefeng Yu
author: Jialin Wang
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-03-27T00:00:00Z
review text: In this paper, the authors study the partial regularity of weak solutions, satisfying a natural growth condition, of a nonlinear sub-elliptic system in divergence form on the Heisenberg group \(\mathbb{H}^n\) of the kind \[ -\sum_{j=1}^{2n}X_jA_j^\alpha(z,u(z),Xu(z))=B^\alpha(z,u(z),Xu(z)),\,\,\,\,z\in\Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{H}^n\), \(\alpha=1,\ldots,N,\) \(X=(X_1,\ldots,X_{2n})\) (the \(X_j\) being the basic vector fields of the Lie algebra of \(\mathbb{H}^n\)), \(u:\Omega\rightarrow\mathbb{R}^N\). The coefficients \(A_j^\alpha\) of the principal part are assumed to be Dini-continuous. Hence the authors weaken the regularity hypothesis from Hölder-continuous to Dini-continuous. Here, they apply the technique of \(\mathcal{A}\)-harmonic approximation adapted to the setting of the Heisenberg group. Their result is optimal, in that when the coefficients are Hölder-continuous they immediately get the optimal Hölder exponent for the horizontal gradient \(Xu\), of a weak solution \(u\), on \(\Omega\setminus\mathrm{Sing}(u)\).
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35R03
Mathematics Subject Classification ID: 35H20
zbMATH DE Number: 6420124
zbMATH Keywords: Dini-continuous coefficients
zbMATH Keywords: natural growth condition
zbMATH Keywords: \(\mathcal A\)-harmonic approximation technique
zbMATH Keywords: system in divergence form
reviewed by: Alberto Parmeggiani
MaRDI profile type: Publication
full work available at URL: j.jmaa.2013.11.002
OpenAlex ID: W2024536384
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DOI: 10.1016/J.JMAA.2013.11.002
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description: scientific article; zbMATH DE number 6420124