Higher integrability for solutions to a system of critical elliptic PDE
type: Item
instance of: scholarly article
title: Higher integrability for solutions to a system of critical elliptic PDE
zbMATH Open document ID: 1315.58013
author: Ben Sharp
published in: Methods and Applications of Analysis
publication date: 2014-11-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1112.1127
reviewed by: Andreas Gastel
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 42B37
zbMATH DE Number: 6366889
zbMATH Keywords: harmonic maps
zbMATH Keywords: regularity
zbMATH Keywords: critical elliptic systems
zbMATH Keywords: Riesz potential estimates
zbMATH Keywords: Morrey-Hardy spaces
MaRDI profile type: Publication
arXiv ID: 1112.1127
OpenAlex ID: W2964169669
Wikidata QID: Q125386250
DOI: 10.4310/MAA.2014.V21.N2.A3
review text: The paper is concerned with critical elliptic systems in the tradition of \textit{T. Rivière} [Invent. Math. 168, No. 1, 1--22 (2007; Zbl 1128.58010)] and \textit{T. Rivière} and \textit{M. Struwe} [Commun. Pure Appl. Math. 61, No. 4, 451--463 (2008; Zbl 1144.58011)] which include, most prominently, the harmonic map system. In the current paper, a right-hand side \(f\) is allowed to appear in the system, which formally reads \(-\Delta u-\Omega\cdot\nabla u=f\), with \(\Omega\) usually depending on \(u\) and \(\nabla u\). The domain dimension is allowed to be critical (\(n=2\)) or super-critical (\(n\geq3\)).NEWLINENEWLINEAssuming \(f\in L^p\) for \(p\in(n/2,n)\) and that both \(\nabla u\) and \(\Omega\) are in the Morrey space \(M^{2,n-2}\) natural for the problem, and that \(\Omega\) is sufficiently small in that space, the author proves higher regularity, namely local control of \(\nabla^2u\) in \(M^{2p/n,n-2}\) and of \(\nabla u\) in \(M^{2p/(n-p),n-2}\).NEWLINENEWLINEThe proof uses Rivière's technique of gauge fixing and requires subtle improvements of Adams-type potential estimates.
description: scientific article; zbMATH DE number 6366889