On a conjecture of J. Serrin
type: Item
instance of: scholarly article
title: On a conjecture of J. Serrin
zbMATH Open document ID: 1197.35079
DOI: 10.4171/RLM/529
author: Haïm Brezis
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2010-09-19T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_abstract.php
review text: Summary: \textit{J. Serrin} [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 18, 385--387 (1964; Zbl 0142.37601)] proposed the following conjecture. Let u \(\in W_{\text{loc}}^{1,1}(\Omega )\) be a weak solution of the second order elliptic equation \[ \sum_{i,j=1}^N \frac{\partial}{\partial x_j} \bigg(a_{ij} \frac{\partial u}{\partial x_i}\bigg)=0\quad\text{in }\Omega, \] in divergence form, with Hölder continuous coefficients \(a_{ij}(x)\); then \(u\) is a ``classical'' solution. We announce a solution of this conjecture assuming only \(u\in BV_{\text{loc}}(\Omega )\) and Dini continuous coefficients.
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35J15
zbMATH DE Number: 5787947
zbMATH Keywords: divergence elliptic equation
zbMATH Keywords: very weak solutions
Wikidata QID: Q123239701
MaRDI profile type: Publication
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OpenAlex ID: W2050299572
description: scientific article; zbMATH DE number 5787947