Nonlinear elliptic systems with measure-valued right hand side
type: Item
instance of: scholarly article
title: Nonlinear elliptic systems with measure-valued right hand side
zbMATH Open document ID: 0895.35029
DOI: 10.1007/PL00004354
author: Georg Dolzmann
author: Norbert Hungerbühler
author: Stefan Müller
published in: Mathematische Zeitschrift
publication date: 1998-09-20T00:00:00Z
review text: The authors consider existence and regularity questions for elliptic system \[ -\text{div} \sigma \bigl(x,u(x), Du(x)\bigr) =\mu \quad\text{in } \Omega, \] with measure-valued right hand side \(\mu\) and Dirichlet boundary conditions on a bounded domain \(\Omega\subset \mathbb{R}^N\). A serious technical obstacle is that the solutions of the system in general do not belong to the Sobolev space \(W^{1,1}\). This fact has led to the use of the weakest notion of solution where the weak derivative \(Du\) is replaced by the approximate derivative \(ap\) \(Du\). Under certain growth, monotonicity and structure conditions on \(\sigma\) the authors prove the existence of a solution satisfying a weak Lebesgue space a priori estimate. The BMO-regularity of the solution is also discussed. The key point in the proof of the theorem is a variant of the ``div-curl inequality'' for Young measures which is given in the paper.
Mathematics Subject Classification ID: 35J55
Mathematics Subject Classification ID: 35R05
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 28A33
zbMATH DE Number: 1121732
zbMATH Keywords: div-curl inequality
zbMATH Keywords: \(BMO\)-regularity
zbMATH Keywords: approximate derivative
zbMATH Keywords: Young measures
reviewed by: Vitaly Moroz
MaRDI profile type: Publication
full work available at URL: pl00004354
OpenAlex ID: W2013858367
description: scientific article; zbMATH DE number 1121732