The \(p\)-harmonic system with measure-valued right hand side
type: Item
instance of: scholarly article
title: The \(p\)-harmonic system with measure-valued right hand side
zbMATH Open document ID: 0879.35052
DOI: 10.1016/S0294-1449(97)80141-2
author: Georg Dolzmann
author: Norbert Hungerbühler
author: Stefan Müller
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1998-01-18T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78415
review text: For \(2-1/n<p<n\) one proves existence of a distributional solution \(u\) of the \(p\)-harmonic system \[ -\text{div}(|\nabla u|^{p-2}\nabla u)= \mu\quad\text{in }\Omega,\quad u=0\quad\text{on }\partial\Omega, \] where \(\Omega\) is an open subset of \(\mathbb{R}^n\) (bounded or unbounded), \(u:\Omega\to\mathbb{R}^n\), and \(\mu\) is an \(\mathbb{R}^m\)-valued Radon measure of finite mass. For the solution \(u\) one establishes the Lorentz space estimate \[ |Du|_{Lq,\infty}+|u|_{Lq^*,\infty}\leq C|\mu|M_{1/(p-1)}, \] \(q=n/(n-1)(p-1)\) and \(q^*=n/(n-p)(p-1)\). The main step of the proof is to show that for suitable approximations the gradients \(Du_k\) converge. This is achieved by the choice of regularized test functions and a localization argument to compensate the fact that in general \(u\not\in W_{1,p}\).
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35R05
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 1019554
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: Radon measure
zbMATH Keywords: Lorentz space estimate
zbMATH Keywords: regularized test function
zbMATH Keywords: localization argument
reviewed by: Bronius Kvedaras
MaRDI profile type: Publication
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OpenAlex ID: W2039749398
description: scientific article; zbMATH DE number 1019554