Local T-sets and renormalized solutions of degenerate quasilinear elliptic equations with an \(L^ 1\)-datum
type: Item
instance of: scholarly article
title: Local T-sets and renormalized solutions of degenerate quasilinear elliptic equations with an \(L^ 1\)-datum
zbMATH Open document ID: 0885.35040
author: Youcef Atik
author: J.-M. Rakotoson
published in: Advances in Differential Equations
publication date: 1998-04-23T00:00:00Z
review text: The questions of uniqueness and stability of renormalized solutions for a boundary value problem associated to the equation \[ Au= -\text{div}(\widehat a(x,u,\nabla u))+ b(x)|u|^{\gamma-1}u=\mu\quad\text{on }\Omega \] with given \(\mu\in L^1(\Omega)\) are studied. Here \(\widehat a\) is a Carathéodory function such that \(\widehat a/a\) satisfies the general conditions of Leray-Lions, and \(\gamma\) is a real number. Because of the degeneracy of \(a\), the authors introduce the notion of local \(T\)-sets associated with the singularity \(S\) of \(\widehat a\). Following the same ideas as for usual Sobolev spaces, some useful properties of these sets and precise formulas and inequalities are given. The notion of renormalized solution is introduced. The uniqueness and stability for \(S\) satisfying a geometrical condition, or \(S\) satisfying an analytic-geometrical one, are given.
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35R05
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 936929
zbMATH Keywords: uniqueness
zbMATH Keywords: stability
reviewed by: Bronius Kvedaras
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 936929