Elliptic obstacle problems with measure data: potentials and low order regularity
type: Item
instance of: scholarly article
title: Elliptic obstacle problems with measure data: potentials and low order regularity
zbMATH Open document ID: 1295.35373
DOI: 10.5565/PUBLMAT_56212_04
author: Christoph Scheven
published in: Publicacions Matemàtiques
publication date: 2014-06-30T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pm/1340127809
review text: The author treats measure data problems of the type \[ -\text{div} a(x,Du) = \mu \qquad \text{on } \Omega \] coupled with an obstacle constraint of the form \(u \geq \psi\), where \(\mu\) a Radon measure and the left-hand side of the equation an elliptic differential operator of \(p\)-Laplace type for \(p > \frac{2n}{n+1}\). Here it is only assumed that \(a: \Omega \times \mathbb{R}^n \to \mathbb{R}^n\) is a Carathéodory function. As the measure data \(\mu\) is only assumed to be a Radon measure, one cannot work with the usual weak formulation of the problem. Instead, the author discusses the concept of a limit \(u\) of (more regular) approximating solutions. In this setting, the author then proves pointwise estimates for the solutions in terms of the Wolff potential. As applications, criteria for Lorentz space regularity, for the continuity and the Hölder continuity of the solutions are derived.
reviewed by: Stephan Fackler
Mathematics Subject Classification ID: 35R06
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35J87
Mathematics Subject Classification ID: 31B35
zbMATH DE Number: 6310394
zbMATH Keywords: measure data problems
zbMATH Keywords: elliptic obstacle problems
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: regularity of solutions
zbMATH Keywords: non-linear potential theory
MaRDI profile type: Publication
cites work: Q5655672
cites work: Q4851005
cites work: Q3935745
cites work: Q4704628
cites work: Nonlinear elliptic and parabolic equations involving measure data
cites work: Q3202751
cites work: Nonlinear Elliptic Equations with Right Hand Side Measures
cites work: Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data
cites work: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations
cites work: Q4487386
cites work: Gradient estimates via non-linear potentials
cites work: Gradient continuity estimates
cites work: Local Lipschitz regularity for degenerate elliptic systems
cites work: Gradient estimates via linear and nonlinear potentials
cites work: Q3151387
cites work: Q4693180
cites work: On the Dirichlet Boundary Value Problem for a Degenerate Parabolic Equation
cites work: Q4033095
cites work: The Wiener test and potential estimates for quasilinear elliptic equations
cites work: Q4415609
cites work: Q3905599
cites work: Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation
cites work: A surprising linear type estimate for nonlinear elliptic equations
cites work: Linear potentials in nonlinear potential theory
cites work: Existence and uniqueness of solutions for nonlinear obstacle problems with measure data
cites work: On the definition and properties of p-superharmonic functions.
cites work: Q5890172
cites work: Q4354419
cites work: Gradient potential estimates
cites work: Existence of localizable solutions to nonlinear parabolic problems with irregular obstacles
cites work: Potential estimates in parabolic obstacle problems
cites work: Gradient potential estimates in non-linear elliptic obstacle problems with measure data
cites work: On the weak continuity of elliptic operators and applications to potential theory
cites work: Weakly Differentiable Functions
OpenAlex ID: W2115302219
description: scientific article