Nonlinear measure data problems
type: Item
instance of: scholarly article
title: Nonlinear measure data problems
zbMATH Open document ID: 1236.35068
author: Giuseppe Mingione
published in: Milan Journal of Mathematics
publication date: 2011-12-20T00:00:00Z
review text: This paper is a survey of some regularity results for solutions to measure data problems, i.~e. solutions of the equation \(-\text{div}\,a(x,Du) = \mu\), where \(\mu\) is a measure on a bounded open set \(\Omega \subset \mathbb{R}^{n}\). After two introductory sections, the basic regularity results are given. These results are new in the sublinear case, \(p<2\) (the case \(p\geq2\) was treated by the author in [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6, No. 2, 195--261 (2007; Zbl 1178.35168)]), and they are proved in the last sections of the paper. Next, based on the concept of Wolff potential, nonlinear extensions of the basic linear estimates via the Riesz potential are presented. Also presented are recently obtained nonlinear potential estimates for parabolic problems \(u_{t}-\text{div}(a(Du))=\mu\).
reviewed by: Mihai Pascu
Mathematics Subject Classification ID: 35J62
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35K59
Mathematics Subject Classification ID: 35-02
zbMATH DE Number: 5990865
zbMATH Keywords: quasilinear degenerate equations
zbMATH Keywords: regularity of solutions
zbMATH Keywords: gradient estimates
zbMATH Keywords: Calderón-Zygmund theory
zbMATH Keywords: \(p\)-Laplacian
MaRDI profile type: Publication
full work available at URL: s00032-011-0168-1
OpenAlex ID: W2066492455
Wikidata QID: Q58880803
cites work: Gradient estimates for thep(x)-Laplacean system
cites work: Gradient estimates for a class of parabolic systems
cites work: Q5624209
cites work: A note on Riesz potentials
cites work: Q5655672
cites work: Q4073917
cites work: Estimates for the gradient of solutions of nonlinear elliptic equations with \(L^1\) data
cites work: Q5812835
cites work: Calderón-Zygmund estimates for parabolic measure data equations
cites work: Q4904966
cites work: Q4851005
cites work: Marcinkiewicz estimates for solutions of some elliptic problems with nonregular data
cites work: Nonlinear parabolic equations with measure data
cites work: Nonlinear elliptic and parabolic equations involving measure data
cites work: Nonlinear Elliptic Equations with Right Hand Side Measures
cites work: Analytical foundations of the theory of quasiconformal mappings in R^n
cites work: On the existence of certain singular integrals
cites work: On Singular Integrals
cites work: Approximated solutions of equations with \(L^ 1\) data. Application to the \(H\)-convergence of quasilinear parabolic equations
cites work: Q4487386
cites work: Q3252874
cites work: Maximal functions measuring smoothness
cites work: C1 + α local regularity of weak solutions of degenerate elliptic equations
cites work: Q3774245
cites work: Degenerate parabolic equations
cites work: Hölder estimates for nonlinear degenerate parabolic sytems.
cites work: Alternative forms of the Harnack inequality for non-negative solutions to certain degenerate and singular parabolic equations
cites work: Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations
cites work: On the Higher Integrability of the Gradient of Weak Solutions of Certain Degenerate Elliptic Systems
cites work: The \(p\)-harmonic system with measure-valued right hand side
cites work: Nonlinear elliptic systems with measure-valued right hand side
cites work: Uniqueness and maximal regularity for nonlinear elliptic systems of n--Laplace type with measure valued right hand side
cites work: The existence of regular boundary points for non-linear elliptic systems
cites work: Gradient estimates in non-linear potential theory
cites work: Gradient estimates via non-linear potentials
cites work: Gradient estimates via linear and nonlinear potentials
cites work: Gradient continuity estimates
cites work: Local Lipschitz regularity for degenerate elliptic systems
cites work: Higher integrability for minimizers of integral functionals with \((p,q)\) growth
cites work: Sharp regularity for functionals with (\(p\),\(q\)) growth
cites work: Q4843565
cites work: Q3151387
cites work: Q3531572
cites work: Regularity of differential forms minimizing degenerate elliptic functionals.
cites work: Thin sets in nonlinear potential theory
cites work: Q4693180
cites work: Projections onto gradient fields and $L^{p}$-estimates for degenerated elliptic operators
cites work: Pathological solutions to elliptic problems in divergence form with continuous coefficients
cites work: Estimates for \(p\)-Poisson equations
cites work: The Wiener test and potential estimates for quasilinear elliptic equations
cites work: Superharmonic functions are locally renormalized solutions
cites work: The singular set of minima of integral functionals
cites work: Boundary regularity in variational problems
cites work: A note on the Wolff potential estimate for solutions to elliptic equations involving measures
cites work: Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations
cites work: Potential estimates and gradient boundedness for nonlinear parabolic systems

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cites work: Universal potential estimates

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cites work: Q3104361

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cites work: Nonlinear potential estimates in parabolic problems

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cites work: The Wolff gradient bound for degenerate parabolic equations

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cites work: A surprising linear type estimate for nonlinear elliptic equations

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cites work: Existence for a degenerate Cauchy problem

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cites work: Regularity results for the gradient of solutions of linear elliptic systems with VMO-coefficients and L 1,λ data

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cites work: Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder

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cites work: Sharp forms of estimates for subsolutions and supersolutions of quasilinear elliptic equations involving measures

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cites work: On the definition and properties of p-superharmonic functions.

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cites work: Wolff potential estimates for elliptic equations with nonstandard growth and applications

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cites work: The singular set of solutions to non-differentiable elliptic systems

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cites work: Regularity of minima: an invitation to the dark side of the calculus of variations.

Nonlinear measure data problems (cites work)
cites work: Q5428774

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cites work: Gradient estimates below the duality exponent

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cites work: Gradient potential estimates

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cites work: Nonlinear aspects of Calderón-Zygmund theory

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cites work: Quasilinear and Hessian equations of Lane-Emden type

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cites work: Singular quasilinear and Hessian equations and inequalities

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cites work: Elliptic obstacle problems with measure data: potentials and low order regularity

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cites work: Gradient potential estimates in non-linear elliptic obstacle problems with measure data

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cites work: Q5518040

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cites work: Local behavior of solutions of quasi-linear equations

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cites work: Q5640160

Nonlinear measure data problems (cites work)
cites work: Q4108681

Nonlinear measure data problems (cites work)
cites work: On the weak continuity of elliptic operators and applications to potential theory

Nonlinear measure data problems (cites work)
cites work: Quasilinear elliptic equations with signed measure

Nonlinear measure data problems (cites work)
cites work: Q5489390

Nonlinear measure data problems (DOI)
DOI: 10.1007/S00032-011-0168-1