The singular set of Lipschitzian minima of multiple integrals
type: Item
instance of: scholarly article
title: The singular set of Lipschitzian minima of multiple integrals
zbMATH Open document ID: 1114.49038
author: Jan Kristensen
author: Giuseppe Mingione
published in: Archive for Rational Mechanics and Analysis
publication date: 2007-05-03T00:00:00Z
review text: This paper deals with the partial regularity of minimizers of integral functional \[ I(u)= \int_{\Omega }F(x,u(x), Du) \,dx \] where \(u: \Omega \rightarrow \mathbb R^N\) and \(F\) is a quasi-convex function in the Morrey's sense. Let \(u\) be a minimizer and denote the singular set of \(u\) with \(\Sigma_u= \Omega-\Omega_u\), where \(\Omega_u\) is the set of \(x \in \Omega\) such that \(u \in C^{1,\sigma}(A,\mathbb R^N)\) for some \(\sigma>0\) and \(A\) is a neighborhood of \(x\). Under suitable assumptions of \(F\) and if \(u \in W^{1, \infty}(\Omega, \mathbb R^N)\) the authors give a precise estimate of the dimension of \(\Sigma_u\) i.e. dim\(_{\mathcal{H}}(\Sigma_u)\leq n-\delta\), with \(\delta>0\) depending on the data. By using the notion and the property of the set porosity, the result gives that the singular set is uniformely porous.
Mathematics Subject Classification ID: 49N60
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 5149167
reviewed by: Elvira Mascolo
MaRDI profile type: Publication
full work available at URL: s00205-006-0036-2
OpenAlex ID: W1996248097
Wikidata QID: Q58880883
cites work: Semicontinuity problems in the calculus of variations
cites work: A regularity theorem for minimizers of quasiconvex integrals
cites work: Convexity conditions and existence theorems in nonlinear elasticity
cites work: Smooth quasiregular mappings with branching
cites work: Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations
cites work: On the singular sets of minimizers of the Mumford-Shah functional
cites work: Q5537959
cites work: Variational methods for crystalline microstructure. Analysis and computation
cites work: A Characterization of Potential Spaces
cites work: Nonlinear elliptic systems with Dini continuous coefficients
cites work: Elliptic Systems, Singular Sets and Dini Continuity
cites work: Regularity of \(\omega\)-minimizers of quasi-convex variational integrals with polynomial growth
cites work: Quasiconvexity and partial regularity in the calculus of variations
cites work: Global regularity for almost minimizers of nonconvex variational problems
cites work: Q3139888
cites work: Q3151387
cites work: Quasimonotonicity, regularity and duality for nonlinear systems of partial differential equations
cites work: Optimal design and relaxation of variational problems, I
cites work: The singular set of \(\omega\)-minima
cites work: The singular set of minima of integral functionals
cites work: Partial regularity of strong local minimizers in the multi-dimensional calculus of variations
cites work: Quasimonotone versus pseudomonotone
cites work: Q4328334
cites work: The singular set of solutions to non-differentiable elliptic systems
cites work: Bounds for the singular set of solutions to non linear elliptic systems
cites work: Regularity of minima: an invitation to the dark side of the calculus of variations.
cites work: Quasi-convexity and the lower semicontinuity of multiple integrals
cites work: Convex integration for Lipschitz mappings and counterexamples to regularity
cites work: Lipschitz regularity of solutions of some asymptotically convex problems
cites work: Uniform partial regularity of quasi minimizers for the perimeter
cites work: On the Minkowski Dimension of Strongly Porous Fractal Sets in R n
cites work: Non-Lipschitz minimizers of smooth uniformly convex functionals
cites work: The regularity of critical points of polyconvex functionals
DOI: 10.1007/S00205-006-0036-2
description: scientific article; zbMATH DE number 5149167