Boundary regularity of minima
type: Item
instance of: scholarly article
title: Boundary regularity of minima
zbMATH Open document ID: 1194.49048
DOI: 10.4171/RLM/524
author: Jan Kristensen
author: Giuseppe Mingione
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2010-09-19T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_abstract.php
review text: Summary: Let u: \(\Omega \rightarrow \mathbb R^N\) be any given solution to the Dirichlet variational problem minw \(\int \Omega F(x,w,Dw)\) \(dx, w\equiv u_{0}\) on \(\partial \Omega \), where the integrand \(F(x,w,Dw)\) is strongly convex in the gradient variable \(Dw\), and suitably Hölder continuous with respect to \((x,u)\). We prove that almost every boundary point, in the sense of the usual surface measure of \(\partial \Omega \), is a regular point for \(u\). This means that \(Du\) is Hölder continuous in a relative neighborhood of the point. The existence of even one of such regular boundary points was an open problem for the general functionals considered here, and known only under certain very special structure assumptions.
Mathematics Subject Classification ID: 49N60
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 49J10
zbMATH DE Number: 5787942
zbMATH Keywords: boundary regularity
zbMATH Keywords: variational problems
zbMATH Keywords: singular sets
MaRDI profile type: Publication
cites work: Gradient estimates for thep(x)-Laplacean system
cites work: BOUNDARY REGULARITY RESULTS FOR VARIATIONAL INTEGRALS
cites work: Q4389910
cites work: Elliptic systems with nonlinearity q greater or equal to two. Regularity of the solution of the Dirichlet problem
cites work: Partial regularity of minimizers of quasiconvex integrals with subquadratic growth
cites work: Maximal functions measuring smoothness
cites work: Nonlinear elliptic systems with Dini continuous coefficients
cites work: The existence of regular boundary points for non-linear elliptic systems
cites work: Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals
cites work: Quasiconvexity and partial regularity in the calculus of variations
cites work: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105)
cites work: Q3331815
cites work: Q3151387
cites work: Boundary regularity for minima of certain quadratic functionals
cites work: Regularity problem for quasilinear elliptic and parabolic systems
cites work: Regularity in oscillatory nonlinear elliptic systems
cites work: The singular set of \(\omega\)-minima
cites work: Non-differentiable functionals and singular sets of minima
cites work: The singular set of minima of integral functionals
cites work: The singular set of Lipschitzian minima of multiple integrals
cites work: Boundary regularity in variational problems
cites work: Boundary regularity for almost minimizers of quasiconvex variational problems
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: Q5428774
cites work: Convex integration for Lipschitz mappings and counterexamples to regularity
cites work: Q4150079
cites work: Non-Lipschitz minimizers of smooth uniformly convex functionals
cites work: Regularity for a class of nonlinear elliptic systems
OpenAlex ID: W1980633130
Recommended article: Boundary regularity for quasiminima
Recommended article: Q3788771
Recommended article: Boundary regularity for minima of certain variational integrals
Recommended article: Boundary regularity for minimizing biharmonic maps
Recommended article: Boundedness of minimizers
Recommended article: Regularity of a minimal surface at its free boundary
Recommended article: Regularity of almost minimizers with free boundary
Recommended article: Boundary regularity of minimizers of \(p(x)\)-energy functionals
Recommended article: Regularity of Lipschitz minima
Recommended article: Boundary regularity for parabolic quasiminima
description: scientific article; zbMATH DE number 5787942