The boundary regularity of non-linear parabolic systems I
type: Item
instance of: scholarly article
title: The boundary regularity of non-linear parabolic systems I
zbMATH Open document ID: 1194.35086
author: Verena Bögelein
author: Frank Duzaar
author: Giuseppe Mingione
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2010-02-09T00:00:00Z
review text: The paper is devoted to the study of boundary regularity for fairly general parabolic systems of the type \(u_t-\text{div}\,a(x,t,u,Du)=0\). Boundary values (given by a continuous function \(g\) with Hölder continuous spatial derivative and time derivative in some Morrey space) are prescribed at both an initial time \(t=0\) and at the spatial boundary of the domain at all times. The function \(a\) is allowed to have \(p\)-growth (\(p\geq2\)) in \(Du\) at infinity, is assumed to be uniformly elliptic, and as a function of \((x,u)\) it is merely assumed to be Hölder continuous. The paper provides a regularity condition under which a boundary point is regular, in the sense that the spatial gradient is Hölder continuous in a relative neighborhood of such a point. More precisely, there are only two ways a boundary point can fail to be regular: Either the liminf of the mean integral of \(|D(u-g)-\overline{D(u-g)}|^p\) over parabolic (half) cylinders is positive or \(\overline{D(u-g)}\) does not stay bounded as the cylinders shrink to the boundary point. In the proof, the weak solutions of the nonlinear system are related to nearby solutions of a some linear parabolic system derived from it. This is done using an appropriate version (including the \(L^p\) case) of what the authors call the ``\(A\)-caloric approximation lemma''; a lemma that states that for every function solving the linear parabolic equation approximately, there is an exact solution within a distance that can be estimated. The regularity criterion proved here does not guarantee the existence of even one regular boundary point, since the assumptions are fairly general. In a sequel to the paper [\textit{V.~Bögelein, F.~Duzaar, G.~Mingione}, Ann.\ Inst. Henry Poincaré, Anal.\ Non Linéaire 27, No.~1, 145--200 (2010; Zbl 1194.35085)], the authors prove dimensional reduction for the singular set under additional hypotheses which allow the conclusion that almost every boundary point is regular.
reviewed by: Andreas Gastel
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35K51
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K59
zbMATH DE Number: 5667951
zbMATH Keywords: Cauchy-Dirichlet problem
zbMATH Keywords: Morrey space
zbMATH Keywords: \(A\)-caloric approximation lemma
zbMATH Keywords: regular boundary point
MaRDI profile type: Publication
full work available at URL: j.anihpc.2009.09.003
OpenAlex ID: W4206521926
Wikidata QID: Q58880847
cites work: Gradient estimates for a class of parabolic systems
cites work: Regularity results for parabolic systems related to a class of non-Newtonian fluids.
cites work: On a partial regularity up to the boundary of weak solutions to quasilinear parabolic systems with quadratic growth
cites work: Partial regularity for weak solutions of nonlinear elliptic systems: the subquadratic case
cites work: Partial regularity and singular sets of solutions of higher order parabolic systems
cites work: The boundary regularity of non-linear parabolic systems. II
cites work: Equazioni paraboliche del secondo ordine e spazi \({\mathfrak L}^{2,\theta} (\Omega,\delta)\)
cites work: On the nonlinear parabolic systems in divergence form. Hölder continuity and partial Hölder continuity of the solutions
cites work: Q4049704
cites work: Q5537959
cites work: Degenerate parabolic equations
cites work: Q3810242
cites work: Elliptic Systems, Singular Sets and Dini Continuity
cites work: Optimal interior partial regularity for nonlinear elliptic systems: The method of \(A\)-harmonic approximation
cites work: Q4665570
cites work: The existence of regular boundary points for non-linear elliptic systems
cites work: The \(p\)-harmonic approximation and the regularity of \(p\)-harmonic maps
cites work: Second order parabolic systems, optimal regularity, and singular sets of solutions
cites work: Harmonic type approximation lemmas
cites work: Parabolic Systems with Polynomial Growth and Regularity
cites work: Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals
cites work: A counter-example to the boundary regularity of solutions to elliptic quasilinear systems
cites work: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105)
cites work: Local existence for quasilinear parabolic systems under nonlinear boundary conditions
cites work: Boundary regularity for nonlinear elliptic systems
cites work: The singular set of minima of integral functionals
cites work: Boundary regularity in variational problems
cites work: Boundary regularity of minima
cites work: Q3593991
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: Compact sets in the space \(L^ p(0,T;B)\)
cites work: Q4896042
cites work: Q3027520
DOI: 10.1016/J.ANIHPC.2009.09.003
description: scientific article; zbMATH DE number 5667951