Parabolic Q-minima and minimal solutions to variational flow
type: Item
instance of: scholarly article
title: Parabolic Q-minima and minimal solutions to variational flow
zbMATH Open document ID: 0674.35042
DOI: 10.1007/BF01171265
author: Wilfried Wieser
published in: Manuscripta Mathematica
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155240
review text: The purpose of the paper is to provide a unified version of some regularity results on parabolic systems of the form \[ u'-\sum D_{x_ i}(D_{p_ i}F(x,u,Du))+D_ uF(x,u,Du)=0. \] To this aim, the notion of parabolic Q-minimum is introduced, where Q is a real number greater than or equal to 1. It is an adaptation to the parabolic case of the notion of (elliptic) Q-minimum introduced in \textit{M. Giaquinta} and \textit{E. Giusti} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 79-104 (1984; Zbl 0541.49008)]. Results on \(L^ p\)-summability of the spatial derivatives of u, Hölder continuity of u (in the scalar case) and partial Hölder continuity of u (in the vector case) are given for parabolic Q-minima. More special results are proved for the so-called minimal solutions, namely Q-minima with \(Q=1\). A characterization of the existence of such minimal solutions is given in terms of the convexity of the associated functional \(\int F(x,u,Du)dx\).
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 4103588
zbMATH Keywords: regularity
zbMATH Keywords: parabolic Q-minimum
zbMATH Keywords: \(L^ p\)-summability
zbMATH Keywords: Hölder continuity
zbMATH Keywords: partial Hölder continuity
zbMATH Keywords: minimal solutions
zbMATH Keywords: existence
zbMATH Keywords: convexity
reviewed by: Marco Degiovanni
MaRDI profile type: Publication
cites work: Equazioni paraboliche del secondo ordine e spazi \({\mathfrak L}^{2,\theta} (\Omega,\delta)\)
cites work: Spazi \({\mathcal L}^{(p,\vartheta)} (\Omega,\delta)\) e loro proprieta
cites work: Q4085497
cites work: The L\(^p\)-integrability of the partial derivatives of a quasiconformal mapping
cites work: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105)
cites work: Q3310473
cites work: Quasi-minima
cites work: Q4196779
cites work: Q3037898
cites work: Q5634306
cites work: Über Flächen konstanter mittlerer Krümmung
description: scientific article; zbMATH DE number 4103588