Variational problems with non-convex obstacles and an integral constraint for vector-valued functions
type: Item
instance of: scholarly article
title: Variational problems with non-convex obstacles and an integral constraint for vector-valued functions
zbMATH Open document ID: 0568.49009
DOI: 10.1007/BF01162348
author: Frank Duzaar
author: M. Fuchs
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173702
review text: We minimize the Dirichlet-integral (or related quadratic functionals) in a class of vector-valued functions \(u: {\mathbb{R}}^ n\supset D\to {\mathbb{R}}^ N\) which is described by an integral constraint and a side condition of the form \(Im(u)\subset \bar M\) and show partial regularity of weak minima up to a set of vanishing \({\mathbb{H}}^{n-2}\)-measure, provided M is a bounded open set with certain smoothness properties. In contrast to the results of Hildebrandt, Tomi and others our theorem is not restricted to special dimensions n or requires convexity of M. The method consists in a suitable linearisation of the problem giving an Euler-inequality with quadratic growth in Du. In combination with certain reverse Hölder-inequalities the techniques of Giaquinta and Giusti imply the partial regularity result. Moreover, their arguments allow to reduce the dimension of the singular set S to \({\mathbb{H}}-\dim (S)\leq n-3\).
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 26B35
zbMATH DE Number: 3907217
zbMATH Keywords: Dirichlet-integral
zbMATH Keywords: vector-valued functions
zbMATH Keywords: integral constraint
zbMATH Keywords: side condition
zbMATH Keywords: partial regularity of weak minima
zbMATH Keywords: Euler-inequality
zbMATH Keywords: reverse Hölder-inequalities
MaRDI profile type: Publication
cites work: Q3714460
cites work: Variational problems with obstacles and integral constraints
cites work: Variational inequalities for vector-valued functions with non-convex obstacles
cites work: Q3331814
cites work: Interior \(C^{1+ \alpha}\)-regularity of solutions of two-dimensional variational problems with obstacles
cites work: On variational problems with obstacles and integral constraints for vector-valued functions
cites work: Variational problems with obstacles and a volume constraint
cites work: Q5518064
cites work: Variationsprobleme vom Dirichlet-Typ mit einer Ungleichung als Nebenbedingung
OpenAlex ID: W2095520073
description: scientific article; zbMATH DE number 3907217