The regularity of minima of variational problems with graph obstacles
type: Item
instance of: scholarly article
title: The regularity of minima of variational problems with graph obstacles
zbMATH Open document ID: 0644.49008
DOI: 10.1007/BF01194875
author: Michael Wiegner
author: M. Fuchs
published in: Archiv der Mathematik
publication date: 1989-01-01T00:00:00Z
review text: \textit{F. Tomi} [Math. Z. 128, 43-74 (1972; Zbl 0243.49023)] showed that continuous minima \(u: {\mathbb{R}}^ n\supset \Omega \to {\mathbb{R}}^ N\), n,N\(\geq 2\), of Dirichlet's integral \[ E(u):=\int _{\Omega}| Du| ^ 2 dx, \] under the side condition \(u^ N(x)\geq f(x,u^ 1(x),...,u^{N-1}(x))\) are of class \(C^{1,\alpha}(\Omega)\) provided \(f: \Omega\) \(\times {\mathbb{R}}^{N-1}\to {\mathbb{R}}\) satisfies some mild regularity assumptions. In our paper we prove that the initial regularity hypothesis of Tomi holds for weak minimizers of Sobolev class \(H^{1,2}\), hence weak solutions of the graph obstacle problem are regular on the whole domain of definition.
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 35J85
zbMATH DE Number: 4050385
zbMATH Keywords: continuous minima
zbMATH Keywords: Dirichlet's integral
zbMATH Keywords: regularity
zbMATH Keywords: weak solutions
zbMATH Keywords: graph obstacle problem
MaRDI profile type: Publication
cites work: Variational inequalities and harmonic mappings.
cites work: Variational problems with non-convex obstacles and an integral constraint for vector-valued functions
cites work: Optimal regularity theorems for variational problems with obstacles
cites work: Variational inequalities for vector-valued functions with non-convex obstacles
cites work: Some remarks on the boundary regularity for minima of variational problems with obstacles
cites work: A note on removable singularities for minima of certain vector-valued obstacle problems
cites work: A regularity theorem for energy minimizing maps of riemannian manifolds
cites work: An elementary partial regularity proof for vector-valued obstacle problems
cites work: Q4123700
cites work: Variationsprobleme vom Dirichlet-Typ mit einer Ungleichung als Nebenbedingung
cites work: Ein optimaler Regularitätssatz für schwache Lösungen gewisser elliptischer Systeme
cites work: A-priori Schranken für Lösungen gewisser elliptischer Systeme
cites work: On minima of variational problems with some nonconvex constraints
full work available at URL: https://doi.org/10.1007/bf01194875
OpenAlex ID: W1984864225
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description: scientific article; zbMATH DE number 4050385