An a priori Campanato type regularity condition for local minimisers in the calculus of variations

Thomas J. Dodd

Research output: Contribution to journalArticle

Abstract

An a priori Campanato type regularity condition is established for a class of W1X local minimisers u of the general variational integral ?O F(? u(x)) dx where O ? Rn is an open bounded domain, F is of class C2, F is strongly quasi-convex and satisfies the growth condition F(E)= c(1+|E|p)for a p>1 and where the corresponding Banach spaces X are the Morrey-Campanato space Lp,µ (O, RN×n)µ<n, Campanato space Lp,n (O,RN×n)$ and the space of bounded mean oscillation BMO (O,RN×n. The admissible maps u O?RN are of Sobolev class W1,p, satisfying a Dirichlet boundary condition, and to help clarify the significance of the above result the sufficiency condition for W1 BMO local minimisers is extended from Lipschitz maps to this admissible class. © 2008 EDP Sciences, SMAI.

Original languageEnglish
Pages (from-to)111-131
Number of pages21
JournalESAIM - Control, Optimisation and Calculus of Variations
Volume16
Issue number1
DOIs
Publication statusPublished - Jan 2010

Fingerprint

Calculus of variations
Regularity Conditions
Campanato Space
Bounded Mean Oscillation
Lipschitz Map
Sobolev Class
Morrey Space
Quasiconvex
Sufficiency
Growth Conditions
Dirichlet Boundary Conditions
Bounded Domain
Banach space
Class

Keywords

  • Calculus of variations
  • Campanato space
  • Extremals
  • Local minimiser
  • Morrey space
  • Morrey-Campanato space
  • Partial regularity
  • Positive second variation
  • Space of bounded mean oscillation
  • Strong quasiconvexity

Cite this

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abstract = "An a priori Campanato type regularity condition is established for a class of W1X local minimisers u of the general variational integral ?O F(? u(x)) dx where O ? Rn is an open bounded domain, F is of class C2, F is strongly quasi-convex and satisfies the growth condition F(E)= c(1+|E|p)for a p>1 and where the corresponding Banach spaces X are the Morrey-Campanato space Lp,µ (O, RN×n)µp,n (O,RN×n)$ and the space of bounded mean oscillation BMO (O,RN×n. The admissible maps u O?RN are of Sobolev class W1,p, satisfying a Dirichlet boundary condition, and to help clarify the significance of the above result the sufficiency condition for W1 BMO local minimisers is extended from Lipschitz maps to this admissible class. {\circledC} 2008 EDP Sciences, SMAI.",
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An a priori Campanato type regularity condition for local minimisers in the calculus of variations. / Dodd, Thomas J.

In: ESAIM - Control, Optimisation and Calculus of Variations, Vol. 16, No. 1, 01.2010, p. 111-131.

Research output: Contribution to journalArticle

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