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

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

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