### Abstract

An a priori Campanato type regularity condition is established for a class of W^{1}X local minimisers u of the general variational integral ?_{O} F(? u(x)) dx where O ? R^{n} is an open bounded domain, F is of class C^{2}, 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 L^{p,µ} (O, R^{N×n})µ<n, Campanato space L^{p,n} (O,R^{N×n})$ and the space of bounded mean oscillation BMO (O,R^{N×n}. The admissible maps u O?R^{N} are of Sobolev class W^{1,p}, satisfying a Dirichlet boundary condition, and to help clarify the significance of the above result the sufficiency condition for W^{1} BMO local minimisers is extended from Lipschitz maps to this admissible class. © 2008 EDP Sciences, SMAI.

Original language | English |
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Pages (from-to) | 111-131 |

Number of pages | 21 |

Journal | ESAIM - Control, Optimisation and Calculus of Variations |

Volume | 16 |

Issue number | 1 |

DOIs | |

Publication status | Published - 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