### 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 |

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### 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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*ESAIM - Control, Optimisation and Calculus of Variations*, vol. 16, no. 1, pp. 111-131. https://doi.org/10.1051/cocv:2008066

**An a priori Campanato type regularity condition for local minimisers in the calculus of variations.** / Dodd, Thomas J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An a priori Campanato type regularity condition for local minimisers in the calculus of variations

AU - Dodd, Thomas J.

PY - 2010/1

Y1 - 2010/1

N2 - 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. © 2008 EDP Sciences, SMAI.

AB - 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. © 2008 EDP Sciences, SMAI.

KW - Calculus of variations

KW - Campanato space

KW - Extremals

KW - Local minimiser

KW - Morrey space

KW - Morrey-Campanato space

KW - Partial regularity

KW - Positive second variation

KW - Space of bounded mean oscillation

KW - Strong quasiconvexity

UR - http://www.scopus.com/inward/record.url?scp=77950252890&partnerID=8YFLogxK

U2 - 10.1051/cocv:2008066

DO - 10.1051/cocv:2008066

M3 - Article

VL - 16

SP - 111

EP - 131

JO - ESAIM - Control, Optimisation and Calculus of Variations

JF - ESAIM - Control, Optimisation and Calculus of Variations

SN - 1292-8119

IS - 1

ER -