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

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 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 W¹ BMO local minimisers is extended from Lipschitz maps to this admissible class. © 2008 EDP Sciences, SMAI.