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

Dodd, Thomas J.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 1, pp. 111-131

An a priori Campanato type regularity condition is established for a class of W¹X local minimisers $\overline{u}$ of the general variational integral ${\int }_{\Omega }F\left(\nabla u\left(x\right)\right)\mathrm{d}x$ where $\Omega \subset {ℝ}^n$ is an open bounded domain, F is of class C², F is strongly quasi-convex and satisfies the growth condition$F\left(\xi \right)\le c(1+|\xi {|}^p)$ for a p > 1 and where the corresponding Banach spaces X are the Morrey-Campanato space ${ℒ}^{p,\mu }(\Omega ,{ℝ}^{N\times n})$, µ < n, Campanato space ${ℒ}^{p,n}(\Omega ,{ℝ}^{N\times n})$ and the space of bounded mean oscillation $\mathrm{BMO}\Omega ,{ℝ}^{N\times n})$. The admissible maps $u:\Omega \to {ℝ}^N$ 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 W¹BMO local minimisers is extended from Lipschitz maps to this admissible class.

Classification : 49N60, 49K10
Keywords: calculus of variations, local minimiser, partial regularity, strong quasiconvexity, Campanato space, Morrey space, Morrey-Campanato space, space of bounded mean oscillation, extremals, positive second variation

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