An a priori Campanato type regularity condition for local minimisers in the calculus of variations
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 1, pp. 111-131.

An a priori Campanato type regularity condition is established for a class of W1X local minimisers u ¯ of the general variational integral Ω F(u(x))dx where Ω n is an open bounded domain, F is of class C2, F is strongly quasi-convex and satisfies the growth conditionF(ξ)c(1+|ξ| p ) for a p > 1 and where the corresponding Banach spaces X are the Morrey-Campanato space p,μ (Ω, N×n ), µ < n, Campanato space p,n (Ω, N×n ) and the space of bounded mean oscillation BMO Ω, N×n ). The admissible maps u:Ω N 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 W1BMO local minimisers is extended from Lipschitz maps to this admissible class.

DOI: 10.1051/cocv:2008066
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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     title = {An a priori {Campanato} type regularity condition for local minimisers in the calculus of variations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {111--131},
     publisher = {EDP-Sciences},
     volume = {16},
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     doi = {10.1051/cocv:2008066},
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Dodd, Thomas J. An a priori Campanato type regularity condition for local minimisers in the calculus of variations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 1, pp. 111-131. doi : 10.1051/cocv:2008066. http://archive.numdam.org/articles/10.1051/cocv:2008066/

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