AN A PRIORI CAMPANATO TYPE REGULARITY CONDITION FOR LOCAL MINIMISERS IN THE CALCULUS OF VARIATIONS

作者：Dodd Thomas J^*

来源：ESAIM: Control, Optimisation and Calculus of Variations , 2010, 16(1): 111-131.

DOI：10.1051/cocv:2008066

• 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

An a priori Campanato type regularity condition is established for a class of (WX)-X-1 local minimisers (u) over bar of the general variational integral
integral F-Omega(del u(x))dx
where Omega subset of R-n is an open bounded domain, F is of class C-2, F is strongly quasi-convex and satisfies the growth condition
F(xi) <= c(1 + |xi|(p))
for a p > 1 and where the corresponding Banach spaces X are the Morrey-Campanato space L-p,L-mu(Omega, R-Nxn), mu < n, Campanato space L-p,L-n(Omega, R-Nxn) and the space of bounded mean oscillation BMO(Omega, R-Nxn). The admissible maps u: Omega -> R-N are of Sobolev class W-1,W-p, satisfying a Dirichlet boundary condition, and to help clarify the significance of the above result the sufficiency condition for (WBMO)-B-1 local minimisers is extended from Lipschitz maps to this admissible class.

• 出版日期2010-3