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

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 16, Issue: 1, page 111-131
- ISSN: 1292-8119

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topDodd, Thomas J.. "An a priori Campanato type regularity condition for local minimisers in the calculus of variations." ESAIM: Control, Optimisation and Calculus of Variations 16.1 (2010): 111-131. <http://eudml.org/doc/250743>.

@article{Dodd2010,

abstract = {
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(\nabla u(x))\,\{\rm d\}x$
where $\Omega \subset \mathbb\{R\}^n$ is an open bounded domain, F is of class C2, F is strongly quasi-convex and satisfies the growth condition$ F(\xi)\leq c(1+|\xi|^p)$
for a p > 1 and where the corresponding Banach spaces X are the Morrey-Campanato space $\mathcal\{L\}^\{p,\mu\} (\Omega,\mathbb\{R\}^\{N\times n\})$, µ < n, Campanato space $\mathcal\{L\}^\{p,n\}(\Omega,\mathbb\{R\}^\{N\times n\})$ and the space of bounded mean oscillation $ \{\rm BMO\} \Omega,\mathbb\{R\}^\{N\times n\})$. The admissible maps $u\colon \Omega \to \mathbb\{R\}^N$ 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.
},

author = {Dodd, Thomas J.},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

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; calculus of variations},

language = {eng},

month = {1},

number = {1},

pages = {111-131},

publisher = {EDP Sciences},

title = {An a priori Campanato type regularity condition for local minimisers in the calculus of variations},

url = {http://eudml.org/doc/250743},

volume = {16},

year = {2010},

}

TY - JOUR

AU - Dodd, Thomas J.

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/1//

PB - EDP Sciences

VL - 16

IS - 1

SP - 111

EP - 131

AB -
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(\nabla u(x))\,{\rm d}x$
where $\Omega \subset \mathbb{R}^n$ is an open bounded domain, F is of class C2, F is strongly quasi-convex and satisfies the growth condition$ F(\xi)\leq c(1+|\xi|^p)$
for a p > 1 and where the corresponding Banach spaces X are the Morrey-Campanato space $\mathcal{L}^{p,\mu} (\Omega,\mathbb{R}^{N\times n})$, µ < n, Campanato space $\mathcal{L}^{p,n}(\Omega,\mathbb{R}^{N\times n})$ and the space of bounded mean oscillation $ {\rm BMO} \Omega,\mathbb{R}^{N\times n})$. The admissible maps $u\colon \Omega \to \mathbb{R}^N$ 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.

LA - eng

KW - 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; calculus of variations

UR - http://eudml.org/doc/250743

ER -

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