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

Thomas J. Dodd

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

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

Abstract

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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 ) ) d x where Ω n is an open bounded domain, F is of class C2, F is strongly quasi-convex and satisfies the growth condition F ( 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.

How to cite

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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 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 -

References

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