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A regularity result for a convex functional and bounds for the singularset

Published online by Cambridge University Press:  11 August 2009

Bruno De Maria
Bruno De Maria*
Affiliation:
Dipartimento di Matematica e Applicazioni “R. Caccioppoli” Università di Napoli “Federico II” Via Cintia, 80126 Napoli, Italy. bruno.demaria@dma.unina.it
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Abstract

In this paper we prove a regularity result for local minimizers of functionals of the Calculus of Variations of the type

$$ \int_{\Omega}f(x, Du)\ {\rm d}x $$

where Ω is a bounded open set in $\mathbb{R}^{n}$, u∈$W^{1,p}_{\rm loc}$(Ω; $\mathbb{R}^{N}$), p> 1, n≥ 2 and N≥ 1. We use the technique of difference quotient without the usual assumption on the growth of the second derivatives of the function f. We apply this result to give a bound on the Hausdorff dimension of the singular set of minimizers.

Keywords

Partial regularitysingular setsfractional differentiabilityvariational integrals
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 4 , October 2010 , pp. 1002 - 1017
Copyright
© EDP Sciences, SMAI, 2009

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