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A regularity result for a convex functionaland 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. [email protected]
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Abstract

In this paper we prove a regularityresult for local minimizers of functionals of the Calculus of Variations of thetype

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

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 givea 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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