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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*
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 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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2009

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