## ESAIM: Control, Optimisation and Calculus of Variations

### A regularity result for a convex functional and bounds for the singular set

Dipartimento di Matematica e Applicazioni “R. Caccioppoli” Università di Napoli “Federico II” Via Cintia, 80126 Napoli, Italy. bruno.demaria@dma.unina.it

Abstract

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

where Ω is a bounded open set in , u (Ω; ), 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.

(Revised May 6 2009)

(Online publication August 11 2009)

Key Words:

• Partial regularity;
• singular sets;
• fractional differentiability;
• variational integrals

Mathematics Subject Classification:

• 35J50;
• 35J60;
• 35B65