ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - October 2010 - Volume 16, Issue 04  

Research Article

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

De Maria, Bruno

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

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

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.

(Received February 4 2009)

(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