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

De Maria, Bruno

ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 4, pp. 1002-1017.

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

$${\int }_{\Omega }f(x,Du)\mathrm{d}x$$

where Ω is a bounded open set in ${ℝ}^n$, u ∈ $W_{\mathrm{loc}}^{1,p}$(Ω; ${ℝ}^N$), 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.

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