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.