*Published Paper*

**Inserted:** 20 mar 2003

**Last Updated:** 24 apr 2004

**Journal:** Arch. Ration. Mech. Anal.

**Volume:** 172

**Number:** 2

**Pages:** 295-312

**Year:** 2004

**Notes:**

Preprint 03-CNA-004, Carnegie Mellon CNA

**Abstract:**

Lack of regularity of local minimizers for convex functionals with
non-standard growth conditions is considered. It is shown
that for every $\varepsilon > 0$ there exists a function $a \in
C^{\alpha}(\Omega)$ such that the functional
$$
\mathcal{F}:u \mapsto \int_{{\Omega}} (

Du^{p} + a(x)

Du^{q)\,} dx
$$
admits a local minimizer $u \in W^{1,p}(\Omega)$ whose set of non-Lebesque
points is a closed set $\Sigma$ with
dim$_{\mathcal H}(\Sigma)>N-p-\varepsilon$,
and where $1< p<N< N+\alpha<q<+\infty$.
\end{abstract}

**Keywords:**
regularity, Hausdorff dimension, Non standard growth conditions