Calculus of Variations and Geometric Measure Theory
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I. Fonseca - J. MalĂ˝ - G. Mingione

Scalar Minimizers with Fractal Singular Sets

created on 20 Mar 2003
modified on 24 Apr 2004


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

Preprint 03-CNA-004, Carnegie Mellon CNA


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 $$ \mathcal{F}:u \mapsto \int{\Omega} (
p + a(x)
q)\, dx $$ 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

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