Scalar minimizers with fractal singular sets


Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
fonseca@andrew.cmu.edu

Jan Maly
Department of Mathematics
Charles University
Prague, Czech Republic
maly@karlin.mff.cuni.cz

Giuseppe Mingione
Department of Mathematics
University of Parma, Italy
giuseppe.mingione@unipr.it


Abstract: Lack of regularity of local minimizers for convex functionals with nonstandard growth conditions is considered. It is shown that for every $\epsilon > 0$ there exists a function $a \in C^{\alpha}(\Omega)$ such that the functional

\begin{displaymath}{\cal F}: u \mapsto \int_{\Omega}(\vert Du\vert^p+a(x) \vert Du\vert^q)dx\end{displaymath}

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

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