Published Paper
Inserted: 1 apr 1996
Last Updated: 23 jun 2012
Journal: Annali della Scuola Normale Superiore di Pisa
Year: 1997
Abstract:
We prove a regularity theorem for minimizers of a class of
free discontinuity problems including the following example:
given $\Omega\subset\rn{n}$ open, $g\in L^\infty(\Omega)$,
consider the functional $$ G(u,K)=\int{\Omega\setminus
K}\bigl \, dx+\h{n-1}{K}, $$ where
$K\subset\Omega$ is relatively closed, $u\in
C^1(\Omega\setminus K)$ and ${\cal H}^{n-1}$ is the
$(n-1)$-dimensional Hausdorff measure in $\rn{n}$. If
$(K,u)$ is a minimizer and $
\nabla u
$ belongs to the Morrey
space $L^{2,\lambda}(\Omega)$ for some $\lambda>n-1$,
then $K$ is a $C^{1,\alpha}$ hypersurface outside a closed
${\cal H}^{n-1}$-negligible singular set.
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