*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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