*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 $\Omega\subset\rn{2}$ open, $g\in L^\infty(\Omega)$,
consider the functional $$ G(u,K)=\int_{{\Omega\setminus
}
K}\bigl \, dx+\h{1}{K}, $$ where
$K\subset\Omega$ is relatively closed, $u\in
C^1(\Omega\setminus K)$ and ${\cal H}^{1}$ is the
$1$-dimensional Hausdorff measure in $\rn{2}$. If $(K,u)$ is
a minimizer then $K$ is a $C^{1,\alpha}$ curve outside a
closed ${\cal H}^{1}$-negligible set.

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