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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