Calculus of Variations and Geometric Measure Theory
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L. Ambrosio - D. Pallara

Partial Regularity of Free Discontinuity Sets I

created on 01 Apr 1996
modified by pallara on 23 Jun 2012


Published Paper

Inserted: 1 apr 1996
Last Updated: 23 jun 2012

Journal: Annali della Scuola Normale Superiore di Pisa
Year: 1997


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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