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{\bf \large On optimal regularity of free boundary problems and a
conjecture of De Giorgi}

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Herbert Koch\\
 Fachbereich Mathematik\\
Universit\"{a}t Dortmund\\
 Dortmund, Germany\\

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Giovanni Leoni\\
Department of Mathematical Sciences\\
Carnegie Mellon University\\
 
and\\

Massimiliano Morini\\
 SISSA/ISAS\\
 via Beirut 4\\
 340 14 Trieste ITALY\\
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\noindent {\bf Abstract}: We present a general theory to study optimal
regularity for a large class of nonlinear elliptic systems satisfying
general boundary conditions and in the presence of a geometric
transmission condition on the free-boundary. As an application we give
a full positive answer to a conjecture of De Giorgi on the analyticity
of local minimizers of the Mumford-Shah functional.


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