On optimal regularity of free boundary problems and a conjecture of De Giorgi

Herbert Koch Fachbereich Mathematik Universität Dortmund Dortmund, Germany

Giovanni Leoni Department of Mathematical Sciences Carnegie Mellon University and Massimiliano Morini SISSA/ISAS via Beirut 4 340 14 Trieste ITALY

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.