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

In the joint work with G. Leoni and M. Morini 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.