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.
On optimal regularity of free boundary problems and a conjecture of De Giorgi / H. Koch; G. Leoni; M. Morini. - In: COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS. - ISSN 0010-3640. - 58(2005), pp. 1051-1076. [10.1002/cpa.20065]