Elliptic and Parabolic Equations - Abstract
De Lellis, Camillo
The Plateau problem consists in finding the surface of least area spanning a given contour. In a celebrated work of the sixties, Federer and Fleming introduced the theory of currents, providing an elegant existence theorem. Following the work of De Giorgi, Almgren, Simons, Federer and Simon (among others), there is a quite satisfactory regularity theory for area-minimizing currents of codimension $1$. In higher codimension the situation is dramatically different, because branching (and hence singularities of codimension $2$) can occur. A deep regularity theory has been developed 30 years ago by Almgren to handle this case and it is contained in a monograph of 950 pages. In some recent works with Emanuele Spadaro we have found much shorter proofs for a good portion of Almgren's monograph. These results come as a combination of his ideas with new techniques, they give some new points of view and provide links to other topics in analysis. Our first work focuses on the theory of harmonic multiple valued functions: useful objects to approximate area-minimizing currents in a neighborhood of a branch point.