Colloquium

Abstract

The Monge-Ampere equation det(D²u)(x) = k(x,u,Du) arises in geometry and mass transportation, and serves as a model for fully nonlinear second order equations. The equation is elliptic if k > 0 and the theory is well-developed in this case due to work of Alexandrov, Pogorelov, Caffarelli, Nirenberg and Spruck, and many others. In the degenerate case where k is permitted to vanish, much less is known regarding smoothness of solutions, apart from the regularity work in two dimensions of Guan and others.

The partial Legendre transform, or semispherical mapping, associated to a convex solution of the Monge-Ampere equation has proved to be a useful tool in the investigation of regularity in two dimensions. We extend this transform to higher dimensions where it satisfies a divergence form quasilinear system of special form, and use this to obtain regularity results for the degenerate Monge-Ampere equation in higher dimensions. A simply stated corollary is that if a smooth k vanishes at a nondegenerate critical point, then a convex solution u is smooth if and only if the (n-1) curvature of u is positive at the critical point. By j curvature we mean the jth symmetric function of the principal curvatures of u. This is joint work with Cristian Rios and Richard Wheeden.