A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations

Abstract

In dimension n ⩾ 3 , we define a generalization of the classical two-dimensional partial Legendre transform, that reduces interior regularity of the generalized Monge–Ampère equation det D 2 u = k ( x , u , Du ) to regularity of a divergence form quasilinear system of special form. This is then used to obtain smoothness of C 2 , 1 solutions, having n - 1 nonvanishing principal curvatures, to certain subelliptic Monge–Ampère equations in dimension n ⩾ 3 . A corollary is that if k ⩾ 0 vanishes only at nondegenerate critical points, then a C 2 , 1 convex solution u is smooth if and only if the symmetric function of degree n - 1 of the principal curvatures of u is positive, and moreover, u fails to be C 3 , 1 - 2 n + ɛ when not smooth.