The Cauchy problem for the homogeneous Monge–Ampère equation, II. Legendre transform

We continue our study of the Cauchy problem for the homogeneous (real and complex) Monge–Ampère equation (HRMA/HCMA). In the prequel (Y.A. Rubinstein and S. Zelditch [27]) a quantum mechanical approach for solving the HCMA was developed, and was shown to coincide with the well-known Legendre transform approach in the case of the HRMA. In this article—that uses tools of convex analysis and can be read independently—we prove that the candidate solution produced by these methods ceases to solve the HRMA, even in a weak sense, as soon as it ceases to be differentiable. At the same time, we show that it does solve the equation on its dense regular locus, and we derive an explicit a priori upper bound on its Monge–Ampère mass. The technique involves studying regularity of Legendre transforms of families of non-convex functions.