Regularity of weak solutions to the Monge–Ampère equation

Abstract:

We study the properties of generalized solutions to the Monge–Ampère equation $\det D^2 u = \nu$, where the Borel measure $\nu$ satisfies a condition, introduced by Jerison, that is weaker than the doubling property. When $\nu = f dx$, this condition, which we call $D_{\epsilon }$, admits the possibility of $f$ vanishing or becoming infinite. Our analysis extends the regularity theory (due to Caffarelli) available when $0 < \lambda \leq f \leq \Lambda < \infty$, which implies that $\nu = f dx$ is doubling. The main difference between the $D_{\epsilon }$ case and the case when $f$ is bounded between two positive constants is the need to use a variant of the Aleksandrov maximum principle (due to Jerison) and some tools from convex geometry, in particular the Hausdorff metric.