Regularity of Weak Solutions to the Monge–ampère Equation

We study the properties of generalized solutions to the Monge– Ampère equation detD2u = ν, where the Borel measure ν satisfies a condition, introduced by Jerison, that is weaker than the doubling property. When ν = f dx, this condition, which we call D , admits the possibility of f vanishing or becoming infinite. Our analysis extends the regularity theory (due to Caffarelli) available when 0 < λ ≤ f ≤ Λ < ∞, which implies that ν = f dx is doubling. The main difference between the D 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.