A property for the Monge-Ampère equation

Abstract

Let Ω ⊆ ℝⁿ be a non-empty open bounded set and h: Ω → ℝ be a non-negative continuous function. We prove that for any u ∈ C²(Ω) ∩ C¹($\overline{\Omega}$) solution of the Monge–Ampère equation

$${\rm{det}}(D^2u)=h\;\;{\rm{in}}\;\Omega,$$

, then ∇u satisfies the convex hull property: ∇u(Ω) ⊆ conv(∇u(∂Ω)).