Monge-Ampère Equation

Abstract

In this work, we focus on studying the Aleksandrov solution of the Monge-Ampère equation. Initially, we develop the notion of a normal mapping and discuss its properties through proving concepts from convex analysis. Moreover, we define the Monge-Ampère measure over a Borel sigma algebra as well as proving the maximum and comparison principles of this equation. We conclude our study with solving the homogeneous and non-homogeneous Dirichlet problems for the Monge-Ampère operator.