Solutions to Monge-Ampère Equations with Low Codimensional singularities

We construct solutions to Monge-Ampère equations whose Monge-Ampère measures contain singular components supported on low codimensional sets. We also study the regularity of such solutions. To motivate our construction, we present examples arising from optimal transport where the potential of optimal transport maps satisfy Monge-Ampère equations similar to the ones we study.