Solutions to Monge-Amp\`ere Equations with Low Codimensional singularities

TL;DR

This paper constructs solutions to Monge-Ampère equations with singular measures on low codimensional sets and explores their regularity, motivated by examples from optimal transport.

Contribution

It introduces new solutions with low codimensional singularities and analyzes their regularity, expanding understanding of Monge-Ampère equations in singular contexts.

Findings

Constructed solutions with singular measures supported on low codimensional sets.

Analyzed the regularity properties of these solutions.

Connected the solutions to optimal transport problems.

Abstract

We construct solutions to Monge-Amp\`ere equations whose Monge-Amp\`ere 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-Amp\`ere equations similar to the ones we study.