On a general class of free boundary Monge-Amp\`ere equations

TL;DR

This paper introduces a broad class of free boundary Monge-Ampère equations, providing solutions and applications to optimal transport, eigenvalue problems, and geometric issues like Minkowski problems and Kähler-Ricci solitons.

Contribution

It develops a general framework for solving free boundary Monge-Ampère equations with diverse applications in geometry and optimal transport.

Findings

Established existence of solutions for the general class of equations.

Connected solutions to problems in optimal transport and geometric analysis.

Extended the understanding of free boundary problems in Monge-Ampère equations.

Abstract

We solve a general class of free boundary Monge-Amp\`ere equations given by \[ \det D^2u = \lambda \dfrac{f(-u)}{g(u^\star)h(\nabla u)}\chi_{\{u<0\}} \; \text{ in } \mathbb{R}^n, \quad \nabla u (\mathbb{R}^n) = P \] where $P$ is a bounded convex set containing the origin, and $h>0$ on $P$. We consider applications to optimal transport with degenerate densities, Monge-Amp\`ere eigenvalue problems, and geometric problems including a hemispherical Minkowski problem and free boundary K\"ahler-Ricci solitons on toric Fano manifolds.