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arXiv · 2508.05551

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

Tristan C. Collins·Benjy Firester
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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.

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Tristan C. Collins, Benjy Firester. 2025-08-07. On a general class of free boundary Monge-Amp\`ere equations. https://arxiv.org/abs/2508.05551

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