On a general class of free boundary Monge-Ampère equations

2025/08/07 by Collins, Tristan C., Firester, Benjy · 1 citation
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.05551

Abstract

We solve a general class of free boundary Monge-Ampère equations given by det D²u = λ\dfracf(-u)g(u^⋆)h(∇ u)χ_\ult;0\ in ℝⁿ, ∇ u (ℝⁿ) = 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-Ampère eigenvalue problems, and geometric problems including a hemispherical Minkowski problem and free boundary Kähler-Ricci solitons on toric Fano manifolds.

Citations

• Boundary regularity of optimal transport maps on convex domains
• A free boundary Monge-Ampère equation and applications to complete Calabi-Yau metrics
• Generalized Monge-Ampère functionals and related variational problems
• Complete Calabi-Yau metrics in the complement of two divisors
• Regularity of free boundary for the Monge-Ampère obstacle problem
• Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems
• Affine hemispheres of elliptic type
• Schauder estimates for degenerate Monge-Ampère equations and smoothness of the eigenfunctions
• Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties
• Interpolations, convexity and geometric inequalities
• The geometry of optimal transportation
• Boundary regularity of maps with convex potentials

Cited by

• Homogeneous optimal transport maps between oblique cones

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• The Dirichlet problem for the Monge-Ampère equation on Hermitian manifolds with boundary
• A free boundary Monge-Ampère equation and applications to complete Calabi-Yau metrics
• Fully Degenerate Monge Ampére Equations
• An optimal transport approach to Monge-Ampère equations on compact Hessian manifolds
• Regularity and classification of the free boundary for a Monge-Ampère obstacle problem