On a general class of free boundary Monge-Ampère equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912525935706112 |
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| author | Collins, Tristan C. Firester, Benjy |
| author_facet | Collins, Tristan C. Firester, Benjy |
| contents | We solve a general class of free boundary Monge-Ampère equations given by \[
\det D^2u = λ\dfrac{f(-u)}{g(u^\star)h(\nabla u)}χ_{\{u<0\}} \; \text{ in } \mathbb{R}^n, \quad \nabla u (\mathbb{R}^n) = P \] 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-Ampère eigenvalue problems, and geometric problems including a hemispherical Minkowski problem and free boundary Kähler-Ricci solitons on toric Fano manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05551 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a general class of free boundary Monge-Ampère equations Collins, Tristan C. Firester, Benjy Analysis of PDEs Differential Geometry We solve a general class of free boundary Monge-Ampère equations given by \[ \det D^2u = λ\dfrac{f(-u)}{g(u^\star)h(\nabla u)}χ_{\{u<0\}} \; \text{ in } \mathbb{R}^n, \quad \nabla u (\mathbb{R}^n) = P \] 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-Ampère eigenvalue problems, and geometric problems including a hemispherical Minkowski problem and free boundary Kähler-Ricci solitons on toric Fano manifolds. |
| title | On a general class of free boundary Monge-Ampère equations |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2508.05551 |