PreprintCC BY 4.0arXiv
(In)finiteness of slopes for Monge-Ampère equations at flat boundary pieces
Abstract
On any bounded convex domain Ω⊂ℝn with a d-dimensional flat boundary piece F⊂∂Ω, we consider convex solutions with zero boundary values to the Monge-Ampère equations detD2 u= dist( · ,Aff(F))α, detD2 u=dist( · ,∂Ω)α, detD2 u=(−u)α, where dist stands for Euclidean distance and Aff(F) is the affine hull of F. We show that the threshold of the exponent α∈ℝ for u to have infinite slope at F is α≤ 2d−n. For the first two equations, we also show that no solution exists when α≤ 2d−2n. Moreover, when Ω satisfies the ``exterior cone condition with a d-dimensional ridge'' at F, both thresholds are sharp for the first equation, and the infinite-slope threshold is sharp for the last equation.
Related
- Boundary Hölder Regularity of a Class of Monge–Ampère Equations at Some Special Points
- Local Well-posedness of Free Boundary Problems for the Euler--Monge--Ampère Equations
- Complex Monge-Ampère equations with boundary values: a variational approach
- Uniqueness and sharp boundary estimates for degenerate Monge-Ampère equations with singular measures
- Regularity of Free Boundary for Monge-Ampère Obstacle Problem up to the Boundary
BibTeX
@misc{nie2026infiniteness,
title = {{(In)finiteness of slopes for Monge-Ampère equations at flat boundary pieces}},
author = {Xin Nie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12165},
eprint = {2610.12165},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12165},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - (In)finiteness of slopes for Monge-Ampère equations at flat boundary pieces
AU - Xin Nie
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.12165
UR - https://arxiv.org/abs/2610.12165
AB - On any bounded convex domain Ω⊂ℝ^n with a d-dimensional flat boundary piece F⊂∂Ω, we consider convex solutions with zero boundary values to the Monge-Ampère equations detD^2 u= dist( · ,Aff(F))^α, detD^2 u=dist( · ,∂Ω)^α, detD^2 u=(−u)^α, where dist stands for Euclidean distance and Aff(F) is the affine hull of F. We show that the threshold of the exponent α∈ℝ for u to have infinite slope at F is α≤ 2d−n. For the first two equations, we also show that no solution exists when α≤ 2d−2n. Moreover, when Ω satisfies the ``exterior cone condition with a d-dimensional ridge'' at F, both thresholds are sharp for the first equation, and the infinite-slope threshold is sharp for the last equation.
ER -