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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.04586 |
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Table of Contents:
- We establish global $C^{1,β}$ and $W^{2, p}$ regularity for singular Monge-Ampère equations of the form \[\det D^2 u \sim \text{dist}^{-α}(\cdot,\partialΩ),\quad α\in (0, 1),\] under suitable conditions on the boundary data and domains. Our results imply that the convex Aleksandrov solution to the singular Monge-Ampère equation \[\det D^2 u=|u|^{-α}\quad \text{in}\quadΩ,\quad u=0\quad \text{in}\quad \partialΩ, \quad α\in (0, 1),\] where $Ω$ is a $C^3$, bounded, and uniformly convex domain, is globally $C^{1,β}$ and belongs to $W^{2, p}$ for all $p<1/α$.