Global $C^{1,β}$ and $W^{2, p}$ regularity for some singular Monge-Ampère equations
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908925221142528 |
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| author | Le, Nam Q. Savin, Ovidiu |
| author_facet | Le, Nam Q. Savin, Ovidiu |
| 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 $p<1/α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04586 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global $C^{1,β}$ and $W^{2, p}$ regularity for some singular Monge-Ampère equations Le, Nam Q. Savin, Ovidiu Analysis of PDEs 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 $p<1/α$. |
| title | Global $C^{1,β}$ and $W^{2, p}$ regularity for some singular Monge-Ampère equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.04586 |