Regularity and classification of the free boundary for a Monge-Ampère obstacle problem
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912354480947200 |
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| author | Jin, Tianling Tu, Xushan Xiong, Jingang |
| author_facet | Jin, Tianling Tu, Xushan Xiong, Jingang |
| contents | We study convex solutions to the Monge-Ampère obstacle problem \[ \operatorname{det} D^2 v=g v^qχ_{\{v>0\}}, \quad v \geq 0, \] where $q \in [0,n)$ is a constant and $g$ is a bounded positive function. This problem emerges from the $L_p$ Minkowski problem. We establish $C^{1, α}$ regularity for the strictly convex part of the free boundary $\partial\{v=0\}$. Furthermore, when $g \in C^α$, we prove a Schauder-type estimate. As a consequence, when $g\equiv 1$, we obtain a Liouville theorem for entire solutions with unbounded coincidence sets $\{v=0\}$. Combined with existing results, this provides a complete classification of entire solutions for the case $q=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21253 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity and classification of the free boundary for a Monge-Ampère obstacle problem Jin, Tianling Tu, Xushan Xiong, Jingang Analysis of PDEs We study convex solutions to the Monge-Ampère obstacle problem \[ \operatorname{det} D^2 v=g v^qχ_{\{v>0\}}, \quad v \geq 0, \] where $q \in [0,n)$ is a constant and $g$ is a bounded positive function. This problem emerges from the $L_p$ Minkowski problem. We establish $C^{1, α}$ regularity for the strictly convex part of the free boundary $\partial\{v=0\}$. Furthermore, when $g \in C^α$, we prove a Schauder-type estimate. As a consequence, when $g\equiv 1$, we obtain a Liouville theorem for entire solutions with unbounded coincidence sets $\{v=0\}$. Combined with existing results, this provides a complete classification of entire solutions for the case $q=0$. |
| title | Regularity and classification of the free boundary for a Monge-Ampère obstacle problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.21253 |