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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.03582 |
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| _version_ | 1866917135596388352 |
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| author | Inauen, Dominik Lewicka, Marta |
| author_facet | Inauen, Dominik Lewicka, Marta |
| contents | We prove that every $\mathcal{C}^1(\barω)$-regular subsolution of the Monge-Ampère system posed on a $2$-dimensional domain $ω$ and with target codimension $2$, can be uniformly approximated by its exact solutions with regularity $\mathcal{C}^{1,α}(\barω)$ for any $α<\min\{1, \frac{s+β}{2}\}$, where $\mathcal{C}^{s,β}$ is the assumed regularity of the system's right hand side. This result suggests the full flexibility of Poznyak's theorem for isometric immersions of $2$d Riemannian manifolds into $\mathbb{R}^4$, and asserts it in the parallel setting of the Monge-Ampère system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03582 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Monge-Ampère system in dimension two is fully flexible in codimension two Inauen, Dominik Lewicka, Marta Analysis of PDEs We prove that every $\mathcal{C}^1(\barω)$-regular subsolution of the Monge-Ampère system posed on a $2$-dimensional domain $ω$ and with target codimension $2$, can be uniformly approximated by its exact solutions with regularity $\mathcal{C}^{1,α}(\barω)$ for any $α<\min\{1, \frac{s+β}{2}\}$, where $\mathcal{C}^{s,β}$ is the assumed regularity of the system's right hand side. This result suggests the full flexibility of Poznyak's theorem for isometric immersions of $2$d Riemannian manifolds into $\mathbb{R}^4$, and asserts it in the parallel setting of the Monge-Ampère system. |
| title | The Monge-Ampère system in dimension two is fully flexible in codimension two |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.03582 |