Differential characterization of quadratic surfaces
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866908765611098112 |
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| author | Zawalski, Bartłomiej |
| author_facet | Zawalski, Bartłomiej |
| contents | Let $f\in W^{3,1}_{\mathrm{loc}}(Ω)$ be a function defined on a connected open subset $Ω\subseteq\mathbb R^2$. We will show that its graph is contained in a quadratic surface if and only if $f$ is a weak solution to a certain system of third-order partial differential equations unless the Hessian determinant of $f$ is non-positive everywhere on $Ω$. Moreover, we will prove that the system is, in a sense, the simplest possible in a wide class of differential equations, which will lead to the classification of all polynomial partial differential equations satisfied by parametrizations of generic quadratic surfaces. Although we will mainly use the tools of linear and commutative algebra, the theorem itself is also somewhat related to holomorphic functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_08073 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Differential characterization of quadratic surfaces Zawalski, Bartłomiej Analysis of PDEs 53A05 (primary), 12H05, 12H20, 13N10 (secondary) Let $f\in W^{3,1}_{\mathrm{loc}}(Ω)$ be a function defined on a connected open subset $Ω\subseteq\mathbb R^2$. We will show that its graph is contained in a quadratic surface if and only if $f$ is a weak solution to a certain system of third-order partial differential equations unless the Hessian determinant of $f$ is non-positive everywhere on $Ω$. Moreover, we will prove that the system is, in a sense, the simplest possible in a wide class of differential equations, which will lead to the classification of all polynomial partial differential equations satisfied by parametrizations of generic quadratic surfaces. Although we will mainly use the tools of linear and commutative algebra, the theorem itself is also somewhat related to holomorphic functions. |
| title | Differential characterization of quadratic surfaces |
| topic | Analysis of PDEs 53A05 (primary), 12H05, 12H20, 13N10 (secondary) |
| url | https://arxiv.org/abs/2304.08073 |