Diffusion orthogonal polynomials in 3-dimensional domains bounded by developable surfaces
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917854831443968 |
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| author | Orevkov, S. Yu. |
| author_facet | Orevkov, S. Yu. |
| contents | The following problem is studied: describe the triplets $(Ω,g,μ)$, $μ=ρ\,dx$, where $g= (g^{ij}(x))$ is the (co)metric associated with the symmetric second order differential operator $L(f) = \frac{1}ρ\sum_{ij} \partial_i (g^{ij} ρ\,\partial_j f)$ defined on a domain $Ω$ of $\mathbb R^n$ and such that there exists an orthonormal basis of $\mathcal L^2(μ)$ made of polynomials which are eigenvectors of $L$, and the basis is compatible with the filtration of the space of polynomials with respect to some weighted degree.
In a joint paper with D. Bakry and M. Zani this problem was solved in dimension 2 for the usual degree. In the author's subsequent paper this problem was solved in dimension 2 for any weighted degree. In the present paper this problem is solved in dimension 3 for the usual degree under the condition that $\partialΩ$ contains a piece of a tangent developable surface. The proof is based on Plücker-like formulas in the form given by Ragni Piene. All the found solutions are generalized for any dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_15100 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Diffusion orthogonal polynomials in 3-dimensional domains bounded by developable surfaces Orevkov, S. Yu. Classical Analysis and ODEs Algebraic Geometry The following problem is studied: describe the triplets $(Ω,g,μ)$, $μ=ρ\,dx$, where $g= (g^{ij}(x))$ is the (co)metric associated with the symmetric second order differential operator $L(f) = \frac{1}ρ\sum_{ij} \partial_i (g^{ij} ρ\,\partial_j f)$ defined on a domain $Ω$ of $\mathbb R^n$ and such that there exists an orthonormal basis of $\mathcal L^2(μ)$ made of polynomials which are eigenvectors of $L$, and the basis is compatible with the filtration of the space of polynomials with respect to some weighted degree. In a joint paper with D. Bakry and M. Zani this problem was solved in dimension 2 for the usual degree. In the author's subsequent paper this problem was solved in dimension 2 for any weighted degree. In the present paper this problem is solved in dimension 3 for the usual degree under the condition that $\partialΩ$ contains a piece of a tangent developable surface. The proof is based on Plücker-like formulas in the form given by Ragni Piene. All the found solutions are generalized for any dimension. |
| title | Diffusion orthogonal polynomials in 3-dimensional domains bounded by developable surfaces |
| topic | Classical Analysis and ODEs Algebraic Geometry |
| url | https://arxiv.org/abs/2210.15100 |