Nodal sets and continuity of eigenfunctions of Krein-Feller operators on Riemannian manifolds
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| Format: | Preprint |
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2024
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| _version_ | 1866915062896132096 |
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| author | Ngai, Sze-Man Zhao, Wen-Quan |
| author_facet | Ngai, Sze-Man Zhao, Wen-Quan |
| contents | Let $d\geq1$, $Ω$ be a bounded domain of a smooth complete Riemannian d-manifold M, and $μ$ be a positive finite Borel measure with compact support in $\overlineΩ$. We prove the Courant nodal domain theorem for the eigenfunctions of Kreĭn-Feller operator $Δ_μ$ under the assumption that such eigenfunctions are continuous on $\overlineΩ$. For $d\geq2$, We prove that on a bounded domain $Ω\subset M$ with smooth boundary and on which the Green's function of the Laplace-Beltrami operator exists, the eigenfunctions of $Δ_μ$ are continuous on $Ω$. We also prove that if M is compact and $\partial M=\emptyset$, then the eigenfuctions of $Δ_μ$ are continuous on M. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10007 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nodal sets and continuity of eigenfunctions of Krein-Feller operators on Riemannian manifolds Ngai, Sze-Man Zhao, Wen-Quan Functional Analysis Let $d\geq1$, $Ω$ be a bounded domain of a smooth complete Riemannian d-manifold M, and $μ$ be a positive finite Borel measure with compact support in $\overlineΩ$. We prove the Courant nodal domain theorem for the eigenfunctions of Kreĭn-Feller operator $Δ_μ$ under the assumption that such eigenfunctions are continuous on $\overlineΩ$. For $d\geq2$, We prove that on a bounded domain $Ω\subset M$ with smooth boundary and on which the Green's function of the Laplace-Beltrami operator exists, the eigenfunctions of $Δ_μ$ are continuous on $Ω$. We also prove that if M is compact and $\partial M=\emptyset$, then the eigenfuctions of $Δ_μ$ are continuous on M. |
| title | Nodal sets and continuity of eigenfunctions of Krein-Feller operators on Riemannian manifolds |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2412.10007 |