Nodal sets and continuity of eigenfunctions of Kre\uı-Feller operators
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916495877996544 |
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| author | Ngai, Sze-Man Zhang, Meng-Ke Zhao, Wen-Quan |
| author_facet | Ngai, Sze-Man Zhang, Meng-Ke Zhao, Wen-Quan |
| contents | Let $μ$ be a compactly supported positive finite Borel measure on $\R^{d}$. Let $0<λ_{1}\leqλ_{2}\leq\ldots$ be eigenvalues of the Kre$\breve{ı}$n-Feller operator $Δ_μ$. We prove that, on a bounded domain, the nodal set of a continuous $λ_{n}$-eigenfunction of a Kre$\breve{ı}$n-Feller operator divides the domain into at least 2 and at most $n+r-1$ subdomains, where $r$ is the multiplicity of $λ_{n}$. This work generalizes the nodal set theorem of the classical Laplace operator to Kre$\breve{ı}$n-Feller operators on bounded domains. We also prove that on bounded domains on which the classical Green function exists, the eigenfunctions of a Kre$\breve{ı}$n-Feller operator are continuous. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14173 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nodal sets and continuity of eigenfunctions of Kre\uı-Feller operators Ngai, Sze-Man Zhang, Meng-Ke Zhao, Wen-Quan Analysis of PDEs 27A78, 28A80 Let $μ$ be a compactly supported positive finite Borel measure on $\R^{d}$. Let $0<λ_{1}\leqλ_{2}\leq\ldots$ be eigenvalues of the Kre$\breve{ı}$n-Feller operator $Δ_μ$. We prove that, on a bounded domain, the nodal set of a continuous $λ_{n}$-eigenfunction of a Kre$\breve{ı}$n-Feller operator divides the domain into at least 2 and at most $n+r-1$ subdomains, where $r$ is the multiplicity of $λ_{n}$. This work generalizes the nodal set theorem of the classical Laplace operator to Kre$\breve{ı}$n-Feller operators on bounded domains. We also prove that on bounded domains on which the classical Green function exists, the eigenfunctions of a Kre$\breve{ı}$n-Feller operator are continuous. |
| title | Nodal sets and continuity of eigenfunctions of Kre\uı-Feller operators |
| topic | Analysis of PDEs 27A78, 28A80 |
| url | https://arxiv.org/abs/2411.14173 |