Electron. J. Differential Equations, Vol. 2025 (2025), No. 12, pp. 1-25.
Nodal sets and continuity of eigenfunctions of Krein-Feller operators
Sze-Man Ngai, Meng-Ke Zhang, Wen-Quan Zhao
Abstract:
Let \(\mu\) be a compactly supported positive finite Borel measure on
\(\mathbb{R}^d\). Let \(0<\lambda_1\leq\lambda_2\leq\cdots\) be eigenvalues
of the Krein-Feller operator \(\Delta_{\mu}\).
We prove that, on a bounded domain, the nodal set of a continuous
\(\lambda_n\)-eigenfunction of a Krein-Feller operator divides the
domain into at least 2 and at most \(n+r_{n}-1\) subdomains, where \(r_{n}\)
is the multiplicity of \(\lambda_n\). This work generalizes the nodal set
theorem of the classical Laplace operator to Krein-Feller operators
on bounded domains. We also prove that on bounded domains on which the
classical Green function exists, the eigenfunctions of a Krein-Feller
operator are continuous.
Submitted April 17, 2024. Published February 5, 2025.
Math Subject Classifications: 35J05, 35B05, 34L10, 28A80, 35J08.
Key Words: Krein-Feller operators; nodal set; continuous eigenfunctions.
DOI: 10.58997/ejde.2025.12
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Sze-Man Ngai
Key Laboratory of High Performance Computing
and Stochastic Information
Processing (HPCSIP)
Hunan Normal University
Changsha, Hunan 410081, China
email: ngai@bimsa.cn
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Meng-Ke Zhang
Key Laboratory of High Performance Computing
and Stochastic Information
Processing (HPCSIP)
Hunan Normal University
Changsha, Hunan 410081, China
email: 2750600901@qq.com
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Wen-Quan Zhao
Key Laboratory of High Performance Computing
and Stochastic Information
Processing (HPCSIP)
Hunan Normal University
Changsha, Hunan 410081, China
email: zhaowq1008@hunnu.edu.cn
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