1. Nodal sets and continuity of eigenfunctions of Kre\u{\i}-Feller operators
- Author
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Ngai, Sze-Man, Zhang, Meng-Ke, and Zhao, Wen-Quan
- Subjects
Mathematics - Analysis of PDEs ,27A78, 28A80 - Abstract
Let $\mu$ be a compactly supported positive finite Borel measure on $\R^{d}$. Let $0<\lambda_{1}\leq\lambda_{2}\leq\ldots$ be eigenvalues of the Kre$\breve{{\i}}$n-Feller operator $\Delta_{\mu}$. We prove that, on a bounded domain, the nodal set of a continuous $\lambda_{n}$-eigenfunction of a Kre$\breve{{\i}}$n-Feller operator divides the domain into at least 2 and at most $n+r-1$ subdomains, where $r$ is the multiplicity of $\lambda_{n}$. This work generalizes the nodal set theorem of the classical Laplace operator to Kre$\breve{{\i}}$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$\breve{{\i}}$n-Feller operator are continuous.
- Published
- 2024