Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 12, pp. 1-25.

Title: Nodal sets and continuity of eigenfunctions of Krein-Feller operators

Authors: Sze-Man Ngai  (Hunan Normal Univ., Changsha, Hunan, China)
         Meng-Ke Zhang (Hunan Normal Univ., Changsha, Hunan, China)
         Wen-Quan Zhao (Hunan Normal Univ., Changsha, Hunan, China)
  
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 05, 2025.
Math Subject Classifications: 35J05, 35B05, 34L10, 28A80, 35J08.
Key Words: Krein-Feller operators; nodal set; continuous eigenfunctions.