Nodal sets and continuity of eigenfunctions of Krein-Feller operators

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. For more information see https://ejde.math.txstate.edu/Volumes/2025/12/abstr.html