Nodal sets and continuity of eigenfunctions of Krein-Feller operators on Riemannian manifolds

Let $d\ge 1$, $\mathrm{\Omega }$ be a bounded domain of a smooth complete Riemannian d-manifold M, and $\mu$ be a positive finite Borel measure with compact support in $\bar{\mathrm{\Omega }}$. We prove the Courant nodal domain theorem for the eigenfunctions of Kreĭn-Feller operator ${\mathrm{\Delta }}_{\mu }$ under the assumption that such eigenfunctions are continuous on $\bar{\mathrm{\Omega }}$. For $d\ge 2$, We prove that on a bounded domain $\mathrm{\Omega }\subset M$ with smooth boundary and on which the Green’s function of the Laplace-Beltrami operator exists, the eigenfunctions of ${\mathrm{\Delta }}_{\mu }$ are continuous on $\mathrm{\Omega }$. We also prove that if M is compact and $\mathrm{\partial }M=\mathrm{\varnothing }$, then the eigenfuctions of ${\mathrm{\Delta }}_{\mu }$ are continuous on M.