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

Abstract

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