Nodal sets and continuity of eigenfunctions of Kre\uı-Feller operators

Abstract

Let $μ$ be a compactly supported positive finite Borel measure on $\R^{d}$. Let $0<λ_{1}\leqλ_{2}\leq\ldots$ be eigenvalues of the Kre$\breve{ı}$n-Feller operator $Δ_μ$. We prove that, on a bounded domain, the nodal set of a continuous $λ_{n}$-eigenfunction of a Kre$\breve{ı}$n-Feller operator divides the domain into at least 2 and at most $n+r-1$ subdomains, where $r$ is the multiplicity of $λ_{n}$. This work generalizes the nodal set theorem of the classical Laplace operator to Kre$\breve{ı}$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$\breve{ı}$n-Feller operator are continuous.