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

2024/12/13 by Ngai, Sze-Man, Zhao, Wen-Quan
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2412.10007

Abstract

Let d≥1, Ω be a bounded domain of a smooth complete Riemannian d-manifold M, and μ be a positive finite Borel measure with compact support in Ω. We prove the Courant nodal domain theorem for the eigenfunctions of Kreĭn-Feller operator Δ_μ under the assumption that such eigenfunctions are continuous on Ω. For d≥2, We prove that on a bounded domain Ω⊂ M with smooth boundary and on which the Green's function of the Laplace-Beltrami operator exists, the eigenfunctions of Δ_μ are continuous on Ω. We also prove that if M is compact and ∂ M=∅, then the eigenfuctions of Δ_μ are continuous on M.

Related

• Nodal sets and continuity of eigenfunctions of Kre\uı-Feller operators
• Nonexistence of Courant-type nodal domain bounds for eigenfunctions of the Dirichlet-to-Neumann operator
• Krein-Feller operators on Riemannian manifolds: compactness of embedding and Hodge's theorem
• Nodal domains a la Courant
• Pleijel nodal domain theorem in non-smooth setting