arXiv subjects

Wen-Quan Zhao

Publications and source records attributed to Wen-Quan Zhao.

2 recordsLinked to original sources

arXiv2024

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

Let $d\geq1$, $Ω$ be a bounded domain of a smooth complete Riemannian d-manifold M, and $μ$ be a positive finite Borel measure with compact support in $\overlineΩ$. 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 $\overlineΩ$. For $d\geq2$, We prove that on a bounded domain $Ω\subset M$ 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 $\partial M=\emptyset$, then the eigenfuctions of $Δ_μ$ are continuous on M.

math.FA↗

arXiv2024

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

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.

math.AP↗