Inequalities between Dirichlet and Neumann Eigenvalues on Surfaces

2024/12/27 by Hua, Bobo, Münch, Florentin, Zhang, Haohang · 1 citation
#35J05 #35P15 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2412.19480

Abstract

For a bounded Lipschitz domain Σ in a Riemannian surface M satisfying certain curvature condition, we prove that μ_3-β1 ≤ λ₁, where μ_k (λ_k resp.) is the k-th Neumann (Dirichlet resp.) Laplacian eigenvalue on Σ and β₁ is the first Betti number of Σ. If Σ is smooth and simply connected, we can further derive the strict inequality μ₃< λ₁. This extends previous results on the Euclidean space to various curved surfaces, including the flat cylinder, the hyperbolic plane, hyperbolic cusp, collar, funnel, and minimal surfaces such as catenoid and helicoid. The novelty of the paper lies in comparing Dirichlet and Neumann Laplacian eigenvalues via the variational principle of the Hodge Laplacian on 1-forms on a surface, extending the variational principle on vector fields in the Euclidean plane as developed by Rohleder. The comparison is reduced to the existence of a distance function with appropriate curvature conditions on its level sets.

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