Spectrum of the Laplacian with weights

2016/06/12 by Bruno Colbois, Colbois, Bruno, Ahmad El Soufi +1 · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1606.04095

openalex publication_date 2016/06/12 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

Given a compact Riemannian manifold (M, g) and two positive functions ρ and σ, we are interested in the eigenvalues of the Dirichlet energy functional weighted by σ, with respect to the L 2 inner product weighted by ρ. Under some regularity conditions on ρ and σ, these eigenvalues are those of the operator ρ^-1 div(σ∇u) with Neumann conditions on the boundary if ∂M = ∅. We investigate the effect of the weights on eigenvalues and discuss the existence of lower and upper bounds under the condition that the total mass is preserved.

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