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Annals of Global Analysis and Geometry

, Volume 55, Issue 2, pp 149–180 | Cite as

Spectrum of the Laplacian with weights

  • Bruno ColboisEmail author
  • Ahmad El Soufi
Article
  • 72 Downloads

Abstract

Given a compact Riemannian manifold (Mg) and two positive functions \(\rho \) and \(\sigma \), we are interested in the eigenvalues of the Dirichlet energy functional weighted by \(\sigma \), with respect to the \(L^2\) inner product weighted by \(\rho \). Under some regularity conditions on \(\rho \) and \(\sigma \), these eigenvalues are those of the operator \( -\rho ^{-1} \text{ div }(\sigma \nabla u) \) with Neumann conditions on the boundary if \(\partial M\ne \emptyset \). 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.

Keywords

Eigenvalue Laplacian Density Cheeger inequality Upper bounds 

Mathematics Subject Classification

35P15 58J50 

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Authors and Affiliations

  1. 1.Laboratoire de MathématiquesUniversité de NeuchâtelNeuchâtelSwitzerland
  2. 2.Laboratoire de Mathématiques et Physique Théorique, UMR-CNRS 7350Université de ToursToursFrance

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