Eigenvalues of the Laplacian with density

2019/08/14 by Kouzayha, Salam, Pétiard, Luc
#FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1908.05051

Abstract

Let (M,g) be a compact Riemannian manifold with a boundary of class \mathscrC¹. We are interested in the spectrum of the weighted Laplacian on M with Neumann boundary conditions. More precisely, given ρ and σ two positive functions on M, we study the eigenvalues of the equation -div(σ∇ u)=λρu. Inspired by a recent work of B. Colbois and A. El Soufi, we investigate upper bounds for the eigenvalues in the case where σ=ρ^α, α>0. We show that α= (n-2)/(n) plays a critical role in the estimation of the spectrum when the total mass of ρ is fixed.

Related

• Spectrum of the Laplacian with weights
• Eigenvalues estimate for the Neumann problem on bounded domains
• Eigenvalue Variations of the Neumann Laplace Operator Due to Perturbed Boundary Conditions
• Eigenvalues of the Neumann Laplacian with density and sharp Sobolev-Orlicz embeddings
• Some inequalities between Laplacian eigenvalues on Riemannian manifolds