Eigenvalues of the Neumann Laplacian with density and sharp Sobolev-Orlicz embeddings

2023/12/20 by Menovschikov, Alexander
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2312.13204

Abstract

We provide the estimates for the constant in the weighted Poincaré inequality for a special class of planar domains and weights. Based on this, we prove the lower bounds for the first non-zero eigenvalue μ_ρ of the Neumann Laplacian with density ρ. These estimates depend on the density function and the geometry of the domain. In particular, it is shown, that μ_ρ can be made arbitrarily large by changing the mass density of the domain.

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• Eigenvalues of the Laplacian with density
• Two new universal inequalities for Neumann eigenvalues of the Laplacian on a planar convex domain
• A universal inequality for Neumann eigenvalues of the Laplacian on a convex domain in Euclidean space
• Inequalities between Neumann and Dirichlet Laplacian eigenvalues on planar domains
• Estimates for the lowest Neumann eigenvalues of parallelograms and domains of constant width