Eigenvalues of the Neumann Laplacian with Density and Sharp Sobolev–Orlicz Embeddings

We provide estimates for the constant in the weighted Sobolev–Poincaré inequality for a special class of planar domains and weights. We obtain lower bounds for the first nonzero eigenvalue μ_ρ of the Neumann Laplacian with density ρ. These estimates depend on the density function and geometry of the domain. We show that μ_ρ can be made arbitrarily large by changing the mass density of the domain.