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

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