Poincaré Inequalities and Neumann Problems for the p-Laplacian

Abstract

We prove an equivalence between weighted Poincar´e inequalities and the existence of weak solutions to a Neumann problem related to a degenerate pLaplacian. The Poincar´e inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the p-Laplacian.