Poincare Inequalities and Neumann Problems for the p-Laplacian

TL;DR

This paper establishes a link between weighted Poincare inequalities and the existence of weak solutions to Neumann problems for degenerate p-Laplacian operators, using degenerate Sobolev spaces.

Contribution

It introduces a novel equivalence between Poincare inequalities and Neumann problem solutions in the context of degenerate Sobolev spaces for the p-Laplacian.

Findings

Proves the equivalence between weighted Poincare inequalities and weak solutions.

Defines degenerate Sobolev spaces based on a quadratic form.

Highlights the role of the matrix causing degeneracy in the p-Laplacian.

Abstract

We prove an equivalence between weighted Poincare inequalities and the existence of weak solutions to a Neumann problem related to a degenerate p- Laplacian. The Poincare 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.