Title:  Poincare inequalities and Neumann problems for the p-Laplacian

Abstract:  I will discuss my recent work with Scott Rodney on the following equivalence:  the existence of solutions to a degenerate p-Laplacian equation and the existence of a weighted (p,p) Poincare inequality.   Our results are in the context of degenerate Sobolev spaces, where the degeneracy is defined in terms of a degenerate elliptic matrix and I will spend some time talking about this general context.    If time permits, I will also discuss the application of similar ideas to show that a local, weak Sobolev inequality implies the existence of a global Sobolev inequality.