Planar Poincaré domains: Geometry and Steiner symmetrization

Abstract

We determine geometric necessary and sufficient conditions on a class of strip-like planar domains in order for them to satisfy the Poincare inequality with exponentp, where 1≤p<∞. The characterization uses hyperbolic geodesics in the domain and a metric which depends onp and generalizes the quasi-hyperbolic metric in the casep=2. As an application, we show that the Poincare inequality is preserved under Steiner symmetrization of these domains but not in general.