A Poincaré Inequality for Orlicz–Sobolev Functions with Zero Boundary Values on Metric Spaces

Abstract

We prove a Poincaré inequality for Orlicz-Sobolev functions with zero boundary values in bounded open subsets of a metric measure space. This result generalizes the ( p, p)-Poincaré inequality for Newtonian functions with zero boundary values in metric measure spaces, as well as a Poincaré inequality for Orlicz-Sobolev functions on a Euclidean space, proved by Fuchs and Osmolovski (J Anal Appl (Z.A.A.) 17 (2): 1998). Using the Poincaré inequality for Orlicz-Sobolev functions with zero boundary values we prove the existence and uniqueness of a solution to an obstacle problem for a variational integral with nonstandard growth.