Complex Anal. Oper. Theory (2011) 5:799–810 DOI 10.1007/s11785-010-0068-3 Complex Analysis and Operator Theory A Poincaré Inequality for Orlicz–Sobolev Functions with Zero Boundary Values on Metric Spaces Marcelina Mocanu Received: 30 December 2009 / Accepted: 30 March 2010 / Published online: 24 April 2010 © Birkhäuser / Springer Basel AG 2010 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 gen- eralizes the ( p, p)-Poincaré inequality for Newtonian functions with zero boundary values in metric measure spaces, as well as a Poincaré inequality for Orlicz–Sobo- lev functions on a Euclidean space, proved by Fuchs and Osmolovski (J Anal Appl (Z.A.A.) 17(2):393–415, 1998). Using the Poincaré inequality for Orlicz–Sobolev functions with zero boundary values we prove the existence and uniqueness of a solu- tion to an obstacle problem for a variational integral with nonstandard growth. Keywords Metric measure space · Orlicz–Sobolev space · Orlicz–Sobolev function with zero boundary values · Poincaré inequality · Obstacle problem Mathematics Subject Classification (2000) 46E30 · 46E35 1 Introduction and Preliminaries The development of analysis on metric measure spaces (MMS) has been partially motivated by questions arising in quasiconformal theory, potential theory and har- monic analysis, fields that are closely related to complex analysis. There are several Communicated by Peter Hasto. This paper is in final form and no version of it will be submitted for publication elsewhere. M. Mocanu (B ) Department of Mathematics and Informatics, “Vasile Alecsandri” University of Bac˘ au, Spiru Haret 8, 600114 Bacau, Romania e-mail: mmocanu@ub.ro