A unified divergent approach to Hardy–Poincaré inequalities in classical and variable Sobolev spaces

Abstract

We present a unified strategy to derive Hardy–Poincaré inequalities on bounded and unbounded domains. The approach allows proving a general Hardy–Poincaré inequality from which the classical Poincaré and Hardy inequalities immediately follow. We extend the idea to the more general context of variable exponent Sobolev spaces. Surprisingly, despite the well-known counterexamples of Fan et al. (2005) [28], we show that a modular form of the Poincaré inequality is actually possible provided one restricts to the class of functions in u∈Cc∞(Ω) such that |u|⩽1. The argument, concise and constructive, does not require a priori knowledge of compactness results and retrieves geometric information on the best constants.