On the Poincaré Inequality on Open Sets in $$\mathbb {R}^n$$

Abstract

We show that the Poincaré inequality holds on an open set \(D\subset \mathbb {R}^n\) if and only if D admits a smooth, bounded function whose Laplacian has a positive lower bound on D. Moreover, we prove that the existence of such a bounded, strictly subharmonic function on D is equivalent to the finiteness of the strict inradius of D measured with respect to the Newtonian capacity. We also obtain a sharp upper bound, in terms of this notion of inradius, for the smallest eigenvalue of the Dirichlet–Laplacian.