On the Poincaré inequality on open sets in ℝⁿ

2023/07/25 by Anne-Katrin Gallagher, Gallagher, A. -K.
Computer Science · Mathematics · #31B99 #32W05 #35P15 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2307.13641

openalex publication_date 2023/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the Poincaré inequality holds on an open set D⊂ℝⁿ 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.

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