Equivalence between validity of the p-Poincaré inequality and finiteness of the strict p-capacitary inradius

2024/04/30 by Gallagher, A. -K.
#31C45 #35J70 #35P30 #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2404.19207

Abstract

It is shown that the p-Poincaré inequality holds on an open set Ω in ℝⁿ if and only if the strict p-capacitary inradius of Ω is finite. To that end, new upper and lower bounds for the infimum for the associated nonlinear Rayleigh quotients are derived.

Related

• On the Poincaré inequality on open sets in ℝⁿ
• Capacitary inradius and Poincaré-Sobolev inequalities
• Variations on the capacitary inradius
• Non-trivial Poincare Constant in any Subset of ℝⁿ
• Maz'ya-type bounds for sharp constants in fractional Poincaré-Sobolev inequalities