An Improved Poincare Inequality

Improved Poincaré inequalities in fractional Sobolev spaces

“…In the case of John domains, a partial converse is also true in the following sense: if Ω has finite measure and satisfies a separation property, then the validity of the Sobolev-Poincaré inequality implies the John condition (see [6]). A possibly incomplete list of references on improved Poincaré inequalities and their generalizations to weighted settings and measure spaces includes [7,8,9,10,14,15,16,17,20,21]. More recently, some authors have turned their attention to fractional generalizations of Poincaré and Sobolev-Poincaré inequalities, where a fractional seminorm appears instead of the norm in W 1,p (Ω).…”

Connections Between Optimal Constants in some Norm Inequalities for Differential Forms

“…In this paper, motivated by the results of [4], we have derived relations between the domain specific improved Poincaré constant and the Friedrichs-Velte constant figuring in the corresponding inequalities for differential forms. This constitutes a generalization of the improved Poincaré inequalities formulated in [9] for the gradient, however, only in case of the L 2 -space on a Lipschitz domain. We have also developed a generalization of the Horgan-Payne type upper estimations for the Friedrichs-Velte constant of a planar [3,8] and of a spatial [11] domain for arbitrary dimensional star-shaped domains, which also constitutes a unification of these estimations.…”

“…Remark 3.6 Setting n ≥ 2 and ℓ = 1 as in the Section 4.1. of [4], we have d = d = grad, d * = d * = − div and the subspace M = (ker grad) ⊥ consists of functions with vanishing integral over each connected component of Ω. In this case, according to [9,10], the improved Poincaré inequality (21) holds for a larger class of domains including John domains. Moreover, as proved in [10], for simply connected planar domains being a John domain is equivalent with the simultaneous finiteness of the investigated constants.…”

Improved Poincaré inequalities and solutions of the divergence in weighted norms

“…If Ω is a John domain one can choose a Whitney decomposition satisfying also the following property (see [H1,DRS]). There exist an open cube Q * 0 (called central cube) that can be connected with every cube Q * by a finite chain of cubes,…”