Solvability of the divergence equation implies John via Poincaré inequality

Abstract

Let Ω ⊂ R 2 be a bounded simply connected domain. We show that, for a fixed (every) p ∈ (1, ∞), the divergence equationif and only if Ω is a John domain, if and only if the weighted Poincaré inequality Ωholds for some (every) q ∈ [1, ∞). In higher dimensions similar results are proved under some additional assumptions on the domain in question.

Hardy and Hardy-Sobolev Spaces on Strongly Lipschitz Domains and Some Applications

“…This type of equations has been studied intensively; see, for instance, [1,3,5,11,14]. Actually, this result even has an application in proving main result of this paper; see Lemma 3.3 …”

“…where C is a positive constant independent of f ; see [1,5,14]. But the same estimate fails when p approaches the limit situation, namely, when p = or p = ∞.…”

Connections Between Optimal Constants in some Norm Inequalities for Differential Forms

“…In inequality (17) we can assume v ∈ M * because for a given u ∈ M the element v ∈ L 2 (Ω, Λ ℓ+1 ) with minimal L 2 -norm that satisfies (16) also satisfies v ∈ M * . Hence similar to (10) we have for such a differential form…”

“…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.…”

Korn inequality on irregular domains

“…The proofs and results in [8] can be easily generalized to our setting, which enables us to deduce a (weighted) Korn inequality; see Section 2 below. For more on the recent progress on the divergence equation, see [3,4,8,18].…”