A note on “Quasihyperbolic boundary conditions and Poincaré domains”

Korn inequality on irregular domains

“…If a = b = 0, then the Poincaré inequality obtained above coincides with[17, Theorem 1]. One can modify[23, Example 5.5] to show that the Poincaré inequality from Theorem 3.3 is sharp, in the sense that the inequalityΩ |u − u Ω,a | q ρ a dxWe have the following Korn inequality for domain satisfying a β-QHBC.Theorem 3.4.…”

“…For p = 1, the same proof as [12, Proof of Theorem 7] applies with Lemma 3.1 replacing the s-John condition there. For p > 1, the proof is similar to the proof of Theorem 3.2 in [23] with small modifications from [17]. We will verify condition (ii) of Theorem 2.3.…”

“…(ii) Ω is a β-QHBC domain if σ ≤ τ , β = 1 2σ−1 and Ω is not a β-QHBC domain for any β > 0 if 1 ≤ τ < σ; see [23,Example 5.5] and [17].…”

Poincaré inequalities in quasihyperbolic boundary condition domains

“…A (q, p)-Poincaré inequality is valid in a β-quasihyperbolic boundary condition domain, if n − nβ < q = p < ∞, see [11,Theorem 1.4], and also [7,Remark 7.11]; and if n − nb < p ≤ q < bnp/(n − p), whenever p < n and b = 2β/(1 + β) [10, Theorem 1]; see also [11,Theorem 1.5], [8,Theorem 1.4]. It is shown in [10] that if 1 ≤ p < n − nb, then there exist β-quasihyperbolic boundary condition domains which do not support the (p, p)-Poincaré inequality. We remark that β-quasihyperbolic boundary condition domains support (1, p)-Poincaré inequality for all p > n − nb by Hölder's inequality while John domains support (1, p)-Poincaré inequality for all 1 ≤ p < ∞.…”

On the smoothness of quasihyperbolic balls

“…Besides quasiconformal mappings, the quasihyperbolic metric has recently found novel and interesting applications in other fields of geometric analysis as well. For example, quasihyperbolic metric has been recently used in study of the Poincaré inequality [12,16].…”