On Korn's first inequality for tangential or normal boundary conditions with explicit constants

Abstract

We will prove that for piecewise C2‐concave domains in double-struckRN Korn's first inequality holds for vector fields satisfying homogeneous normal or tangential boundary conditions with explicit Korn constant 2. Copyright © 2016 John Wiley & Sons, Ltd.

Poincare-Friedrichs Type Constants for Operators Involving grad, curl, and div: Theory and Numerical Experiments

“…being valid for all vector fields E ∈ H Γ (grad, Ω), the closure of Ω-compactly supported test fields, see (15). Using a more sophisticated integration by parts formula from [12,Corollary 6], which has been proved already in, e.g., [28,Theorem 2.3] for the case of full boundary conditions, we see that (26) remains true for polyhedral domains Ω and for vector fields…”

“…Now, Theorem 2.14 implies that the crucial assumption (13) holds for the operators A n of the de Rham complex (14), cf. the general complex (12). More precisely, by Theorem 2.14…”

“…Now, we specialise to linear acoustics and electromagnetics in 3D, i.e., to the classical operators of the 3D-de Rham complex, cf. (12),…”

On Korn’s first inequality for mixed tangential and normal boundary conditions on bounded Lipschitz domains in $$\mathbb {R}^N$$ R N

“…Throughout this paper and unless otherwise explicitly stated, let Ω ⊂ R N , N ≥ 2, be a bounded domain with strong Lipschitz boundary Γ := ∂Ω, i.e., locally Γ can be represented as a graph of a Lipschitz function. As in [2], we introduce the standard scalar valued Lebesgue and Sobolev spaces by L 2 (Ω) and H 1 (Ω) as well as , respectively, where • C ∞ (Ω) denotes the test functions yielding the usual Sobolev space • H 1 (Ω) with zero boundary traces. These definitions extend component-wise to vector or matrix, or more general tensor fields and we will use the same notations for these spaces.…”

Friedrichs inequality in irregular domains

“…Gaffney inequality can be deduced from the Friedrichs inequality. To our knowledge, such inequalities hold for convex and Lipschitz domains; among the others, we refer to [2,16,15,1], see also [4] and the references listed in. In this paper, we first prove Friedrichs inequality for (ε, δ) domains, and then prove Gaffney inequality by adapting the methods of [5] (developed for Korn inequality) to this framework.…”