On Constants in Maxwell Inequalities for Bounded and Convex Domains

Abstract

For a bounded and convex domain in three dimensions we show that the Maxwell constants are bounded from below and above by Friedrichs' and Poincaré's constants.

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

“…While a lot of our theoretical findings hold for domains Ω in arbitrary dimensions, we restrict numerical experiments to the 2D and 3D cases. Moreover, we verify various theoretical results established in the last few years in [37,38,39,41]. There is a recent interest in these eigenvalues, see, e.g., [17,18,29] and related contributions [19,20,26,27,28,30,45], but little results for mixed boundary conditions are known in the literature, except for, e.g., [39,41].…”

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

“…Now, follows immediately by combining and if Ω is bounded and either or convex with because in this case, and . In the bounded and convex case, there are even no Neumann fields, that is, , and holds; see, for example, for the cases N = 2 or N = 3, which follows essentially by Corollary and uniform approximation of a convex domain Ω by a sequence of smooth and convex domains. The Neumann fields generally vanish if and only if Ω is simply connected.…”

A Global div-curl-Lemma for Mixed Boundary Conditions in Weak Lipschitz Domains

“…see [3,Theorem 5.1]. Recent estimates for the Maxwell constant c m can be found in [14,15,16]. Analogously, Rellich's selection theorem (1) shows the Friedrichs/Poincaré estimate, i.e., there exists c f,p > 0 such that for all u ∈ D(…”