Gaffney's Inequality and the Closed Range Property of the de Rham Complex in Unbounded Domains
Dirk Pauly, Marcus Waurick
TL;DR
This work addresses closed-range properties of rot- and div-type operators in unbounded domains within the de Rham complex, linking closedness to the directions in which a domain is bounded. It develops a simple, self-contained approach based on Gaffney's inequality and Friedrichs estimates, then extends cuboid results to global Lipschitz domains via bi-Lipschitz transformations, establishing sharp criteria in terms of the directional boundedness $\mathop{\mathrm{d}}_{\Omega}$. A key application is the Maxwell operator, where a spectral gap near $0$ yields exponential stability for damped Maxwell equations, illustrating the practical impact for wave propagation and stability analyses. The paper also provides explicit mixed-boundary examples and a transformation theorem that preserves closed-range properties under admissible mappings, making the results broadly applicable to electromagnetic problems in waveguides and related PDE contexts.
Abstract
The classical Poincaré estimate establishes closedness of the range of the gradient in unweighted $L^2(Ω)$-spaces as long as $Ω\subseteq\mathbb{R}^3$ is contained in a slab, that is, $Ω$ is bounded in one direction. Here, as a main observation, we provide closed range results for the $\operatorname{rot}$-operator, if (and only if) $Ω$ is bounded in two directions. Along the way, we characterise closed range results for all the differential operators of the primal and dual de Rham complex in terms of directions of boundedness of the underlying domain. As a main application, one obtains the existence of a spectral gap near the $0$ of the Maxwell operator allowing for exponential stability results for solutions of Maxwell's equations with sufficient damping in the conductivity. Our results are based on the validity of Gaffney's (in)equality and the transition of the same to unbounded (simple) domains as well as on the stability of closed range results under bi-Lipschitz regular transformations. The latter technique is well-known and detailed in the appendix; for the results concerning Gaffney's estimate, we shall provide accessible, simple proofs using mere standard results. Moreover, we shall present non-trivial examples and a closed range result for $\operatorname{rot}$ with mixed boundary conditions on a set bounded in one direction only.