Gaffney's Inequality and the Closed Range Property of the De Rham Complex in Unbounded Domains

TL;DR

This paper extends Gaffney's inequality and the closed range property of the de Rham complex to unbounded domains, characterizing conditions based on domain boundedness and applying results to Maxwell's equations.

Contribution

It provides new closed range results for differential operators in unbounded domains based on domain boundedness and applies these to spectral gaps in Maxwell's equations.

Findings

Closed range results for the rot-operator when the domain is bounded in two directions.

Characterization of closed range conditions for all de Rham complex operators based on domain boundedness.

Existence of spectral gaps near zero for the Maxwell operator enabling exponential stability.

Abstract

The classical Poincar\'e estimate establishes closedness of the range of the gradient in unweighted $L^2(\Omega)$-spaces as long as $\Omega\subseteq\mathbb{R}^3$ is contained in a slab, that is, $\Omega$ is bounded in one direction. Here, as a main observation, we provide closed range results for the $\operatorname{rot}$-operator, if (and only if) $\Omega$ 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.