A global div-curl-lemma for mixed boundary conditions in weak Lipschitz domains and a corresponding generalized A 0 * \mathrm{A}_{0}^{*} - A 1 \mathrm{A}_{1} -lemma in Hilbert spaces

Abstract

We prove global and local versions of the so-called div-curl-lemma, a crucial result in the homogenization theory of partial differential equations, for mixed boundary conditions on bounded weak Lipschitz domains in 3D with weak Lipschitz interfaces. We will generalize our results using an abstract Hilbert space setting, which shows corresponding results to hold in arbitrary dimensions as well as for various differential operators. The crucial tools and the core of our arguments are Hilbert complexes and related compact embeddings.We will prove a global version of the div-curl-lemma stating that under certain (mixed tangential and normal) boundary conditions and (very weak) regularity assumptions on a domain Ω ⊂ R 3 , see Section 2, the following holds:Theorem II (global div-curl-lemma). Let Ω ⊂ R 3 be a bounded weak Lipschitz domain with boundary Γ and weak Lipschitz boundary parts Γ t and Γ n . Let (E n ) and (H n ) be two sequences bounded in L 2 (Ω), such that (curl E n ) and (div H n ) are also bounded in L 2 (Ω) and ν × E n = 0 on Γ t and ν · H n = 0 on Γ n . Then there exist subsequences, again denoted by (E n ) and (H n ), such that (E n ), (curl E n ) and (H n ), (div H n ) converge weakly to E, curl E and H, div H in L 2 (Ω), respectively, and the inner products converge as well, i.e.,A local version similar to the classical div-curl-lemma from Theorem I (distributional like convergence for arbitrary domains and no boundary conditions needed) is then immediately implied.Corollary III (local div-curl-lemma). Let Ω ⊂ R 3 be an open set. Let (E n ) and (H n ) be two sequences bounded in L 2 (Ω), such that (curl E n ) and (div H n ) are also bounded in L 2 (Ω). Then there exist subsequences, again denoted by (E n ) and (H n ), such that (E n ), (curl E n ) and (H n ), (div H n ) converge weakly to E, curl E and H, div H in L 2 (Ω), respectively, and the inner products converge in the distributional sense as well, i.e., for all ϕ ∈C ∞ (Ω) it holds

3. Weck’s selection theorem: The Maxwell compactness property for bounded weak Lipschitz domains with mixed boundary conditions in arbitrary dimensions

“…The central role of compact embeddings of this type can for example be seen in connection with Hilbert space complexes, where the compact embeddings immediately provide closed ranges, solution theories by continuous inverses, Friedrichs/Poincaré-type estimates, and access to Hodge-Helmholtz-type decompositions, Fredholm theory, div-curl-type lemmas, and a-posteriori error estimation, see [20,19,21]. In exterior domains, where local versions of the compact embeddings hold, one obtains radiation solutions (scattering theory) with the help of Eidus' limiting absorption principle [4,5,6], see [13,14,15,17,16,18].…”

The index of some mixed order Dirac type operators and generalised Dirichlet–Neumann tensor fields

“…Rather recently, the notion of Hilbert complexes (reusing the idea of writing the kernel of differential operators by means of other operators) has found applications in the context of homogenisation theory of partial differential equations. More precisely, it was possible to derive a certain operator-theoretic version of the so-called div-curl lemma (see [26,48]), which implied a whole family of div-curl lemma-type results, see [35,50].…”

“…Proof Utilising the 'FA-ToolBox' from, e.g., [34][35][36][38][39][40][41], and Lemma 3.7 we observe that both ranges ran D and ran D * are closed and that both kernels ker D and ker D * are finite-dimensional. Therefore, both D and D * are Fredholm operators.…”

Nonlocal H-convergence

“…The proof of Theorem 6.2 needs some prerequisites. The first one is a global div-curl type result, see [49,Theorem 2.4]; see also [26] for several applications and [6, Theorem 3.1] for a Banach space setting. We shall furthermore refer to [22] and the references therein for a guide to the literature for other results and approaches to the div-curl lemma.…”