A compactness result for the div-curl system with inhomogeneous mixed boundary conditions for bounded Lipschitz domains and some applications

Abstract

For a bounded Lipschitz domain with Lipschitz interface we show the following compactness theorem: Any $$\mathsf {L}_{}^{2}$$ L 2 -bounded sequence of vector fields with $$\mathsf {L}_{}^{2}$$ L 2 -bounded rotations and $$\mathsf {L}_{}^{2}$$ L 2 -bounded divergences as well as $$\mathsf {L}_{}^{2}$$ L 2 -bounded tangential traces on one part of the boundary and $$\mathsf {L}_{}^{2}$$ L 2 -bounded normal traces on the other part of the boundary, contains a strongly $$\mathsf {L}_{}^{2}$$ L 2 -convergent subsequence. This generalises recent results for homogeneous mixed boundary conditions in Bauer et al. (SIAM J Math Anal 48(4):2912-2943, 2016) Bauer et al. (in: Maxwell’s Equations: Analysis and Numerics (Radon Series on Computational and Applied Mathematics 24), De Gruyter, pp. 77-104, 2019). As applications we present a related Friedrichs/Poincaré type estimate, a div-curl lemma, and show that the Maxwell operator with mixed tangential and impedance boundary conditions (Robin type boundary conditions) has compact resolvents.

M-dissipative boundary conditions and boundary tuples for Maxwell operators

“…Theorem 3.1 (cf. [3,51]). (i) The Hilbert space L 2 ξ (Ω, C 3 ) admits the following orthogonal decomposition…”

“…The operator M imp,a is m-dissipative in L 2 ε,µ (Ω) if (1.7)-(1.8) hold true [16] (for the case of uniformly positive a(•), see [37]). Note that, if a(•) vanishes on a set of positive surface measure or is unbounded, the aforementioned interpretation of (1.6) does not necessarily lead to an mdissipative operator [16] (in the cases of degenerate or singular impedance coefficients a, other analytic interpretations of (1.6) are needed, see [45,3,16,51] and references therein).…”

“…The pioneering analytic result of S.G. Krein & Kulikov [36] states that σ(M ) \ {0} = σ disc (M ) in the particular case where a ≡ c with a constant c ∈ C right and ∂Ω is smooth. The recent paper [51] considers the case of mixed boundary condition with a constant impedance coefficient a ≡ c > 0 for a Leontovich boundary condition on an 'absorbing part' Γ 1 of the boundary ∂Ω, while the remaining part of the boundary Γ 0 = ∂Ω \ Γ 1 is equipped with the 'perfect metal condition' n × E = 0. We were not able to find results on the the discreteness of the nonzero spectrum of Maxwell-Leontovich operators M imp,a in the case where the impedance coefficient is not a constant function on the absorbing part of the boundary.…”

“…The extension to general impedance coefficients a(•) of the Krein-Kulikov proof for the discreteness of σ(M ) \ {0} requires not only the compact embedding for the space H imp (curl, Ω) 2 ∩ S ε,µ (Ω) into S ε,µ (Ω) (see [3,51] and (3.7)), but also an additional step in the form of the continuous embedding…”