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

For a bounded Lipschitz domain with Lipschitz interface we show the following compactness theorem: Any L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {L}_{}^{2}$$\end{document}-bounded sequence of vector fields with L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {L}_{}^{2}$$\end{document}-bounded rotations and L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {L}_{}^{2}$$\end{document}-bounded divergences as well as L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {L}_{}^{2}$$\end{document}-bounded tangential traces on one part of the boundary and L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {L}_{}^{2}$$\end{document}-bounded normal traces on the other part of the boundary, contains a strongly L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {L}_{}^{2}$$\end{document}-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.


Introduction
Let Ω ⊂ R 3 be open with boundary Γ, composed of the boundary parts Γ 0 (tangential) and Γ 1 (normal).In [2,Theorem 4.7] the following version of Weck's selection theorem has been shown: Theorem 1.1 (compact embedding for vector fields with homogeneous mixed boundary conditions).Let (Ω, Γ 0 ) be a bounded strong Lipschitz pair and let ε be admissible.Then Here, cpt ֒→ denotes a compact embedding, and -in classical terms and in the smooth case -we have for a vector field E (n denotes the exterior unit normal at Γ) Date: January 5, 2022.Key words and phrases.compact embeddings, div-curl system, mixed boundary conditions, inhomogeneous boundary conditions.
We gratefully acknowledge ISem23 (23rd Internet Seminar 2019/2020, Evolutionary Equations, http://www.mat.tuhh.de/isem23)for providing the platform to start this research.The second author has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 765579.
In this paper, we shall generalise Theorem 1.1 to the case of inhomogeneous boundary conditions, i.e., we will show that the compact embedding in Theorem 1.1 still holds if the space where in classical terms and in the smooth case The main result (compact embedding) is formulated in Theorem 4.1.As applications we show in Theorem 5.1 that the compact embedding implies a related Friedrichs/Poincaré type estimate, showing well-posedness of related systems of partial differential equations.Moreover, in Theorem 5.3 we prove that Theorem 4.1 yields a div-curl lemma.Note that corresponding results for exterior domains are straight forward using weighted Sobolev spaces, see [11,12].Another application is presented in Section 5.3 where we show that our compact embedding result implies compact resolvents of the Maxwell operator with inhomogeneous mixed boundary conditions, even of impedance type.We finally note in Section 5.4 that the corresponding result holds (in the simpler situation) for the impedance wave equation (acoustics) as well.
The usual Lebesgue and Sobolev Hilbert spaces (of scalar or vector valued fields) are denoted by L 2 (Ω), H 1 (Ω), H(curl, Ω), H(div, Ω), and by H 0 (curl, Ω) and H 0 (div, Ω) we indicate the spaces with vanishing curl and div, respectively.Homogeneous boundary conditions are introduced in the strong sense as closures of respective test fields from and we set H 1 ∅ (Ω) := H 1 (Ω), H ∅ (curl, Ω) := H(curl, Ω), and H ∅ (div, Ω) := H(div, Ω).Spaces with vanishing curl and div are again denoted by H Γ0,0 (curl, Ω) and H Γ0,0 (div, Ω), respectively.Moreover, we introduce the cohomology space of Dirichlet/Neumann fields (generalised harmonic fields) The L 2 (Ω)-inner product and norm (of scalar or vector valued L 2 (Ω)-spaces) will be denoted by • , • L 2 (Ω) and • L 2 (Ω) , respectively, and the weighted Lebesgue space L 2 ε (Ω) is defined as L 2 (Ω) (of vector fields) but being equipped with the weighted L 2 (Ω)-inner product and norm with norms given by, e.g., E 2 HΓ 0 (curl,Ω) . The definitions of the latter Hilbert spaces need some explanations: (i) The tangential trace of a vector field E ∈ H(curl, Ω) is a well-defined tangential vector field τ Γ E ∈ H −1/2 (Γ) generalising the classical tangential trace τ Γ E = −n × n × E| Γ for smooth vector fields E. By the notation τ Γ0 E ∈ L 2 (Γ 0 ) we mean, that there exists a tangential vector field Then we set Here and in the following, the twisted tangential trace of the smooth vector field Φ is given by the tangential vector field Here, the well-known scalar trace of the smooth function φ is given by σ Remark 2.2 (L 2 -traces).Analogously to Definition 2.1 (i) and as holds on Γ 0 for smooth vector fields E, H, we can define the twisted tangential trace for all vector fields Φ ∈ H 1 Γ1 (Ω).

Preliminaries
In [4, Theorem 5.5], see [3,Theorem 7.4] for more details and compare to [2], the following theorem about the existence of regular potentials for the rotation with homogeneous mixed boundary conditions has been shown.Theorem 3.1 (regular potential for curl with homogeneous mixed boundary conditions).
Combining Theorem 3.1 and Theorem 3.2 shows immediately the following.

Compact Embeddings
Our main result reads as follows: Theorem 4.1 (compact embedding for vector fields with inhomogeneous mixed boundary conditions).

Friedrichs/Poincaré Type Estimates.
A first application is the following estimate: Theorem 5.1 (Friedrichs/Poincaré type estimate for vector fields with inhomogeneous mixed boundary conditions).There exists a positive constant c such that for all vector fields E in For a proof we use a standard compactness argument using Theorem 4.1.If the estimate was wrong, then there exists a sequence (E ℓ ) ∈ H Γ0 (curl, Ω) ∩ ε −1 H Γ1 (div, Ω) ∩ H Γ0,Γ1,ε (Ω) Thus, by Theorem 4.1 (after extracting a subsequence) and curl E = 0 and div εE = 0 (by testing).Moreover, for all Φ ∈ C ∞ Γ1 (Ω) and for all φ [4,Theorem 4.7] (weak and strong homogeneous boundary conditions coincide).This shows Proof.We follow in closed lines the proof of [14,Theorem 3.1].Let (E n ) and (H n ) be as stated.
First, we pick subsequences, again denoted by (E n ) and (H n ), and E and H, such that E n ⇀ E in H Γ0 (curl, Ω) and H n ⇀ H in H Γ1 (div, Ω).In particular, To see (1) is uniformly bounded with respect to n we obtain (1).
As in Remark 4.2 and Remark 5.2 there are corresponding generalised div-curl lemmas for weaker boundary data, where the L 2 (Γ 0/1 )-spaces and norms are replaced by H −s (Γ 0/1 )-spaces and norms.

5.3.
Maxwell's Equations with Mixed Impedance Type Boundary Conditions.Let ε, µ be admissible and time-independent matrix fields, and let T, k ∈ R + .In I × Ω with I := (0, T ) we consider Maxwell's equations with mixed tangential and impedance boundary conditions (magnetic initial value) (2i) Note that the impedance boundary condition, also called Leontovich boundary condition, is of Robin type and that the impedance is given by λ Despite of other recent and very powerful approaches such as the concept of "evolutionary equations", see the pioneering work of Rainer Picard, e.g., [20,10], one can use classical semigroup theory for solving the Maxwell system (2).
We will split the system (2) into two static systems and a dynamic system.For simplicity we set ε = µ = 1 and F = G = 0.The static systems are where g is any suitable tangential vector field in L 2 (Γ 1 ).For simplicity we put g = 0, then these two systems are solvable by [2,Theorem 5.6].However, the same result also gives conditions for which g = 0 this system is solvable.The dynamic system is ν Γ0 H = 0, (4e) The initial conditions for the dynamic system are E(0) = E 0 − E stat and H(0) = H 0 − H stat , where E stat and H stat are the solutions of the two static systems (3).We can write (4a) and (4b) as and the boundary conditions (4f) and (4g) shall be covered by the domain of A 0 : Here, we did ignore the equations div E = 0, div H = 0 and ν Γ0 H = 0.However, A 0 is a generator of a C 0 -semigroup by [22,Example 8.10] or [25,Section 5], where the input function is u = 0. (In these sources they regard boundary control systems and system nodes, respectively.One condition of those concepts is that the system with u = 0 is described by a generator of a C 0 -semigroup).The next lemma provides a tool to respect the remaining conditions of (4) as well.
Lemma 5.5.Let T(•) be a C 0 -semigroup on a Banach space X, and let A be its generator.Then every subspace V ⊇ ran A is invariant under T(•).Moreover, A V generates the strongly continuous semigroup T Proof.Let t ≥ 0 and let x ∈ V .Then ran A ∋ A t 0 T(s)x ds = T(t)x − x and hence T(t)x ∈ V .The remaining assertion follows from [6, Chapter II, Section 2.3].❑ Therefore, it is left to show that the remaining conditions establish a closed and invariant subspace under the semigroup T 0 generated by A 0 or contains ran A 0 .Note that by Theorem 3.1 This space is closed as the intersection of kernels of closed operators.Clearly, H Γ,∅ (Ω) × H Γ1,Γ0 (Ω) is invariant under T 0 , since every (E, H) ∈ H Γ,∅ (Ω) × H Γ1,Γ0 (Ω) is a constant in time solution of the system (4), i.e., T 0 (t) and Lemma 5.5 we have that also curl H(curl, Ω) × curl H Γ0 (curl, Ω) is invariant under T 0 .Hence, Lemma 5.5 and Theorem 4.1 imply the next theorem.
Theorem 5.6.A := A 0 S is a generator of a C 0 -semigroup and Consequently, every resolvent operator of A is compact. ).This would also match our separation of static solutions and dynamic solutions, since solutions with initial condition in H Γ,∅ (Ω) × H Γ1,Γ0 (Ω) are constant in time.

5.4.
Wave Equation with Mixed Impedance Type Boundary Conditions.For the scalar wave equation the situation is even simpler since traces of H 1 (Ω)-functions already belong to L 2 (Γ), even to H 1/2 (Γ).In I × Ω we consider the wave equation in first order form (linear acoustics) with mixed scalar and impedance boundary conditions We write the system as with dom A 0 := (w, v) ∈ H 1 Γ0 (Ω) × H Γ1 (div, Ω) σ Γ1 w + kν Γ1 = 0 .As before, by [8,Theorem 4.4] or [22,Example 8.9], A 0 is a generator of C 0 -semigroup.Again, we want to separate the static solutions from the dynamic system.The static solutions are given by ker A 0 , which can be characterise by ker A 0 = {0} × H Γ1,0 (div, Ω), where we assumed Γ 0 = ∅, otherwise the first component can also be constant and the second component would be in H Γ1,0 (div, Ω).By Theorem 3.2, the orthogonal complement of ker A 0 is Note that S contains ran A 0 and is therefore (by Lemma 5.5) an invariant subspace under the semigroup generated by A 0 .Moreover, note that ∇ H 1 Γ0 (Ω) ⊆ H Γ0,0 (curl, Ω) and that S is closed.Hence, Lemma 5.5 and Theorem 4.1 imply the next theorem.Consequently, every resolvent operator of A is compact.
Alternatively, we can also regard the classical formulation of the wave equation and see that it is necessary for the second component in our formulation to be in ∇ H 1 Γ0 (Ω), if we want the solutions to correspond.