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The Maxwell Compactness Property in Bounded Weak Lipschitz Domains with Mixed Boundary Conditions

For a bounded weak Lipschitz domain we show the so called `Maxwell compactness property', that is, the space of square integrable vector fields having square integrable weak rotation and divergence and satisfying mixed tangential and normal boundary conditions is compactly embedded into the space of square integrable vector fields. We will also prove some canonical applications, such as Maxwell estimates, Helmholtz decompositions and a static solution theory. Furthermore, a Fredholm alternative for the underlying time-harmonic Maxwell problem and all corresponding and related results for exterior domains formulated in weighted Sobolev spaces are straight forward.


Introduction
One of the main and most important tools in the theory of Maxwell's equations is the compact embedding of vector fields with weak divergence and rotation in L 2 , subject to appropriate possibly mixed boundary conditions, into L 2 .
A first result for non-smooth, more precisely cone-like, i.e., more or less strong Lipschitz, domains in R N or even Riemannian manifolds, was obtained by Weck [24] in the case of full homogeneous boundary conditions, i.e., Γ ν = ∅ or Γ τ = ∅, using the general setting provided by the calculus of alternating differential forms. Weber [22] found a new proof for bounded domains in R 3 satisfying the uniform conecondition, which is again more or less strong Lipschitz. This result was improved upon by Witsch [26], who showed that the compact embedding is valid for bounded domains of R 3 satisfying merely the p-cusp condition for 1 < p < 2. An elementary proof for weak Lipschitz domains, which even holds for weak Lipschitz manifolds, was given by Picard [17]. Costabel [2] proved the compact embedding by means of a weak regularity result, i.e., there holds the continuous embedding into H 1 /2 and therefore the compact embedding into H t for t < 1/2 and into H 0 = L 2 in particular. All these results have been obtained for full boundary conditions. Kuhn [7], using the methods developed by Weck [24] and comparable assumptions about the regularity of the boundary and the interface, obtained the compact embedding for mixed boundary conditions and even in the general setting of differential forms. Based on the techniques developed by Weber [22], Jochmann [6] showed the compact embedding to hold for vector fields satisfying mixed boundary conditions, if Ω has a (strong) Lipschitz continuous boundary Γ with a (strong) Lipschitz continuous interface Γ τ ∩ Γ ν . More precisely, the boundary Γ and the interface can be locally represented as graphs of Lipschitz functions.
In this paper it is shown that the assumptions in [6] can be weakened to include domains Ω with weak Lipschitz boundaries and weak Lipschitz interfaces, i.e., the boundary Γ is assumed to be a Lipschitz manifold and the interface a Lipschitz submanifold of Γ. This assumption is weaker than the assumptions in [6]. Moreover, weak Lipschitz domains are relevant in various applications. A prominent example is the so-called two brick domain.
Our paper closely follows [6] and hence [22]. Conveniently, the proofs from [6] carry over practically verbatim. However, a modification of [6, Theorem 1] (see our Theorem 3.8) allows for a shorter and more straightforward proof of the compact embedding. Moreover, we will use different transformations (charts), which reduces the compact embedding to the flat situation of a half-cube.
The main result of this contribution is the compact embedding stated in Theorem 4.7. In the last chapter we present applications of the main theorem, i.e. the Maxwell estimate, Helmholtz decompositions and, following the approach developed by Picard in [15] and [16], a solution theory for the static Maxwell problem involving mixed boundary conditions.
Another application is the proof of Fredholm's alternative for the time-harmonic Maxwell's equations. This is straightforward but would exceed the frame of this paper.
Other important applications can be mentioned in connection with the treatment of static or timeharmonic Maxwell's equations in exterior domains Ω ext ⊂ R 3 , i.e., domains with compact complement. In this case the compact embedding (1) no longer holds. On the other hand (1) immediately implies the so-called local Maxwell compactness property, i.e., the compactness of the embedding (2) is the main and most important tool for showing, e.g., polynomially weighted Maxwell inequalities and Helmholtz decompositions or the validity of Eidus' principles of limiting absorption and limiting amplitude [3,4,5], which are fundamental in the theory of static or time-harmonic Maxwell's equations in exterior domains, see e.g. [9,15,16,18,19,20,8,10,11,13,12,14].

Notation, preliminaries and outline of the proof
Let Ω ⊂ R 3 be a domain, i.e., an open and connected set. We introduce the function spaces where Θ ⋐ Ω means Θ is compact and a subset of Ω. The usual Lebesgue and Sobolev spaces will be denoted by where ⊥ means orthogonality in L 2 (Ω). We also introduce the Sobolev (Hilbert) spaces in the distributional sense and define the test functions or vector fields Note that as closures of test functions respectively fields. For Γ τ = Γ (resp. Γ ν = Γ) we set Moreover, we define the closed subspaces and again for Γ τ = Γ (resp. Γ ν = Γ) we set In (3) and (4)  (Ω), respectively. In the special case of no boundary conditions, we need for this continuity argument the density of C ∞ (Ω) resp. C 0,1 (Ω) in D(Ω), R(Ω) and H 1 (Ω), respectively. These densities hold e.g.
if Ω has the segment property, which is very weak and basically means that Γ is just continuous.
Note that by switching Γ τ and Γ ν we can define the respective boundary conditions on the other part of the boundary as well.
Lemma 2.2. The following inclusions hold: Later we will show that in fact for all these spaces the strong and weak definitions of the boundary conditions coincide, i.e., which is an important feature of these Sobolev spaces. In case of full boundary conditions this can be seen from the following perspective: Define the unbounded linear rotation operator Rot : By its closure Rot : the differential operator 'rot' is extended to elements of • R(Ω) and its adjoint further generalizes the operator to the larger space R(Ω). We have Rot ⊂ Rot ⊂ Rot * . Moreover (Rot * ) * : without any assumptions about the regularity (or boundedness) of Ω or ∂ Ω. Analogously we have Grad, Grad, Grad * and Grad = (Grad * ) * for the gradient operator '∇' as well as Div, Div, Div * and Div = (Div * ) * for the divergence operator 'div'. This way we also get In case of mixed boundary conditions we may consider the operator Rot : its closure Rot : Figure 1. Mappings φ k and ψ k between a ball U k and the cube B. and its adjoint Rot * : Then the second adjoint is again the rotation rot. Its domain of definition D((Rot * ) * ) is the space of all E ∈ R(Ω), such that for which it is not clear if it coincides with • R Γτ (Ω), since the test fields are a priori allowed to be taken not only from • R Γν (Ω) but also from the possibly larger space • R Γν (Ω). The same problem occurs for the operators '∇' and 'div'. Therefore, the equalities (5) have to be shown by hand.

Lipschitz domains.
Let Ω ⊂ R 3 be a bounded domain with boundary Γ := ∂Ω. We introduce the setting we will be working in. Define (cf. Figure 2 there is an open covering U 1 , . . . , U k ⊂ R 3 of Γ and vector fields φ k := U k → B, such that for k = 1, . . . , K and in addition to (i), (ii) in Definition 2.3 one of holds. We define Γ ν := Γ \ Γ τ to be the relatively open complement of Γ τ . Definition 2.6. A pair (Ω, Γ τ ) conforming to Definitions 2.3 and 2.5 will be called weak Lipschitz.
For later purposes we introduce special notations for the half-cube domain and its relatively open boundary parts γ τ and γ ν := γ \ γ τ . We will only consider the cases and we note that Ξ and γ τ are strong Lipschitz.
• As a first step, we show with elementary arguments • In the second and essential step, we construct various H 1 -potentials on simple domains, mainly for the half-cube Ξ from (6) with the special boundary conditions (7), i.e., • In the third step it is shown that the strong and weak definitions of the boundary conditions also coincide for the divergence and rotation spaces on the half-cube Ξ from (6) with the special boundary conditions (7), i.e., • The fourth step proves the compact embedding on the half-cube Ξ from (6) with the special boundary conditions (7), i.e., • In the last step, we finally prove the compact embedding (9) for weak Lipschitz pairs, i.e., is compact.

H 1 -potentials
In this section H 1 -potentials for irrotational or solenoidal L 2 -vector fields or L 2 -functions are obtained. For illustration we will first give the proofs for the half-cube Ξ from (6) with the special boundary conditions (7) which will also later be used as the image of the coordinate transformation that flattens out the boundary of a weak Lipschitz pair, and then show how to adjust them for more general domains.
We start out with a density result for H 1 -functions, i.e., the strong and weak definitions of the boundary conditions coincide for H 1 -functions, which is first proved for a flat boundary and then generalized to weak Lipschitz pairs. The proof can be found in [6, Lemma 2, Lemma 3]. For the convenience of the reader we repeat and present a simplified proof, using our notation, in the appendix.
Let Ω ⊂ R 3 be a bounded domain and (Ω, Γ τ ) be a weak Lipschitz pair as well as 3.1. H 1 -potentials without boundary conditions. The next three lemmas ensure the existence of H 1 -potentials without boundary conditions. Suppose Ω ⊂ R 3 to be a bounded domain.

Lemma 3.2.
Let Ω be strong Lipschitz and simply connected. Then there exists a continuous linear operator Especially R 0 (Ω) = ∇H 1 (Ω) = ∇H 1 ⊥ (Ω) and the potential depends continuously on the data. In particular these are closed subspaces of L 2 (Ω).
A proof can be found in [6, Lemma 1]. For the convenience of the reader we repeat the proof using our notation.

Proof.
Let Ω be a subset of B ρ (0), the ball with radius ρ > 0 centered at the origin. Define Θ := B ρ (0)\Ω and let E : which by the Lax-Milgram lemma is well defined, linear and continuous. Next define for H ∈ L 2 (Ω) the linear and continuous operator B : 1 Let X, Y, Z be normed spaces. We call an operator T : holds and thusĤ ∈ L 2 (R 3 ) withĤ is a linear and continuous Using the same method, a divergence potential for an L 2 function can be obtained.
Proof. Leth be the extension of h into R 3 by zero. For the Fourier transform F holds and thusĤ ∈ L 2 (R 3 ) withĤ Remark 3.5. Let Θ ⊂ R 3 be a domain with Ω ⊂ Θ. Using a cutting technique we can choose continuous

3.2.
H 1 -potentials with boundary conditions. Now we start constructing H 1 -potentials with boundary conditions. Let us recall our special setting on the half-cube Theorem 3.6. There exists a continuous linear operator and the potential depends continuously on the data. In particular these are closed subspaces of L 2 (Ω).
Remark 3.7. The latter theorem is also from [6, Lemma 4]. Nevertheless, we will give a modified and simplified proof. Moreover, as the proof will show the result holds for more general domains. Let Ω ⊂ R 3 be a bounded and simply connected domain and let (Ω, Γ ν ) be a weak Lipschitz pair, such that Γ ν is connected. Then a potential operator S ∇ : If Ω is even strong Lipschitz, a continuous potential operator S ∇ : The other two cases can be treated together. We have and need to show It follows rotẼ = 0 inΞ, as for any Φ ∈ which meansẼ ∈ R 0 (Ξ). BecauseΞ is simply connected, there exists a potential u ∈ H 1 (Ξ) with E = ∇u inΞ.
In particular ∇u = 0 inΞ which implies u = c inΞ for some constant c ∈ R. Definẽ Note thatũ = 0 inΞ, soũ| γν = 0 follows, which meansũ ∈ • H 1 γν (Ξ) and with Lemma 3.1 (Ω = Ξ, By Poincaré's inequality the potential u depends continuously on the data E. Using Calderon's extension theorem we can extend any potential Next up is the existence of an H 1 -potential for divergence free fields subject to the special normal boundary condition. This Theorem is a modification of [6, Theorem 1], where the potential is only in Theorem 3.8. There exists a continuous linear operator -potential depends continuously on the data. In particular these are closed subspaces of L 2 (Ω).
Proof. We start with the case γ ν = ∅. By Lemma 3.3 D 0 (Ξ) = D 0 (Ξ) = rot H 1 (Ξ) and we can set The other two cases can be treated together. Suppose H ∈ • D γν ,0 (Ξ) and defineH ∈ L 2 (Ξ) byH It follows divH = 0 inΞ, as for any ψ ∈ be a continuous, linear extension operator, for example Calderon's. Then is linear and continuous. Since S r H = 0 inΞ, we have S r H| γν = 0, which means S r H ∈ Remark 3.9. Inspection of the above proof shows that the latter theorem holds for more general domains.
Theorem 3.10. There exists a continuous linear operator -potential depends continuously on the data. In particular these are closed subspaces of L 2 (Ω).

3.3.
Weak equals strong for the half-cube in terms of boundary conditions. Now the two main density results immediately follow. We note that this has already been proved for the H 1 (Ω)-spaces in  Theorem 4.1. The embedding Proof. The cases of full boundary conditions, i.e., γ τ = γ or γ τ = ∅, are well known, see the introduction.

4.2.
The compact embedding for weak Lipschitz domains. The aim of this section is to transfer Theorem 4.1 to arbitrary weak Lipschitz pairs (Ω, Γ τ ). We need a technical lemma, our subsequent both for functions and for vector fields. We set, identify and note From now on, we make the following  Then Υ =Γ k andΥ = φ k (Γ k ) = γ as well as (depending on the respective case) Remark 4.3. Theorems 3.6, 3.8, 3.10 and Remarks 3.7, 3.9, 3.11 as well as Theorems 3.12, 4.1 hold for γ ν = B 0,− without any (substantial) modification as well.
Hence, by definition there exists a sequence (Ã k,ℓ ) ⊂ Then with Lemma 4.2 . Now the compact embedding for weak Lipschitz pairs (Ω, Γ τ ) can be proved.
(Ω) for their extensions by zero to Ω. Finally It is clear, that by the same but much simpler arguments Rellich's selection theorem holds for weak Lipschitz domains Ω resp. weak Lipschitz pairs (Ω, Γ τ ). Therefore, also Poincaré's estimate holds in this case by a standard indirect argument.

Applications
From now on let Ω ⊂ R 3 be a bounded domain and let (Ω, Γ τ ) be a weak Lipschitz pair as well as ε ∈ L ∞ (Ω) be admissible.

The Maxwell estimate.
A first consequence of the compact embedding Theorem 4.7, i.e., is that the space of so-called 'Dirichlet-Neumann fields' is finite dimensional because the unit ball in H ε (Ω) is compact. By a standard indirect argument Theorem 4.7 immediately implies the so-called Maxwell estimate: Here we introduce L 2 ε (Ω) := L 2 (Ω) equipped with the scalar product · , · L 2 ε (Ω) := ε · , · L 2 (Ω) . Proof. Suppose the estimate does not hold. Then there exists a sequence Since (E n ) is bounded in R(Ω) ∩ ε −1 D(Ω), by Theorem 4.7 there exists a L 2 (Ω)-converging subsequence, again denoted by (E n ), with E n → E ∈ L 2 (Ω). But then E n → E in R(Ω) ∩ ε −1 D(Ω) and we see Helmholtz decompositions. Applying the projection theorem to the linear and closed operator ∇ τ : and • D Γν ,0 (Ω) = • D Γν ,0 (Ω) follows by Theorem 3.8. Here ⊕ ε denotes the orthogonal sum in L 2 ε (Ω). On the other hand, for the closed linear operator ε −1 rot ν : and again and (14) yield and hence by (13) or (14) the refined decomposition follows. Again from (13) and (14) we obtain and thus the further refinements i.e., (E n ) is a Cauchy sequence in Analogously we show that div Altogether we obtain Theorem 5.3. The following orthogonal decompositions hold: Moreover, the scalar ∇and vector rot-, div-potentials are uniquely determined in (i) If Ω is simply connected and (Ω, Γ τ ) a weak Lipschitz pair, such that Γ τ is connected, then, since the Poincaré-Friedrichs estimate holds by Theorem 4.8 (ii), a linear and continuous potential operator S ∇ : Again, if Γ τ = ∅, we have to replace If Ω is a strong Lipschitz domain, such that R 3 \ Ω is connected (i.e. Γ is connected), and if Γ ν = K k=1 Γ ν,k , K ∈ N, with disjoint, relatively open and simply connected strong Lipschitz surface patches Γ ν,k ⊂ Γ satisfying dist(Γ ν,k , Γ ν,ℓ ) > 0 for all 1 ≤ k = ℓ ≤ K, then there exists a linear and continuous potential operator S r : where dist(Γ ν,k , Γ ν,ℓ ) > 0 for all 1 ≤ k = ℓ ≤ K and Γ ν,k are C 3 -boundary patches allowing for C 3 -regular extensionsΩ ν,k having connected complements R 3 \Ω ν,k , then there exists a linear and continuous potential operator As noted earlier at the end of Section 3.2, the results of (iii) are not optimal. Using different techniques from [21, Lemma 2.1.1], it has been shown in [1,Lemma 3.2], that the assertions of (iii) hold for any strong Lipschitz pair (Ω, Γ ν ).
We will point out, how the potentials in the latter Helmholtz decompositions can be computed. By Theorem 5.3 any vector field E ∈ L 2 (Ω) can be written as Interchanging the roles of Γ ν und Γ τ in the decompositions of Theorem 5.3 yields (with ε = id) H ∈ X(Ω), i.e., E can be written as In order to calculate u we test (18) with ∇ϕ, ϕ ∈ • H 1 Γτ (Ω), and due to orthogonality we get In case Γ τ = ∅ we set again To calculate H ∈ X(Ω) we test (18) with rot Φ, Φ ∈ X(Ω), and get again by orthogonality Due to the Maxwell estimate, i.e., Theorem 5.1, ε −1 rot · , rot · L 2 (Ω) is a coercive bilinear form or even a scalar product on X(Ω) and Lax-Milgram's lemma or Riesz' representation theorem yields a unique H ∈ X(Ω), satisfying (20) and ||H|| R(Ω) ≤ c ||E|| L 2 (Ω) .
Remark 5.5. (19) is the variational formulation of the classical boundary value problem div ε∇u = div εE in Ω, because in the smooth case (19) yields for all ϕ ∈ Here,  which gives a meaning to the Neumann boundary condition (24) in this dual space as well.
For uniqueness, we require the additional conditions where d is the dimension and {D ℓ } an ε-orthonormal basis of H ε (Ω). The boundary values on Γ τ and Γ ν , respectively, are realized by the given fields E τ and E ν , respectively.
Let us solve the problem: Theorem 5.3 yields The decomposition E τ = E τ,r + E τ,d with E τ,r ∈ • R Γτ ,0 (Ω) and E τ,d ∈ R(Ω) ∩ ε −1 which means the field E τ realizing the boundary values on Γ τ can w.l.o.g. be chosen from the more regular so the field E ν can w.l.o.g. be chosen from ε −1 D(Ω) ∩ • R Γτ ,0 (Ω) ∩ H ε (Ω) ⊥ε . Therefore we will work with for the boundary conditions and note , div εE ν,r = div εE ν . We first note that the system admits at most one solution, as for the homogeneous problem E ∈ H ε (Ω) together with (29) yield E = 0. Turning to existence the conditions are necessary and from now on shall be assumed. By setting the problem is transformed into a problem with homogenous boundary conditions, i.e.,Ẽ must solve Necessary conditions for the existence of solutions arẽ which have been assumed above. The conditions are already sufficient, as Theorem 5.3 shows the existence of fieldsẼ with rotẼ r =F and div εẼ d =g. Theñ solves the system (30)-(32) withẼ ⊥ ε H ε (Ω). Therefore i.e., (29) with α = 0. Finally solves (25)-(29). Furthermore, E depends continuously on the data by Theorem 5.1. More precisely, equipping R(Ω) and ε −1 D(Ω) with the norms and by the Maxwell estimate Theorem 5.1 ||Ẽ|| L 2 ε (Ω) ≤ c m ||F || L 2 (Ω) + ||g|| L 2 (Ω) ≤ c m ||F || L 2 (Ω) + ||g|| L 2 (Ω) + || rot E τ || L 2 (Ω) + || div εE ν || L 2 (Ω) . Thus there exists a constantc m > 0, just depending on c m , such that Our latter results show that the linear solution operator with The solution E ∈ R(Ω)∩ε −1 D(Ω) can be chosen in a way, such that condition (29) with α ∈ R d is fulfilled, which then uniquely determines the solution. Furthermore the solution depends linearly and continuously on the data.
We note that with Theorem 5.3 the ranges can be described by is a topological isomorphism. Its inverse M −1 maps not only continuously onto We will show the density result for H 1 -functions, Lemma 3.1, which is first proved for a flat boundary in Lemma A.1 and then generalized to weak Lipschitz pairs in Lemma 3.1. Although both proofs can be found in [6, Lemma 2,3], we repeat them here, using our notation and with some major simplifications. Let us introduce the following notations: with compact support and u − | R 3 0,− = 0 in the sense of traces. Then a. x ∈ R 3 − , the translation of u − in the direction of x 1 . Then u t − ∈ H 1 (R 3 − ) and its trace satisfies u t − (x) = 0 f.a.a. x ∈ R 3 0 with x 1 ≤ t. Since the translation is continuous, there exists t > 0 with ||u t − − u − || H 1 (R 3 − ) < δ/3. The reflection u t + of u t − defined by u t + (x) := u t − (x 1 , x 2 , −x 3 ) for x ∈ R 3 + defines an element of H 1 (R 3 + ) with u t + (x) = u t − (x) f.a.a. x ∈ R 3 0 and hence u t + (x) = u t − (x) = 0 f.a.a. x ∈ R 3 0 with x 1 ≤ t.
Since supp u t,s ⋐ R 3 , Friedrichs' mollification yields some u ∈ , completing the proof.