Well-posedness theory for electromagnetic obstacle problems

This paper develops a well-posedness theory for hyperbolic Maxwell obstacle problems generalizing the result by Duvaut and Lions (1976) [5]. Building on the recently developed result by Yousept (2020) [30], we prove an existence result and study the uniqueness through a local H(curl)-regularity analysis with respect to the constraint set. More precisely, every solution is shown to locally satisfy the Maxwell-Ampère equation (resp. Faraday equation) in the region where no obstacle is applied to the electric field (resp. magnetic field). By this property, along with a structural assumption on the feasible set, we are able to localize the obstacle problem to the underlying constraint regions. In particular, the resulting localized problem does not employ the electric test function (resp. magnetic test function) in the area where the L2-regularity of the rotation of the electric field (resp. magnetic field) is not a priori guaranteed. This localization strategy is the main ingredient for our uniqueness proof. After establishing the well-posedness, we consider the case where the electric permittivity is negligibly small in the electric constraint region and investigate the corresponding eddy current obstacle problem. Invoking the localization strategy, we derive an existence result under an L2-boundedness assumption for the electric constraint region along with a compatibility assumption on the initial data. The developed theoretical results find applications in electromagnetic shielding. © 2020 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
More than four decades ago, Duvaut and Lions [5,Chapter 7,Section 8] proposed and analyzed a (hyperbolic) Maxwell obstacle problem describing the propagation of electromagnetic waves in a polarizable medium with an obstacle constraint on the electric field of the form |E(x, t)| ≤ d(x) a.e. in × (0, T ) (1.1) for some d : → [0, ∞]. Based on the method of vanishing (curl-curl) viscosity and constraint penalization, they proved a global well-posedness result for the proposed obstacle problem [5,Theorem 8.1]. Some years later, Milani [15,16] extended their theory to the case of a timedependent upper bound d = d (x, t). See also Miranda and Santos [19] for the non-Hilbertian extension of the electromagnetic antenna problem in [5,Chapter 7]. Maxwell (quasi)-variational inequalities play as well a profound role in the mathematical modeling of type-II superconductivity. See Bossavit [2], Prigozhin [22], Barrett and Prigozhin [1], Elliott and Kashima [6], Jochmann [11], Rodrigues and Santos [24,25], Pan [20,21], Yin et al. [29], and our previous works [27,32] for results in this direction. From among many other contributions towards obstacle problems, we refer to the monographs [9,12,23] and the pioneering works by Fichera [7,8], Brézis and Stampacchia [3], and Lions and Stampacchia [13,14]. Quite recently, the author [30] examined the mathematical analysis for Maxwell variational inequalities of the second kind. Based on a local boundedness assumption for the governing subdifferential, [30,Theorem 3.3] proved a global well-posedness result. However, as shown in [30,Example 3.6], the local boundedness assumption fails to hold for indicator functionals. As a remedy, through the use of the minimal section operator and the Nemytskii operator of the governing subdifferential, a more refined existence result was derived in [30,Theorem 3.11]. Though this result applies to a wider class of nonlinearities, including indicator functionals, it merely affirms the existence of a rather weak solution, which does not necessarily belong to the effective domain of the nonlinearity. More crucially, the uniqueness of the solution is not guaranteed.
This paper discusses and explores obstacle problems for Maxwell's equations with a general feasible set structure. Our study is mainly motivated by electromagnetic shielding applications to block or redirect undesired electromagnetic fields in a certain domain of interest by means of barriers (obstacles) made of conductive or magnetic materials. Typical materials used for the barriers in electric shielding include highly conductive sheet metals and metallic alloys, while materials with high magnetic permeability such as ferromagnetic materials are typically used for magnetic obstacles (see [4]). From the mathematical point of view, electromagnetic shielding phenomena fall into the class of obstacle problems: In the free region, the electromagnetic fields satisfy the fundamental Maxwell equations, whereas in the shielded area obstacle constraints are applied to the electric field (electric shielding) or the magnetic field (magnetic shielding).
Let ⊂ R 3 be an open set (not necessarily bounded, Lipschitz, or connected) representing an anisotropic medium where the electromagnetic fields are acting. Inside , we consider two open subsets c E , c H ⊂ . The set c E (resp. c H ) represents the region where an obstacle constraint is imposed on the electric field E (resp. magnetic field H ). The free regions for the electric and magnetic fields are denoted, respectively, by The exact mathematical formulation of the Maxwell obstacle problem under consideration is presented in (P). The novelties of this paper include existence and uniqueness results for (P) (Theorems 1 and 2) generalizing [5, Theorem 8.1] since [5] solely considers the case (1.1) and c H = ∅, i.e., in the absence of an obstacle for the magnetic field. Furthermore, our results hold true without the higher regularity assumption on the initial data [5,Eq. (89), p. 370] and without the piecewise constant assumption on the material parameters [5,Eq. (3.19), p. 338]. Differently from [5], our analysis is not based on the method of vanishing viscosity. Here, the proof of the existence result is built on the recently developed result [30,Theorem 3.11]. Along with the existence result, we show that every solution locally satisfies (i) the Maxwell-Ampère equation in the electric free region E , (ii) the Faraday equation in the magnetic free region H .
On this basis, the uniqueness analysis is studied. First, in the case where the obstacle constraint is applied only either to the magnetic field H or to the electric field E, i.e., c E = ∅ or c H = ∅, a uniqueness result is obtained from (i)-(ii) (see Theorem 2). The uniqueness question becomes more challenging if both c E and c H have positive measure. To tackle this case, we propose structural assumptions on the constraint set ( c E ∩ c H = ∅) and the tangential components across the interfaces between the free and obstacle regions (Assumption 1.1). From the physical point of view, the proposed separation assumption c E ∩ c H = ∅ is reasonable since the electric and magnetic fields are coupled to each other by Maxwell's equations. See Fig. 1 and Example 1.1 for an exemplary physical model satisfying Assumption 1.1 related to electromagnetic shielding. Making use of Assumption 1.1, we are able to localize (P) into the constraint regions c E and c H . In particular, the resulting localized problem does not employ the magnetic test function (resp. electric test function) in c E (resp. c H ), i.e., in the region where the L 2 -regularity of the rotation of the corresponding field is not a priori guaranteed. This localization strategy is the central ingredient of our uniqueness proof under Assumption 1.1.
The final part of this paper considers the case where the electric permittivity is negligibly small in the electric constraint region c E . We investigate the resulting eddy current obstacle problem (1.19) by neglecting in c E and derive an existence result for (1.19) (Theorem 3). The proof is based on the above-mentioned localization strategy together with an L 2 -boundedness assumption for the constraint set on c E and a compatibility assumption on the initial data (Assumption 1.2). For earlier contributions towards eddy current (semistatic) approximations for nonlinear Maxwell's equations, we refer to Milani and Picard [17], Jochmann [10], and Yin [28].
The remainder of this paper is organized as follows. In the upcoming subsection, we introduce our notation and formulate the Maxwell obstacle problem (P) under investigation. The first two main results are Theorems 1 and 2 regarding existence and uniqueness results for the Maxwell obstacle problem (P). The final main result is Theorem 3 concerning an existence result for the eddy current obstacle problem (1.19). The proofs for these three theorems are presented, respectively, in Sections 2, 3, and 4.

Problem formulation and main results
For a given Hilbert space V , we use the notation · V and (·, ·) V for a standard norm and a standard scalar product in V . A bold typeface is used to indicate a three-dimensional vector function or a Hilbert space of three-dimensional vector functions.
The electric permittivity and the magnetic permeability in the medium are matrix-valued functions: , μ : → R 3×3 . Moreover, they are assumed to be of class L ∞ ( ) 3×3 , symmetric, and uniformly positive definite in the sense that there exist positive constants , μ > 0 such that ξ T (x)ξ ≥ |ξ | 2 and ξ T μ(x)ξ ≥ μ|ξ | 2 for a.e. x ∈ and all ξ ∈ R 3 . (1.2) Given a symmetric and uniformly positive definite matrix-valued function α ∈ L ∞ ( ) 3×3 , let L 2 α ( ) denote the weighted L 2 ( )-space endowed with the weighted scalar product (α·, ·) L 2 ( ) . The pivot Hilbert space for our analysis is where the operator curl is understood in the sense of distributions. As usual, C ∞ 0 (O) stands for the space of all infinitely differentiable three-dimensional vector functions with compact support contained in O. We denote the closure of . It is well known that the Hilbert space H 0 (curl, O) satisfies In the following, let K ⊂ X be a closed and convex set containing the origin (0, 0) ∈ K. This set denotes the feasible set for the electromagnetic fields E and H . As pointed out in the introduction, the open subsets E , H ⊂ represent the free region for E and H , respectively. In other words, K satisfies Having introduced all the required function spaces, let us now formulate the Maxwell obstacle problem under investigation: Let T ∈ (0, ∞). Given initial data By (1.5) and (1.3), the variational inequality in (P) can be compactly written as (1.9) Let us point out that Theorem 1 yields a rather weak existence result for (P), which does not necessarily satisfy the prescribed electric boundary condition and the global D(A)-regularity, i.e., (E, H ) ∈ L 2 ((0, T ), D(A)) is not guaranteed by Theorem 1. This is the reason why it is difficult to derive a uniqueness result in Theorem 1 since classical energy arguments cannot be directly applied here. Note that Theorem 1 affirms that the electric boundary condition E ∈ L ∞ ((0, T ), H 0 (curl)) holds in the case of H = , i.e., if there is no obstacle constraint imposed on the magnetic field. Of course, if K is assumed to additionally satisfy K ⊂ D(A), then the prescribed electric boundary condition and the global D(A)-regularity are pointwisely satisfied. However, the additional assumption K ⊂ D(A) is rather restrictive since our analysis requires that K is closed in X. Note that a closed set in D(A) is not necessarily closed in X. For this reason, we do not focus on the additional restrictive assumption K ⊂ D(A) in our analysis.
As pointed out earlier, if the obstacle constraint is applied only either to the magnetic field or to the electric field, i.e., if E = or H = , then (1.9) leads to a uniqueness result for (P) (see Theorem 2). For a more general case, we propose the following structural assumption on the feasible set: There is no particular physical reason why the magnetic constraint region c H ⊂ is chosen just to be open (not necessarily Lipschitz). This choice is considered solely to make our result mathematically more general. On the other hand, the Lipschitz regularity for the domain U E = U \ c E ⊂ E is required for the application of the extension theorem [10,Appendix] in (3.6). In the real applications, both electric and magnetic constraint regions c E , c H are typically given by bounded Lipschitz polyhedral domains.
We note that the assumption (1.11) is also related to obstacle problems with curl constraints (see Miranda et al. [18] for recent mathematical results on parabolic nonlinear obstacle problems with curl constraints). An example for such a set is given in Example 1.2. As a consequence of Assumption 1.1, if ∂ ⊂ ∂ c H , then the solution to (P) satisfies the magnetic boundary condition, i.e., H ∈ H 0 (curl, c H ). Fig. 1) is closed, convex and satisfies (0, 0) ∈ K, (1.6), and Assumption 1.1. In this case, c H models a medium with a magnetic shielding property, and c E describes an electromagnetic coil with electric insulation on ∂ c E (for δ ≈ 0).
By the above construction, we see that (1.11) is readily satisfied, and K c is a closed and convex subset of H 0 (curl, c E ) × H 0 (curl, c H ). Moreover, it is also obvious that K ⊂ X is convex and contains (0, 0). Let us now show that K ⊂ X is closed. To this aim, let {v n , w n } ∞ n=1 ⊂ K be a strongly converging sequence in X, i.e., By the definition of K and in view of In conclusion, K ⊂ X is closed, convex and satisfies (0, 0) ∈ K, (1.6), and Assumption 1.1.

Theorem 2 (Uniqueness
Our final theoretical finding is an existence result for the eddy current approximation to (P) in the case where the electric permittivity is negligibly small in the electric constraint region c E . As pointed out in the introduction, our result relies on the following additional assumption: with an open set c E ⊂ satisfying |∂ c E | = 0 and E = \ c E . If Assumption 1.1 holds, then the initial data (E 0 , H 0 ) ∈ D(A) ∩ K is assumed to satisfy the following compatibility condition: is assumed to satisfy the following compatibility condition: Let us remark that the L 2 -boundedness assumption (1.14) is reasonable since c E is exactly the region where the obstacle constraint is applied to the electric field.

Preliminaries
Let us first make a preparation by recalling some well-known results. For a nonempty, convex, and closed subset K ⊂ X, let I K : X → R := (−∞, ∞] denote the indicator functional of K: By definition, for every (p, q) ∈ X, the subdifferential ∂I K (p, q) is given by Furthermore, for every λ > 0, let J λ : X → X and λ : X → X denote, respectively, the resolvent and the Yosida approximation of the subdifferential ∂I K , i.e., where I d : X → X denotes the identity operator. Since K ⊂ X is nonempty, convex and closed, the indicator functional I K : X → R is proper, convex, and lower semicontinuous. As a consequence (see [26, Proposition 1.5, p. 157]), the subdifferential ∂I K : X → 2 X is m-accretive, which implies that J λ : X → X is non-expansive, and λ : X → X is m-accretive and Lipschitzcontinuous with the Lipschitz constant L λ = λ −1 (see [26, Theorem 1.1, p. 161]). We also make use of the Hilbert projection operator on K denoted by P K : X → K, i.e., for every (p, q) ∈ X, P K (p, q) ∈ K is given by the unique minimizer of (1.23) Lemma 4. Let K ⊂ X be nonempty, convex and closed. Then, it holds that J λ = P K for all λ > 0.

Proof of Theorem 1
We split the proof into two parts. 1.
Step: Existence for (P). Let {λ n } ∞ n=1 ⊂ (0, ∞) be a null sequence. For every n ∈ N, let To this aim, we make use the fact that A is skew adjoint and (2.1) to deduce that By the monotonicity of λ n , it holds for all t ∈ [0, T ], n ∈ N and (v, w) ∈ D(A) that Using again the fact that A is skew-adjoint, we obtain from (2.1) that (2.7) On the other hand, from Lemma 4, we know that J λ n = P K ∀n ∈ N. (2.8) Altogether, due to (2.3), (2.5), (2.7) and (2.8), passing to the limit n → ∞ in (2.6) leads to Consequently, as D(A) ⊂ X is dense and (I d − P K ) : X → X is continuous, we obtain that τ (p, q)), (p, q)) X ≥ 0.

Inserting (v, w) = (E, H )(t) + τ (p, q) with τ > 0 and (p, q) ∈ X, we have
Then, letting τ → 0, we deduce from the continuity of (I d − P K ) : X → X that from which it follows that

Proof of Theorem 2
We split the proof into three steps.

Consequently, by the local regularity property (3.3), (1.4) implies
Let us underline that in (3.5) the test function w (resp. v) does not appear in the region c E (resp. c H ), i.e., in the region where the L 2 -regularity of curl H (resp. curl E) is not guaranteed. This particular structure allows us to deduce that the third integral in (3.5)  We modify the extended vector field intõ The modified vector field η belongs as well to H (curl). Indeed, as η ∈ H (curl), the distributional definition of the curl-operator yields Thus, from which it follows that η ∈ H (curl) and curlη = H (curlη | H ) where H : L 2 ( H ) → L 2 ( ) denotes the zero extension operator (2.9). As a consequence, since (0, 0) ∈ K, (1.6) ensures that (0, η) ∈ D(A) ∩ K. Therefore, for arbitrarily fixed τ ∈ [0, T ) and h ∈ (0, T − τ ), we may insert Since η was chosen arbitrarily, similarly to the proof of Theorem 1 (2. Step), it follows that for a.e. τ ∈ (0, T ) and all η ∈ H (curl, E ) with η | c H ∈ H 0 (curl, c H ). (3.8) In particular, thanks to (3.

T ), D(A))
and (v, w)(t) ∈ K for all t ∈ [0, T ]. Therefore, considering this specific test function in (3.11) leads to from which it follows, after passing to the limit h ↓ 0, that  Furthermore, thanks to (3.14), we may insert (y, H 2 )) to obtain after adding the resulting inequalities that  ∈ (0, T ).
Multiplying the second equation in (3.16) with h and then integrating the resulting equality over E ∩ H , we obtain from the above identity that ∈ (0, T ).
Step 3: Uniqueness for (P) in the case of H = or E = . We only consider the case H = . The proof for the case E = is completely analogous. In view of H = , we have that K = K E × L 2 μ ( ) for some closed and convex subset K E ⊂ L 2 ( ) containing the zero element. Let (E, H ) ∈ W 1,∞ ((0, T ), X) denote a solution to (P). According to Theorem 1, since holds for a.e. t ∈ (0, T ). Then, applying (3.17) to (P) leads to the following variational inequality for the electric field E: with τ ∈ (0, T ), h ∈ (0, T − τ ) and z ∈ H 0 (curl) ∩ K E , we get the following variational inequality for E without the time integration: for a.e. t ∈ (0, T ) and all z ∈ H 0 (curl) ∩ K E . (3.18) Suppose that (E j , H j ) ∈ W 1,∞ ((0, T ), X), j = 1, 2, are solutions to (P). We set (e, h) : in the variational inequality (3.18) for E = E 1 (resp. E = E 2 ) and adding the resulting inequalities, we obtain μ∂ t e(t) · e(t) − h(t) · curl e(t) dx ≤ 0 for a.e. t ∈ (0, T ).
On the other hand, inserting w = H 2 (t) (resp. w = H 1 (t)) in the variational equality (3.17) for H = H 1 (resp. H = H 2 ) and adding the resulting equalities, we obtain

μ∂ t h(t) · h(t) = −curl e(t) · h(t) for a.e. t ∈ (0, T ).
In conclusion, it holds that d dt e(t) 2 μ ( ) ≤ 0 for a.e. t ∈ (0, T ), which yields that (P) admits at most only one solution. This completes the proof. In the following let (P n ) denote (P) with replaced by n .
Applying this equality to the previous inequality leads to 1 2n ∈ (0, T ).

Conclusion
In this paper, we developed an existence and uniqueness theory for the electromagnetic obstacle problem (P). While the existence is guaranteed for a general closed and convex set K ⊂ X containing the origin, the uniqueness is shown under two different assumptions. The general one is based on a localization strategy, leading to a localized variational inequality (3.15) on the electric and magnetic constraint regions c E and c H . The established well-posedness result finds applications in the mathematical modeling of electromagnetic shielding. Therefore, it serves as a fundament for the numerical simulation and shape design of electromagnetic shielding materials that requires a substantial extension of the developed techniques [31,33]. In particular, the numerical analysis of (P) requires a Sobolev regularity property for the electric field of the type E ∈ L 1 ((0, T ), H s ( )) for some s > 0. We aim at investigating this Sobolev regularity issue in our future research related to the numerical analysis of (P).