Consistency of a phase field regularisation for an inverse problem governed by a quasilinear Maxwell system

An inverse problem of reconstructing the magnetic reluctivity in a quasilinear magnetostatic Maxwell system is studied. To overcome the ill-posedness of the inverse problem, we propose and investigate two regularisations posed as constrained minimisation problems. The first uses the total variation (perimeter) regularisation, and the second makes use of the phase field regularisation. Existence of minimisers, sequential stability with respect to data perturbation, and consistency as the regularisation parameters tending to zero are rigorously analysed. Under some regularity assumption, we infer a relation between the regularisation parameters that allows one to recover a solution to the original inverse problem from the phase field regularised problem. The second focus of the paper is set on the first-order analysis of both regularisation approaches. For the phase field approach, two types of optimality systems are derived through a weak directional differentiability result and the domain variation technique of shape calculus. As a final result, we show the convergence of the optimality conditions obtained from shape calculus, leading to a necessary optimality system for the total variation inverse problem.


Introduction
Let Ω ⊂ R 3 be a bounded domain with a connected Lipschitz boundary containing two different types of materials. The first is a ferromagnetic material (iron, nickel, cobalt, etc.) which responses strongly to an external magnetic field, and the second is a nonmagnetic material (copper, aluminium, silver etc.) which is significantly less responsive to the magnetic field. Denoting the region occupied by the nonmagnetic material as Ω 0 and the complement region occupied by the ferromagnetic material as Ω 1 , so that Ω = Ω 0 ∪ Ω 1 and Ω 0 ∩ Ω 1 = ∅, our interest lies in solving the inverse problem of determining the location of Ω 1 based on measurements of certain quantities.
Electromagnetic phenomena can be well-described with the help of Maxwell's equations. In this paper, we focus on the magnetostatic setting, for which the governing partial differential equations (PDEs) derived from Maxwell's equations reduce to curl H = J, H = νB, div B = 0. (1.1) Here, H, B and J denote the magnetic field, the magnetic induction, and the applied current density, respectively. The magnetic reluctivity ν is the reciprocal of the magnetic permeability μ, so that the second equation in (1.1) can be equivalently expressed as the more standard form B = μH. Denoting by ν 0 as the vacuum magnetic reluctivity, for nonmagnetic materials under consideration, we assume that the corresponding magnetic reluctivity is equal to ν 0 . This is reasonable as many nonmagnetic materials exhibit a permeability close to the vacuum permeability μ 0 = 1/ν 0 . On the other hand, for the ferromagnetic material, the reluctivity ν highly depends on the magnitude of the magnetic induction B, i.e. ν = ν 1 (|B|) for some strictly positive function ν 1 ∈ C 1 (R). We refer to Kaltenbacher et al [28] concerning the identification of the nonlinear reluctivity ν 1 (|B|) based on magnetic induction measurements and its numerical realisation. Introducing the overall reluctivity as the function ν : In essence, complete knowledge of u in Ω allows us to determine the location of Ω 0 and Ω 1 as the sets {u = 0} and {u = 1}, respectively. Therefore, the function u plays the role of the solution to the inverse problem we introduce below. The equation (1.2) gives rise to a quasilinear saddle point structure. Under appropriate assumptions for the magnetic reluctivity, the well-posedness of (1.2) can be deduced. This leads to the solution operator S : u → y assigning every function u taking values in [0, 1] to the magnetic vector potential y posed on Ω. For the moment, let Z denote the solution space in which y belongs to (see (2.3) for its precise definition). Furthermore, let y m ∈ O denote a measurement of the vector potential y, where O is a Hilbert space with norm · O , and G : Z → O a bounded continuous operator. Then, our inverse problem read as (1.5) The prototype setting for this paper is O = L 2 (D) and G( y) = y| D where D ⊂ Ω is an open subset, and y m is a measurement of the magnetic vector potential in D induced by the applied current density J. Both y m ∈ O and J ∈ L 2 (Ω) are given data for the inverse problem (1.5).
Under stronger assumptions such as ∂Ω is of class C 1,1 or ∂Ω is convex, one may even consider y m to be a boundary measurement with O = L 2 (Σ) and G( y) = y| Σ , where Σ can be a part of the boundary (Σ ⊂ ∂Ω) or the entire boundary (Σ = ∂Ω).
Due to the compactness of the embedding Z ⊂⊂ L 2 (Ω) (see (2.5)), the inverse problem (1.5) is ill-posed. The standard approach to tackle the ill-posedness is to employ the Tikhonov regularisation [16,39]. From a theoretical point of view, perhaps the most intuitive is to penalise the perimeter of Ω 1 = {u = 1}, leading to the total variation inverse problem (TVIP): Here, α > 0 is a regularisation parameter, TV(·) denotes the total variation functional and BV(Ω, {0, 1}) is the set of functions of bounded variations with values in {0, 1} (see section 2.2 for more details). On the other hand, for numerical purposes, the non-convexity of BV(Ω, {0, 1}) introduces challenging issues for implementation. One remedy is to rephrase TVIP as a shape optimization problem and derive optimality conditions using shape calculus [24,25,27]. The second is to consider a further regularisation in which the total variation functional TV(·) is approximated by the Ginzburg-Landau functional E ε (·), where ε > 0 is a small parameter. This has been employed to great effect in various shape and topology optimisation problems, see [4-7, 21, 22, 25] and the references cited therein. In our present setting, it leads to the phase field inverse problem (PFIP): with a double well potential Ψ(s) and a nonempty closed convex set K ⊂ H 1 (Ω) (see (5.2) and (5.3) for their definitions).
The goal of this paper is to examine the mathematical analysis of both regularisation approaches TVIP and PFIP. More precisely, we show the existence of minimisers and investigate desirable properties such as the sequential stability with respect to perturbations in the data y m , and asymptotic limits α → 0 and ε → 0. To the best of the authors' knowledge, this paper is the first contribution towards TVIP and PFIP for the quasilinear Maxwell system (1.2) with two main novelties: the first is the convergence analysis for PFIP involving their respective minimisers and optimality conditions (theorems 5.2 and 6.11). Here, two types of optimality systems for PFIP (theorems 6.6 and 6.10) are established based on a weak directional differentiability result (lemma 6.3) and the domain variation technique of shape calculus (lemma 6.9). Based on the latter technique, we prove the convergence of the optimality conditions for PFIP as ε → 0, leading to an optimality system for TVIP. The second main novelty, which seems not to have received much attention in the literature, is the convergence of solutions for PFIP to a solution of the original inverse problem (theorem 5.3), and inferring (under ideal conditions) a relation of the form α = O(ε d ) with d < 2 (see (5.11)) between parameters α and ε that could serve as a useful guideline for numerical implementations. Let us mention that there are several possible numerical strategies to solve the phase field optimality conditions to numerically realise our solution, namely a parabolic variational inequality approach [4][5][6]14] or the VMPT method [7,8]. The analysis of these strategies, together with a finite element approximation of the forward model and adjoint system in a fashion similar to [43], would be the next step of our investigation. However, due to the length of the paper, we choose to defer the numerical investigations of the inverse problem (1.5) and its regularisations to future work.
Identification problems of material parameters in linear Maxwell's equations have been extensively investigated in many contributions, including [19,29,33,35,41]. On the other hand, the mathematical analysis for inverse problems governed by nonlinear Maxwell's equations is still in the early stages of development. In the context of the optimal control, stationary and evolutionary nonlinear Maxwell's equations were investigated in [32,[42][43][44]. More recently, [12] analysed ill-posed backward nonlinear Maxwell's equations and derived a variational source condition for the convergence rate of the corresponding Tikhonov regularisation. We believe that our present results may lead to further progresses in the mathematical analysis of nonlinear electromagnetic inverse problems.
The paper is structured as follows: in section 2 we introduce several useful auxiliary results and preliminaries. In section 3 we prove the well-posedness of (1.2) and continuity properties of the solution operator S. The existence of minimisers, sequential stability with respect to data perturbation, and asymptotic behaviour as α → 0 and ε → 0 are discussed in sections 4 and 5, respectively. In section 6 the first-order necessary optimality conditions are derived, and we discuss the convergence of the phase field optimality conditions to the total variation optimality conditions. In the appendix we state a useful result involving the Γ-convergence of functionals.

Function spaces for electromagnetic problems
For any open set Ω ⊂ R 3 , the Hilbert spaces where curl f and div f are to be understood as the weak curl and weak divergence of f ∈ L 2 (Ω), respectively. We define the subspace H 0 (curl) ⊂ H(curl) as the completion of C ∞ c (Ω; R 3 ) with respect to the H(curl)-topology, which admits the following characterisation (cf. appendix A of [45].) Our solution space, consisting of divergence-free H 0 (curl) functions, is denoted as and is equipped with the H(curl)-norm (2.1). We now state several well-known results: • The continuous embedding We refer to ( [13], theorem 2) for the first case and to ([2], theorems 2.12 and 2.17) for the second case. • An immediate consequence is the Maxwell compactness property first attributed to Weck [40] H 0 (curl) ∩ H(div) ⊂⊂ L 2 (Ω). (2.5) • The Poincaré-Friedrich-type inequality: there exists a positive constant C such that

Functions of bounded variations
We review basic properties for functions of bounded variations that are sufficient for our analysis. For a more detailed introduction we point to [1,18,23]. We say that u ∈ L 1 (Ω) is a function of bounded variation in Ω if its distributional gradient Du is a finite Radon measure. The space of all such functions is denoted as BV(Ω) and is endowed with the norm · BV(Ω) = · L 1 (Ω) + TV(·), where for u ∈ BV(Ω), the total variation TV(u) is defined as The space BV(Ω, {0, 1}) denotes the space of all BV(Ω) functions taking values in {0, 1}. We say that a set E ⊂ Ω is a set of finite perimeter if χ E ∈ BV(Ω, {0, 1}), where for a set A, is not constant, then there exists a measurable set of finite perimeter E u defined as in Ω, where B δ (x) denotes the ball centred at x with radius δ, and |B δ (x)| its Lebesgue measure. The perimeter of a subset E ⊂ Ω of finite perimeter is defined as is a bounded sequence, then there exists a subsequence (k n ) n∈N and a limit u ∈ BV(Ω) such that u k n → u in L 1 (Ω) and TV(u) ≤ lim inf n→∞ TV(u k n ).

Saddle point problems
The (d) ∃k > 0 s.t. B q V * ≥ k q W and V admits a direct decomposition with V 0 := KerB and V ⊥ 0 . Then, for any ( f, g) ∈ V * × W * , the nonlinear saddle point problem

Analysis of the forward model
For a fixed function u : Ω → [0, 1], we define the operator A u : H 0 (curl) → H 0 (curl) * and the bilinear form b : Then, a mixed formulation of (1.2) reads as The function φ is referred to as the Lagrange multiplier associated with (1.2). If holds, then choosing v = ∇ψ in the first equality yields b(∇ψ, φ) = 0 for all ψ ∈ H 1 0 (Ω), i.e. φ is a weak solution to the homogeneous Dirichlet Laplace problem, hence φ = 0.
To analyse the forward model (3.1), we make the following assumptions (cf. [28] for their physical justification), which we assume to hold throughout the rest of the paper. Assumption 3.1. Let ν 0 > 0 denote the vacuum magnetic reluctivity. We assume that (A3) There exists a constantν ∈ [ν 0 , ∞) such that the mapping s → ν 1 (s)s satisfies The observation operator G : Z → O is bounded and continuous.
Furthermore, there exists a positive constant C depending only on ν 0 , ν ,ν and Ω such that We stress that the above estimate is independent of u ∈ L 1 (Ω; [0, 1]) thanks to (3.7). The well-posedness of (3.2) allows us to define a solution operator and the next result shows a continuity property.
Proof. Let (u k ) k∈N be a sequence satisfying the hypothesis. From the estimate stated in theorem 3.1, we can extract a non-relabelled subsequence and limit functions ( y, φ) ∈ Z × H 1 0 (Ω) such that Settingŷ k = y k − y as the difference, and upon subtracting the term Ω [ν 0 (1 − u k ) + ν 1 (|curl y|)u k ]curl y · curl v dx from both sides of (3.2) leads to Substituting v =ŷ k ∈ Z and employing the bounds ν ≤ ν 1 (·) ≤ ν 0 , the strong monotonicity The right-hand side of (3.12) converges to zero thanks to the L 2 (Ω)-weak convergenceŝ y k 0, curlŷ k 0 and the L 2 (Ω)-strong convergence ν i u k curl y → ν i u curl y for i = 0, 1. Then, (2.5) and (2.7) imply Thus, after extracting a non-relabelled subsequence we obtain due to (A1) and (A2) that In conjunction with u k curl y k → u curl y in L 2 (Ω) derived from the generalized dominating convergence theorem and the facts u k curl y k → u curl y a.e. in Ω, |u k curl y k | ≤ |curl y k | and (3.13), it follows that Hence, passing to the limit k → ∞ in (3.2) for ( y k , φ k ) with data (u k , J) shows that the limiting pair ( y, φ) satisfies (3.2) with data (u, J). Now, since the unique solution of (3.2) with data (u, J) is independent of the choice of the extracted subsequence ( y k , φ k ) k∈N , classical arguments yield that the convergence properties (3.11) and (3.13) hold true for the whole sequence. This completes the proof.

Total variation inverse problem
Throughout this section let J ∈ L 2 (Ω), y m ∈ O and α > 0 be fixed. The total variation regularised inverse problem (TVIP) reads as The following theorem shows that (4.1) exhibits desirable properties, such as existence of a solution and being (sequentially) stable with respect to data perturbations. Furthermore, under suitable conditions, a minimum-variation solution to the original inverse problem (1.5) can be obtained (provided the solution set is non-empty) from (4.1) as α → 0.
• If ( y n m ) n∈N ⊂ O is a sequence such that y n m → y m in O, and u α n denotes a solution to (4.1) with data y n m , then along a non-relabelled subsequence it holds that u α n converges to a solution u α to (4.1) with data y m in the sense of intermediate convergence (2.9). . Although our present setting allows for a more abstract measurement space O and measurement operator G, we mention that, for u α n → u α in L 1 (Ω), theorem 3.2 together with the boundedness and continuity of G implies . These observations, in conjunction with the arguments in [5] are sufficient to infer the assertions of theorem 4.1. Hence, we omit the proof.

Phase field inverse problem
In all what follows, let J ∈ L 2 (Ω), y m ∈ O and α, ε > 0 be fixed. The phase field regularised inverse problem (PFIP) reads as In the setting of (5.1), Ψ is a nonnegative double well potential with minima at 0 and 1, while γ is a constant depending only on Ψ. It is clear from the definition that for E ε to be well-defined, we must expand the solution space K from BV(Ω, {0, 1}) to subsets of H 1 (Ω). If Ψ is defined everywhere on R, such as the smooth double well potential Ψ(s) = s 2 (1 − s) 2 , we may choose K as the whole of H 1 (Ω). Alternatively we can consider the double obstacle potential [9] Ψ(s) = that is only finite over the interval [0, 1], so that the solution space for PFIP can be taken to be the following closed and convex set In this setting, , then by minor modifications of ( [9], theorem 3.7) detailed in appendix, we find that the following extended functionals which motivates the investigation of (5.1). For the rest of the paper, we consider K as defined in (5.3) and take Ψ as the double obstacle potential (5.2) with γ = 8 π .

Properties of solutions
Theorem 5.1. The following assertions hold: • There exists at least one solution u α ε ∈ K to (5.1). • If ( y n m ) n∈N ⊂ O is a sequence such that y n m → y m in O, and u α ε,n ∈ K denotes a solution to (5.1) with data y n m , then along a non-relabelled subsequence it holds that u α ε,n → u α ε in H 1 (Ω) where u α ε ∈ K is a solution to (5.1) with data y m . Let us point out the analogue of intermediate convergence for H 1 (Ω)-functions would be the norm convergence of the gradient ∇u α ε,n L 2 (Ω) → ∇u α ε L 2 (Ω) . Furthermore, since the arguments to prove theorem 5.1 is somewhat standard in the literature, we will omit the proof of existence (which is shown via the direct method) and sketch the details for sequential stability.
and by definition, Choosing, for instance, v = 1 yields the boundedness of (u α ε,n ) n∈N in K. Then, the compact embedding H 1 (Ω) ⊂ L 2 (Ω) and theorem 3.2 give along a non-relabelled subsequence with limit u α ε ∈ K due to K being convex and closed. Passing to the limit n → ∞ in (5.5) and employing weak lower semicontinuity, we arrive at J ε (u α ε ) ≤ J ε (v) for all v ∈ K, and so u α ε is a solution to (5.1). Meanwhile, passing to the limit n → ∞ in the inequality (5.5) with the choice v = u α ε ∈ K yields

Convergence of solutions
An immediate consequence of the Gamma convergence (5.4) is the following result concerning the asymptotic behaviour of minimisers (u α ε ) ε>0 to (5.1) as ε → 0. Theorem 5.2. Let (u α ε ) ε>0 ⊂ K denote a sequence of solutions to PFIP (5.1). Then, there exist a non-relabelled subsequence and a limit u α ∈ BV(Ω, {0, 1}) such that lim ε→0 u α ε = u α in L 1 (Ω), lim ε→0 J ε (u α ε ) = J(u α ), and u α is a solution to TVIP (4.1). Proof. While the argument is somewhat standard, see for instance ([21], proof of theorem 2), nevertheless we briefly sketch the details, as some of the elements of the proof will be used later. Let w ∈ BV(Ω, {0, 1}) be arbitrary, then by (3.10) it is clear that J(w) < ∞. We define the set E w := {w = 1} so that w = χ E w . Our aim is to construct a sequence (w ε ) ε>0 ⊂ K such that w ε − w L 1 (Ω) → 0 and lim sup ε→0 J ε (w ε ) ≤ J(w). In the trivial case where w ≡ 0 (resp. w ≡ 1), which corresponds to E w = ∅ (resp. E w = Ω), we can choose w ε = w for all ε > 0 so that E ε (w ε ) = 0 and In the non-trivial case where 0 < |E w | < |Ω|, using ( [30], lemma 1), we can approximate E w ⊂ Ω by a sequence (E k ) k∈N of open bounded sets in R 3 with smooth boundaries such that where AΔB denotes the symmetric difference between the sets A and B, and H 2 denotes the two-dimensional Hausdorff measure. Then, setting w k = χ E k ∩Ω leads to We apply item (b) of lemma A.1 with v 0 = w k and A = E k , so that for each k there exists a sequence (w k For this particular sequence (ε k ) k∈N , setting y ε k := S(w k ε k ) and y w := S(w), by theorem 3.2 we have y ε k → y w in Z as k → ∞. Continuity of G : Next, by definition of u α ε as a solution to (5.1), it holds that This implies that sup k∈N E ε k (u α ε k ) < ∞ and by item (c) of lemma A.1, there exists a nonrelabelled subsequence and a limit u α ∈ BV(Ω, {0, 1}) such that u α ε k → u α strongly in L 1 (Ω). Furthermore, theorem 3.2 asserts S(u α ε k ) → S(u α ) in Z and so lim k→∞ J f (u α ε k ) = J f (u α ), while when we invoke item (a) of lemma A.1, (5.6) and (5.7), we obtain As w ∈ BV(Ω, {0, 1}) is arbitrary this implies that u α is a solution to (4.1). Now, following the start of the proof, we construct a sequence (v k ε k ) k∈N satisfying (5.6) with u α in place of w, and observe that which implies lim k→∞ J ε k (u α ε k ) = J(u α ). Let us now address the convergence as α → 0 and ε → 0.  (a) We mention that the obvious choice for (w k ) k∈N is the sequence constructed in the proof of theorem 5.2 which satisfies (5.6) (with u * in place of w). This fixes the null sequence (ε k ) k∈N subordinate to u * . In particular, since (w k ) k∈N always exists, the statement of condition (b) can always be reduced to 'the inverse problem (1.5) has a solution u * ∈ BV(Ω, {0, 1})'. However, in order to define the null sequence (α k ) k∈N subordinate to (ε k ) k∈N and u * , it is necessary to state condition (b) as it is presented. (b) If u * is a minimum-variation solution to the inverse problem (1.5), then the inequality (5.9) implies that u is a minimum-variation solution to (1.5) as well.
Proof. For each k ∈ N, let u α k ε k ∈ K denote a solution to min v∈K J k (v) where For the sequence (w k ) k∈N in the hypothesis, we set z k = S(w k ). Then, from the inequality since for y * = S(u * ) it holds that G( y * ) = y m in O. Lipschitz continuity of G : Z → O and the estimate (3.14) imply that The right-hand side is non-negative and its limit superior as k → ∞ is bounded by the hypothesis. Hence, it is clear that Invoking the compactness property (c) of lemma A.1 leads to the existence of a non-relabelled subsequence (k → ∞) and a limit u ∈ BV(Ω, {0, 1}) such that u Taking limit superior on both sides and employing the hypothesis lim sup k→∞ E ε k (w k ) ≤ TV(u * ) leads to From the equality J f (u) = 0 we infer that G • S(u) = y m , i.e. u is a solution to the inverse problem (1.5). Moreover, as u α k ε k → u in L 1 (Ω), by the hypothesis, (5.10) and the item (a) of lemma A.1, This completes the proof.
Let us point out that in corollary 5.4 the null sequence (ε k ) k∈N does not need to be subordinate to the true solution u * (or its approximating sequence) as in theorem 5.3, which gives greater flexibility at a cost of assuming more regularity on the true solution. Furthermore, we note from the condition (5.11) and the estimate w ε k − u * L 1 (Ω) = O(ε k ) that ε k plays a similar role to the parameter δ in theorem 4.1.

Optimality system via directional differentiability
In this section, we establish a necessary optimality system for (5.1) through the use of the weak directional differentiability of the solution operator S : K → Z. To show this property, let us first discuss the linearised equation associated with the saddle point system (3.2). In the following, letū ∈ K, h ∈ L ∞ (Ω) andȳ = S(ū). We seek a unique solution (z, θ) ∈ Z × H 1 0 (Ω) to the linear saddle point problem where the bilinear form aȳ : H 0 (curl) × H 0 (curl) → R and the linear form f h,ȳ : H 0 (curl) → R are given as for all z, v ∈ H 0 (curl). Strictly speaking, the Lagrange multiplier θ is zero as f h,ȳ (∇ψ) = 0 for all ψ ∈ H 1 0 (Ω). But we include it to retain the saddle point structure. Lemma 6.2. Letū ∈ K, h ∈ L ∞ (Ω) andȳ = S(ū). Then, there exists a unique solution z = z(ū, h) ∈ Z, θ = 0 ∈ H 1 0 (Ω) to (6.5) satisfying for a positive constant C depending only on ν 0 , ν and Ω.

Optimality system via shape calculus and its convergence
In this section, we derive an alternative optimality system for PFIP (5.1) via the domain variation technique of shape calculus. Our main result is the convergence of this system as ε → 0, which leads to an optimality system for TVIP (4.1).

Assumption 6.2.
In addition to assumption 6.1, we further assume (A7) The domain Ω is either a convex domain or a domain with C 1,1 -boundary. (A8) The prescribed current density J satisfies J ∈ H 1 (Ω) and the measurement vector potential y m additionally satisfies y m ∈ H 1 (Ω).
We remark that (A8) is required due to the domain variation methodology we employ to derive optimality conditions. Moreover, (A7) together with (2.4) and (2.6) implies that and hence S(u) ∈ H 1 (Ω) for any u ∈ L 1 (Ω; [0, 1]). This improved regularity is needed to prove the differentiability of certain transformed solutions (see lemma 6.9) in preparation for the main result (theorem 6.11).
The optimality conditions derived in this section involves domain variations, which is performed with admissible transformations and their corresponding velocity fields.
and there exists C > 0 such that Then, the space T ad of admissible transformations is defined as the set of corresponding solutions to the ordinary differential equation for V ∈ V ad , which yields a mapping T: [−τ 0 , τ 0 ] × Ω → Ω, T t (x) := T(t, x), for some τ 0 ∈ (0, τ) sufficiently small.
as t → 0, where I is the identity matrix, DT t is the Jacobian matrix of T t and ξ(t, While we recognise T t (Ω) = Ω for all t ∈ [−τ 0 , τ 0 ] due to (V2), in some of the calculations below we use the notation Ω t = T t (Ω) for better clarity. Through the relation where curl is the curl operator with respect to the transformed variables x = T t (x) and the tensor product A × B for two second order tensors A = (a kl ) 1≤k,l≤3 and B = (b kl ) 1≤k,l≤3 is defined as A × B = 3 j,k=1 ijk 3 l=1 a kl b lj with the antisymmetric tensor ijk . In particular, y ∈ H(curl; Ω) may not imply y • T t ∈ H(curl; Ω t ). Although via (6.2) the solution to the state equation belongs to H 1 (Ω), the expression (A7) for the curl operator is still difficult to work with. On the other hand, the transformation is curl preserving, i.e. it holds that [see ( [31], corollary 3.58)] ( curl y) and so curl y ∈ L 2 (Ω) if and only if curl y ∈ L 2 (Ω t ). Another advantage of (6.23) is that, for given y, z ∈ H(curl) with y : ∈ H 0 (curl; Ω t ) and y ∈ H 0 (curl; Ω t ) iff DT t y • T t ∈ H 0 (curl; Ω). Therefore, in the sequel we focus mainly on the transformation (6.23) for the vector potential y. Furthermore, we use the following notations Then, by remark 6.1, there exists a positive constant C such that for all t ∈ (−τ 0 , τ 0 ) with τ 0 sufficiently small, we infer, similar to the proof of lemma 6.7, that . The second term on the right-hand side tends to zero as t → 0 by remark 6.1, while the first term tends to zero as t → 0 by (6.25) and the dominated convergence theorem. Lemma 6.9. Let V ∈ V ad be an admissible velocity with the corresponding transformation T ∈ T ad . Let u ∈ L 1 (Ω; [0, 1]) and define u t : 2) corresponding to data (u t , J), and we denote (u 0 , y 0 , φ 0 ) simply by (u, y, φ). Then, there exists τ 0 > 0 such that the mappings where a y (·, ·) is defined as in (6.6), while G u,y,φ : H 0 (curl) → R and H y : |curl y| (curl y · B (0)curl y)curl y · curl v dx (6.28) Proof. Thanks to the smoothness of V ∈ V ad , we can choose τ 0 sufficiently small so that there exist positive constants c 0 , . . . , c 4 , independent of t such that hold for all t ∈ [−τ 0 , τ 0 ] and for all non-zero ζ ∈ R 3 . We consider the functional t, ( y, φ)), F 2 (t, ( y, φ))) ∈ H 0 (curl) * × H −1 (Ω), defined as 1
Since DT t y t • T t ∈ H 1 (Ω), see the discussion after remark 6.1, it holds that (I − A −1 ) Y t ∈ H 1 (Ω), which implies the relation divΥ t = div((I − A −1 ) y t ) ∈ L 2 (Ω) holds with the estimate divΥ t 2 L 2 (Ω) ≤ C I − A −1 2 L ∞ (Ω) + div(I − A −1 ) 2 L ∞ (Ω) , (6.36) where C > 0 depends only on y t H 1 (Ω) (which is bounded by J L 2 (Ω) due to (6.30)) and DT t W 1,∞ (Ω) (which is bounded in t due to remark 6.1). Hence, substituting (6.35) into (6.34), and adding (6.36) to the resulting inequality, by Young's inequality applied to the terms involving the L 2 (Ω)-norm of Υ t , we arrive at for some positive constant K > 0 that can be chosen as small as one desires, with the constant C appearing twice above depending on K. Invoking (2.6) and choosing K sufficiently small, together with (2.4), (6.25), (6.33), and lemma 6.8 we then infer the uniform bounds so that along a non-relabelled subsequence for some functionsẎ[V] ∈ H 1 (Ω) andφ[V] ∈ H 1 0 (Ω). Let us remark divΥ t = 0 due to the definition of y t , and so the Poincaré inequality (2.7) for Z-valued functions cannot be used to control the L 2 (Ω)-norm of Υ t arising on the right-hand side of (6.34). Therefore, it is necessary to derive an estimate for divΥ t and in turn we require the regularity y t ∈ H 1 (Ω t ) for all t ∈ [−τ 0 , τ 0 ], which is guaranteed with the assumption (A7) for the domain Ω and the continuous embedding (2.4).