A note on $3$d-$1$d dimension reduction with differential constraints

Starting from three-dimensional variational models with energies subject to a general type of PDE constraint, we use Γ-convergence methods to derive reduced limit models for thin strings by letting the diameter of the cross section tend to zero. A combination of dimension reduction with homogenization techniques allows for addressing the case of thin strings with fine heterogeneities in the form of periodically oscillating structures. Finally, applications of the results in the classical gradient case, corresponding to nonlinear elasticity with Cosserat vectors, as well as in micromagnetics are discussed.

1. Introduction. In [1], Acerbi, Buttazzo & Percivale give the first rigorous derivation of a lower dimensional theory for thin (almost one-dimensional) strings using an ansatz-free approach based on variational methods. The work by Le Dret & Raoult [25], which can be viewed as the analogue for objects that are thin in one space dimension, relies on Γ-convergence techniques as well, and has brought up a two-dimensional nonlinear membrane model. This article focuses on strings and provides an abstract dimension reduction result, which has the potential of covering a number of different applications in continuum mechanics and electromagnetism, like hyperelasticity, micromagnetics or magnetoelasticity. Mathematically, this is achieved by working with energy functionals of integral form whose admissible functions solve a PDE constraint conveyed by a first-order differential operator made precise in (3). For a discussion of the analogous problem in the context of thin films we refer to [19,20,22]. The literature on 3d-1d dimension reduction, also with different scalings of the energy leading to models for beams and rods, includes e.g. [24,29,28,34,33,11].
Regarding the integrands, we require g ε : Ω ε × R m → [0, ∞) to be Caratheodory (i.e. g ε ( · , ξ) is measurable for all ξ ∈ R m and g ε (y, · ) is continuous for almost all y ∈ Ω ε ) with uniform (in ε) p-growth and p-coercivity. Moreover, let A be a linear first-order differential operator with constant coefficients that can be represented as and satisfies Murat's constant-rank property, which has its origins in the theory of compensated compactness [31,32,35], and says that for the symbol A of A, If, for example, A = curl (and Ω ε is simply connected, so that curl-free fields on Ω ε have a gradient representation), then I ε in (2) can be interpreted as an integral functional depending on gradients, and hence, as the stored energy of a deformation, such as stretching or shearing, acting on an elastic string. Solenoidal fields are admissible for (1) if one takes A = div. A connection to ferromagnetism becomes apparent by using A = A mag as defined later in (24). The variational theory of integral problems with differential constraints as above was essentially established by Fonseca & Müller [14], building amongst others on work by Dacorogna [9]. Further problems in the A-free setting, including relaxation and homogenization, are investigated for example in [7,12,13,3].
A suitable rescaling of (1) helps to loose the parameter-dependence of Ω ε , and thus of X ε . By choosing the change of variables with y = (y 1 , . . . , y d−1 ) and defining u( where the transformed density functions f ε : with constants c, C independent of ε. The rescaled differential operator in (6) reads Note that the parameter transformation (5) makes the operator in the PDE constraint ε-dependent (see Figure 1). This amounts to a technical difficulty, since the classical projection onto A ε -fields [14, Lemma 2.14] may lead to diverging projection errors as ε gets small. Figure 1. Transformation of Ω ε into Ω 1 and rescaling of the differential constraints.
The main theorem of this paper on a reduced model for homogeneous strings is formulated in the framework of Γ-convergence, for an introduction the reader is referred to [10,6]. We remark that the following statement requires the operator A to fulfill the two assumptions A1 and A2, which are made precise in Section 2.1 and hold for all the examples mentioned above. Theorem 1.1 (Dimension reduction). Let A be a constant-rank operator according to (4) such that Assumptions A1 and A2 are satisfied, and let f ε = f for all ε > 0 with a Caratheodory function f : Ω 1 × R m → [0, ∞) of p-growth and p-coercivity.
(i) If f is convex in the second variable, then The Γ-limit is taken with respect to weak convergence in L p (Ω 1 ; R m ), and A 0 as in (12) is the limit operator of (A ε ) for ε → 0.
(ii) If f is only continuous in the second variable, there is the following upper bound: Here, Q A f denotes the A-quasiconvex envelope of f regarding the second variable (cf. [14]), i.e.
Here and in the following, T d is the d-dimensional torus as it results from glueing opposite edges of Q d , and by Av = 0 in T d , or equivalently v ∈ ker T d A, we understand that Similarly, by v ∈ ker Ω1 A, we mean that (9) holds for all test functions ϕ ∈ C ∞ c (Ω 1 ; R l ). Remark 1. a) The essential step in the proof is to handle the parameter dependence in the differential constraint. In particular, this requires the characterization of a suitable limit operator A 0 of (A ε ), see Section 2. Then, the remaining relaxation argument follows along the lines of [19]. b) A lower bound for continuous (in the second argument) f can be derived in analogy to the case of thin films [19, Theorem 1.1], using a Young measure approach based on localization by blow-up. If A = div, then Q div f = f * * with f * * the convexification of f with respect to the second argument, and the matching lower bound is trivial to prove. In general, though, the two bounds will not coincide, so that the characterization of the Γ-limit remains an open problem. Theorem 1.1 is the basis of further implications regarding thin strings with heterogeneities, see Section 3. In Section 4.1, we apply the abstract results of this paper to the gradient case, discussing applications in elasticity with Cosserat vectors. Finally, Section 4.2 provides a new model for thin ferromagnetic strings.
2. Characterization of the limit operator. This section is devoted to the derivation of the correct limit operator for (A ε ) as ε → 0, i.e. we are looking for a map A 0 : L p (Ω 1 ; R m ) → W −1,p (Ω 1 ; R l ) whose kernel coincides with the set of weak L p -limits of A ε -free sequences in Ω 1 .
2.1. Assumptions on A. Throughout this paper, we always assume without further mentioning that the constant-rank operator A has the representation where + ∈ R s×m , and s = rank A . In other words, the number of nonzero rows in A is supposed to equal rank A . In terms of symbols, (10) says that c) We point out that (10) is not restrictive for the set of A-free fields in view of Gauss elimination (switching rows and adding rows multiplied by a nonzero factor).
where e k is the kth standard unit vector. Then, m = d and l = s = r = 1, so that A = A + ∈ R 1×d(d−1) and A (d) = A Since s = rank A = 3, (10) is realized with A = A + . c) For general A = curl, defined for a function U : one obtains m = nd and l = n d(d−1)

2
. Notice that the requirement i < j is usually not found in the standard definition of curl. Here it is used to eliminate expressions redundant for the characterization A-free fields, cf. [19,Section 2.7.2]. A careful analysis of (A (k) ) hij,qp = δ ki δ qh δ pj − δ kj δ qh δ pi , i, j, p, k = 1, . . . , d, i < j, h, q = 1, . . . , n, shows that r = n(d − 1) and s = rank A = l. Thus, here as well, we find that A = A + , so that curl as in (11) fits into the framework of (10).
Provided (10) holds, the limit operator of (A ε ) can be defined row-wise, precisely If (10) is violated, A 0 may not capture the full asymptotic properties of A ε -free fields, as [19, Example 2.1] with exchanged roles of x 1 and x 2 demonstrates. In the special case A = A + , as for instance for A = div and A = curl (cf. Example 2.1), formula (12) reduces to A 0 = A .
In order to prove that A 0 is in fact the correct limit operator for (A ε ), two more conditions are needed. The first one regards the extension of A 0 -free fields from Ω 1 to T d , while preserving the A 0 -freeness. Recall that we always assume ω ⊂⊂ Q d−1 .
Assumption 1 (Extension of A 0 -free fields). Let A and A 0 be as in (10) and (12), If A 0 = A , which is the most relevant case for the applications we have in mind, it is sufficient that A , interpreted as a constant-rank operator in (d−1) dimensions, admits a suitable extension.
gives rise to a well-defined linear, bounded operator. Settingū = T u immediately yields Assumption A1.
Remark 3. Lemma 2.2 essentially requires ω to be an A -free extension domain in the sense of [17,Definition 4.3]. The example of the differential operator associated with the Cauchy-Riemann equations (d = m = 2, C ∼ = R 2 ) makes clear that an extension operator as in Lemma 2.2 may not always exist; for more details see [17,Section 4.2].
it follows that the dth column of W , denoted by W d , is constant, and that

bounded linear extension operator for Sobolev spaces, defining
The second assumption is a structural property on the operator A.

2.2.
A characterization result and the proof of Theorem 1.1. The following proposition specifies the statement that A 0 is the suitable limit operator for (A ε ) as ε → 0.
Proposition 1 (Limit operator A 0 ). Let A be a constant-rank operator satisfying Assumptions A1 and A2, and let A 0 be defined in (12).
Once Proposition 1 is proven, Theorem 1.1 follows essentially as a corollary.
Proof of Theorem 1.1. (i) In view of Proposition 1 (i), the lower bound becomes trivial and follows immediately from the weak lower semicontinuity of integral functionals with convex integrands.
For a given u ∈ by Lebesgue's dominated convergence, which is applicable due to the continuity and the p-growth of f (in the second argument).
(ii) Let u ∈ L p (Ω 1 ; R m ) ∩ ker Ω1 A 0 be given, and (u ε ) be the corresponding sequence resulting from Proposition 1 (ii). By [7, Theorem 1.1], one finds for every We observe that Q A f = Q Aε f for every ε > 0, as a consequence of a change of variables in the spirit of [19,Lemma 2.12]. Besides, Q A f is upper semicontinuous in the second variable by [14,Proposition 3.4]. Hence, we may conclude by selecting a diagonal sequence (u j(ε) ε ) ε with the required properties. Remark 5. Even if A and its rescaled versions A ε have constant rank, the limit operator A 0 : L p (Ω 1 ; R m ) → W −1,p (Ω 1 ; R l ) as defined in (12) in general does not. In fact, if A = A + (or s = l), then A 0 = A , and rank A (e 1 ) = rank We remark that A is of constant rank, though, when interpreted as an operator

2.3.
Proofs. The proof of Proposition 1 is split in two parts, separated by a paragraph on technical tools regarding projections onto A ε -free fields.
Proof of Proposition 1 [Part I]. (i) As u ε ∈ ker Ω1 A ε for ε > 0, one obtains for all Multiplying with 1/ε and letting ε tend to zero entails A + u = 0 in Ω 1 . Similarly, we argue that A Step 1: Extension and mollifications. By Assumption A1 there existsū ∈ L p (T d ; R m ) ∩ ker T d A 0 such thatū = u in Ω 1 . Mollifyingū with standard unit kernels is an operation compatible with the A 0 -free constraint and hence, gives rise to a smooth approximating sequence (ū j ) ⊂ C ∞ (T d ; R m )∩ker T d A 0 . In view of a diagonalization procedure, it is sufficient to prove (ii) with Ω 1 = Q d for eachū j . In the following, we may therefore assume that u is smooth and that A 0 u = 0 in T d .
Step 2: Special case u = u(x d ). In this case, an explicit construction gives a suitable A ε -free approximation of u.
where S (k) are the matrices resulting from Assumption A2, has the desired properties. By construction, u ε ∈ C ∞ (Q d ; R m ) and it is immediate to see that u ε → u in L p (Q d ; R m ). Next, we show that Indeed, since u ε is smooth, we compute for every x ∈ Q d , For the last equality, we exploited Assumption A2, as well as A For the remaining steps of the proof of Proposition 1 (ii), projections play a decisive role. The heuristic idea is to define u ε := P Aε u for all u ∈ ker A 0 not covered by Step 2 of the previous proof, where P Aε is a suitable projection operator onto A ε -free fields. Then, in a sense to be made precise later.
To make these considerations rigorous, we take a new viewpoint by switching to the Fourier space setting. There, projections can conveniently be defined algebraically. For η ∈ R d \ {0} and a differential operator A of the form (3) we consider the orthogonal projection onto ker A(η) ⊂ R m , calling it P A (η) ∈ Lin(R m ; R m ). Let us recall that a map m : R d \ {0} → R is called a L p -Fourier multiplier, if the operator T m defined by T m := F −1 (m F) on the Schwartz space S(R d ), where F denotes the Fourier transform and F −1 its inverse, extends to a bounded linear operator on L p (R d ). The space of L p -Fourier multipliers will be denoted by M p (R d ), and a norm is defined by m Mp(R d ) = T m Lin(L p (R d );L p (R d )) .

Lemma 2.5 (Projections as Fourier multipliers). Let
A be an operator of constant rank. (i) The mapping P A : R d \{0} → Lin(R m ; R m ) is an L p -Fourier multiplier, i.e. P A ∈ M p (R d ; Lin(R m ; R m )).
(ii) There is a constant C > 0 (independent of ε) such that for every ε > 0.
Proof. The proof of (i) can be found in [14, Lemma 2.14], where it is employed that P A is 0-homogeneous and smooth owing to the constant-rank property of A. To prove (ii), one possibility is to invoke the Mihlin multiplier theorem along with a scaling argument for Fourier multipliers in the spirit of [19, Lemma 2.7], leading to P Aε Mp(R d ;Lin(R m ;R m )) = P A Mp(R d ;Lin(R m ;R m )) < ∞. Alternatively, we infer from the Lizorkin multiplier theory (see [26], also cf. [19, Remark 2.9]) that P Aε Mp(R d ;Lin(R m ;R m )) ≤ c C L (P Aε ) with c > 0 depending on d and p, and Defining η ε := (η /ε, η d ) for η ∈ R d , we compute . Since η = 0 if and only if η ε = 0, this shows that C L (P Aε ) = C L (P A ) for all ε > 0.

Remark 6.
As an immediate consequence of the ε-independence of C L (P Aε ), we observe that P Aε L ∞ (R d ;Lin(R m ;R m )) is uniformly bounded with respect to ε.
In terms of the discrete Fourier multiplier operators associated with P Aε one obtains the following: Lemma 2.6 (Projection onto A ε -free fields). Let A be a constant-rank operator. Then there are linear, uniformly (in ε > 0) bounded operators P Aε : Proof. Let P Aε be the discrete multiplier operator corresponding to (P Aε (η)) η∈Z d \{0} , precisely, for u ∈ L p (T d ; R m ), withû the discrete Fourier coefficients of u. In view of (15), there is c > 0 such that P Aε u L p (T d ;R m ) ≤ c u L p (T d ;R m ) for all u ∈ L p (T d ; R m ). Proving the estimate of the projection error follows closely along the lines of [19, Theorem 2.8], which in turn is a modification of [14, Lemma 2.14]. Here, again the Lizorkin multiplier theorem is needed to get C > 0 independent of ε.

Remark 7.
Notice that the constant-rank property of A is essential here. In particular, a projection operator "P A0 " (cf. (14)) with the properties of Lemma 2.6 does not exist in general, as the following counterexample (slightly adapted from an idea by Krömer [21]) illustrates: Let A = div with d = m = 2, l = 1, and consider the corresponding limit operator div 0 with div 0 u = div u = ∂ 1 u 1 . We observe that the orthogonal projection onto div 0 -free fields is given by For the sequence (u j ) ⊂ L 2 (T d ; R 2 ) defined by The next lemma deals with the convergence of the kernels of the symbols A ε , formulated in terms of the corresponding projections P Aε .

Lemma 2.7 (Convergence of the projection operators in Fourier space). Suppose
A is a constant-rank operator. If η ∈ R d is such that η = 0, then P Aε (η) → P A0 (η) pointwise.
To prove (16) for s < l, we start by setting It is immediate to see that B ε (η) → A 0 (η) as ε → 0. Moreover, we assert that rank B ε (η) = r for small ε, which follows from rank A ε (η) + = r for small ε.
Step 2: Convergence of rows of A ε . The kernel of the symbol of A 0 can be expressed in terms of r linearly independent rows of A 0 (η), i.e.
. . , l} such that I 0 = r, and U ⊥ denoting the orthogonal complement of a subspace U of R m . Since the rows of A (d) − η d can be written as linear combinations of the rows of A + (η ) by Step 1, we may actually take I 0 ⊂ {1, . . . , s}. Suppose that ε > 0 is small enough, so that e T i A ε (η) = 0 for all i ∈ I 0 . Then, As a consequence, the r vectors e T i A ε (η) with i ∈ I 0 are linearly independent, and ker A ε (η) = span{e T i A ε (η) : i ∈ I 0 } ⊥ for ε sufficiently small.

Proof of Proposition 1 [Part II].
After discussing another special case in Step 3, the general situation u ∈ C ∞ (T d ; R m ) ∩ ker T d A 0 (see Step 1) is addressed in Step 4.

By
Step 1, may assume that A 0 u = 0 in T d , hence, A 0 (η)û(η) = 0 for all η ∈ Z d . According to Lemma 2.5 (and a transference argument), the function is well-defined for every ε > 0 and lies in L p (Q d ; R m ). Thus, as ε → 0. To show the convergence in the previous line we argue with Lebesgue's convergence theorem for series, making use of the pointwise convergence of the projections P Aε to P A0 outside the linear subspace {η ∈ R d : η = 0} by Lemma 2.7, and employing that the matrices P Aε (η) are uniformly bounded due to Remark 6.
Step 4: General case. We split u into two parts to which the results of Step 2 and 3 can be applied. Precisely, for x ∈ Q d . Note that this decomposition preserves A 0 -freeness, i.e., A 0 u (1) = A 0 u (2) = 0 in Q d in view of A 0 u = 0 in Q d . This follows immediately from A + u (2) = 0 in Q d and 0 u = 0 in Q d . To conclude, we define u ε := u ε ) and (u (2) ε ) are the A ε -free corresponding to u (1) and u (2) by Step 3 and 2, respectively.
3. Strings with heterogeneities. We continue by investigating the asymptotic behavior of (6) with where α > 0, and f : R d × R m → [0, ∞) is supposed to be measurable and Q dperiodic in its first variable, convex and continuously differentiable in the second argument ξ. Moreover, f has p-growth and p-coercivity, so that (7) is fulfilled. Studying this problem contributes to a better understanding of thin strings with periodically oscillating structures such as they arise from heterogeneities in material composition. The parameter α describes the relative magnitude between the thickness of the string and the characteristic length scale of the fine substructures, see Figure 2.
The results of this section are closely related to the analogous statements for thin films as presented in detail in [20]. Essentially, once Proposition 1 is proven, that is the limit operator A 0 is characterized, and the projection result Lemma 2.6 is available, the difference between films and strings comes down to interchanging the roles of the variables x and x d . Therefore, we will be content with sketching the ideas of the proofs and pointing out the necessary modifications. From now on, let us focus on the situation where the heterogeneities are fine compared to the diameter of the cross section, i.e. α > 1 (see Remark 9 for a comment on α ≤ 1).
Theorem 3.1 (Local Γ-limit for α > 1). Let A be a constant-rank operator satisfying Assumptions A1 and A2. Suppose that f ε : Ω 1 × R m → [0, ∞) as in (19). Then, regarding weak convergence in L p (Ω 1 ; R m ). The homogenized energy density is defined as see (8) for the definition of V A .
Along with this Γ-convergence result comes the compactness of sequences of bounded energy, i.e. a sequence (u ε ) ⊂ L p (Ω 1 ; R m ) with sup ε>0 E ε (u ε ) < ∞ is relatively sequentially compact with respect to the weak L p (Ω 1 ; R m )-topology. This is an immediate consequence of the p-coercivity of f .
The proof of the lower bound is based on multiscale convergence methods, precisely, on Proposition 2 regarding necessary conditions for a multiscale limit (suitably adapted to the context under consideration, see the definition below) of A ε -free fields.
Definition 3.2 (A notion of reduced weak multiscale convergence). A sequence (u ε ) ⊂ L p (Ω 1 ; R m ) is said to converge weakly to w ∈ L p (Ω 1 × Q d ; R m ) in the this effect can be compensated, if u ε features sufficiently fast oscillations along the string, i.e. in x d -direction. The analogue of [20,Lemma 5.9] for thin films (with interchanged roles of x and x d ) is the following: Then, for every cut-off function ψ : whereū ε is defined as In the proof of Theorem 3.1, this lemma is applied with τ ε = ε α . Notice that only for α > 1, the necessary requirement τ ε ε −1 → 0 is satisfied.
Remark 9. The previous reasoning does not carry over to the case α ≤ 1, meaning to strings with coarse heterogeneities. In fact, the counterexample of [20, Section 6.2] (by exchanging the roles of the variables x and x d ) indicates that the Γ-limit of (E ε ) (if existent) may even fail to be local in the sense that it need not have an integral representation with respect to the Lebesgue measure.

4.
Applications. In this last section we implement the statements of Theorems 1.1 and 3.1 for two relevant applications and give a brief interpretation of the resulting limit models for strings in R 3 .

Elastic strings and Cosserat vectors.
Let Ω 1 ⊂ R 3 be a bounded and simply connected Lipschitz domain, modeling the rescaled (see (5)) reference configuration of a thin string with cross section εω made of hyperelastic (possibly heterogeneous) material. Let u : Ω 1 → R 3 stand for the rescaled deformation. Since energies in finite elasticity commonly depend on the deformation gradient, the admissible vector fields (in the rescaled setting) have the form The functional E el ε : with density function f ε : Ω 1 × R 3×3 → [0, ∞) represents a typical (rescaled) elastic energy. We observe that choosing A = curl as in Example 2.1 c) with n = d = 3 leads to the equality E el ε = E ε with E ε defined in (6). Hence, the following Γ-convergence result is an immediate consequence of Theorems 1.1 and 3.1, considering that for U ∈ L p (Ω 1 ; R 3×3 ), if and only if, for almost every x ∈ Ω 1 , with z ∈ W 1,p (0, 1; R 3 ), w ∈ L p (0, 1; W 1,p (ω; R 3 )). Note that z ∈ L p (0, 1; R 3 ) denotes the weak derivative of z.
Corollary 1. Let the assumptions of either Theorem 1.1 (homogeneous case) or Theorem 3.1 (heterogeneous case) hold for the density function f ε : (22), where the Γ-limit is taken with respect to weak L p -convergence.
In the homogeneous case, one has h = f , and in the heterogeneous case, h(x, ξ) = h(ξ) = f hom (ξ) for x ∈ Ω 1 and ξ ∈ R 3×3 with Remark 10. a) The homogenized energy density f hom in (23) is the classical cell formula, which naturally emerges in homogenization problems with convex integrands in the gradient setting [27,30], and coincides with f curl hom as defined in (20). b) For a closely related result on the asymptotics of (heterogeneous) thin films in hyperelasticity, we refer to [20, Section 1].
The function z in (22), with one independent real variable, describes the deformation of the mid-fiber, while the two columns of U = (U 1 |U 2 ) capture the deformation of the cross section in the limit problem and can be interpreted as Cosserat vector fields.
Indeed, let (U ε ) be a bounded energy sequence for (E el ε ), i.e. E el ε (U ε ) < C for all ε > 0. Then, (after passing to a subsequence) the weak L p -limit of (U ε ), called U , has exactly the form (22). In particular, U keeps track of the expression ( 1 ε ∇ u ε ) in the weak limit as ε → 0, and therefore contains valuable information about the limit behavior of (U ε ), which will be nontrivial in general, even though (∇ u ε ) has to vanish asymptotically regarding weak convergence.
As a consequence, the Γ-limit of Corollary 1 is not purely one-dimensional as in [1]. A similar effect appears in the context of membrane theory. In contrast to [25], the model studied by Bouchitté, Fonseca & Mascarenhas in [5,4] involves an additional Cosserat vector or bending moment, leading to a limit energy that depends nontrivially on the direction orthogonal to the film.

4.2.
Thin strings in micromagnetics. In analogy to [18] for thin films, we establish here a model for ferromagnetic strings suitable for small samples in three space dimensions. With ω ⊂ R 2 smooth and simply connected, letm : Ω ε → R 3 be the magnetization (uniformly saturated with |m| = 1), andh : R 3 → R 3 the induced magnetic field. To encode the relation betweenm andh, which is governed by the static Maxwell equations, we use the operator A mag m h = div(m +h) curlh .
As pointed out in [14], A mag has the constant-rank property (4).

3D-1D DIMENSION REDUCTION WITH CONSTRAINTS 71
Let m and h denote the rescaled versions (cf. (5)) ofm andh, respectively, and identify m with its trivial extension to R 3 by zero. The micromagnetic energy comprises an exchange, an anisotropy and an induced energy contribution, see e.g. [8,23]. After rescaling one obtains for (m, h) ∈ V mag : |m| = 1 in Ω 1 }, and E mag ε = ∞ otherwise in W 1,2 (Ω 1 ; R m ) × L 2 (R 3 ; R 3 ). Notice that (m, h) ∈ ker R 3 A mag if the equations div ε (m+h) = 0 and curl ε h = 0, or equivalently, hold in R 3 in the sense of distributions. In (25), α > 0 is a material constant and the continuous function ϕ : R m → R favors the easy axes of magnetization.
In view of Proposition 1 and (12), the limit operator A mag 0 for (A mag ε ) is represented by which leads to the following result.
Proof. The lower bound is immediate, since for any bounded energy sequence (m ε , h ε ) of (E mag ε ) it follows that ∇ m ε → 0 in L 2 (Ω 1 ; R 3×2 ). Moreover, after passing to a subsequence, m ε → m in L 2 (Ω 1 ; R 3 ) for some m ∈ W 1,2 (Ω 1 ; R 3 ), ensuring that m meets the nonconvex constraint |m| = 1. Exploiting Proposition 1 (i) and the lower semicontinuity of the norm concludes the proof of the lower bound.
The construction of a recovery sequence is based on Proposition 1 (ii) in a version on the whole space R 3 . This modification simply requires to replace Fourier series by Fourier transforms in the proof, cf. [18,Section 4]. Hence, for every (m, h) ∈ V mag 0 there exists (m ε ,ĥ ε ) ∈ [L 2 (R 3 ; R 3 ) × L 2 (R 3 ; R 3 )] ∩ ker R 3 A mag ε such thatm ε → m andĥ ε → h both in L 2 (R 3 ; R 3 ). Notice, though, thatm ε cannot be expected to admissible for E mag ε , as, in particular, the nonconvex constraint may be violated. Therefore, in analogy to [18,Proposition 4.7], we set m ε = m h ε = P curlε (ĥ ε − m +m ε ) (27) to obtain the sought recovery sequence. Indeed, the properties of the projection operator P curlε imply that h ε ∈ L 2 (R 3 ; R 3 ) ∩ ker R 3 curl ε for all ε > 0 and h ε → h in L 2 (R 3 ; R 3 ). Let us point out that, with F the notation for the Fourier transform, P curlε in (27) is to be understood as P curlε := F −1 (P curlε F).
Remark 11. We observe that m in (26) describes the magnetization of the onedimensional center line of the string.