A variational approach to the emergence of domain structures in magnetostrictive solids

Abstract

We consider a frame-indifferent variational model for magnetoelastic solids in the large-strain setting with the magnetization field defined on the unknown deformed configuration. Through a simultaneous linearization of the deformation and sharp-interface limit of the magnetization with respect to the easy axes, performed in terms of Γ-convergence, we identify a variational model describing the formation of domain structures and accounting for magnetostriction in the small-strain setting. Our analysis incorporates the effect of magnetic fields and, for uniaxial materials, ensures the regularity of minimizers of the effective energy.

1. Introduction

1.1. Motivation

In the modeling of deformable ferromagnets, magnetostriction refers to the effect for which mechanical and magnetic responses of materials influence each other. Nowadays, this effect is largely exploited in the design of many technological devices such as sensors and actuators [1, 2]. In this regard, specific rare-earth and shape-memory alloys (e.g., TerFeNOL and NiMnGa) have been found of great interest in view of their ability of undergoing remarkably large deformations upon the application of relatively moderate magnetic field, which is often referred to as giant magnetostriction [3].

Magnetocrystalline anisotropy, i.e., the existence of preferred directions of magnetization (called easy axes) determined by the underlying lattice structure of materials, plays a key role in the mechanism behind magnetostriction. In the absence of external stimuli, magnetic materials spontaneously arrange themselves in regions of uniform magnetization (called Weiss domains) separated by thin transition layers (called Bloch walls) [4]. The application of an external magnetic field leads to a reorganization of the domain structure mainly due to the more favorable orientation of certain easy axes with respect to the magnetic field. The movement of domains results in a macroscopic strain, thus inducing a deformation of the material. For a comprehensive treatment of the physics of magnetoelastic interactions, we refer, e.g., to the books of Brown [5], Landau and Lifshitz [6].

Providing an effective description for the emergence of domain structures accounting for mechanical stresses in precise mathematical terms appears to be a very challenging task. This study constitutes an attempt in this direction by means of variational methods. Starting from a large-strain model for magnetostrictive solids, we study the asymptotic behavior of the energy for infinitesimal strains and for magnetization aligning with the easy axes. In this way, we identify an effective model accounting for the linearized elastic energy and the formation of domain structures. Our main results are outlined in subsection 1.4 below.

1.2. The variational model

In this work, we adopt the variational model of magnetoelasticity due to Brown [5, 7]. A magnetoelastic body is subjected to a deformation $\mathit{y}:\mathrm{\Omega }\to {\mathrm{ℝ}}^3$ , where $\mathrm{\Omega }\subset {\mathrm{ℝ}}^3$ represents its reference configuration, and a magnetization field $\mathit{m}:\mathit{y}(\mathrm{\Omega })\to {\mathrm{ℝ}}^3$ . The latter is understood as the average of magnetic dipoles per unit volume. Hence, the saturation constraint [5, p. 73] requires $|\mathit{m}|=1$ in $\mathit{y}(\mathrm{\Omega })$ . The magnetoelastic energy reads

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{(\mathit{y},\mathit{m})\mapsto }& \multicolumn{1}{c}{\underset{\mathrm{\Omega }}{\int }\mathrm{\Psi }(D\mathit{y},\mathit{m}\circ \mathit{y})\mathrm{d}\mathit{x}+\underset{\mathit{y}(\mathrm{\Omega })}{\int }\mathrm{\Phi }\left(\left({(D\mathit{y})}^{\mathrm{\top }}\circ {\mathit{y}}^{-1}\right)\mathit{m}\right)\mathrm{d}\mathit{\xi }}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{+\underset{\mathit{y}(\mathrm{\Omega })}{\int }|D\mathit{m}{\mathrm{|}}^2\mathrm{d}\mathit{\xi }+\underset{{\mathrm{ℝ}}^3}{\int }|{\mathit{h}}^{\mathit{y},\mathit{m}}{\mathrm{|}}^2\mathrm{d}\mathit{\xi }.}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

(1)

The first term in (1) accounts for the elastic energy and involves a material density Ψ which exhibits the coupling between magnetization and deformation gradient. The prototypical example takes the form

$$\mathrm{\Psi }(\mathit{F},\mathit{z})=W({\mathbf{\Lambda }}^{-1}(\mathit{z})\mathit{F}),$$

(2)

where W is an auxiliary density and $\mathbf{\Lambda }(\mathit{z})$ represents the spontaneous strain corresponding to z . For instance, magnetostriction often induces an elongation along the magnetization direction [8, 9], which can be modeled as

$$\mathbf{\Lambda }(\mathit{z}):=a\mathit{z}\otimes \mathit{z}+b(\mathit{I}-\mathit{z}\otimes \mathit{z})$$

(3)

by choosing $a\gg b>0$ .

The second term in (1) stands for the anisotropy energy. The density Φ is nonnegative and vanishes exactly in the direction of the easy axes, thus favoring their attainment. For instance, in the case of uniaxial materials,

$$\mathrm{\Phi }(\mathit{z})=\kappa \left(1-{(\mathit{a}\cdot \mathit{z})}^2\right)$$

(4)

with $\mathit{a}\in {\mathbb{S}}^2$ being the easy axis and $\kappa >0$ , while for cubic materials,

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\mathrm{\Phi }(\mathit{z})=}& \multicolumn{1}{c}{{\kappa }_1\left({(\mathit{z}\cdot {\mathit{a}}_1)}^2{(\mathit{z}\cdot {\mathit{a}}_2)}^2+{(\mathit{z}\cdot {\mathit{a}}_1)}^2{(\mathit{z}\cdot {\mathit{a}}_3)}^2+{(\mathit{z}\cdot {\mathit{a}}_2)}^2{(\mathit{z}\cdot {\mathit{a}}_3)}^2\right)}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{+{\kappa }_2{(\mathit{z}\cdot {\mathit{a}}_1)}^2{(\mathit{z}\cdot {\mathit{a}}_2)}^2{(\mathit{z}\cdot {\mathit{a}}_3)}^2,}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

(5)

where ${\mathit{a}}_1,{\mathit{a}}_2,{\mathit{a}}_3\in {\mathbb{S}}^2$ are the mutually orthogonal easy axes, ${\kappa }_1>0$ , and ${\kappa }_2\ge 0$ . The easy axes are determined by the undistorted lattice structure in the reference configuration. In this model, these axes are transported to the deformed configuration via the Cauchy–Born rule [10]. For densities Φ depending on the direction cosines, such as those in (4)–(5), this assumption explains the term ${(D\mathit{y})}^{\mathrm{\top }}\circ {\mathit{y}}^{-1}$ multiplying m in (1). The third term in (1) constitutes the exchange energy which favors a constant magnetization of the specimen. More generally, the norm squared could be replaced by a quadratic form in the magnetization gradient.

Eventually, the fourth term in (1) stands for the magnetostatic self-energy related to the long-range interactions. In absence of external currents, the stray field ${\mathit{h}}^{\mathit{y},\mathit{m}}$ solves the Maxwell system

$$\{\begin{array}{cc}\multicolumn{1}{c}{\mathrm{curl}{\mathit{h}}^{\mathit{y},\mathit{m}}=\mathbf{0}}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{\mathrm{div}{\mathit{h}}^{\mathit{y},\mathit{m}}=-\mathrm{div}\left({\chi }_{\mathit{y}(\mathrm{\Omega })}\mathit{m}\det D{\mathit{y}}^{-1}\right)}& \multicolumn{1}{c}{}\end{array}\mathrm{in}{\mathrm{ℝ}}^3.$$

We remark that, when Ψ is invariant under left-multiplication by rotation matrices (see, e.g., [8, Equation (20)]), the energy in (1) fulfills the principle of frame indifference. In particular, this happens for (2)–(3) when the same property is satisfied by W. In this regard, we highlight that the dependence of the anisotropy energy on the deformation gradient (see, e.g., [8, Equation (21)]) remains crucial for frame indifference. Additionally, the energy is invariant upon replacing m with $-\mathit{m}$ .

Equilibrium configurations correspond to minimizers of the energy (1). The minimization becomes nontrivial once the deformation y is prescribed on a portion of the boundary or applied loads are considered. In presence of an external field f , the corresponding energy contribution reads

$$(\mathit{y},\mathit{m})\mapsto \underset{\mathit{y}(\mathrm{\Omega })}{\int }\mathit{f}\cdot \mathit{m}\mathrm{d}\mathit{\xi }.$$

1.3. Review of the most relevant literature

To put our findings into context, we briefly review the most relevant literature. Without any claim of completeness, we only mention a few works concerning the model outlined in subsection 1.2 that are related to our work.

The existence of minimizers for the functional (1) has been investigated in [11–15] under various regularity assumptions on admissible deformations. The papers [12, 13, 15] address also the existence of quasistatic evolution in the rate-independent setting. Other results of this kind have been achieved in previous works [16–18] for a similar model of nematic elastomers. Note that in all these studies the anisotropy energy is either assumed to be independent on the deformation gradient [13, Equation (1.2)] or entirely neglected. Indeed, according to (1), the anisotropy contribution constitutes a leading-order term of the functional whose lower semicontinuity cannot be guaranteed when Φ is nonconvex such as in (4)–(5).

The linearization of the energy (1) has been rigorously performed in the work by Almi et al. [19], thus recovering a well-established magnetoelasticity model for the small-strain setting [20]. For rigid bodies, the sharp-interface limit of magnetizations toward the easy axes has been addressed by Anzellotti et al. [21]. The case of deformable bodies has been tackled in the papers by Grandi and co-authors [22, 23]. More precisely, these works concern multiphase solids with Eulerian interfaces but, as mentioned in [23, p. 411], their analysis can be reformulated in terms of the model in subsection 1.2 by interpreting the diffuse phase indicators as magnetization fields. Still, the dependence of the anisotropy energy on the deformation gradient is dropped in these two works, so that the model does not comply with frame indifference. All above-mentioned results for the linearization and the sharp-interface limit are stated in terms of Γ-convergence [24, 25].

Although not directly relevant for our study, we also mention the derivation of effective models, still by Γ-convergence, in the direction of dimension reduction [26, 27] and relaxation [28, 29].

1.4. Contributions of the paper

In this paper, we rigorously derive a variational model describing the formation of domain structures and accounting for magnetostriction in the small-strain regime. The derivation is performed in the sense of Γ-convergence through a simultaneous linearization of deformations and sharp-interface limit of magnetizations with respect to the easy axes. Our main results are given in Theorem 2.1, Theorem 2.2, and Theorem 2.4. Here, we limit ourselves in briefly outlining their content and we refer to section 2 for their precise statement.

We adopt the functional setting of Barchiesi et al. [11]. Deformations are modeled as maps $\mathit{y}\in W^{1,p}(\mathrm{\Omega };{\mathrm{ℝ}}^3)$ with $p>2$ satisfying the divergence identities (see Definition 3.3 below), which is equivalent to excluding cavitation. Thus, by replacing the deformed configuration $\mathit{y}(\mathrm{\Omega })$ with the topological image (see Definition 3.2 below), we can define magnetizations as Sobolev maps on this set, which in turn allows for a meaningful definition of the magnetoelastic energy (1).

For a small parameter $\epsilon >0$ , we consider a rescaled version ${\mathcal{E}}_{\epsilon }$ of the magnetoelastic energy (see (7)–(9) beylow). Specifically, the elastic term is rescaled by ${\epsilon }^2$ and the elastic density takes the form in (2) with the spontaneous strain of magnetizations (i.e., the term ${\mathbf{\Lambda }}_{\epsilon }(\mathit{m}\circ \mathit{y})$ in (8) below) given by a perturbation of the identity of order ε (see (10) below); anisotropy and exchange term are rescaled similarly to the classical Modica–Mortola functional [30] in terms of a power ${\epsilon }^{\beta }$ with

$$0<\beta <\frac{2(q-1)}{q}\wedge 1,$$

(6)

where $q>1$ specifies the growth of W with respect to the determinant of its argument (see assumption (W5) below); eventually, we neglect the stray-field term. Condition (6) is commented in subsection 1.5 below.

Theorem 2.1 shows that the sequence ${({\mathcal{E}}_{\epsilon })}_{\epsilon >0}$ enjoys suitable compactness properties or, in the terminology of Γ-convergence, that this sequence is equi-coercive (see, e.g., [25, Definition 7.6]). Limiting states are determined by infinitesimal displacements $\mathit{u}:\mathrm{\Omega }\to {\mathrm{ℝ}}^3$ and magnetizations $\mathit{m}:\mathrm{\Omega }\to {\mathbb{S}}^2$ which attain only the easy axes (up to sign). Thus, each limiting magnetization determines a domain structure.

Theorem 2.2 provides the Γ-limit $\mathcal{E}$ of the sequence ${({\mathcal{E}}_{\epsilon })}_{\epsilon >0}$ , as $\epsilon \to 0^{+}$ . The functional $\mathcal{E}$ is the sum of two contributions (see (12)–(14) below): a linearized elastic energy where the strain corresponding to stress-free configuration is quadratic in the magnetization, and an energy measuring the area of the interfaces between different magnetic domains that resemble the classical Γ-limit of the vectorial Modica–Mortola functional [30]. The elastic contribution coincides with the one commonly adopted in small-strain models [20], while the magnetic one agrees with the sharp-interface limit identified in [21].

From Theorem 2.1 and Theorem 2.2, we immediately deduce the convergence of almost minimizers by standard Γ-convergence arguments (see, e.g., [25, Theorem 7.8]). As the contributions given by stray and external fields are both continuous with respect to the relevant notion of convergence, these are easily incorporated (see, e.g., [24, Remark 1.7]). The convergence result for almost minimizers is given in Corollary 2.3.

Eventually, Theorem 2.4 addresses the regularity of minimizers of the functional $\mathcal{E}$ in the case of uniaxial materials as in (4). For sufficiently regular boundary datum and external field, we prove the regularity of the infinitesimal displacement and the boundaries of magnetic domains. This result provides a slight extension of [21, Theorem 5.1] which accounts for magnetostriction in the small-strain setting.

1.5. Comments and outlook

Our main findings represent the first results combining linearization and sharp-interface limit for Eulerian-Lagrangian energies, i.e., energies comprising both terms in the reference configuration and in the unknown deformed configuration.

The expression of the spontaneous strain of magnetizations postulated here (see (10) below) is evidenced in the literature (see, e.g., [8, Equation (90)]) and seems more natural compared with the one in previous work [19], even though the two are asymptotically equivalent. In contrast with previous works [22, 23], the anisotropy energy depends on the deformation gradient, so that our diffuse-interface model complies with frame indifference. In this regard, the rigidity of the model arising from the linearization process is steadily exploited while handling this additional dependence on the deformation gradient.

From a technical point of view, we observe that $p\ge 3$ is assumed in previous works [19, 22, 23], while here we only require $p>2$ . Thus, admissible deformations are possibly discontinuous in our setting. The consequent difficulties are overcome by means of the techniques developed in previous works [11, 12, 14]. In this regard, excluding cavitation is essential not only for a consistent definition of energy functionals on the deformed configuration, but also for analytical reasons. Indeed, this setting ensures that geometric image (see Definition 3.1 below) and topological image of deformations coincide up to negligible set (see Proposition 3.4 below), which justifies the change-of-variable formulas used throughout the proofs.

Condition (6) identifies the regime where no interaction between elastic and magnetic energy terms occurs during the process of Γ-limit. As a result, the magnetic term of the limiting energy is independent of the displacement. This decoupling simplifies the proofs significantly. Different regimes potentially leading to a Γ-limit where the magnetic term depends also on the displacement will be the subject of future investigations.

Other interesting and ambitious directions might concern the linearization for spontaneous strains of magnetizations that are not perturbations of the identity, and the sharp-interface limit at large strains with the anisotropy energy depending on the deformation gradient.

1.6. Structure of the paper

The paper is organized into four sections. Section 1 constitutes this introduction. In section 2, we specify the setting of our problem and we state our main findings. Section 3 collects results from the literature that will be needed for the proofs of our main results presented in section 4.

1.7. Notation

Given $a,b\in \mathrm{ℝ}$ , we employ the notation $a\wedge b:=\min \{a,b\}$ and $a\vee b:=\max \{a,b\}$ . We work in ${\mathrm{ℝ}}^3$ with ${\mathbb{S}}^2$ denoting the unit sphere centered at the origin. The class ${\mathrm{ℝ}}^{3\times 3}$ of square matrices is endowed with the Frobenius norm. The symbol id is used for the identity map on ${\mathrm{ℝ}}^3$ , while I and O denote the identity and the null matrix in ${\mathrm{ℝ}}^{3\times 3}$ . Given $\mathit{A}\in {\mathrm{ℝ}}^{3\times 3}$ , we write $\det \mathit{A}$ for its determinant and ${\mathit{A}}^{\mathrm{\top }}$ for its transpose. $G{L}_{+}(3)$ is the set of matrices with positive determinant, while $\mathrm{SO}(3)$ is the set of rotations. $\mathrm{Sym}(3)$ stands for the class of symmetric matrices and $\mathrm{sym}\mathit{A}:=(\mathit{A}+{\mathit{A}}^{\mathrm{\top }})∕2$ is the symmetric part of $\mathit{A}\in {\mathrm{ℝ}}^{3\times 3}$ . We write $E△F$ for the symmetric difference of two sets $E,F\subset {\mathrm{ℝ}}^3$ .

The Lebesgue measure and the two-dimensional Hausdorff measure on ${\mathrm{ℝ}}^3$ are denoted by ${ℒ}^3$ and ${\mathscr{H}}^2$ , respectively. The expressions “almost everywhere” (abbreviated as “a.e.” within the formulas) as well as the adjective “negligible” always refer to the Lebesgue measure. The symbol ${\chi }_A$ is employed for the indicator function of a set $A\subset {\mathrm{ℝ}}^3$ . We use standard notation for Lebesgue spaces ( $L^p$ ), spaces of differentiable and Hölder functions ( $C^k$ and $C^{k,\alpha }$ ), Sobolev spaces ( $W^{k,p}$ ), and spaces of functions with bounded variation (BV). The subscripts “loc” and “c” are used for their local and compactly supported versions. Domain and codomain are separated by a semicolon, and the codomain is omitted for scalar-valued maps. We write $g\mathrm{d}\mathit{x}$ for the measure with density g with respect to ${ℒ}^3$ . Lebesgue spaces on boundaries are always meant with respect to ${\mathscr{H}}^2$ . Given $1<p<\infty$ , we write $p^{\prime }:=p∕(p-1)$ . Distributional gradients are indicated by D, while E is used for symmetrized gradients. Boundary conditions for Sobolev functions are always understood in the sense of traces. Given a map w with bounded variation, we write $|\mathrm{Dw}|$ and $J_w$ for its total variation and jump set, respectively. Eventually, the class of sets of locally finite perimeter in ${\mathrm{ℝ}}^3$ is written as $\mathcal{P}({\mathrm{ℝ}}^3)$ with ${\partial }^{\ast }E$ denoting the reduced boundary of $E\in \mathcal{P}({\mathrm{ℝ}}^3)$ . Thus, $\mathrm{Per}(E;\mathrm{\Omega })={\mathscr{H}}^2({\partial }^{\ast }E\cap \mathrm{\Omega })$ stands for the relative perimeter of E in an open set $\mathrm{\Omega }\subset {\mathrm{ℝ}}^3$ .

We adopt the usual convention of denoting by C a positive constant depending on known quantities (in particular, independent on ε) whose value can change from line to line.

2. Setting and main results

2.1. Setting

Throughout the paper, we fix

$\begin{array}{cccc}\multicolumn{1}{c}{}& \multicolumn{1}{c}{\mathrm{\Omega }\subset {\mathrm{ℝ}}^3\mathrm{bounded}\mathrm{domain}\mathrm{with}\mathrm{Lipschitz}\mathrm{boundary,}p>2,q>1,}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\mathrm{\Delta }\subset \partial \mathrm{\Omega }\mathrm{with}\mathrm{Lipschitz}\mathrm{boundary}\mathrm{in}\partial \mathrm{\Omega },{\mathscr{H}}^2(\mathrm{\Delta })>0,\mathit{d}\in W^{1,\infty }(\mathrm{\Omega };{\mathrm{ℝ}}^3).}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\end{array}$

The fact that Δ has Lipschitz boundary in $\partial \mathrm{\Omega }$ is understood in the sense of [31, Definition 2.1].

Fix $\beta >0$ . For each $\epsilon >0$ , we consider the functional

$${\mathcal{E}}_{\epsilon }(\mathit{y},\mathit{m}):={\mathcal{E}}_{\epsilon }^e(\mathit{y},\mathit{m})+{\mathcal{E}}_{\epsilon }^m(\mathit{y},\mathit{m})$$

(7)

with

$$\begin{array}{ccc}\multicolumn{1}{c}{{\mathcal{E}}_{\epsilon }^e(\mathit{y},\mathit{m})}& \multicolumn{1}{c}{:=\frac{1}{{\epsilon }^2}\underset{\mathrm{\Omega }}{\int }W({\mathbf{\Lambda }}_{\epsilon }^{-1}(\mathit{m}\circ \mathit{y})D\mathit{y})\mathrm{d}\mathit{x},}& \multicolumn{1}{c}{}\end{array}$$

(8)

$$\begin{array}{ccc}\multicolumn{1}{c}{{\mathcal{E}}_{\epsilon }^m(\mathit{y},\mathit{m})}& \multicolumn{1}{c}{:=\frac{1}{{\epsilon }^{\beta }}\underset{{\mathrm{im}}_T(\mathit{y},\mathrm{\Omega })}{\int }\mathrm{\Phi }\left(\left({(D\mathit{y})}^{\mathrm{\top }}\circ {\mathit{y}}^{-1}\right)\mathit{m}\right)\mathrm{d}\mathit{\xi }+{\epsilon }^{\beta }\underset{{\mathrm{im}}_T(\mathit{y},\mathrm{\Omega })}{\int }|D\mathit{m}{\mathrm{|}}^2\mathrm{d}\mathit{\xi },}& \multicolumn{1}{c}{}\end{array}$$

(9)

defined on the class of admissible states

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{{\mathcal{A}}_{\epsilon }:=\{(\mathit{y},\mathit{m}):\mathit{y}\in {\mathcal{Y}}_{p,q}(\mathrm{\Omega }),\mathit{y}=\mathit{id}+\epsilon \mathit{d}\mathrm{on}\mathrm{\Delta },\mathit{m}\in W^{1,2}({\mathrm{im}}_T(\mathit{y},\mathrm{\Omega });{\mathbb{S}}^2)}& \multicolumn{1}{c}{\},}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

where

$${\mathcal{Y}}_{p,q}(\mathrm{\Omega }):=\left\{\mathit{y}\in W^{1,p}(\mathrm{\Omega };{\mathrm{ℝ}}^3):\det D\mathit{y}\in L^q(\mathrm{\Omega };(0,+\infty )),\mathit{y}\mathrm{satisfies}\mathrm{(DIV)},\mathit{y}\mathrm{a.e.}\mathrm{injective}\right\}.$$

The divergence identities (DIV), introduced in [32], are given in Definition 3.3 below. Intuitively, the validity of these identities excludes the phenomenon of cavitation [33]. Recall that y is termed almost everywhere injective whenever the restriction of y to the complement of a negligible subset of Ω is an injective map [34].

The elastic energy (8) depends on a material density $W:{\mathrm{ℝ}}^{3\times 3}\to [0,+\infty ]$ on which we make the following assumptions:

• (W1) Orientation preservation: $W(\mathit{A})<+\infty$ if and only if $\mathit{A}\in G{L}_{+}(3)$ ;

• (W2) Regularity: W is continuous on ${\mathrm{ℝ}}^{3\times 3}$ and of class $C^2$ in a neighborhood of $\mathrm{SO}(3)$ ;

• (W3) Frame indifference: W is frame indifferent, i.e.,

  $$W(\mathit{Q}\mathit{A})=W(\mathit{A})\mathrm{for}\mathrm{all}\mathit{Q}\in \mathrm{SO}(3)\mathrm{and}\mathit{A}\in G{L}_{+}(3)\mathrm{;}$$

• (W4) Ground state: $W(\mathit{I})=0$ ;

• (W5) Growth: there exist a constant $c_W>0$ such that

  $$W(\mathit{A})\ge c_W\left(g_p(\mathrm{dist}(\mathit{A};\mathrm{SO}(3)))+h_q(\det \mathit{A})\right)\mathrm{for}\mathrm{all}\mathit{A}\in G{L}_{+}(3),$$

  where $g_p:[0,+\infty )\to [0,+\infty )$ and $h_q:(0,+\infty )\to [0,+\infty )$ are defined as

  $$g_p(t):=\{\begin{array}{cc}\multicolumn{1}{c}{\frac{t^2}{2}}& \multicolumn{1}{c}{\mathrm{if}0\le t\le 1,}\\ \multicolumn{1}{c}{\frac{t^p}{p}+\frac{1}{2}-\frac{1}{p}}& \multicolumn{1}{c}{\mathrm{if}t>1,}\end{array}$$

  and

  $$h_q(t):=\frac{t^q}{q}+\frac{1}{t}-\frac{q+1}{q}\mathrm{for}\mathrm{all}t>0.$$

The specific expressions of the functions $g_p$ and $h_q$ are not relevant as long as $g_p(t)$ is bounded from below by $t^2\vee t^p$ , and $h_q(t)$ goes as $t^q$ for $t\gg 1$ and controls $1∕t$ for $t\ll 1$ . The tensor field ${\mathbf{\Lambda }}_{\epsilon }:{\mathbb{S}}^2\to {\mathrm{ℝ}}^{3\times 3}$ appearing in (8) is defined as

$${\mathbf{\Lambda }}_{\epsilon }(\mathit{z}):=a_{\epsilon }\mathit{z}\otimes \mathit{z}+b_{\epsilon }\left(\mathit{I}-\mathit{z}\otimes \mathit{z}\right)\mathrm{for}\mathrm{all}\mathit{z}\in {\mathbb{S}}^2$$

(10)

with $a_{\epsilon }:=1+\mathrm{\epsilon a}$ and $b_{\epsilon }:=1+\mathrm{\epsilon b}$ , where $a,b\in \mathrm{ℝ}$ . This tensor is invertible and its inverse reads as

$${\mathbf{\Lambda }}_{\epsilon }^{-1}(\mathit{z})=\frac{1}{a_{\epsilon }}\mathit{z}\otimes \mathit{z}+\frac{1}{b_{\epsilon }}\left(\mathit{I}-\mathit{z}\otimes \mathit{z}\right)\mathrm{for}\mathrm{all}\mathit{z}\in {\mathbb{S}}^2.$$

(11)

The magnetic energy (9) is given by the sum of two integrals on the topological image ${\mathrm{im}}_T(\mathit{y},\mathrm{\Omega })$ of Ω under y . This set was introduced in [11] and it is here recalled in Definition 3.2 below. The first integral features the function $\mathrm{\Phi }:{\mathrm{ℝ}}^3\to [0,+\infty )$ which is assumed to satisfy:

• (Φ1) Multiple wells: There exist $M\in \mathrm{\mathbb{N}}$ with $M\ge 2$ and distinct elements ${\mathit{b}}_1,\dots ,{\mathit{b}}_M\in {\mathbb{S}}^2$ such that $\{\mathit{z}\in {\mathbb{S}}^2:\mathrm{\Phi }(\mathit{z})=0\}=\{{\mathit{b}}_1,\dots ,{\mathit{b}}_M\}$ ;

• (Φ2) Regularity: Φ is continuous on ${\mathrm{ℝ}}^3$ and of class $C^1$ in a neighborhood of ${\mathbb{S}}^2$ .

From the modeling point of view, in (Φ1) we should take $M=2N$ with $N\in \mathrm{\mathbb{N}}$ , ${\mathit{b}}_i:={\mathit{a}}_i$ for $i=1,\dots ,N$ , and ${\mathit{b}}_i:=-{\mathit{a}}_{i-N}$ for $i=N+1,\dots ,M$ where ${\mathit{a}}_1,\dots ,{\mathit{a}}_N\in {\mathbb{S}}^2$ denote the easy axes. However, this fact is not relevant for the present analysis.

By passing, we also remark that the energy in (7)–(9) is frame indifferent in view of (W3) and the objectivity of ${\mathbf{\Lambda }}_{\epsilon }^{-1}$ , that is,

$${\mathbf{\Lambda }}_{\epsilon }^{-1}(\mathit{Q}\mathit{z})=\mathit{Q}{\mathbf{\Lambda }}_{\epsilon }^{-1}(\mathit{z}){\mathit{Q}}^{\mathrm{\top }}\mathrm{for}\mathrm{all}\mathit{Q}\in \mathrm{SO}(3)\mathrm{and}\mathit{z}\in {\mathbb{S}}^2.$$

The limit functional

$$\mathcal{E}(\mathit{u},\mathit{m}):={\mathcal{E}}^e(\mathit{u},\mathit{m})+{\mathcal{E}}^m(\mathit{u},\mathit{m})$$

(12)

with

$$\begin{array}{ccc}\multicolumn{1}{c}{{\mathcal{E}}^e(\mathit{u},\mathit{m})}& \multicolumn{1}{c}{:=\frac{1}{2}\underset{\mathrm{\Omega }}{\int }{Q}_W\left(E\mathit{u}-\mathbf{\Lambda }(\mathit{m})\right)\mathrm{d}\mathit{x},}& \multicolumn{1}{c}{}\end{array}$$

(13)

$$\begin{array}{ccc}\multicolumn{1}{c}{{\mathcal{E}}^m(\mathit{m})}& \multicolumn{1}{c}{:=\frac{1}{2}\sum _{\begin{array}{c}\multicolumn{1}{c}{i,j=1,\dots ,M}\\ \multicolumn{1}{c}{i\ne j}\end{array}}{\sigma }_{\mathrm{\Phi }}^{i,j}{\mathscr{H}}^2\left({\partial }^{\ast }\{\mathit{m}={\mathit{b}}_i\}\cap {\partial }^{\ast }\{\mathit{m}={\mathit{b}}_j\}\right),}& \multicolumn{1}{c}{}\end{array}$$

(14)

is defined on the class

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{c}\multicolumn{1}{c}{\mathcal{A}:=\{(\mathit{u},\mathit{m}):\mathit{u}\in W^{1,2}(\mathrm{\Omega };{\mathrm{ℝ}}^3),\mathit{u}=\mathit{d}\mathrm{on}\mathrm{\Delta },\mathit{m}\in \mathrm{BV}(\mathrm{\Omega };\{{\mathit{b}}_1,\dots ,{\mathit{b}}_M\})\}}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

In (13), the quadratic form $Q_W:{\mathrm{ℝ}}^{3\times 3}\to [0,+\infty )$ is defined as

$$Q_W(\mathit{B}):={\mathbf{C}}_W(\mathit{B}):\mathit{B}\mathrm{for}\mathrm{all}\mathit{B}\in {\mathrm{ℝ}}^{N\times N},$$

where the elasticity tensor ${\mathbf{C}}_W:{\mathrm{ℝ}}^{N\times N}\to {\mathrm{ℝ}}^{N\times N}$ is defined as the differential of W at the identity, $E\mathit{u}:=\mathrm{sym}D\mathit{u}$ is the symmetrized gradient of u , and, as in (3), we define $\mathbf{\Lambda }:{\mathbb{S}}^2\to {\mathrm{ℝ}}^{3\times 3}$ by setting

$$\mathbf{\Lambda }(\mathit{z}):=a\mathit{z}\otimes \mathit{z}+b\left(\mathit{I}-\mathit{z}\otimes \mathit{z}\right)\mathrm{for}\mathrm{all}\mathit{z}\in {\mathbb{S}}^2.$$

(15)

Moreover, we set

$${\sigma }_{\mathrm{\Phi }}^{i,j}:=d_{\mathrm{\Phi }}({\mathit{b}}_i,{\mathit{b}}_j)\mathrm{for}\mathrm{all}i,j=1,\dots ,M,$$

where $d_{\mathrm{\Phi }}:{\mathbb{S}}^2\times {\mathbb{S}}^2\to [0,+\infty )$ denotes the geodesic distance defined by

$$d_{\mathrm{\Phi }}({\mathit{z}}_0,{\mathit{z}}_1):=\inf \left\{\underset{0}{\overset{1}{\int }}\sqrt{\mathrm{\Phi }(\mathit{\gamma }(t))}|{\mathit{\gamma }}^{\prime }(t)|\mathrm{d}t:\mathit{\gamma }\in C^1([0,1];{\mathbb{S}}^2),\mathit{\gamma }(0)={\mathit{z}}_0,\mathit{\gamma }(1)={\mathit{z}}_1\right\}$$

(16)

for all ${\mathit{z}}_0,{\mathit{z}}_1\in {\mathbb{S}}^2$ .

2.2. Main results

Our main results show that the asymptotic behavior of the functionals ${({\mathcal{E}}_{\epsilon })}_{\epsilon >0}$ , as $\epsilon \to 0^{+}$ , is effectively described by the functional $\mathcal{E}$ in the regime (6). In this setting, each of the two functionals in (13) and (14) is the Γ-limit of the corresponding term in (8) and (9).

The first result ensures that the sequence ${({\mathcal{E}}_{\epsilon })}_{\epsilon >0}$ enjoys suitable compactness properties.

Theorem 2.1 (Compactness). Assume (W1)–(W5), (Φ1)–(Φ2), and (6). Let ${(({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon }))}_{\epsilon >0}$ be a sequence with $({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })\in {\mathcal{A}}_{\epsilon }$ for all $\epsilon >0$ . Suppose that

$$\underset{\epsilon >0}{\sup }{\mathcal{E}}_{\epsilon }({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })<+\infty .$$

(17)

Then, there exists $(\mathit{u},\mathit{m})\in \mathcal{A}$ such that, up to subsequences, we have as $\epsilon \to 0^{+}$

$$\begin{array}{ccc}\multicolumn{1}{c}{{\mathit{y}}_{\epsilon }\to \mathit{id}}& \multicolumn{1}{c}{\mathrm{in}{W}^{1,p}(\mathrm{\Omega };{\mathrm{ℝ}}^3),}& \multicolumn{1}{c}{}\end{array}$$

(18)

$$\begin{array}{ccc}\multicolumn{1}{c}{\frac{1}{\epsilon }\left({\mathit{y}}_{\epsilon }-\mathit{id}\right)\rightharpoonup \mathit{u}}& \multicolumn{1}{c}{\mathrm{in}{W}^{1,2}(\mathrm{\Omega };{\mathrm{ℝ}}^3),}& \multicolumn{1}{c}{}\end{array}$$

(19)

$$\begin{array}{ccc}\multicolumn{1}{c}{{\chi }_{{\mathrm{im}}_T({\mathit{y}}_{\epsilon },\mathrm{\Omega })}{\mathit{m}}_{\epsilon }\to {\chi }_{\mathrm{\Omega }}\mathit{m}}& \multicolumn{1}{c}{\mathrm{in}{L}^1({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3),}& \multicolumn{1}{c}{}\end{array}$$

(20)

$$\begin{array}{ccc}\multicolumn{1}{c}{{\mathit{m}}_{\epsilon }\circ {\mathit{y}}_{\epsilon }\to \mathit{m}}& \multicolumn{1}{c}{\mathrm{in}{L}^1(\mathrm{\Omega };{\mathrm{ℝ}}^3).}& \multicolumn{1}{c}{}\end{array}$$

(21)

The second result shows that the functional $\mathcal{E}$ provides an optimal lower bound for ${({\mathcal{E}}_{\epsilon })}_{\epsilon >0}$ . For the notion of Γ-convergence, we refer to [24, 25].

Theorem 2.2 (Γ -convergence). Assume (W1)–(W5), (Φ1)–(Φ2), and (6). Then, the sequence ${({\mathcal{E}}_{\epsilon })}_{\epsilon >0}\mathrm{\Gamma }$ -converges to the functional $\mathcal{E}$ , as $\epsilon \to 0^{+}$ , with respect to the convergences in Theorem 2.1. Namely, the following two conditions hold:

• (i) Lower bound: For any sequence ${\left(({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })\right)}_{\epsilon >0}$ with $({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })\in {\mathcal{A}}_{\epsilon }$ for all $\epsilon >0$ and any $(\mathit{u},\mathit{m})\in \mathcal{A}$ for which (18) – (21) hold true, we have

  $$\underset{\epsilon \to 0^{+}}{\lim \inf }{\mathcal{E}}_{\epsilon }({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })\ge \mathcal{E}(\mathit{u},\mathit{m});$$

  (22)

• (ii) Optimality of the lower bound: For any $(\mathit{u},\mathit{m})\in \mathcal{A}$ , there exists a sequence ${\left(({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })\right)}_{\epsilon >0}$ with $({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })\in {\mathcal{A}}_{\epsilon }$ for all $\epsilon >0$ for which (18) – (21) hold true and we have

  $$\underset{\epsilon \to 0^{+}}{\lim }{\mathcal{E}}_{\epsilon }({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })=\mathcal{E}(\mathit{u},\mathit{m}).$$

  (23)

The effect of magnetic fields can be easily incorporated within our analysis. Following [7], we define the stray field corresponding to a deformation $\mathit{y}\in {\mathcal{Y}}_{p,q}(\mathrm{\Omega })$ and a magnetization $\mathit{m}\in L^1({\mathrm{im}}_T(\mathit{y},\mathrm{\Omega });{\mathbb{S}}^2)$ as the unique distributional solution ${\mathit{h}}^{\mathit{y},\mathit{m}}\in L^2({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)$ to the Maxwell system

$$\{\begin{array}{cc}\multicolumn{1}{c}{\mathrm{curl}{\mathit{h}}^{\mathit{y},\mathit{m}}=\mathbf{0}}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{\mathrm{div}{\mathit{h}}^{\mathit{y},\mathit{m}}=-\mathrm{div}\left({\chi }_{{\mathrm{im}}_T(\mathit{y},\mathrm{\Omega })}\mathit{m}\det D{\mathit{y}}^{-1}\right)}\end{array}\mathrm{in}{\mathrm{ℝ}}^3,$$

(24)

where ${\mathit{y}}^{-1}:{\mathrm{im}}_T(\mathit{y},\mathrm{\Omega })\to {\mathrm{ℝ}}^3$ denotes the inverse of y as given in Proposition 3.5 below. As a consequence of Proposition 4.1 below, existence and uniqueness of the stray field are guaranteed in our setting, and the associated energy contribution

$$\mathcal{H}(\mathit{y},\mathit{m}):=\underset{{\mathrm{ℝ}}^3}{\int }|{\mathit{h}}^{\mathit{y},\mathit{m}}{\mathrm{|}}^2\mathrm{d}\mathit{\xi }$$

is well defined. Additionally, the work of the applied magnetic field $\mathit{f}\in L^1({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)$ is accounted by the Zeeman energy functional

$$\mathcal{F}(\mathit{y},\mathit{m}):=\underset{{\mathrm{im}}_T(\mathit{y},\mathrm{\Omega })}{\int }\mathit{f}\cdot \mathit{m}\mathrm{d}\mathit{\xi }.$$

For a fixed constant $\lambda \ge 0$ , we define the total energies

$${\mathcal{G}}_{\epsilon }(\mathit{y},\mathit{m}):={\mathcal{E}}_{\epsilon }(\mathit{y},\mathit{m})+\lambda \mathcal{H}(\mathit{y},\mathit{m})-\mathcal{F}(\mathit{y},\mathit{m})\mathrm{for}\mathrm{all}\epsilon >0\mathrm{and}(\mathit{y},\mathit{m})\in {\mathcal{A}}_{\epsilon }$$

and

$$\mathcal{G}(\mathit{u},\mathit{m}):=\mathcal{E}(\mathit{u},\mathit{m})+\lambda \mathcal{H}(\mathit{id},\mathit{m})-\mathcal{F}(\mathit{id},\mathit{m})\mathrm{for}\mathrm{all}(\mathit{u},\mathit{m})\in \mathcal{A}.$$

It turns out that the functionals $\mathcal{H}$ and $\mathcal{F}$ constitute continuous perturbations with respect to the relevant topology. Therefore, standard Γ-convergence arguments (see, e.g., [24, Remark 1.7]) yield the following statement for sequences of almost minimizers.

Corollary 2.3 (Convergence of almost minimizers). Assume (W1)–(W5), (Φ1)–(Φ2), and (6). Let ${\left(({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })\right)}_{\epsilon >0}$ be a sequence with $({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })\in {\mathcal{A}}_{\epsilon }$ for all $\epsilon >0$ satisfying

$$\underset{\epsilon \to 0^{+}}{\lim }\left({\mathcal{G}}_{\epsilon }({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })-G_{\epsilon }\right)=0,$$

(25)

where we set $G_{\epsilon }:=\inf \{{\mathcal{G}}_{\epsilon }(\mathit{w},\mathit{n}):(\mathit{w},\mathit{n})\in {\mathcal{A}}_{\epsilon }\}$ . Then, there exists $(\mathit{u},\mathit{m})\in \mathcal{A}$ such that, up to subsequences, (18)–(21) hold true and $(\mathit{u},\mathit{m})$ is a minimizer of $\mathcal{G}$ with

$$\underset{\epsilon \to 0^{+}}{\lim }{\mathcal{G}}_{\epsilon }({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })=\mathcal{G}(\mathit{u},\mathit{m}).$$

(26)

Eventually, we discuss the regularity of minimizers for the limiting energy in a special case. For simplicity, as in [21], we discuss only the interior regularity in the case $M=2$ . This setting can describe uniaxial materials with Φ as in (4) by choosing ${\mathit{b}}_1:=\mathit{a}$ and ${\mathit{b}}_2:=-\mathit{a}$ . In this situation, the limiting magnetic energy (14) can be trivially rewritten as

$${\mathcal{E}}^m(\mathit{m})={\sigma }_{\mathrm{\Phi }}^{1,2}\mathrm{Per}(\{\mathit{m}={\mathit{b}}_1\};\mathrm{\Omega }).$$

(27)

Therefore, we can resort to the regularity theory for sets with minimal perimeter [35].

Theorem 2.4 (Regularity of minimizers). Let $(\mathit{u},\mathit{m})\in \mathcal{A}$ be a minimizer of the functional $\mathcal{G}$ . Then, $\mathit{u}\in W^{1,r}(\mathrm{\Omega };{\mathrm{ℝ}}^3)$ for all $r\in [1,\infty )$ . Additionally, assume $M=2$ . If $\mathit{f}\in L^{s_{\mathrm{f}}}({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)$ with $s_f>3$ , then the set $J_{\mathit{m}}$ is a surface of class $C^{1,(s_f-3)∕(2s_f)}$ in Ω. Furthermore, if $\mathit{d}\in W_{\mathrm{loc}}^{2,s_d}(\mathrm{\Omega };{\mathrm{ℝ}}^3)$ with $s_d>3$ , then $\mathit{u}\in C^{1,1-3∕s_d}(\mathrm{\Omega }\setminus J_{\mathit{m}};{\mathrm{ℝ}}^3)$ .

3. Preliminary results

In this section, we collect some preliminary notions and facts that will be instrumental for the proofs of the results in section 2.

3.1. Geometric and topological image

We introduce two concepts of image for deformations, namely that of geometric and topological images. For the notions of approximate differentiability and density of a set, we refer to [36].

Definition 3.1 (Geometric image). Let $\mathit{y}\in W^{1,p}(\mathrm{\Omega };{\mathrm{ℝ}}^3)$ with $\det D\mathit{y}(\mathit{x})>0$ for almost all $\mathit{x}\in \mathrm{\Omega }$ . The geometric domain ${\mathrm{dom}}_G(\mathit{y},\mathrm{\Omega })$ of y is defined as the set of points $\mathit{x}\in \mathrm{\Omega }$ at which y is approximately differentiable with $\det D\mathit{y}(\mathit{x})>0$ and there exist a compact set $K\subset \mathrm{\Omega }$ with density one at x and a map $\mathit{w}\in C^1({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)$ such that $\mathit{w}{\mathrm{|}}_K=\mathit{y}{\mathrm{|}}_K$ and $(D\mathit{w}){\mathrm{|}}_K=(D\mathit{y}){\mathrm{|}}_K$ . Moreover, for any measurable set $A\subset \mathrm{\Omega }$ , the geometric image of A under y is defined as ${\mathrm{im}}_G(\mathit{y},A):=\mathit{y}\left(A\cap {\mathrm{dom}}_G(\mathit{y},\mathrm{\Omega })\right)$ .

From [37, pp. 582–583], we known that ${\mathrm{dom}}_G(\mathit{y},\mathrm{\Omega })$ has full measure in Ω. The technical condition expressed in terms of w is included in the definition just for convenience as it ensures that $\mathit{y}{\mathrm{|}}_{{\mathrm{dom}}_G(\mathit{y},\mathrm{\Omega })}$ is (everywhere) injective [37, Lemma 3]. Hence, we can consider the inverse of y as the map ${\mathit{y}}^{-1}:{\mathrm{im}}_G(\mathit{y},\mathrm{\Omega })\to {\mathrm{dom}}_G(\mathit{y},\mathrm{\Omega })$ .

Below, ${\mathit{y}}^{\ast }:\mathrm{\Omega }\to {\mathrm{ℝ}}^3$ will denote the precise representative of the map $\mathit{y}\in W^{1,p}(\mathrm{\Omega };{\mathrm{ℝ}}^3)$ defined as

$${\mathit{y}}^{\ast }(\mathit{x}):=\underset{r\to 0^{+}}{\lim \sup }{\sout{\int }}_{B(\mathit{x},r)}\mathit{y}(\mathit{\chi })\mathrm{d}\mathit{\chi }\mathrm{for}\mathrm{all}\mathit{x}\in \mathrm{\Omega }.$$

In loose terms, the restriction of ${\mathit{y}}^{\ast }$ to the boundary of almost every set $U\subset \subset \mathrm{\Omega }$ is continuous as a consequence of the assumption $p>2$ and Morrey’s embedding. This observation allows for a notion of topological degree for Sobolev mappings. For the degree of continuous functions, we refer to [38]. The next definition exploits the sole dependence of the degree on boundary values.

Definition 3.2 (Topological image). Let $\mathit{y}\in W^{1,p}(\mathrm{\Omega };{\mathrm{ℝ}}^3)$ and $U\subset \subset \mathrm{\Omega }$ be a domain such that ${\mathit{y}}^{\ast }{\mathrm{|}}_{\mathrm{\partial U}}\in C^0(\mathrm{\partial U};{\mathrm{ℝ}}^3)$ . The topological image of U under y is defined as

$${\mathrm{im}}_T(\mathit{y},U):=\left\{\mathit{\xi }\in {\mathrm{ℝ}}^3\setminus {\mathit{y}}^{\ast }(\mathrm{\partial U}):\mathrm{deg}({\mathit{y}}^{\ast },U,\mathit{\xi })\ne 0\right\},$$

where $\mathrm{deg}({\mathit{y}}^{\ast },U,\mathit{\xi })$ denotes the topological degree of any extension of ${\mathit{y}}^{\ast }{\mathrm{|}}_{\mathrm{\partial U}}$ to $\overline{U}$ at the point ξ . Moreover, the topological image of Ω under y is defined as

$${\mathrm{im}}_T(\mathit{y},\mathrm{\Omega }):=\bigcup \left\{{\mathrm{im}}_T(\mathit{y},U):U\subset \subset \mathrm{\Omega }\mathrm{domain}\mathrm{with}{\mathit{y}}^{\ast }{\mathrm{|}}_{\mathrm{\partial U}}\in C^0(\mathrm{\partial U};{\mathrm{ℝ}}^3)\right\}.$$

By classical properties of the topological degree (see, e.g., [14, Remark 2.17]), ${\mathrm{im}}_T(\mathit{y},U)$ is a bounded open set. The set ${\mathrm{im}}_T(\mathit{y},\mathrm{\Omega })$ is thus open but possibly unbounded.

3.2. Divergence identities

The divergence identities are a generalization of the identity $\mathrm{Det}D\mathit{y}=\det D\mathit{y}$ , asserting that the distributional and pointwise determinant coincide [39], which has been investigated in connection with the weak continuity of the Jacobian determinant [32].

Definition 3.3 (Divergence identities). Let $\mathit{y}\in W^{1,p}(\mathrm{\Omega };{\mathrm{ℝ}}^3)$ . The map y is termed to satisfy the divergence identities (DIV) whenever

$$\mathrm{div}\left((\mathrm{adj}D\mathit{y})\mathit{\psi }\circ \mathit{y}\right)=(\mathrm{div}\mathit{\psi })\circ \mathit{y}\det D\mathit{y}\mathrm{for}\mathrm{all}\mathit{\psi }\in \underset{c}{\overset{1}{C}}({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3),$$

where the divergence on the left-hand side and the equality are understood in the sense of distributions over Ω. Namely,

$$-\underset{\mathrm{\Omega }}{\int }\left((\mathrm{adj}D\mathit{y})\mathit{\psi }\circ \mathit{y}\right)\cdot \mathrm{D\phi }\mathrm{d}\mathit{x}=\underset{\mathrm{\Omega }}{\int }(\mathrm{div}\mathit{\psi })\circ \mathit{y}(\det D\mathit{y})\phi \mathrm{d}\mathit{x}$$

for all $\phi \in C_c^1(\mathrm{\Omega })$ and $\mathit{\psi }\in C_c^1({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)$ .

Let us remark that satisfying the divergence identities is equivalent to having zero surface energy in the setting of [11, 40].

We state two consequences of the divergence identities. The first result shows that geometric and topological images actually coincide and, in turn, rule out the phenomenon of cavitation.

Proposition 3.4 (Excluding cavitation). Let $\mathit{y}\in W^{1,p}(\mathrm{\Omega };{\mathrm{ℝ}}^3)$ with $\det D\mathit{y}\in L^1(\mathrm{\Omega };(0,+\infty ))$ satisfy (DIV). Then, ${\mathrm{im}}_G(\mathit{y},\mathrm{\Omega })={\mathrm{im}}_T(\mathit{y},\mathrm{\Omega })$ up to negligible sets. That is, ${ℒ}^3\left({\mathrm{im}}_G(\mathit{y},\mathrm{\Omega })△{\mathrm{im}}_T(\mathit{y},\mathrm{\Omega })\right)=0$ .

Proof. See [11, Lemma 5.18(c)]. □

The second result concerns the inverse of deformations. As already noted, if y is almost everywhere injective, then the restriction $\mathit{y}{\mathrm{|}}_{{\mathrm{dom}}_G(\mathit{y},\mathrm{\Omega })}$ is actually (everywhere) injective. Thus, the inverse ${\mathit{y}}^{-1}:{\mathrm{im}}_T(\mathit{y},\mathrm{\Omega })\to {\mathrm{dom}}_G(\mathit{y},\mathrm{\Omega })$ yields an almost everywhere defined map on the open set ${\mathrm{im}}_T(\mathit{y},\mathrm{\Omega })$ in view of Proposition 3.4. The next results assert its Sobolev regularity.

Proposition 3.5 (Regularity of the inverse). Let $\mathit{y}\in W^{1,p}(\mathrm{\Omega };{\mathrm{ℝ}}^3)$ with $\det D\mathit{y}\in L^1(\mathrm{\Omega };(0,+\infty ))$ be almost everywhere injective and satisfy (DIV). Then, ${\mathit{y}}^{-1}\in W^{1,1}({\mathrm{im}}_T(\mathit{y},\mathrm{\Omega });{\mathrm{ℝ}}^3)$ with $D{\mathit{y}}^{-1}={(D\mathit{y})}^{-1}\circ {\mathit{y}}^{-1}$ almost everywhere in ${\mathrm{im}}_T(\mathit{y},\mathrm{\Omega })$ .

Proof. From [11, Proposition 5.3], we know that ${\mathit{y}}^{-1}\in W_{\mathrm{loc}}^{1,1}({\mathrm{im}}_T(\mathit{y},\mathrm{\Omega });{\mathrm{ℝ}}^3)$ and the inverse formula holds. Moreover,

$$\underset{{\mathrm{im}}_T(\mathit{y},\mathrm{\Omega })}{\int }|{\mathit{y}}^{-1}|\mathrm{d}\mathit{\xi }=\underset{\mathrm{\Omega }}{\int }|\mathit{id}|\det D\mathit{y}\mathrm{d}\mathit{x}\le \parallel \mathit{id}{\parallel }_{L^{\infty }(\mathrm{\Omega };{\mathrm{ℝ}}^3)}\parallel \det D\mathit{y}{\parallel }_{L^1(\mathrm{\Omega })}$$

and

$$\underset{{\mathrm{im}}_T(\mathit{y},\mathrm{\Omega })}{\int }|D{\mathit{y}}^{-1}|\mathrm{d}\mathit{\xi }=\underset{\mathrm{\Omega }}{\int }|\mathrm{adj}D\mathit{y}|\mathrm{d}\mathit{x}$$

thanks to Proposition 3.4, the inverse formula, and the change-of-variable formula, where the last integral is finite since $p>2$ . □

3.3. Least upper bound of positive measures

Let $U\subset {\mathrm{ℝ}}^3$ be an open set and let $({\mu }_1,\dots ,{\mu }_M)$ be a finite family of positive Borel measures on U. The least upper bound ${\mathrm{\vee }}_{i=1}^M{\mu }_i$ of $({\mu }_1,\dots ,{\mu }_M)$ is defined for every Borel set $E\subset U$ by setting

$$\left(\underset{i=1}{\overset{M}{\vee }}{\mu }_i\right)(E):=\sup \left\{\sum _{i=1}^M{\mu }_i(E_i):(E_1,\dots ,E_M)\mathrm{Borel}\mathrm{partition}\mathrm{of}E\right\}.$$

This is also a positive Borel measure, namely, the smallest positive Borel measure on U with the property that $\max \{{\mu }_i(E):i=1,\mathrm{\dots },M\}\le ({\mathrm{\vee }}_{i=1}^M{\mu }_i)(E)$ for any Borel set $E\subset U$ [36, p. 31]. In particular, as noted in [36, Remark 1.69], for absolutely continuous measures, one has

$$\underset{i=1}{\overset{M}{\vee }}(g_i\mathrm{d}\mathit{x})=\max \{g_i:i=1,\mathrm{\dots },M\}\mathrm{d}\mathit{x}\mathrm{for}\mathrm{all}{g}_1,\mathrm{\dots },g_M\in L^1(U;(0,+\infty )).$$

(28)

The following lower semicontinuity result will be employed in the proof of Theorem 2.2(i).

Lemma 3.6 (Lower semicontinuity of the least upper bound of total variations). Let $U\subset {\mathrm{ℝ}}^3$ be an open set and, for each $i=1,\dots ,M$ , let ${(w_{i,\epsilon })}_{\epsilon >0}$ be a sequence in $\mathrm{BV}(U)$ and $w_i\in \mathrm{BV}(U)$ such that $w_{i,\epsilon }\to w_i$ in $L_{\mathrm{loc}}^1(U)$ , as $\epsilon \to 0^{+}$ . Then,

$$\left(\underset{i=1}{\overset{M}{\vee }}|D{w}_i|\right)(U)\le \underset{\epsilon \to 0^{+}}{\lim \inf }\left(\underset{i=1}{\overset{M}{\vee }}|D{w}_{i,\epsilon }|\right)(U).$$

Proof. Let $(E_1,\dots ,E_M)$ be a Borel partition of U. By the lower semicontinuity of the total variation,

$$|D{w}_i|(E_i)\le \underset{\epsilon \to 0^{+}}{\lim \inf }|D{w}_{i,\epsilon }|(E_i)\mathrm{for}\mathrm{all}i=1,\mathrm{\dots },M.$$

Thus

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\sum _{i=1}^M|D{w}_i|(E_i)}& \multicolumn{1}{c}{\le \sum _{i=1}^M\underset{\epsilon \to 0^{+}}{\lim \inf }|D{w}_{i,\epsilon }|(E_i)}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le \underset{\epsilon \to 0^{+}}{\lim \inf }\left(\sum _{i=1}^M|D{w}_{i,\epsilon }|(E_i)\right)\le \underset{\epsilon \to 0^{+}}{\lim }\left(\underset{i=1}{\overset{M}{\vee }}|D{w}_{i,\epsilon }|\right)(U).}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

The claim follows by taking the supremum among all Borel partitions $(E_1,\dots ,E_M)$ of U. □

4. Proofs

In this section, we prove the main results stated in section 2.

4.1. Compactness result

We begin by proving our compactness result. Below, we will consider the first-order Taylor expansion of Φ. For fixed ${\mathit{z}}_0\in {\mathbb{S}}^2$ , we have

$$\mathrm{\Phi }({\mathit{z}}_0+\mathit{z})=\mathrm{\Phi }(\mathit{w})+D\mathrm{\Phi }({\mathit{z}}_0)\cdot \mathit{z}+{\eta }_{\mathrm{\Phi }}(\mathit{z};{\mathit{z}}_0)\mathrm{for}\mathrm{all}\mathit{z}\in {\mathrm{ℝ}}^3\mathrm{with}|\mathit{z}|\ll 1,$$

(29)

where ${\eta }_{\mathrm{\Phi }}(\mathit{z};{\mathit{z}}_0)=\mathrm{o}(|\mathit{z}|)$ , as $|\mathit{z}|\to 0^{+}$ . For $t>0$ , we define

$${\zeta }_{\mathrm{\Phi }}(t):=\sup \left\{\frac{|{\eta }_{\mathrm{\Phi }}(\mathit{z};{\mathit{z}}_0)|}{|\mathit{z}|}:{\mathit{z}}_0\in {\mathbb{S}}^2,\mathit{z}\in {\mathrm{ℝ}}^3,|\mathit{z}|\le t\right\},$$

so that

$$\underset{t\to 0^{+}}{\lim }{\zeta }_{\mathrm{\Phi }}(t)=0$$

(30)

in view of (Φ2). Additionally, for $i=1,\dots ,M$ , we define $f_i:{\mathbb{S}}^2\to [0,+\infty )$ by setting

$$f(\mathit{z}):=d_{\mathrm{\Phi }}(\mathit{z},{\mathit{b}}_i)\mathrm{for}\mathrm{all}\mathit{z}\in {\mathbb{S}}^2,$$

(31)

where $d_{\mathrm{\Phi }}$ is given by (16).

Proof of Theorem 2.1. We subdivide the proof into three steps.

Step 1 (Compactness of deformations and displacements). For convenience, we set ${\mathit{F}}_{\epsilon }:=D{\mathit{y}}_{\epsilon },{\mathit{L}}_{\epsilon }:={\mathbf{\Lambda }}_{\epsilon }({\mathit{m}}_{\epsilon }\circ {\mathit{y}}_{\epsilon })$ , and ${\mathit{A}}_{\epsilon }:={\mathit{L}}_{\epsilon }^{-1}{\mathit{F}}_{\epsilon }$ , where we adopt the notation in (10). From (17) and assumption (W5), we obtain

$$\underset{\mathrm{\Omega }}{\int }\left(g_p(\mathrm{dis}({\mathit{A}}_{\epsilon };\mathrm{SO}(3)))+h_q(\det {\mathit{A}}_{\epsilon })\right)\mathrm{d}\mathit{x}\le \frac{{\epsilon }^2}{c_W}{\mathcal{E}}_{\epsilon }^e({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })\le C{\epsilon }^2.$$

(32)

Given that $t^p∕p\le g_p(t)$ for all $t\ge 0$ , the previous estimate implies $\mathrm{dist}({\mathit{A}}_{\epsilon };\mathrm{SO}(3))\to 0$ in $L^p(\mathrm{\Omega })$ . Thus, the sequence ${({\mathit{A}}_{\epsilon })}_{\epsilon >0}$ is bounded in $L^p(\mathrm{\Omega };{\mathrm{ℝ}}^{3\times 3})$ . Noting that ${\mathbf{\Lambda }}_{\epsilon }(\mathit{z})=\mathit{I}+\epsilon \mathbf{\Lambda }(\mathit{z})$ for all $\mathit{z}\in {\mathbb{S}}^2$ , where we employ (10) and (15), and computing

$$|\mathbf{\Lambda }(\mathit{z})|=\ell :=\sqrt{a^2+2b^2}\mathrm{for}\mathrm{all}\mathit{z}\in {\mathbb{S}}^2,$$

(33)

the triangle inequality gives

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\mathrm{dist}({\mathit{F}}_{\epsilon };\mathrm{SO}(3))}& \multicolumn{1}{c}{=\mathrm{dist}({\mathit{L}}_{\epsilon }{\mathit{A}}_{\epsilon };\mathrm{SO}(3))}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le |{\mathit{L}}_{\epsilon }-\mathit{I}||{\mathit{A}}_{\epsilon }|+\mathrm{dist}({\mathit{A}}_{\epsilon };\mathrm{SO}(3))}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{=\mathrm{\ell \epsilon }|{\mathit{A}}_{\epsilon }|+\mathrm{dist}({\mathit{A}}_{\epsilon };\mathrm{SO}(3)).}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

(34)

The estimates (32) and (34) together with the monotonicity of $g_p$ and the elementary inequality

$$g_p(t+\tilde{t})\le C\left(t^p+g_p(\tilde{t})\right)\mathrm{for}\mathrm{all}t,\tilde{t}\ge 0,$$

yield

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\underset{\mathrm{\Omega }}{\int }{g}_p(\mathrm{dist}({\mathit{F}}_{\epsilon };\mathrm{SO}(3)))\mathrm{d}\mathit{x}}& \multicolumn{1}{c}{\le C\left({\ell }^p{\epsilon }^p\underset{\mathrm{\Omega }}{\int }|{\mathit{A}}_{\epsilon }{\mathrm{|}}^{p}d\mathit{x}+\underset{\mathrm{\Omega }}{\int }{g}_p(\mathrm{dist}({\mathit{A}}_{\epsilon };\mathrm{SO}(3)))\mathrm{d}\mathit{x}\right)\le C{\epsilon }^2}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

given that $p>2$ . As

$$\frac{1}{2p}(t^2+t^p)\le \frac{1}{p}(t^2\vee t^p)\le g_p(t)\mathrm{for}\mathrm{all}t\ge 0,$$

from the last estimate we obtain

$$\underset{\mathrm{\Omega }}{\int }{\mathrm{dist}}^2({\mathit{F}}_{\epsilon };\mathrm{SO}(3))\mathrm{d}\mathit{x}+\underset{\mathrm{\Omega }}{\int }{\mathrm{dist}}^p({\mathit{F}}_{\epsilon };\mathrm{SO}(3))\le C{\epsilon }^2.$$

(35)

Let ${\mathit{u}}_{\epsilon }:={\epsilon }^{-1}({\mathit{y}}_{\epsilon }-\mathit{id})$ and ${\mathit{G}}_{\epsilon }:=D{\mathit{u}}_{\epsilon }$ . First, we show that ${({\mathit{G}}_{\epsilon })}_{\epsilon >0}$ is bounded in $L^2(\mathrm{\Omega };{\mathrm{ℝ}}^{3\times 3})$ . Then, by applying the Poincaré inequality with trace term, we find $\mathit{u}\in W^{1,2}(\mathrm{\Omega };{\mathrm{ℝ}}^3)$ for which (19) holds along a not relabeled subsequence. In particular, $\mathit{u}=\mathit{d}$ on Δ owing to the weak continuity of the trace operator. From (35), using the classical rigidity estimate [41, Theorem 3.1] (see also [26, Remark 2.3] and [42, Section 2.4]), we identify a sequence ${({\mathit{R}}_{\epsilon })}_{\epsilon >0}$ in $\mathrm{SO}(3)$ satisfying

$$\underset{\mathrm{\Omega }}{\int }|{\mathit{F}}_{\epsilon }-{\mathit{R}}_{\epsilon }{\mathrm{|}}^2\mathrm{d}\mathit{x}+\underset{\mathrm{\Omega }}{\int }|{\mathit{F}}_{\epsilon }-{\mathit{R}}_{\epsilon }{\mathrm{|}}^p\mathrm{d}\mathit{x}\le C{\epsilon }^2.$$

(36)

Then, by arguing as in [43, Proposition 3.4], taking advantage of the boundary condition ${\mathit{u}}_{\epsilon }=\mathit{d}$ on Δ, we establish

$$|{\mathit{R}}_{\epsilon }-\mathit{I}{\mathrm{|}}^2\le C{\epsilon }^2\left(1+\underset{\mathrm{\Gamma }}{\int }|\mathit{d}{\mathrm{|}}^2\mathrm{d}{\mathscr{H}}^2\right)\le C{\epsilon }^2$$

(37)

and, in turn,

$$\underset{\mathrm{\Omega }}{\int }|{\mathit{G}}_{\epsilon }{\mathrm{|}}^2\mathrm{d}\mathit{x}=\frac{1}{{\epsilon }^2}\underset{\mathrm{\Omega }}{\int }|{\mathit{F}}_{\epsilon }-\mathit{I}{\mathrm{|}}^2\mathrm{d}\mathit{x}\le C.$$

This proves the claim.

Next, we check that $\epsilon {\mathit{G}}_{\epsilon }\to \mathit{O}$ in $L^p(\mathrm{\Omega };{\mathrm{ℝ}}^{3\times 3})$ . Then, (18) follows once again thanks to the Poincaré inequality. We argue as in [19, Lemma 3.3] and we estimate

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\underset{\mathrm{\Omega }}{\int }|\epsilon {\mathit{G}}_{\epsilon }{\mathrm{|}}^p\mathrm{d}\mathit{x}}& \multicolumn{1}{c}{\le 2^{p-1}\left(\underset{\mathrm{\Omega }}{\int }|{\mathit{F}}_{\epsilon }-{\mathit{R}}_{\epsilon }{\mathrm{|}}^p\mathrm{d}\mathit{x}+\underset{\mathrm{\Omega }}{\int }|{\mathit{R}}_{\epsilon }-\mathit{I}{\mathrm{|}}^p\mathrm{d}\mathit{x}\right).}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

The first integral on the right-hand side is bounded by (36), while for the second one we estimate

$$|{\mathit{R}}_{\epsilon }-\mathit{I}{\mathrm{|}}^p={\left(|{\mathit{R}}_{\epsilon }-\mathit{I}{\mathrm{|}}^2\right)}^{p∕2}\le {(C{\epsilon }^2)}^{p∕2}=C{\epsilon }^p$$

thanks to (37). Altogether, we find

$$\underset{\mathrm{\Omega }}{\int }|\epsilon {\mathit{G}}_{\epsilon }{\mathrm{|}}^p\mathrm{d}\mathit{x}\le C({\epsilon }^2+{\epsilon }^p)$$

which yields the desired convergence.

Furthermore, we claim that

$$\det D{\mathit{y}}_{\epsilon }\to 1\mathrm{in}{L}^1(\mathrm{\Omega }),$$

(38)

so that, by applying [12, Proposition 2.23(ii)], we obtain

$${\chi }_{{\mathrm{im}}_T({\mathit{y}}_{\epsilon },\mathrm{\Omega })}\to {\chi }_{\mathrm{\Omega }}\mathrm{in}{L}^1({\mathrm{ℝ}}^3).$$

(39)

Note that

$$\det {\mathbf{\Lambda }}_{\epsilon }(\mathit{z})=a_{\epsilon }{b}_{\epsilon }^2=:c_{\epsilon }\mathrm{for}\mathrm{all}\mathit{z}\in {\mathbb{S}}^2,$$

(40)

where $c_{\epsilon }\to 1$ . Hence, as $\det {\mathit{F}}_{\epsilon }=(\det {\mathit{L}}_{\epsilon })(\det {\mathit{A}}_{\epsilon })$ , we have

$$(1∕C)\det {\mathit{A}}_{\epsilon }\le \det {\mathit{F}}_{\epsilon }\le C\det {\mathit{A}}_{\epsilon }.$$

(41)

Using the elementary estimate

$$(q+1)(h_q(t)+1)\ge q{h}_q(t)+(q+1)=t^q+\frac{q}{t}\ge t^q\mathrm{for}\mathrm{all}t>0$$

together with (32), we deduce

$$\underset{\mathrm{\Omega }}{\int }{(\det {\mathit{F}}_{\epsilon })}^q\mathrm{d}\mathit{x}\le C\underset{\mathrm{\Omega }}{\int }{(\det {\mathit{A}}_{\epsilon })}^q\mathrm{d}\mathit{x}\le C\left(\underset{\mathrm{\Omega }}{\int }{h}_q(\det {\mathit{A}}_{\epsilon })\mathrm{d}\mathit{x}+1\right)\le C.$$

(42)

Thus, ${(\det {\mathit{F}}_{\epsilon })}_{\epsilon >0}$ is bounded in $L^q(\mathrm{\Omega })$ . In view of (18), we also know that $\det {\mathit{F}}_{\epsilon }\to 1$ almost everywhere in Ω for a not relabeled subsequence. Therefore, (38) follows by Vitali’s convergence theorem and Urysohn’s property.

Step 2 (Compactness of magnetizations). We fix an exponent $\alpha >0$ satisfying

$$\frac{\mathrm{\beta q}}{2(q-1)}<\alpha <1,$$

(43)

noting that such a choice is always possible in view of (6). Let ${\mathrm{\Omega }}_{\epsilon }:=\{\mathit{x}\in \mathrm{\Omega }:|{\mathit{G}}_{\epsilon }(\mathit{x})|<{\epsilon }^{-\alpha }\}$ . By Chebyshev’s inequality,

$${ℒ}^3(\mathrm{\Omega }\setminus {\mathrm{\Omega }}_{\epsilon })\le {\epsilon }^{2\alpha }\underset{\mathrm{\Omega }}{\int }|D{\mathit{u}}_{\epsilon }{\mathrm{|}}^2\mathrm{d}\mathit{x}\le C{\epsilon }^{2\alpha }\mathrm{for}\mathrm{all}\epsilon >0.$$

(44)

Recall that

$$\det \left(\mathit{I}+\epsilon \mathit{G}\right)=1+\sum _{k=1}^3{\epsilon }^k{P}_k(\mathit{G})\mathrm{for}\mathrm{all}\mathit{G}\in {\mathrm{ℝ}}^{3\times 3},$$

(45)

where each $P_k(\mathit{G})$ is a homogeneous polynomial of degree k in the entries of G and, as such,

$$|P_k(\mathit{G})|\le C|\mathit{G}{\mathrm{|}}^k\mathrm{for}\mathrm{all}k=1,2,3\mathrm{and}\mathit{G}\in {\mathrm{ℝ}}^{3\times 3}.$$

(46)

Thus, recalling (43), we have

$$\parallel \det {\mathit{F}}_{\epsilon }{\parallel }_{L^{\infty }({\mathrm{\Omega }}_{\epsilon })}\le 1+C\sum _{k=1}^3{\epsilon }^{k(1-\alpha )}\le C\mathrm{for}\mathrm{all}\epsilon >0.$$

(47)

By considering the Taylor expansion of Φ around ${\mathit{m}}_{\epsilon }$ as in (29) and Proposition 3.4, we write

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\frac{1}{{\epsilon }^{\beta }}\underset{{\mathrm{im}}_T({\mathit{y}}_{\epsilon },\mathrm{\Omega })}{\int }\mathrm{\Phi }}& \multicolumn{1}{c}{\left(\left({\mathit{F}}_{\epsilon }^{\mathrm{\top }}\circ {\mathit{y}}_{\epsilon }^{-1}\right){\mathit{m}}_{\epsilon }\right)\mathrm{d}\mathit{\xi }}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\ge \frac{1}{{\epsilon }^{\beta }}\underset{{\mathrm{im}}_G({\mathit{y}}_{\epsilon },{\mathrm{\Omega }}_{\epsilon })}{\int }\mathrm{\Phi }\left(\left({\mathit{F}}_{\epsilon }^{\mathrm{\top }}\circ {\mathit{y}}_{\epsilon }^{-1}\right){\mathit{m}}_{\epsilon }\right)\mathrm{d}\mathit{\xi }}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{=\frac{1}{{\epsilon }^{\beta }}\underset{{\mathrm{im}}_G({\mathit{y}}_{\epsilon },{\mathrm{\Omega }}_{\epsilon })}{\int }\mathrm{\Phi }({\mathit{m}}_{\epsilon })\mathrm{d}\mathit{\xi }}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{+{\epsilon }^{1-\beta }\underset{{\mathrm{im}}_G({\mathit{y}}_{\epsilon },{\mathrm{\Omega }}_{\epsilon })}{\int }D\mathrm{\Phi }({\mathit{m}}_{\epsilon })\cdot \left(\left({\mathit{G}}_{\epsilon }^{\mathrm{\top }}\circ {\mathit{y}}_{\epsilon }^{-1}\right){\mathit{m}}_{\epsilon }\right)\mathrm{d}\mathit{\xi }}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{+\frac{1}{{\epsilon }^{\beta }}\underset{{\mathrm{im}}_G({\mathit{y}}_{\epsilon },{\mathrm{\Omega }}_{\epsilon })}{\int }{\eta }_{\mathrm{\Phi }}\left(\epsilon \left({\mathit{G}}_{\epsilon }^{\mathrm{\top }}\circ {\mathit{y}}_{\epsilon }^{-1}\right){\mathit{m}}_{\epsilon };{\mathit{m}}_{\epsilon }\right)\mathrm{d}\mathit{\xi }.}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

(48)

For the last two integrals on the right-hand side of (48), we observe that

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{|{\epsilon }^{1-\beta }\underset{{\mathrm{im}}_G({\mathit{y}}_{\epsilon },{\mathrm{\Omega }}_{\epsilon })}{\int }D\mathrm{\Phi }({\mathit{m}}_{\epsilon })\cdot }& \multicolumn{1}{c}{\left(\left({\mathit{G}}_{\epsilon }^{\mathrm{\top }}\circ {\mathit{y}}_{\epsilon }^{-1}\right){\mathit{m}}_{\epsilon }\right)\mathrm{d}\mathit{\xi }|}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le {\epsilon }^{1-\beta }\parallel D\mathrm{\Phi }{\parallel }_{L^{\infty }({\mathbb{S}}^2;{\mathrm{ℝ}}^3)}\underset{{\mathrm{im}}_G({\mathit{y}}_{\epsilon },{\mathrm{\Omega }}_{\epsilon })}{\int }|{\mathit{G}}_{\epsilon }^{\mathrm{\top }}\circ {\mathit{y}}_{\epsilon }^{-1}|\mathrm{d}\mathit{\xi }}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le C{\epsilon }^{1-\beta }\underset{{\mathrm{\Omega }}_{\epsilon }}{\int }|{\mathit{G}}_{\epsilon }|\det {\mathit{F}}_{\epsilon }\mathrm{d}\mathit{x}\le C{\epsilon }^{1-\beta }}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

and

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{|\frac{1}{{\epsilon }^{\beta }}\underset{{\mathrm{im}}_G({\mathit{y}}_{\epsilon },{\mathrm{\Omega }}_{\epsilon })}{\int }{\eta }_{\mathrm{\Phi }}(\epsilon }& \multicolumn{1}{c}{({\mathit{G}}_{\epsilon }^{\mathrm{\top }}\circ {\mathit{y}}_{\epsilon }^{-1}){\mathit{m}}_{\epsilon };{\mathit{m}}_{\epsilon })\mathrm{d}\mathit{\xi }|}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le {\zeta }_{\mathrm{\Phi }}({\epsilon }^{1-\alpha }){\epsilon }^{1-\beta }\underset{{\mathrm{im}}_G({\mathit{y}}_{\epsilon },{\mathrm{\Omega }}_{\epsilon })}{\int }|D{\mathit{G}}_{\epsilon }^{\mathrm{\top }}\circ {\mathit{y}}_{\epsilon }^{-1}|\mathrm{d}\mathit{\xi }}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{={\zeta }_{\mathrm{\Phi }}({\epsilon }^{1-\alpha }){\epsilon }^{1-\beta }\underset{{\mathrm{\Omega }}_{\epsilon }}{\int }|{\mathit{G}}_{\epsilon }|\det {\mathit{F}}_{\epsilon }\mathrm{d}\mathit{x}\le C{\epsilon }^{1-\beta }}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

thanks to the change-of-variable formula, (30), the boundedness of ${({\mathit{G}}_{\epsilon })}_{\epsilon >0}$ in $L^1(\mathrm{\Omega };{\mathrm{ℝ}}^{3\times 3})$ , and (47). Thus, from (48), we deduce

$$\frac{1}{{\epsilon }^{\beta }}\underset{{\mathrm{im}}_T({\mathit{y}}_{\epsilon },\mathrm{\Omega })}{\int }\mathrm{\Phi }\left(\left({\mathit{F}}_{\epsilon }^{\mathrm{\top }}\circ {\mathit{y}}_{\epsilon }^{-1}\right){\mathit{m}}_{\epsilon }\right)\mathrm{d}\mathit{\xi }\ge \frac{1}{{\epsilon }^{\beta }}\underset{{\mathrm{im}}_G({\mathit{y}}_{\epsilon },{\mathrm{\Omega }}_{\epsilon })}{\int }\mathrm{\Phi }({\mathit{m}}_{\epsilon })\mathrm{d}\mathit{\xi }-C{\epsilon }^{1-\beta }.$$

(49)

Exploiting again Proposition 3.4, the integral on the right-hand side can be rewritten as

$$\frac{1}{{\epsilon }^{\beta }}\underset{{\mathrm{im}}_G({\mathit{y}}_{\epsilon },{\mathrm{\Omega }}_{\epsilon })}{\int }\mathrm{\Phi }({\mathit{m}}_{\epsilon })\mathrm{d}\mathit{\xi }=\frac{1}{{\epsilon }^{\beta }}\underset{{\mathrm{im}}_T({\mathit{y}}_{\epsilon },\mathrm{\Omega })}{\int }\mathrm{\Phi }({\mathit{m}}_{\epsilon })\mathrm{d}\mathit{\xi }-\frac{1}{{\epsilon }^{\beta }}\underset{{\mathrm{im}}_G({\mathit{y}}_{\epsilon },\mathrm{\Omega }\setminus {\mathrm{\Omega }}_{\epsilon })}{\int }\mathrm{\Phi }({\mathit{m}}_{\epsilon })\mathrm{d}\mathit{\xi }.$$

Using (42), (44), the change-of-variable formula, and Hölder’s inequality, the second term can be estimated as

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\left|\frac{1}{{\epsilon }^{\beta }}\underset{{\mathrm{im}}_G({\mathit{y}}_{\epsilon },\mathrm{\Omega }\setminus {\mathrm{\Omega }}_{\epsilon })}{\int }\mathrm{\Phi }({\mathit{m}}_{\epsilon })\mathrm{d}\mathit{\xi }\right|}& \multicolumn{1}{c}{\le \frac{\parallel \mathrm{\Phi }{\parallel }_{L^{\infty }({\mathbb{S}}^2)}}{{\epsilon }^{\beta }}{ℒ}^3({\mathrm{im}}_G({\mathit{y}}_{\epsilon },\mathrm{\Omega }\setminus {\mathrm{\Omega }}_{\epsilon }))=\frac{C}{{\epsilon }^{\beta }}\underset{\mathrm{\Omega }\setminus {\mathrm{\Omega }}_{\epsilon }}{\int }\det {\mathit{F}}_{\epsilon }\mathrm{d}\mathit{x}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le \frac{C}{{\epsilon }^{\beta }}\parallel \det {\mathit{F}}_{\epsilon }{\parallel }_{L^q(\mathrm{\Omega })}{ℒ}^3{(\mathrm{\Omega }\setminus {\mathrm{\Omega }}_{\epsilon })}^{1∕q^{\prime }}\le C{\epsilon }^{\frac{2\alpha (q-1)}{q}-\beta }.}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

Thus, from (49), we obtain

$$\frac{1}{{\epsilon }^{\beta }}\underset{{\mathrm{im}}_T({\mathit{y}}_{\epsilon },\mathrm{\Omega })}{\int }\mathrm{\Phi }\left(\left({\mathit{F}}_{\epsilon }^{\mathrm{\top }}\circ {\mathit{y}}_{\epsilon }^{-1}\right){\mathit{m}}_{\epsilon }\right)\mathrm{d}\mathit{\xi }\ge \frac{1}{{\epsilon }^{\beta }}\underset{{\mathrm{im}}_T({\mathit{y}}_{\epsilon },\mathrm{\Omega })}{\int }\mathrm{\Phi }({\mathit{m}}_{\epsilon })\mathrm{d}\mathit{\xi }-C{\epsilon }^{\gamma },$$

(50)

where

$$\gamma :=\left(\frac{2\alpha (q-1)}{q}\wedge 1\right)-\beta .$$

(51)

Noting that $\gamma >0$ in view of (6) and (43), from (17) and (50) we obtain

$$\underset{\epsilon >0}{\sup }\left\{\frac{1}{{\epsilon }^{\beta }}\underset{{\mathrm{im}}_T({\mathit{y}}_{\epsilon },\mathrm{\Omega })}{\int }\mathrm{\Phi }({\mathit{m}}_{\epsilon })\mathrm{d}\mathit{\xi }+{\epsilon }^{\beta }\underset{{\mathrm{im}}_T({\mathit{y}}_{\epsilon },\mathrm{\Omega })}{\int }|D{\mathit{m}}_{\epsilon }{\mathrm{|}}^2\mathrm{d}\mathit{\xi }\right\}\le C_0,$$

(52)

where $C_0>0$ is a constant depending on the supremum in (17).

Given (18), by applying [11, Lemma 2.24 and Lemma 3.6(a)], we find a Lipschitz domain $U\subset \subset \mathrm{\Omega }$ such that $U={\mathrm{im}}_T(\mathit{id},U)\subset \subset {\mathrm{im}}_T({\mathit{y}}_{\epsilon },\mathrm{\Omega })$ along a not relabeled subsequence. Hence, (52) yields

$$\underset{\epsilon >0}{\sup }\left\{\frac{1}{{\epsilon }^{\beta }}\underset{U}{\int }\mathrm{\Phi }({\mathit{m}}_{\epsilon })\mathrm{d}\mathit{x}+{\epsilon }^{\beta }\underset{U}{\int }|D{\mathit{m}}_{\epsilon }{\mathrm{|}}^2\mathrm{d}\mathit{x}\right\}\le C_0.$$

(53)

Let $i=1,\dots ,M$ . Recall (31) and define $w_{i,\epsilon }:=f_i\circ {\mathit{m}}_{\epsilon }$ . By [21, Lemma 4.2], we infer that $w_{i,\epsilon }\in W^{1,2}(U)$ with

$$|D{w}_{i,\epsilon }(\mathit{x})|\le \sqrt{\mathrm{\Phi }({\mathit{m}}_{\epsilon }(\mathit{x}))}|D{\mathit{m}}_{\epsilon }(\mathit{x})|\mathrm{for}\mathrm{almost}\mathrm{all}\mathit{x}\in U.$$

(54)

From (53)–(54) and Young’s inequality, we find

$$\underset{\epsilon >0}{\sup }\underset{U}{\int }|D{w}_{i,\epsilon }|\mathrm{d}\mathit{x}\le C_0,$$

while, trivially,

$$\underset{\epsilon >0}{\sup }\parallel w_{i,\epsilon }{\parallel }_{L^{\infty }(U)}\le \parallel f_i{\parallel }_{L^{\infty }({\mathbb{S}}^2)}.$$

We deduce that the sequence ${(w_{i,\epsilon })}_{\epsilon >0}$ is bounded in $\mathrm{BV}(U)$ . Hence, there exists $w_i\in \mathrm{BV}(U)$ such that, up to subsequences,

$$w_{i,\epsilon }\to w_i\mathrm{in}{L}^1(U),$$

(55)

as $\epsilon \to 0^{+}$ . Since the constant $C_0$ in (53) does not depend on U, by applying [11, Lemma 2.24 and Lemma 3.6(a)] with a sequence of Lipschitz domains invading Ω together with a diagonal argument, we realize that $w_1,\dots ,w_M\in \mathrm{BV}(\mathrm{\Omega })$ and we select a not relabeled subsequence for which (55) holds true for any domain $U\subset \subset \mathrm{\Omega }$ .

Let $E_i:=\{\mathit{x}\in \mathrm{\Omega }:w_i(\mathit{x})=0\}$ and define $\mathit{m}:\mathrm{\Omega }\to {\mathbb{S}}^2$ by setting $\mathit{m}:={\mathrm{\sum }}_{i=1}^M{\mathit{b}}_i{\chi }_{E_i}$ . We claim that

$${\mathit{m}}_{\epsilon }\to \mathit{m}\mathrm{in}{L}^1(U)\mathrm{for}\mathrm{any}\mathrm{domain}U\subset \subset \mathrm{\Omega }.$$

(56)

From (53), we have $\mathrm{\Phi }({\mathit{m}}_{\epsilon })\to 0$ in $L^1(U)$ . Hence, we can select a not relabeled subsequence with the property that, for almost every $\mathit{x}\in U$ , there exists $\underset{\epsilon \to 0^{+}}{\lim }{\mathit{m}}_{\epsilon }(\mathit{x})\in \{{\mathit{b}}_1,\mathrm{\dots },{\mathit{b}}_M\}$ . Let $i,j=1,\dots ,M$ and recall (31). For almost every $\mathit{x}\in U\cap E_i$ , if ${\mathit{m}}_{\epsilon }(\mathit{x})\to {\mathit{b}}_j$ , then $w_{i,\epsilon }(\mathit{x})=f_i({\mathit{m}}_{\epsilon }(\mathit{x}))\to f_i({\mathit{b}}_j)$ but also $w_{i,\epsilon }(\mathit{x})=f_i({\mathit{m}}_{\epsilon }(\mathit{x}))\to w_i(\mathit{x})=0$ , as $\epsilon \to 0^{+}$ , in view of (55). Thus, $f_i({\mathit{b}}_j)=0$ which entails $i=j$ . Therefore, ${\mathit{m}}_{\epsilon }(\mathit{x})\to \mathit{m}(\mathit{x})$ for almost all $\mathit{x}\in U$ and, by applying the dominated convergence theorem, we establish (56). Additionally, this argument ensures that $E_i\cap E_j=\varnothing$ for all $i\ne j$ as well as ${ℒ}^3(\mathrm{\Omega }\setminus {\mathrm{\bigcup }}_{i=1}^M{E}_i)=0$ . Given the continuity of $f_i$ , by applying once again the dominated convergence theorem and comparing the result with (55), we realize that $w_i=f_i\circ \mathit{m}$ almost everywhere in Ω. Therefore, we deduce that each $E_i$ has finite perimeter in Ω and, in turn, $\mathit{m}\in \mathrm{BV}(\mathrm{\Omega };\{{\mathit{b}}_1,\dots ,{\mathit{b}}_M\})$ . In particular, $(\mathit{u},\mathit{m})\in \mathcal{A}$ . Eventually, the combination of (39) and (56) easily gives (20).

Step 3 (Convergence of compositions). We move to the proof of (21). From (32) and the trivial inequality

$$h_q(t)+\frac{q+1}{q}=\frac{t^q}{q}+\frac{1}{t}\ge \frac{1}{t}\mathrm{for}\mathrm{all}t>0,$$

we have

$$\underset{\mathrm{\Omega }}{\int }\frac{1}{\det {\mathit{A}}_{\epsilon }}\mathrm{d}\mathit{x}\le \underset{\mathrm{\Omega }}{\int }{h}_q(\det {\mathit{A}}_{\epsilon })\mathrm{d}\mathit{x}+\frac{q+1}{q}{ℒ}^3(\mathrm{\Omega })\le C.$$

Then, thanks to (41), we find

$$\underset{\mathrm{\Omega }}{\int }\frac{1}{\det {\mathit{F}}_{\epsilon }}\mathrm{d}\mathit{x}\le C\underset{\mathrm{\Omega }}{\int }\frac{1}{\det {\mathit{A}}_{\epsilon }}\mathrm{d}\mathit{x}\le C.$$

By [44, Remark 3.5], we infer the equi-integrability of the sequence ${({\chi }_{{\mathrm{im}}_G({\mathit{y}}_{\epsilon },\mathrm{\Omega })}\det D\underset{\epsilon }{\overset{-1}{\mathit{y}}})}_{\epsilon >0}$ , while the same property trivially holds for ${({\mathit{m}}_{\epsilon }\circ {\mathit{y}}_{\epsilon })}_{\epsilon >0}$ . Therefore, given (18) and (20), the convergence in (21) is achieved by applying [14, Proposition 3.4] and the dominated convergence theorem. □

4.2. Gamma-convergence

Next, we move to the proof of the Γ-convergence result. We will consider the second-order Taylor expansion of W near the identity. Taking into account (W2)–(W5), we have

$$W(\mathit{I}+\mathit{B})=\frac{1}{2}{Q}_W(\mathit{B})+{\eta }_W(\mathit{B})\mathrm{for}\mathrm{all}\mathit{B}\in {\mathrm{ℝ}}^{3\times 3},$$

(57)

where ${\eta }_W(\mathit{B})=\mathrm{o}(|\mathit{B}{\mathrm{|}}^2)$ , as $|\mathit{B}|\to 0^{+}$ . In particular, by (W3)–(W5), W has a global minimum at the identity, so that $Q_W$ is a positive-semidefinite quadratic form and, hence, convex. Additionally, in view of the minor symmetries of the elasticity tensor [45, p. 195], ${\mathbf{C}}_W(\mathit{B})\in \mathrm{Sym}(3)$ and ${\mathbf{C}}_W(\mathit{B})={\mathbf{C}}_W(\mathrm{sym}\mathit{B})$ for all $\mathit{B}\in {\mathrm{ℝ}}^{3\times 3}$ . Therefore, we have

$$Q_W(\mathit{B})=Q_W(\mathrm{sym}\mathit{B})\mathrm{for}\mathrm{all}\mathit{B}\in {\mathrm{ℝ}}^{3\times 3}.$$

(58)

For $t>0$ , we define

$${\zeta }_W(t):=\sup \left\{\frac{|{\eta }_W(\mathit{B})|}{|\mathit{B}{\mathrm{|}}^2}:\mathit{B}\in {\mathrm{ℝ}}^{3\times 3},|\mathit{B}|\le t\right\},$$

so that assumption (W2) yields

$$\underset{t\to 0^{+}}{\lim }{\zeta }_W(t)=0.$$

(59)

We begin by proving the lower bound.

Proof of claim (i) of Theorem 2.2. For convenience, the proof is split into two steps.

Step 1 (Lower bound for the elastic energy). As in the proof of Theorem 2.1, we set ${\mathit{F}}_{\epsilon }:=D{\mathit{y}}_{\epsilon }$ , ${\mathit{G}}_{\epsilon }:=D{\mathit{u}}_{\epsilon }$ , and ${\mathit{L}}_{\epsilon }:={\mathbf{\Lambda }}_{\epsilon }({\mathit{m}}_{\epsilon }\circ {\mathit{y}}_{\epsilon })$ . Additionally, we let ${\mathit{K}}_{\epsilon }:=\mathbf{\Lambda }({\mathit{m}}_{\epsilon }\circ {\mathit{y}}_{\epsilon })$ . Given (11), we have

$${\mathit{L}}_{\epsilon }^{-1}=\mathit{I}-\epsilon {\mathit{K}}_{\epsilon }+{\epsilon }^2{\mathit{J}}_{\epsilon },$$

where

$${\mathit{J}}_{\epsilon }:=\frac{a^2}{a_{\epsilon }}{\mathit{m}}_{\epsilon }\circ {\mathit{y}}_{\epsilon }\otimes {\mathit{m}}_{\epsilon }\circ {\mathit{y}}_{\epsilon }+\frac{b^2}{b_{\epsilon }}\left(\mathit{I}-{\mathit{m}}_{\epsilon }\circ {\mathit{y}}_{\epsilon }\otimes {\mathit{m}}_{\epsilon }\circ {\mathit{y}}_{\epsilon }\right).$$

Similar to (33), we note that

$$\parallel {\mathit{J}}_{\epsilon }{\parallel }_{L^{\infty }(\mathrm{\Omega };{\mathrm{ℝ}}^{3\times 3})}=\sqrt{{\left(\frac{a^2}{a_{\epsilon }}\right)}^2+2{\left(\frac{b^2}{b_{\epsilon }}\right)}^2}\le \sqrt{a^4+2b^4}=:\tilde{\ell }.$$

(60)

As in Step 2 of the proof of Theorem 2.1, define ${\mathrm{\Omega }}_{\epsilon }:=\{\mathit{x}\in \mathrm{\Omega }:|{\mathit{G}}_{\epsilon }(\mathit{x})|<{\epsilon }^{-\alpha }\}$ for $\alpha >0$ satisfying (43). From (19), we recover (44). Thus, ${\chi }_{{\mathrm{\Omega }}_{\epsilon }}\to \mathrm{\Omega }$ in measure and, from (19) and (21), we infer

$$\begin{array}{ccc}\multicolumn{1}{c}{{\chi }_{{\mathrm{\Omega }}_{\epsilon }}{\mathit{G}}_{\epsilon }\rightharpoonup D\mathit{u}}& \multicolumn{1}{c}{\mathrm{in}{L}^2(\mathrm{\Omega };{\mathrm{ℝ}}^{3\times 3}),}& \multicolumn{1}{c}{}\end{array}$$

(61)

$$\begin{array}{ccc}\multicolumn{1}{c}{{\chi }_{{\mathrm{\Omega }}_{\epsilon }}{\mathit{K}}_{\epsilon }\to \mathbf{\Lambda }(\mathit{m})}& \multicolumn{1}{c}{\mathrm{in}{L}^1(\mathrm{\Omega };{\mathrm{ℝ}}^{3\times 3}).}& \multicolumn{1}{c}{}\end{array}$$

(62)

For convenience, let ${\mathit{H}}_{\epsilon }:={\mathit{J}}_{\epsilon }+(\epsilon {\mathit{J}}_{\epsilon }-{\mathit{K}}_{\epsilon }){\mathit{G}}_{\epsilon }$ . From (33) and (60), we easily obtain

$$\epsilon \parallel {\mathit{H}}_{\epsilon }{\parallel }_{L^{\infty }({\mathrm{\Omega }}_{\epsilon };{\mathrm{ℝ}}^{3\times 3})}\le \epsilon \tilde{\ell }+{\epsilon }^{1-\alpha }(\epsilon \tilde{\ell }+\ell ),$$

(63)

where the right-hand side goes to zero because of (43). Using the Taylor expansion (57), we write

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{{\mathcal{E}}_{\epsilon }^e({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })}& \multicolumn{1}{c}{\ge \frac{1}{{\epsilon }^2}\underset{{\mathrm{\Omega }}_{\epsilon }}{\int }W({\mathit{L}}_{\epsilon }^{-1}{\mathit{F}}_{\epsilon })\mathrm{d}\mathit{x}=\frac{1}{{\epsilon }^2}\underset{{\mathrm{\Omega }}_{\epsilon }}{\int }W\left(\mathit{I}+\epsilon ({\mathit{G}}_{\epsilon }-{\mathit{K}}_{\epsilon })+{\epsilon }^2{\mathit{H}}_{\epsilon }\right)\mathrm{d}\mathit{x}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{=\frac{1}{2}\underset{{\mathrm{\Omega }}_{\epsilon }}{\int }{Q}_W({\mathit{G}}_{\epsilon }-{\mathit{K}}_{\epsilon }+\epsilon {\mathit{H}}_{\epsilon })\mathrm{d}\mathit{x}+\frac{1}{{\epsilon }^2}\underset{{\mathrm{\Omega }}_{\epsilon }}{\int }{\eta }_W\left(\epsilon ({\mathit{G}}_{\epsilon }-{\mathit{K}}_{\epsilon })+{\epsilon }^2{\mathit{H}}_{\epsilon }\right)\mathrm{d}\mathit{x}.}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

For the first integral on the right-hand side, owing to the convexity of $Q_W$ and (58), we have

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\underset{\epsilon \to 0^{+}}{\lim \inf }\underset{{\mathrm{\Omega }}_{\epsilon }}{\int }{Q}_W\left({\mathit{G}}_{\epsilon }-{\mathit{K}}_{\epsilon }+\epsilon {\mathit{H}}_{\epsilon }\right)\mathrm{d}\mathit{x}}& \multicolumn{1}{c}{\ge \underset{\mathrm{\Omega }}{\int }{Q}_W(D\mathit{u}-\mathbf{\Lambda }(\mathit{m}))\mathrm{d}\mathit{x}=\underset{\mathrm{\Omega }}{\int }{Q}_W(E\mathit{u}-\mathbf{\Lambda }(\mathit{m}))\mathrm{d}\mathit{x}}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

thanks to (61)–(63). For the second one, we estimate

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{|\frac{1}{{\epsilon }^2}\underset{{\mathrm{\Omega }}_{\epsilon }}{\int }{\eta }_W}& \multicolumn{1}{c}{(\epsilon ({\mathit{G}}_{\epsilon }-{\mathit{K}}_{\epsilon })+{\epsilon }^2{\mathit{H}}_{\epsilon })\mathrm{d}\mathit{x}|}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le \underset{{\mathrm{\Omega }}_{\epsilon }}{\int }{\zeta }_W\left(\epsilon \left(|{\mathit{G}}_{\epsilon }|+|{\mathit{K}}_{\epsilon }|\right)+{\epsilon }^2|{\mathit{H}}_{\epsilon }|\right)|{\mathit{G}}_{\epsilon }-{\mathit{K}}_{\epsilon }+\epsilon {\mathit{H}}_{\epsilon }{\mathrm{|}}^2\mathrm{d}\mathit{x}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le 3{\zeta }_W\left({\epsilon }^{1-\alpha }+\mathrm{\epsilon \ell }+\epsilon \parallel {\mathit{H}}_{\epsilon }{\parallel }_{L^{\infty }({\mathrm{\Omega }}_{\epsilon };{\mathrm{ℝ}}^{3\times 3})}\right)\underset{\mathrm{\Omega }}{\int }\left(|{\mathit{G}}_{\epsilon }{\mathrm{|}}^2+|{\mathit{K}}_{\epsilon }{\mathrm{|}}^2+|\epsilon {\mathit{H}}_{\epsilon }{\mathrm{|}}^2\right)\mathrm{d}\mathit{x}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le C{\zeta }_W\left({\epsilon }^{1-\alpha }+\mathrm{\epsilon \ell }+\epsilon \parallel {\mathit{H}}_{\epsilon }{\parallel }_{L^{\infty }({\mathrm{\Omega }}_{\epsilon };{\mathrm{ℝ}}^{3\times 3})}\right),}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

thanks to the boundedness of ${({\mathit{G}}_{\epsilon })}_{\epsilon >0}$ in $L^2(\mathrm{\Omega };{\mathrm{ℝ}}^{3\times 3})$ coming from (19), and to (33) and (63). Hence, recalling (43), we obtain

$$\underset{\epsilon \to 0^{+}}{\lim \inf }{\mathcal{E}}_{\epsilon }^e({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })\ge {\mathcal{E}}^e(\mathit{u},\mathit{m}).$$

(64)

Step 2 (Lower bound for the magnetic energy). Without loss of generality, we assume (17) and we select a not relabeled subsequence for which the inferior limit of ${({\mathcal{E}}_{\epsilon }^m({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon }))}_{\epsilon >0}$ is attained as a limit.

By arguing as in Step 1 of the proof of Theorem 2.1, we conclude that the sequence ${(\det D{\mathit{y}}_{\epsilon })}_{\epsilon >0}$ is equi-integrable. Additionally, we recover (50) with $\gamma >0$ defined as in (51) for some $0<\alpha <1$ and, in turn,

$$\frac{1}{{\epsilon }^{\beta }}\underset{{\mathrm{im}}_T({\mathit{y}}_{\epsilon },\mathrm{\Omega })}{\int }\mathrm{\Phi }({\mathit{m}}_{\epsilon })\mathrm{d}\mathit{\xi }+{\epsilon }^{\beta }\underset{{\mathrm{im}}_T({\mathit{y}}_{\epsilon },\mathrm{\Omega })}{\int }|D{\mathit{m}}_{\epsilon }{\mathrm{|}}^2\mathrm{d}\mathit{\xi }\le {\mathcal{E}}_{\epsilon }^m({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })+C{\epsilon }^{\gamma }.$$

(65)

Let $U\subset \subset \mathrm{\Omega }$ be arbitrary. Given (18), by [11, Lemma 2.24 and Lemma 3.6(a)], we deduce that $U={\mathrm{im}}_T(\mathit{id},U)\subset \subset {\mathrm{im}}_T({\mathit{y}}_{\epsilon },\mathrm{\Omega })$ for a not relabeled subsequence. Hence, (20) yields (56). Recalling (31), again by [30, Lemma 4.2] we have that $w_{i,\epsilon }:=f_i\circ {\mathit{m}}_{\epsilon }\in W^{1,2}(U)$ satisfies (54) for all $i=1,\dots ,M$ . Using (65) and Young’s inequality, we obtain

$$\underset{E}{\int }|D{w}_{i,\epsilon }|\mathrm{d}\mathit{x}\le {\mathcal{E}}_{\epsilon }^m({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })+C{\epsilon }^{\gamma }$$

for any measurable subset $E\subset U$ , and, in turn,

$$\left(\underset{i=1}{\overset{M}{\vee }}\left(|D{w}_{i,\epsilon }|\mathrm{d}\mathit{x}\right)\right)(U)=\underset{U}{\int }\max \{|D{w}_{i,\epsilon }|:i=1,\mathrm{\dots },M\}\mathrm{d}\mathit{x}\le {\mathcal{E}}_{\epsilon }^m({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })+C{\epsilon }^{\gamma },$$

where the equality is justified by (28). From (56), we obtain (55) for $w_i:=f_i\circ \mathit{m}$ , where we employ (31). Therefore, taking the inferior limit, as $\epsilon \to 0^{+}$ , in the previous inequality by means of Lemma 3.6, then the supremum among all domains $U\subset \subset \mathrm{\Omega }$ , and applying [30, Proposition 2.2], we obtain

$$\underset{\epsilon \to 0^{+}}{\lim \inf }{\mathcal{E}}_{\epsilon }^m({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })\ge \left(\underset{i=1}{\overset{M}{\vee }}|D{w}_i|\right)(\mathrm{\Omega })={\mathcal{E}}^m(\mathit{m}).$$

(66)

Eventually, the combination of (64) and (66) proves (22). □

Proof of claim (ii) of Theorem 2.2. We subdivide the proof into three steps.

Step 1 (Reduction to Lipschitz displacements). We argue that it is sufficient to prove the claim for Lipschitz displacements. Given our assumptions on Δ and d , in view of [31, Proposition A.2], there exists a sequence ${({\mathit{u}}_n)}_{n\in \mathrm{\mathbb{N}}}$ in $W^{1,\infty }(\mathrm{\Omega };{\mathrm{ℝ}}^3)$ such that

$${\mathit{u}}_n\to \mathit{u}\mathrm{in}{W}^{1,2}(\mathrm{\Omega };{\mathrm{ℝ}}^3),$$

so that, by dominated convergence,

$$\underset{n\to \infty }{\lim }\mathcal{E}({\mathit{u}}_n,\mathit{m})=\mathcal{E}(\mathit{u},\mathit{m}).$$

Hence, if we show that there exists a recovery sequence for each $({\mathit{u}}_n,\mathit{m})$ , then the claim for u can be established by means of a standard diagonal argument (see, e.g., [24, Remark 1.29]). Henceforth, to avoid cumbersome notation, we will simply assume that $\mathit{u}\in W^{1,\infty }(\mathrm{\Omega };{\mathrm{ℝ}}^3)$ and we will always consider its Lipschitz representative.

Step 2 (Construction of the recovery sequence). Setting $L:=\parallel D\mathit{u}{\parallel }_{L^{\infty }(\mathrm{\Omega };{\mathrm{ℝ}}^{3\times 3})}\vee 1$ , we consider $0<\epsilon \le {(2L)}^{-1}$ . We define the Lipschitz map ${\mathit{y}}_{\epsilon }:=\mathit{id}+\epsilon \mathit{u}$ and we check that it is injective. Let ${\mathit{x}}_1,{\mathit{x}}_2\in \overline{\mathrm{\Omega }}$ with ${\mathit{x}}_1\ne {\mathit{x}}_2$ and, by contradiction, suppose that ${\mathit{y}}_{\epsilon }({\mathit{x}}_1)={\mathit{y}}_{\epsilon }({\mathit{x}}_2)$ . Then

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{|{\mathit{x}}_1-{\mathit{x}}_2|}& \multicolumn{1}{c}{=|{\mathit{y}}_{\epsilon }({\mathit{x}}_1)-{\mathit{y}}_{\epsilon }({\mathit{x}}_2)-({\mathit{x}}_1-{\mathit{x}}_2)|=\epsilon |\mathit{u}({\mathit{x}}_1)-\mathit{u}({\mathit{x}}_2)|}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le \epsilon L|{\mathit{x}}_1-{\mathit{x}}_2|\le \frac{1}{2}|{\mathit{x}}_1-{\mathit{x}}_2|}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

which provides a contradiction. A similar argument shows that the inverse deformation ${\mathit{y}}_{\epsilon }^{-1}:{\mathit{y}}_{\epsilon }(\overline{\mathrm{\Omega }})\to {\mathrm{ℝ}}^3$ is also Lipschitz: given ${\mathit{x}}_1,{\mathit{x}}_2\in \overline{\mathrm{\Omega }}$ , we have

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{|{\mathit{x}}_1-{\mathit{x}}_2|}& \multicolumn{1}{c}{\le |{\mathit{y}}_{\epsilon }({\mathit{x}}_1)-{\mathit{y}}_{\epsilon }({\mathit{x}}_2)|+\epsilon |\mathit{u}({\mathit{x}}_1)-\mathit{u}({\mathit{x}}_2)|}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le |{\mathit{y}}_{\epsilon }({\mathit{x}}_1)-{\mathit{y}}_{\epsilon }({\mathit{x}}_2)|+\mathrm{\epsilon L}|{\mathit{x}}_1-{\mathit{x}}_2|}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

so that

$$|{\mathit{x}}_1-{\mathit{x}}_2|\le \frac{1}{1-\mathrm{\epsilon L}}|{\mathit{y}}_{\epsilon }({\mathit{x}}_1)-{\mathit{y}}_{\epsilon }({\mathit{x}}_2)|\le 2|{\mathit{y}}_{\epsilon }({\mathit{x}}_1)-{\mathit{y}}_{\epsilon }({\mathit{x}}_2)|$$

since $\epsilon \le {(2L)}^{-1}$ . This proves the claim and shows that

$$\parallel D{\mathit{y}}_{\epsilon }^{-1}{\parallel }_{L^{\infty }({\mathit{y}}_{\epsilon }(\mathrm{\Omega });{\mathrm{ℝ}}^{3\times 3})}\le 2\mathrm{for}\mathrm{all}\epsilon \le {(2L)}^{-1}.$$

We observe that $\mathit{y}\in {\mathcal{Y}}_{p,q}(\mathrm{\Omega })$ , ${\mathrm{im}}_T({\mathit{y}}_{\epsilon },\mathrm{\Omega })={\mathit{y}}_{\epsilon }(\mathrm{\Omega })$ , and ${ℒ}^3({\mathrm{im}}_G(\mathit{y},\mathrm{\Omega }))={ℒ}^3({\mathit{y}}_{\epsilon }(\mathrm{\Omega }))$ . Also, (18) and (19) trivially hold.

Let $\mathit{m}\in \mathrm{BV}(\mathrm{\Omega };\{{\mathit{b}}_1,\dots ,{\mathit{b}}_M\})$ . From [21, Theorem 2.6], we infer the existence of a sequence ${({\mathit{\mu }}_{\epsilon })}_{\epsilon >0}$ in $W^{1,2}(\mathrm{\Omega };{\mathbb{S}}^2)$ satisfying

$${\mathit{\mu }}_{\epsilon }\to \mathit{m}\mathrm{in}{L}^1(\mathrm{\Omega };{\mathrm{ℝ}}^3),$$

(67)

and

$$\underset{\epsilon \to 0^{+}}{\lim }\left\{\frac{1}{{\epsilon }^{\beta }}\underset{\mathrm{\Omega }}{\int }\mathrm{\Phi }({\mathit{\mu }}_{\epsilon })\mathrm{d}\mathit{x}+{\epsilon }^{\beta }\underset{\mathrm{\Omega }}{\int }|D{\mathit{\mu }}_{\epsilon }{\mathrm{|}}^2\mathrm{d}\mathit{x}\right\}={\mathcal{E}}^m(\mathit{m}).$$

(68)

We set ${\mathit{m}}_{\epsilon }:={\mathit{\mu }}_{\epsilon }\circ {\mathit{y}}_{\epsilon }^{-1}\in W^{1,2}({\mathit{y}}_{\epsilon }(\mathrm{\Omega });{\mathbb{S}}^2)$ . To prove (20), fix $r>1$ . By the area formula,

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\underset{\epsilon \to 0^{+}}{\lim }\underset{{\mathrm{ℝ}}^3}{\int }|{\chi }_{{\mathit{y}}_{\epsilon }(\mathrm{\Omega })}{\mathit{m}}_{\epsilon }{\mathrm{|}}^r\mathrm{d}\mathit{\xi }}& \multicolumn{1}{c}{=\underset{\epsilon \to 0^{+}}{\lim }{ℒ}^3({\mathit{y}}_{\epsilon }(\mathrm{\Omega }))=\underset{\epsilon \to 0^{+}}{\lim }\underset{\mathrm{\Omega }}{\int }\det D{\mathit{y}}_{\epsilon }\mathrm{d}\mathit{x}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{={ℒ}^3(\mathrm{\Omega })=\underset{{\mathrm{ℝ}}^3}{\int }|{\chi }_{\mathrm{\Omega }}\mathit{m}{\mathrm{|}}^r\mathrm{d}\mathit{x},}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

while, for any $\mathit{\psi }\in C_c^0({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)$ , we have

$$\underset{\epsilon \to 0^{+}}{\lim }\underset{{\mathit{y}}_{\epsilon }(\mathrm{\Omega })}{\int }{\mathit{m}}_{\epsilon }\cdot \mathit{\psi }\mathrm{d}\mathit{x}=\underset{\epsilon \to 0^{+}}{\lim }\underset{\mathrm{\Omega }}{\int }{\mathit{\mu }}_{\epsilon }\cdot \mathit{\psi }\circ {\mathit{y}}_{\epsilon }\det D{\mathit{y}}_{\epsilon }\mathrm{d}\mathit{x}=\underset{\mathrm{\Omega }}{\int }\mathit{m}\circ \mathit{\psi }\mathrm{d}\mathit{x},$$

thanks to the change-of-variable formula and the dominated convergence theorem with (18) and (67). Hence, ${\chi }_{{\mathit{y}}_{\epsilon }(\mathrm{\Omega })}{\mathit{m}}_{\epsilon }\to {\chi }_{\mathrm{\Omega }}\mathit{m}$ in $L^r({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)$ . As ${\mathit{y}}_{\epsilon }\to \mathit{id}$ uniformly in Ω, the set ${\mathit{y}}_{\epsilon }(\mathrm{\Omega })$ is contained in any compact neighborhood of Ω, at least for $\epsilon \ll 1$ , so that the previous convergence implies (20).

Step 3 (Attainment of the lower bound). First, we prove that

$$\underset{\epsilon \to 0^{+}}{\lim }{\mathcal{E}}_{\epsilon }^e({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })={\mathcal{E}}^e(\mathit{u},\mathit{m}).$$

(69)

For convenience, noting that ${\mathit{m}}_{\epsilon }\circ {\mathit{y}}_{\epsilon }={\mathit{\mu }}_{\epsilon }$ , we set ${\mathit{F}}_{\epsilon }:=D{\mathit{y}}_{\epsilon }$ , ${\mathit{L}}_{\epsilon }:={\mathbf{\Lambda }}_{\epsilon }({\mathit{\mu }}_{\epsilon })$ , and ${\mathit{K}}_{\epsilon }:=\mathbf{\Lambda }({\mathit{\mu }}_{\epsilon })$ . Clearly, (67) yields

$${\mathit{K}}_{\epsilon }\to \mathbf{\Lambda }(\mathit{m})\mathrm{in}{L}^2(\mathrm{\Omega };{\mathrm{ℝ}}^{3\times 3})$$

(70)

by the dominated convergence theorem. Similar to Step 1 in the proof of Theorem 2.2(i), we define

$${\mathit{J}}_{\epsilon }:=\frac{a^2}{a_{\epsilon }}{\mathit{\mu }}_{\epsilon }\otimes {\mathit{\mu }}_{\epsilon }+\frac{b^2}{b_{\epsilon }}\left(\mathit{I}-{\mathit{\mu }}_{\epsilon }\otimes {\mathit{\mu }}_{\epsilon }\right)$$

which still fulfills (60). Also, we let ${\mathit{H}}_{\epsilon }:={\mathit{J}}_{\epsilon }+(\epsilon {\mathit{J}}_{\epsilon }-{\mathit{K}}_{\epsilon })D\mathit{u}$ and, similar to (63), we check that

$$\parallel {\mathit{H}}_{\epsilon }{\parallel }_{L^{\infty }(\mathrm{\Omega };{\mathrm{ℝ}}^{3\times 3})}\le \tilde{\ell }+(\epsilon \tilde{\ell }+\ell )L\le \tilde{\ell }+(\tilde{\ell }+\ell )L=:\widehat{\ell }.$$

(71)

By considering the Taylor expansion (57), we write

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{{\mathcal{E}}_{\epsilon }^e({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })}& \multicolumn{1}{c}{=\frac{1}{{\epsilon }^2}\underset{\mathrm{\Omega }}{\int }W({\mathit{L}}_{\epsilon }^{-1}{\mathit{F}}_{\epsilon })\mathrm{d}\mathit{x}=\frac{1}{{\epsilon }^2}\underset{\mathrm{\Omega }}{\int }W(\mathit{I}+\epsilon (D\mathit{u}-{\mathit{K}}_{\epsilon })+{\epsilon }^2{\mathit{H}}_{\epsilon })\mathrm{d}\mathit{x}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{=\frac{1}{2}\underset{\mathrm{\Omega }}{\int }{Q}_W(D\mathit{u}-{\mathit{K}}_{\epsilon }+\epsilon {\mathit{H}}_{\epsilon })\mathrm{d}\mathit{x}+\frac{1}{{\epsilon }^2}\underset{\mathrm{\Omega }}{\int }{\eta }_W(\epsilon (D\mathit{u}-{\mathit{K}}_{\epsilon })+{\epsilon }^2{\mathit{H}}_{\epsilon })\mathrm{d}\mathit{x}.}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

Using (58), (70) and (71), and the dominated convergence theorem, we compute

$$\underset{\epsilon \to 0^{+}}{\lim }\frac{1}{2}\underset{\mathrm{\Omega }}{\int }{Q}_W(D\mathit{u}-{\mathit{K}}_{\epsilon }+\epsilon {\mathit{H}}_{\epsilon })\mathrm{d}\mathit{x}=\frac{1}{2}\underset{\mathrm{\Omega }}{\int }{Q}_W(D\mathit{u}-\mathbf{\Lambda }(\mathit{m}))\mathrm{d}\mathit{x}={\mathcal{E}}^e(\mathit{u},\mathit{m}).$$

As

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\frac{1}{{\epsilon }^2}\underset{\mathrm{\Omega }}{\int }{\eta }_W(\epsilon (D\mathit{u}-{\mathit{K}}_{\epsilon })+{\epsilon }^2{\mathit{H}}_{\epsilon })\mathrm{d}\mathit{x}\mathrm{d}\mathit{x}}& \multicolumn{1}{c}{\le {\zeta }_W(\epsilon (L+\ell )+{\epsilon }^2\widehat{\ell })\underset{\mathrm{\Omega }}{\int }|D\mathit{u}-{\mathit{K}}_{\epsilon }+\epsilon {\mathit{H}}_{\epsilon }{\mathrm{|}}^2\mathrm{d}\mathit{x}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le 3{\zeta }_W(\epsilon (L+\ell )+{\epsilon }^2\widehat{\ell })\underset{\mathrm{\Omega }}{\int }\left(|D\mathit{u}{\mathrm{|}}^2+|{\mathit{K}}_{\epsilon }{\mathrm{|}}^2+{\epsilon }^2|{\mathit{H}}_{\epsilon }{\mathrm{|}}^2\right)\mathrm{d}\mathit{x}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le 3\left(L^2+{\ell }^2+{\epsilon }^2{\widehat{\ell }}^2\right){ℒ}^3(\mathrm{\Omega }){\zeta }_W(\epsilon (L+\ell )+{\epsilon }^2\widehat{\ell })}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

in view of (33) and (71), then (69) follows thanks to (59).

Next, we prove that

$$\underset{\epsilon \to 0^{+}}{\lim }{\mathcal{E}}_{\epsilon }^m({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })={\mathcal{E}}^m(\mathit{m}).$$

(72)

Having (68) in mind, it is sufficient to check that

$$\underset{\epsilon \to 0^{+}}{\lim }\frac{1}{{\epsilon }^{\beta }}\left|\underset{{\mathit{y}}_{\epsilon }(\mathrm{\Omega })}{\int }\mathrm{\Phi }\left((D{\mathit{y}}_{\epsilon }^{\mathrm{\top }}\circ {\mathit{y}}_{\epsilon }^{-1}){\mathit{m}}_{\epsilon }\right)\mathrm{d}\mathit{\xi }-\underset{\mathrm{\Omega }}{\int }\mathrm{\Phi }({\mathit{\mu }}_{\epsilon })\mathrm{d}\mathit{x}\right|=0$$

(73)

and

$$\underset{\epsilon \to 0^{+}}{\lim }{\epsilon }^{\beta }\left|\underset{{\mathit{y}}_{\epsilon }(\mathrm{\Omega })}{\int }|D{\mathit{m}}_{\epsilon }{\mathrm{|}}^2\mathrm{d}\mathit{\xi }-\underset{\mathrm{\Omega }}{\int }|D{\mathit{\mu }}_{\epsilon }{\mathrm{|}}^2\mathrm{d}\mathit{x}\right|=0.$$

(74)

For (73), recalling (45)–(46), we observe that $\det {\mathit{F}}_{\epsilon }=1+\epsilon g_{\epsilon }$ for a sequence ${(g_{\epsilon })}_{\epsilon >0}$ bounded in $L^{\infty }(\mathrm{\Omega })$ . By applying the change-of-variable formula and considering the Taylor expansion of Φ in (29) around ${\mathit{\mu }}_{\epsilon }$ , we write

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\underset{{\mathit{y}}_{\epsilon }(\mathrm{\Omega })}{\int }}& \multicolumn{1}{c}{\mathrm{\Phi }\left((D{\mathit{y}}_{\epsilon }^{\mathrm{\top }}\circ {\mathit{y}}_{\epsilon }^{-1}){\mathit{m}}_{\epsilon }\right)\mathrm{d}\mathit{\xi }-\underset{\mathrm{\Omega }}{\int }\mathrm{\Phi }({\mathit{\mu }}_{\epsilon })\mathrm{d}\mathit{x}=\underset{\mathrm{\Omega }}{\int }\left\{\mathrm{\Phi }\left({\mathit{F}}_{\epsilon }^{\mathrm{\top }}{\mathit{\mu }}_{\epsilon }\right)\det {\mathit{F}}_{\epsilon }-\mathrm{\Phi }({\mathit{\mu }}_{\epsilon })\right\}\mathrm{d}\mathit{x}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{=\epsilon \underset{\mathrm{\Omega }}{\int }D\mathrm{\Phi }({\mathit{\mu }}_{\epsilon })\cdot \left({(D\mathit{u})}^{\mathrm{\top }}{\mathit{\mu }}_{\epsilon }\right)\mathrm{d}\mathit{x}+\underset{\mathrm{\Omega }}{\int }{\eta }_{\mathrm{\Phi }}\left(\epsilon {(D\mathit{u})}^{\mathrm{\top }}{\mathit{\mu }}_{\epsilon }\right)\mathrm{d}\mathit{x}+\epsilon \underset{\mathrm{\Omega }}{\int }{g}_{\epsilon }\mathrm{\Phi }\left({\mathit{F}}_{\epsilon }^{\mathrm{\top }}{\mathit{\mu }}_{\epsilon }\right)\mathrm{d}\mathit{x}.}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

Thus, given a neighborhood V of ${\mathbb{S}}^2$ , for $\epsilon \ll 1$ we find

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\frac{1}{{\epsilon }^{\beta }}|\underset{{\mathit{y}}_{\epsilon }(\mathrm{\Omega })}{\int }\mathrm{\Phi }}& \multicolumn{1}{c}{((D{\mathit{y}}_{\epsilon }^{\mathrm{\top }}\circ {\mathit{y}}_{\epsilon }^{-1}){\mathit{m}}_{\epsilon })\mathrm{d}\mathit{\xi }-\underset{\mathrm{\Omega }}{\int }\mathrm{\Phi }({\mathit{\mu }}_{\epsilon })\mathrm{d}\mathit{x}|}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le \left\{{\epsilon }^{1-\beta }\left(L\parallel D\mathrm{\Phi }{\parallel }_{L^{\infty }({\mathbb{S}}^2;{\mathrm{ℝ}}^3)}+\parallel g_{\epsilon }{\parallel }_{L^{\infty }(\mathrm{\Omega })}\parallel \mathrm{\Phi }{\parallel }_{L^{\infty }(V)}\right)+{\epsilon }^{1-\beta }L{\zeta }_{\mathrm{\Phi }}\left(\mathrm{\epsilon L}\right)\right\}{ℒ}^3(\mathrm{\Omega }),}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

so that (73) follows from (6) and (30).

For (74), we observe that ${\mathit{F}}_{\epsilon }^{-1}=\mathit{I}+\epsilon {\mathit{E}}_{\epsilon }$ for a sequence ${({\mathit{E}}_{\epsilon })}_{\epsilon >0}$ bounded in $L^{\infty }(\mathrm{\Omega };{\mathrm{ℝ}}^{3\times 3})$ . Indeed, using Neumann series,

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{{\mathit{F}}_{\epsilon }^{-1}}& \multicolumn{1}{c}{=\mathit{I}+\sum _{k=1}^{\infty }{(-\epsilon )}^k{(D\mathit{u})}^k=\mathit{I}+\epsilon \sum _{k=1}^{\infty }{(-1)}^k{\epsilon }^{k-1}{(D\mathit{u})}^k=\mathit{I}+\epsilon \sum _{l=0}^{\infty }{(-1)}^{l+1}{\epsilon }^l{(D\mathit{u})}^{l+1},}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

where, for $\epsilon \le {(2L)}^{-1}$ ,

$$\left|\sum _{l=0}^{\infty }{(-1)}^{l+1}{\epsilon }^l{(D\mathit{u})}^{l+1}\right|\le L\sum _{l=0}^{\infty }|\mathrm{\epsilon L}{\mathrm{|}}^l\le L\sum _{l=0}^{\infty }\frac{1}{2^l}=2L.$$

Additionally, $D{\mathit{m}}_{\epsilon }=(D{\mathit{\mu }}_{\epsilon })\circ {\mathit{y}}_{\epsilon }^{-1}{(D{\mathit{y}}_{\epsilon })}^{-1}\circ {\mathit{y}}_{\epsilon }^{-1}$ almost everywhere in ${\mathit{y}}_{\epsilon }(\mathrm{\Omega })$ thanks to the chain rule, and (68) entails

$$\underset{\epsilon >0}{\sup }{\epsilon }^{\beta }\underset{\mathrm{\Omega }}{\int }|{\mathbf{M}}_{\epsilon }{\mathrm{|}}^2\mathrm{d}\mathit{x}\le C_0,$$

(75)

where we set ${\mathit{M}}_{\epsilon }:=D{\mathit{\mu }}_{\epsilon }$ . By applying the change-of-variable formula, we write

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\underset{{\mathit{y}}_{\epsilon }(\mathrm{\Omega })}{\int }|D{\mathit{m}}_{\epsilon }{\mathrm{|}}^2\mathrm{d}\mathit{\xi }-\underset{\mathrm{\Omega }}{\int }|D{\mathit{\mu }}_{\epsilon }{\mathrm{|}}^2\mathrm{d}\mathit{x}}& \multicolumn{1}{c}{=\underset{\mathrm{\Omega }}{\int }\left\{|{\mathit{M}}_{\epsilon }{\mathit{F}}_{\epsilon }^{-1}{\mathrm{|}}^2\det {\mathit{F}}_{\epsilon }-|{\mathit{M}}_{\epsilon }{\mathrm{|}}^2\right\}\mathrm{d}\mathit{x}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{=\epsilon \underset{\mathrm{\Omega }}{\int }\left(g_{\epsilon }|{\mathbf{M}}_{\epsilon }{\mathrm{|}}^2+2\left({\mathbf{M}}_{\epsilon }:({\mathbf{M}}_{\epsilon }{\mathit{E}}_{\epsilon })\right)(1+\epsilon g_{\epsilon })\right)\mathrm{d}\mathit{x}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{+{\epsilon }^2\underset{\mathrm{\Omega }}{\int }|{\mathbf{M}}_{\epsilon }{\mathit{E}}_{\epsilon }{\mathrm{|}}^2(1+\epsilon g_{\epsilon })\mathrm{d}\mathit{x}}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

Therefore, (75) and the boundedness of the sequences ${(g_{\epsilon })}_{\epsilon >0}$ and ${({\mathit{E}}_{\epsilon })}_{\epsilon >0}$ in $L^{\infty }(\mathrm{\Omega })$ and $L^{\infty }(\mathrm{\Omega };{\mathrm{ℝ}}^{3\times 3})$ , respectively, yield

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{{\epsilon }^{\beta }\left|\underset{{\mathit{y}}_{\epsilon }(\mathrm{\Omega })}{\int }|D{\mathit{m}}_{\epsilon }{\mathrm{|}}^2\mathrm{d}\mathit{\xi }-\underset{\mathrm{\Omega }}{\int }|D{\mathit{\mu }}_{\epsilon }{\mathrm{|}}^2\mathrm{d}\mathit{x}\right|}& \multicolumn{1}{c}{\le C_0\left(\parallel g_{\epsilon }{\parallel }_{L^{\infty }(\mathrm{\Omega })}+2\parallel {\mathit{E}}_{\epsilon }{\parallel }_{L^{\infty }(\mathrm{\Omega };{\mathrm{ℝ}}^{3\times 3})}\parallel 1+\epsilon g_{\epsilon }{\parallel }_{L^{\infty }(\mathrm{\Omega })}\right)\epsilon }\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{+C_0\parallel 1+\epsilon g_{\epsilon }{\parallel }_{L^{\infty }(\mathrm{\Omega })}\parallel {\mathit{E}}_{\epsilon }{\parallel }_{L^{\infty }(\mathrm{\Omega };{\mathrm{ℝ}}^{3\times 3})}^2{\epsilon }^2,}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

and, in turn, (74) follows.

This concludes the proof of (72) which, together with (69), gives (23). □

4.3. Convergence of almost minimizers

Before presenting the proof of Corollary 2.3, we discuss the well-posedness of the steady Maxwell system (24). More generally, given $s\in (1,\infty )$ and $\mathit{\zeta }\in L^s({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)$ , we consider the system

$$\{\begin{array}{c}\multicolumn{1}{c}{\mathrm{curl}\mathit{h}=\mathbf{0}}\\ \multicolumn{1}{c}{\mathrm{div}\mathit{h}=\mathrm{div}\mathit{\zeta }}\end{array}\mathrm{in}{\mathrm{ℝ}}^3.$$

(76)

Solutions to the previous system will be considered in the distributional sense. Below, we refer to the homogeneous Sobolev space (sometimes named after Beppo Levi or Deny-Lions [46]) defined as

$$V^{1,s}({\mathrm{ℝ}}^3):=\left\{v\in W_{\mathrm{loc}}^{1,s}({\mathrm{ℝ}}^3):\mathrm{Dv}\in L^s({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)\right\}.$$

Proposition 4.1 (Well-posedness of the Maxwell system). Let $\mathit{\zeta }\in L^s({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)$ with $s\in (1,\infty )$ . Then, there exists a unique distributional solution ${\mathit{h}}_{\mathit{\zeta }}\in L^s({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)$ to (76). Moreover, the solution operator $\mathit{\zeta }\mapsto {\mathit{h}}_{\mathit{\zeta }}$ is linear and bounded on $L^s({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)$ .

Proof. In view of the first equation, we can write $\mathit{h}=-\mathrm{Dv}$ for some $v\in V^{1,s}({\mathrm{ℝ}}^3)$ . Hence, the system (76) is equivalent to the single equation

$$\mathrm{div}\left(-\mathrm{Dv}+\mathit{\zeta }\right)=0\mathrm{in}{\mathrm{ℝ}}^3.$$

By [47, Theorem 5.1], the latter admits a unique distributional solution $v_{\mathit{\zeta }}\in V^{1,s}({\mathrm{ℝ}}^3)$ up to additive constants. The same result shows that the map $\mathit{\zeta }\mapsto {\mathit{h}}_{\mathit{\zeta }}:=-D{v}_{\mathit{\zeta }}$ defines a bounded linear operator on $L^s({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)$ . □

For the proof of Corollary 2.3, we need the following result.

Proposition 4.2 (Additional compactness). Under the same assumptions of Theorem 2.1, we additionally have

$$\begin{array}{ccc}\multicolumn{1}{c}{\frac{1}{\det D{\mathit{y}}_{\epsilon }}\to 1}& \multicolumn{1}{c}{\mathrm{in}{L}^1(\mathrm{\Omega }),}& \multicolumn{1}{c}{}\end{array}$$

(77)

$$\begin{array}{ccc}\multicolumn{1}{c}{{\chi }_{{\mathrm{im}}_T({\mathit{y}}_{\epsilon },\mathrm{\Omega })}{\mathit{m}}_{\epsilon }\det D\underset{\epsilon }{\overset{-1}{\mathit{y}}}\to {\chi }_{\mathrm{\Omega }}\mathit{m}}& \multicolumn{1}{c}{\mathrm{in}{L}^2({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3).}& \multicolumn{1}{c}{}\end{array}$$

(78)

Proof. First, we prove (77). For convenience, let again ${\mathit{F}}_{\epsilon }:=D{\mathit{y}}_{\epsilon }$ and ${\mathit{A}}_{\epsilon }:={\mathbf{\Lambda }}_{\epsilon }^{-1}({\mathit{m}}_{\epsilon }\circ {\mathit{y}}_{\epsilon }){\mathit{F}}_{\epsilon }$ . By (W5), we have

$$\underset{\mathrm{\Omega }}{\int }{h}_q(\det {\mathit{A}}_{\epsilon })\mathrm{d}\mathit{x}\le \frac{1}{c_W}\underset{\mathrm{\Omega }}{\int }W({\mathit{A}}_{\epsilon })\mathrm{d}\mathit{x}\le C{\epsilon }^2,$$

so that $h_q(\det {\mathit{A}}_{\epsilon })\to 0$ in $L^1(\mathrm{\Omega })$ . Recalling that $\det {\mathit{F}}_{\epsilon }=c_{\epsilon }\det {\mathit{A}}_{\epsilon }$ by (40), we estimate

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\underset{\mathrm{\Omega }}{\int }{h}_q(\det {\mathit{F}}_{\epsilon })\mathrm{d}\mathit{x}}& \multicolumn{1}{c}{=\underset{\mathrm{\Omega }}{\int }\left\{\frac{c_{\epsilon }^q{(\det {\mathit{A}}_{\epsilon })}^q}{q}+\frac{1}{c_{\epsilon }\det {\mathit{A}}_{\epsilon }}-\frac{q+1}{q}\right\}\mathrm{d}\mathit{x}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le \left(c_{\epsilon }^q\vee c_{\epsilon }^{-1}\right)\underset{\mathrm{\Omega }}{\int }\left\{\frac{{(\det {\mathit{A}}_{\epsilon })}^q}{q}+\frac{1}{\det {\mathit{A}}_{\epsilon }}\right\}\mathrm{d}\mathit{x}-\frac{q+1}{q}{ℒ}^3(\mathrm{\Omega })}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{=\left(c_{\epsilon }^q\vee c_{\epsilon }^{-1}\right)\underset{\mathrm{\Omega }}{\int }{h}_q(\det {\mathit{A}}_{\epsilon })\mathrm{d}\mathit{x}+\left(\left(c_{\epsilon }^q\vee c_{\epsilon }^{-1}\right)-1\right)\frac{q+1}{q}{ℒ}^3(\mathrm{\Omega }),}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

where the right-hand side goes to zero as $c_{\epsilon }\to 1$ . Thus,

$$h_q(\det {\mathit{F}}_{\epsilon })\to 0\mathrm{in}{L}^1(\mathrm{\Omega }).$$

(79)

As in Step 1 of Theorem 2.1, we have (38) and, from the convergence in measure, we obtain

$${\chi }_{\{\det {\mathit{F}}_{\epsilon }>\frac{q}{q+1}\}}\to 1\mathrm{in}{L}^1(\mathrm{\Omega }),{\chi }_{\{\det {\mathit{F}}_{\epsilon }\le \frac{q}{q+1}\}}\to 0\mathrm{in}{L}^1(\mathrm{\Omega }).$$

(80)

Using (38) and dominated convergence, we easily see that

$$\underset{\epsilon \to 0^{+}}{\lim }\underset{\{\det {\mathit{F}}_{\epsilon }>\frac{q}{q+1}\}}{\int }\left|\frac{1}{\det {\mathit{F}}_{\epsilon }}-1\right|\mathrm{d}\mathit{x}=0.$$

Then,

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\underset{\{\det {\mathit{F}}_{\epsilon }\le \frac{q}{q+1}\}}{\int }|\frac{1}{\det {\mathit{F}}_{\epsilon }}-1|\mathrm{d}\mathit{x}}& \multicolumn{1}{c}{=\underset{\{\det {\mathit{F}}_{\epsilon }\le \frac{q}{q+1}\}}{\int }\left(\frac{1}{\det {\mathit{F}}_{\epsilon }}-1\right)\mathrm{d}\mathit{x}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{=\underset{\{\det {\mathit{F}}_{\epsilon }\le \frac{q}{q+1}\}}{\int }\left(\frac{1}{\det {\mathit{F}}_{\epsilon }}-\frac{q+1}{q}\right)\mathrm{d}\mathit{x}+\frac{1}{q}{ℒ}^3(\{\det {\mathit{F}}_{\epsilon }\le \frac{q}{q+1}\})}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le \underset{\mathrm{\Omega }}{\int }{h}_q(\det {\mathit{F}}_{\epsilon })\mathrm{d}\mathit{x}+\frac{1}{q}{ℒ}^3(\{\det {\mathit{F}}_{\epsilon }\le \frac{q}{q+1}\}),}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

where the right-hand side goes to zero in view of (79) and (80). Therefore, (77) follows.

Next, we look at (78). For convenience, we write ${\mathit{\zeta }}_{\epsilon }:={\chi }_{{\mathrm{im}}_T({\mathit{y}}_{\epsilon },\mathrm{\Omega })}{\mathit{m}}_{\epsilon }\det D\underset{\epsilon }{\overset{-1}{\mathit{y}}}$ . By applying the change-of-variable formula, we compute

$$\underset{{\mathrm{ℝ}}^3}{\int }|{\mathit{\zeta }}_{\epsilon }{\mathrm{|}}^2\mathrm{d}\mathit{\xi }=\underset{{\mathrm{im}}_T({\mathit{y}}_{\epsilon },\mathrm{\Omega })}{\int }|\det D\underset{\epsilon }{\overset{-1}{\mathit{y}}}{\mathrm{|}}^2\mathrm{d}\mathit{\xi }=\underset{\mathrm{\Omega }}{\int }\frac{1}{\det D{\mathit{y}}_{\epsilon }}\mathrm{d}\mathit{x}.$$

Given (77), letting $\epsilon \to 0^{+}$ in the previous equation yields

$$\underset{\epsilon \to 0^{+}}{\lim }\parallel {\mathit{\zeta }}_{\epsilon }{\parallel }_{L^2({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)}^2=\underset{\epsilon \to 0^{+}}{\lim }{\parallel \frac{1}{\det D{\mathit{y}}_{\epsilon }}\parallel }_{L^1(\mathrm{\Omega })}={ℒ}^3(\mathrm{\Omega })=\parallel {\chi }_{\mathrm{\Omega }}\mathit{m}{\parallel }_{L^2({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)}^2.$$

(81)

Let $\mathit{\psi }\in C_c^0({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)$ . By employing once again the change-of-variable formula, with the aid of the dominated convergence theorem, using (18) and (20)–(21), we obtain

$$\underset{\epsilon \to 0^{+}}{\lim }\underset{{\mathrm{ℝ}}^3}{\int }{\mathit{\zeta }}_{\epsilon }\cdot \mathit{\psi }\mathrm{d}\mathit{\xi }=\underset{\epsilon \to 0^{+}}{\lim }\underset{\mathrm{\Omega }}{\int }{\mathit{m}}_{\epsilon }\circ {\mathit{y}}_{\epsilon }\cdot \mathit{\psi }\circ {\mathit{y}}_{\epsilon }\mathrm{d}\mathit{x}=\underset{\mathrm{\Omega }}{\int }\mathit{m}\cdot \mathit{\psi }\mathrm{d}\mathit{x}.$$

Thus, ${\mathit{\zeta }}_{\epsilon }\rightharpoonup {\chi }_{\mathrm{\Omega }}\mathit{m}$ in $L^2({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)$ which, together with (81), gives (78). □

With Proposition 4.2 at hand, the result for almost minimizers follows from Theorem 2.1 and Theorem 2.2 in a standard manner.

Proof of Corollary 2.3. We consider a sequence ${(({\mathit{w}}_{\epsilon },{\mathit{n}}_{\epsilon }))}_{\epsilon >0}$ with $({\mathit{w}}_{\epsilon },{\mathit{n}}_{\epsilon })\in {\mathcal{A}}_{\epsilon }$ for all $\epsilon >0$ such that

$$\underset{\epsilon >0}{\sup }\mathcal{E}({\mathit{w}}_{\epsilon },{\mathit{n}}_{\epsilon })<+\infty .$$

By applying Theorem 2.2(ii) for limit $(\mathit{d},\mathit{n})\in \mathcal{A}$ with $\mathit{n}:\mathrm{\Omega }\to {\mathbb{S}}^2$ constantly equal to ${\mathit{b}}_1$ , we see that such sequences exist. From Proposition 4.2 to ${(({\mathit{w}}_{\epsilon },{\mathit{n}}_{\epsilon }))}_{\epsilon >0}$ , we infer that the sequence ${({\mathit{\zeta }}_{\epsilon })}_{\epsilon >0}$ with ${\mathit{\zeta }}_{\epsilon }:={\chi }_{{\mathrm{im}}_T({\mathit{w}}_{\epsilon },\mathrm{\Omega })}{\mathit{n}}_{\epsilon }\det D\underset{\epsilon }{\overset{-1}{\mathit{w}}}$ is bounded in $L^2({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)$ . Thus,

$$\underset{\epsilon >0}{\sup }\mathcal{H}({\mathit{w}}_{\epsilon },{\mathit{n}}_{\epsilon })<+\infty$$

in view of Proposition 4.1. Using Hölder’s inequality, we immediately find

$$\underset{\epsilon >0}{\sup }|\mathcal{F}({\mathit{w}}_{\epsilon },{\mathit{n}}_{\epsilon })|\le \parallel \mathit{f}{\parallel }_{L^1({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)}.$$

Hence,

$$\underset{\epsilon >0}{\sup }{G}_{\epsilon }\le \underset{\epsilon >0}{\sup }{\mathcal{G}}_{\epsilon }({\mathit{w}}_{\epsilon },{\mathit{n}}_{\epsilon })<+\infty ,$$

so that (25) entails

$$\underset{\epsilon >0}{\sup }{\mathcal{G}}_{\epsilon }({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })<+\infty .$$

From this, we easily get

$$\underset{\epsilon >0}{\sup }\mathcal{E}({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })\le \underset{\epsilon >0}{\sup }\left\{{\mathcal{G}}_{\epsilon }({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })+\mathcal{F}({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })\right\}<+\infty .$$

By applying Theorem 2.1 and Proposition 4.2 to the sequence ${(({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon }))}_{\epsilon >0}$ , we identify $(\mathit{u},\mathit{m})\in \mathcal{A}$ for which (18)–(21) and (77)–(78) hold true. Proposition 4.1 ensures that ${\mathit{h}}_{\epsilon }:={\mathit{h}}^{{\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon }}$ and $\mathit{h}:={\mathit{h}}^{\mathit{id},\mathit{m}}$ are well-defined and belong to $L^2({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)$ . In particular, linearity and boundedness of the solution operator of the Maxwell system yield the estimate

$$\parallel {\mathit{h}}_{\epsilon }-\mathit{h}{\parallel }_{L^2({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)}\le C\parallel {\chi }_{{\mathrm{im}}_T(\mathit{y},\mathrm{\Omega })}{\mathit{m}}_{\epsilon }\det D\underset{\epsilon }{\overset{-1}{\mathit{y}}}-{\chi }_{\mathrm{\Omega }}\mathit{m}{\parallel }_{L^2({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)}$$

and, in turn,

$$\underset{\epsilon \to 0^{+}}{\lim }\mathcal{H}({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })=\mathcal{H}(\mathit{id},\mathit{m})$$

thanks to (78). Having in mind (20), we obtain

$$\underset{\epsilon \to 0^{+}}{\lim }\mathcal{F}({\mathit{y}}_{\epsilon },{\mathit{m}}_{\epsilon })=\mathcal{F}(\mathit{id},\mathit{m}).$$

At this point, the minimality of $(\mathit{u},\mathit{m})$ and (26) follow from Theorem 2.2 by standard Γ-convergence arguments (see, e.g., [24, Remark 1.7]). □

4.4. Regularity of the limiting problem

Recalling that for $M=2$ we have (27), our problem is related to the regularity of perimeter minimizers. Specifically, we will refer to the notion of sets of almost minimal boundary [35].

Definition 4.3 (Set of almost minimal boundary). Let $\varkappa \in (0,1)$ . A set $E\in \mathcal{P}({\mathrm{ℝ}}^3)$ has ϰ-almost minimal boundary in Ω if, for every set $A\subset \subset \mathrm{\Omega }$ , there exist $c_A,R_A>0$ such that, for every $\mathit{a}\in A$ , $\rho \in \left(0,R_A\right)$ , and $F\in \mathcal{P}({\mathrm{ℝ}}^3)$ with $E△F\subset \subset B(\mathit{a},\rho )$ , there holds

$$\mathrm{Per}(E;B(\mathit{a},\rho ))\le \mathrm{Per}(F;B(\mathit{a},\rho ))+c_A{\rho }^{2(1+\varkappa )}.$$

In view of [35, Theorem 1], we known that ${\partial }^{\ast }E\cap \mathrm{\Omega }=\mathrm{\partial E}\cap \mathrm{\Omega }$ is a surface of class $C^{1,\varkappa }$ whenever E has ϰ-almost minimal boundary in Ω.

We now present the proof of the regularity result. For convenience, we fix ${\omega }_3:=\frac{4\pi }{3}$ .

Proof of Theorem 2.4. We subdivide the proof into three steps.

Step 1 (Regularity of the displacement). Let $\mathit{v}\in W^{1,2}(\mathrm{\Omega };{\mathrm{ℝ}}^3)$ with $\mathit{v}=\mathit{d}$ on Δ. Testing the minimality of $(\mathit{u},\mathit{m})$ by taking $(\mathit{v},\mathit{m})\in \mathcal{A}$ as a competitor, we realize that u is a minimizer of $\mathit{v}\mapsto \mathcal{G}(\mathit{v},\mathit{m})$ . Thus, $\mathit{u}=\mathit{d}+\widehat{\mathit{u}}$ , where $\widehat{\mathit{u}}\in W^{1,2}(\mathrm{\Omega };{\mathrm{ℝ}}^3)$ is the unique weak solution of the boundary value problem

$$\{\begin{array}{cc}\multicolumn{1}{c}{-\mathrm{div}\left({\mathbf{C}}_W(E\widehat{\mathit{u}})\right)=-\mathrm{div}\left({\mathbf{C}}_W\left(\mathbf{\Lambda }(\mathit{m})-E\mathit{d}\right)\right)}& \multicolumn{1}{c}{\mathrm{in}\mathrm{\Omega },}\\ \multicolumn{1}{c}{\widehat{\mathit{u}}=\mathbf{0}}& \multicolumn{1}{c}{\mathrm{on}\mathrm{\Delta },}\\ \multicolumn{1}{c}{{\mathbf{C}}_W(E\widehat{\mathit{u}}){\mathit{\nu }}_{\mathrm{\Omega }}=-{\mathbf{C}}_W(E\mathit{d}){\mathit{\nu }}_{\mathrm{\Omega }}}& \multicolumn{1}{c}{\mathrm{on}\partial \mathrm{\Omega }\setminus \mathrm{\Delta }.}\end{array}$$

Here, ${\mathbf{C}}_W:{\mathrm{ℝ}}^{3\times 3}\to {\mathrm{ℝ}}^{3\times 3}$ and ${\mathit{\nu }}_{\mathrm{\Omega }}:\partial \mathrm{\Omega }\to {\mathbb{S}}^2$ denote the elasticity tensor of W and the outer unit normal to Ω, respectively. Note that ${\mathbf{C}}_W\left(\mathbf{\Lambda }(\mathit{m})-E\mathit{d}\right)\in L^{\infty }(\mathrm{\Omega };{\mathrm{ℝ}}^{3\times 3})$ . Therefore, standard elliptic regularity theory (see, e.g., [48, Theorem 16.4]) yields $\widehat{\mathit{u}}\in W^{1,r}(\mathrm{\Omega };{\mathrm{ℝ}}^3)$ and, in turn, $\mathit{u}\in W^{1,r}(\mathrm{\Omega };{\mathrm{ℝ}}^3)$ for all $r\in [1,\infty )$ .

Step 2 (Regularity of the jump set). Let $E:=\{\mathit{m}={\mathit{b}}_1\}$ and $\varkappa :=(s_f-3)∕(2s_f)\in (0,1∕2)$ . We claim that E is a set of ϰ-almost minimal boundary in Ω. To see this, we consider $A\subset \subset \mathrm{\Omega }$ and we set $R_A:=\mathrm{dist}(A;\partial \mathrm{\Omega })\wedge 1$ . Let $\mathit{a}\in A$ , $\rho \in (0,R_A)$ , and $F\in \mathcal{P}({\mathrm{ℝ}}^3)$ with $E△F\subset \subset B(\mathit{a},\rho )$ . We need to show that

$$\mathrm{Per}(E;B(\mathit{a},\rho ))\le \mathrm{Per}(F;B(\mathit{a},\rho ))+c_A{\rho }^{2(1+\varkappa )}$$

(82)

for some constant $c_A>0$ to be determined. Define $\mathit{n}\in \mathrm{BV}(\mathrm{\Omega };\{{\mathit{b}}_1,{\mathit{b}}_2\})$ by setting $\mathit{n}:={\mathit{b}}_1$ in F and $\mathit{n}:={\mathit{b}}_2$ in $\mathrm{\Omega }\setminus F$ . Then, $\mathit{n}=\mathit{m}$ in $\mathrm{\Omega }\setminus B(\mathit{a},\rho )$ . By minimality, $\mathcal{G}(\mathit{u},\mathit{m})\le \mathcal{G}(\mathit{u},\mathit{n})$ from which we obtain

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\mathrm{Per}(E;B(\mathit{a},\rho ))}& \multicolumn{1}{c}{\le \mathrm{Per}(F;B(\mathit{a},\rho ))+\left({\mathcal{E}}^e(\mathit{u},\mathit{n})-{\mathcal{E}}^e(\mathit{u},\mathit{m})\right)}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{+\lambda (\mathcal{H}(\mathit{id},\mathit{n})-\mathcal{H}(\mathit{id},\mathit{m}))+(\mathcal{F}(\mathit{id},\mathit{n})-\mathcal{F}(\mathit{id},\mathit{m})).}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

(83)

We estimate the three differences on the right-hand side of (83). We have

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{{\mathcal{E}}^e(\mathit{u},\mathit{n})-{\mathcal{E}}^e(\mathit{u},\mathit{m})}& \multicolumn{1}{c}{=\frac{1}{2}\underset{\mathrm{\Omega }}{\int }\left(Q_W(\mathbf{\Lambda }(\mathit{n}))-Q_W(\mathbf{\Lambda }(\mathit{m}))\right)\mathrm{d}\mathit{x}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{+\underset{\mathrm{\Omega }}{\int }{\mathbf{C}}_W(E\mathit{u}):\left(\mathbf{\Lambda }(\mathit{m})-\mathbf{\Lambda }(\mathit{n})\right)\mathrm{d}\mathit{x}.}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

Recalling (33), the first summand on the right-hand side is easily estimated as

$$\left|\underset{\mathrm{\Omega }}{\int }\left(Q_W(\mathbf{\Lambda }(\mathit{n}))-Q_W(\mathbf{\Lambda }(\mathit{m}))\right)\mathrm{d}\mathit{x}\right|\le 2\max \{Q_W(\mathit{E}):|\mathit{E}|=\ell \}{\omega }_3{\rho }^3=C{\rho }^3,$$

(84)

while for the second one we find

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\left|\underset{\mathrm{\Omega }}{\int }{\mathbf{C}}_W(E\mathit{u}):\left(\mathbf{\Lambda }(\mathit{m})-\mathbf{\Lambda }(\mathit{n})\right)\mathrm{d}\mathit{x}\right|}& \multicolumn{1}{c}{\le 2\ell |{\mathbf{C}}_W|\underset{B(\mathit{a},\rho )}{\int }|E\mathit{u}|\mathrm{d}\mathit{x}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le 2\ell |{\mathbf{C}}_W|\parallel E\mathit{u}{\parallel }_{L^r(\mathrm{\Omega };{\mathrm{ℝ}}^{3\times 3})}{\omega }_3^{1∕r^{\prime }}{\rho }^{3∕r^{\prime }}\le C{\rho }^{3∕r^{\prime }}}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

(85)

for any $r\in [1,\infty )$ thanks to Hölder’s inequality. Writing $\mathit{k}:={\mathit{h}}^{\mathit{id},\mathit{n}}$ and $\mathit{h}:={\mathit{h}}^{\mathit{id},\mathit{m}}$ , the second difference on the right-hand side of (83) is bounded by

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{|\mathcal{H}(\mathit{id},\mathit{n})-\mathcal{H}(\mathit{id},\mathit{m})|}& \multicolumn{1}{c}{=\left|\underset{{\mathrm{ℝ}}^3}{\int }|\mathit{k}{\mathrm{|}}^2\mathrm{d}\mathit{\xi }-\underset{{\mathrm{ℝ}}^3}{\int }|\mathit{h}{\mathrm{|}}^2\mathrm{d}\mathit{\xi }\right|=\left|\underset{{\mathrm{ℝ}}^3}{\int }(\mathit{k}-\mathit{h})\cdot (\mathit{k}+\mathit{h})\mathrm{d}\mathit{\xi }\right|}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le \parallel \mathit{k}-\mathit{h}{\parallel }_{L^s({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)}\left(\parallel \mathit{k}{\parallel }_{L^{s^{\prime }}({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)}+\parallel \mathit{h}{\parallel }_{L^{s^{\prime }}({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)}\right)}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

for any $s\in (1,\infty )$ . Then, from Proposition 4.1, we infer the estimates

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\parallel \mathit{k}-\mathit{h}{\parallel }_{L^s({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)}}& \multicolumn{1}{c}{\le C\parallel {\chi }_{\mathrm{\Omega }}\mathit{m}-{\chi }_{\mathrm{\Omega }}\mathit{m}{\parallel }_{L^s({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)}=C\parallel {\chi }_{B(\mathit{a},\rho )}(\mathit{n}-\mathit{m}){\parallel }_{L^s({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le 2^{1∕s}C{\omega }_3^{1∕s}{\rho }^{3∕s}\le C{\rho }^{3∕s}}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

and

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{\parallel \mathit{k}{\parallel }_{L^{s^{\prime }}({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)}+\parallel \mathit{h}{\parallel }_{L^{s^{\prime }}({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)}}& \multicolumn{1}{c}{\le C\left(\parallel \mathit{n}{\parallel }_{L^{s^{\prime }}(\mathrm{\Omega };{\mathrm{ℝ}}^3)}+\parallel \mathit{m}{\parallel }_{L^{s^{\prime }}(\mathrm{\Omega };{\mathrm{ℝ}}^3)}\right)=2C{ℒ}^3{(\mathrm{\Omega })}^{1∕s^{\prime }}\le C.}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

Thus,

$$|\mathcal{H}(\mathit{id},\mathit{n})-\mathcal{H}(\mathit{id},\mathit{m})|\le C{\rho }^{3∕s}.$$

(86)

Eventually, for the last difference on the right-hand side of (83), we have

$$\begin{array}{cc}\multicolumn{1}{c}{\begin{array}{cc}\multicolumn{1}{c}{|\mathcal{F}(\mathit{id},\mathit{n})-\mathcal{F}(\mathit{id},\mathit{m})|}& \multicolumn{1}{c}{\le \underset{\mathrm{\Omega }}{\int }|\mathit{f}||\mathit{n}-\mathit{m}|\mathrm{d}\mathit{x}\le 2\underset{B(\mathit{a},\rho )}{\int }|\mathit{f}|\mathrm{d}\mathit{x}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\le 2\parallel \mathit{f}{\parallel }_{L^{s_f}({\mathrm{ℝ}}^3;{\mathrm{ℝ}}^3)}{\omega }_3^{1∕s_f^{\prime }}{\rho }^{3∕s_f^{\prime }}\le C{\rho }^{3∕s_f^{\prime }}.}\end{array}}& \multicolumn{1}{c}{}\end{array}$$

(87)

Altogether, (83)–(87) yield the estimate

$$\mathrm{Per}(E;B(\mathit{a},\rho ))\le \mathrm{Per}(F;B(\mathit{a},\rho ))+C\left({\rho }^{3∕r^{\prime }}+{\rho }^{3∕s}+{\rho }^{3∕s_f^{\prime }}\right).$$

In particular, choosing $r=s_f$ and $s=s_f^{\prime }$ , by noting that $3∕s_f^{\prime }=2(1+\varkappa )$ , we obtain the desired estimate (82) with $c_A=C$ . Therefore, E is a set of ϰ-almost minimal boundary in Ω, so that $J_{\mathit{m}}={\partial }^{\ast }E\cap \mathrm{\Omega }$ is a surface of class $C^{1,\varkappa }$ in Ω.

Step 3 (Improved regularity for the displacement). Observe that $J_{\mathit{m}}$ is a closed subset of Ω. Let U be one of the connected components of $\mathrm{\Omega }\setminus J_{\mathit{m}}$ . As m is constant on U, the displacement $\widehat{\mathit{u}}=\mathit{u}-\mathit{d}$ is a weak solution to the equation

$$-\mathrm{div}\left({\mathbf{C}}_W(E\widehat{\mathit{u}})\right)=\mathrm{div}\left({\mathbf{C}}_W(E\mathit{d})\right)\mathrm{in}U.$$

Hence, if $\mathit{d}\in W_{\mathrm{loc}}^{2,s_d}(\mathrm{\Omega };{\mathrm{ℝ}}^3)$ with $s_d>3$ , elliptic regularity theory (see, e.g., [48, Remark 15.16]) yields $\widehat{\mathit{u}}\in W_{\mathrm{loc}}^{2,s_d}(U;{\mathrm{ℝ}}^3)$ and, by Morrey’s embedding, $\mathit{u}=\widehat{\mathit{u}}+\mathit{d}\in C^{1,1-3∕s_d}(U;{\mathrm{ℝ}}^3)$ . As the connected component U is arbitrary, we conclude $\mathit{u}\in C^{1,1-3∕s_d}(\mathrm{\Omega }\setminus J_{\mathit{m}};{\mathrm{ℝ}}^3)$ . □