Homogenization of high-contrast composites under differential constraints

We derive, by means of variational techniques, a limiting description for a class of integral functionals under linear differential constraints. The functionals are designed to encode the energy of a high-contrast composite, that is, a heterogeneous material which, at a microscopic level, consists of a periodically perforated matrix whose cavities are occupied by a filling with very different physical properties. Our main result provides a $\Gamma$-convergence analysis as the periodicity tends to zero, and shows that the variational limit of the functionals at stake is the sum of two contributions, one resulting from the energy stored in the matrix and the other from the energy stored in the inclusions. As a consequence of the underlying high-contrast structure, the study is faced with a lack of coercivity with respect to the standard topologies in $L^p$, which we tackle by means of two-scale convergence techniques. In order to handle the differential constraints, instead, we establish new results about the existence of potentials and of constraint-preserving extension operators for linear, $k$-th order, homogeneous differential operators with constant coefficients and constant rank.


Introduction
The identification of materials capable to switch, enhance, or tune their properties according to external stimuli is one of the key catalysts for the creation of new artificially engineered composites: the metamaterials.In the present paper, we contribute to the mathematical study of metamaterials by providing a unified limiting description for a class of energies that are inspired by the physics of high-contrast composites and are evaluated on fields subject to suitable partial differential constraints.
High-contrast materials are characterized by the property that their microscopic physical features may change abruptly from point to point.In the case of a binary periodic medium of this kind, we have two relevant microscales: the periodicity of the microstructure and the high-contrast parameter, which encodes the strong (high-contrast) difference in the properties of the two components.Typically, the scale at which periodicity is observed is much smaller than the size of the specimen.It is hence natural, from a mathematical viewpoint, to study the asymptotics of the quantities that describe the system in the limit of vanishing period.This procedure is called homogenization.It allows to make predictions on the effective character of heterogeneous media when no experiments are available, and eases numerical simulations by reducing the degrees of freedom of the problem.Therefore, mathematical homogenization has been a lively field of investigation for many decades.We refer e.g. to the monographs [10,34,84] for a thorough introduction to the subject.
In this paper, we tackle the case in which both the periodicity and the high-contrast are quantified by a small number ε > 0 (see (1.3) and Remark 2.1).We assume that the vector fields that encode the significant quantities of the system satisfy a linear first-order differential constraint.The rationale behind this choice becomes transparent if one focuses on specific theories: in large strain elasticity the deformation gradient is a significant strain measure, while electromagnetism deals with the differential operators encoded by the Maxwell system.In other settings, the elastic behavior of some materials can be influenced by electric currents or magnetic fields and vice versa (see e.g.[80] for a recent contribution).In all these cases, the key quantities lie in the kernel of a suitable differential operator.We therefore see that the need of combining homogenization and differential constraints naturally arises and leads to generalized notions of convexity (see [13] for an overview).It is well-known that all the differential constraints above (and many more) can be treated in a unified way in the framework of A -quasiconvexity (see [50] and references therein).Such a notion allows to extend various results which were originally available only for Sobolev maps and their gradients to the setting where admissible fields belong to the kernel of a linear first-order differential operator A of constant rank.
The novelty of our contribution consists in tackling homogenization of high-contrast materials in the setting of A -quasiconvexity.In order to describe our main results, we need to introduce some basic notation.
For d ∈ N, d ≥ 2, let Ω ⊂ R d be a bounded, connected, open set with Lipschitz boundary.We regard Ω as a reference configuration for a composite with a high-contrast microstructure obtained by periodically inserting inclusions of a specific kind in a surrounding matrix.For instance, in the context of elasticity, M. Cherdantsev & K. D. Cherednichenko [25] considered a stiff matrix with embedded soft fillings.In general, applications require that the properties of the inclusions and of the matrix are tuned so that the resulting effective behavior of the composite is the desired one.
To depict the fine texture of the material in mathematical terms, we consider the periodicity cell Q := (0, 1) d ⊂ R d and an open connected subset with Lipschitz boundary D 0 ⋐ Q (see Figure 1), where the symbol ⋐ indicates that D 0 is compactly contained in Q.The high-contrast behavior of the composite sitting in Ω is identified by means of the sets D 0 and D 1 := Q \ D0 in the following way: we introduce a small parameter ε > 0 representing the periodicity scale and we define the sets and which correspond respectively to the inclusions and to the surrounding matrix (see Figure 2).
Hereafter, we will systematically adopt the adjectives 'soft' and 'stiff' to refer respectively to Ω 0,ε and to Ω 1,ε , as well as to their related quantities.This use is informal and just meant to convey ideas, The reference set Ω and its microscopic structure.The collection of the 'soft' inclusions Ω 0,ε is depicted in white, whereas the matrix Ω 1,ε is in grey.ε though being justified by the possible interpretation of our model in the framework of elasticity.The reader should however bear in mind that our treatment is suited for a wider range of applications.
The main goal of this contribution is to characterize the asymptotic behavior as ε → 0 in the sense of Γ-convergence [15,38] of the energy functionals evaluated on fields u ∈ L p (Ω; R N ) satisfying A u = 0, where A is a suitable linear first-order partial differential operator.The precise class of constraints is presented in Subsection 2.2 below and further investigated in Section 7. In the expression above, the family { f 0,ε } represents the energy densities stored in the 'soft' inclusions, while f 1 encodes the behavior of the 'stiff' matrix.The exact assumptions on { f 0,ε } ε>0 and f 1 are collected in Subsection 2.1.We point out that considering a sequence of ε-dependent energy densities for the 'soft' component is not merely a mathematical mannerism: on the contrary, it is a modeling necessity that arises when the admissible maps u are assumed to have the form u = ∇v for suitable fields v, i.e., when A = curl.We refer to [25,Remark 2] for a discussion on this point.When f 1 does not oscillate and the parameter ε is fixed, determining an optimal shape D = { x ∈ Ω : χ(x) = 1 } for the functionals under a volume constraint amounts to solve a shape-optimization problem.In this case, the aim is, for instance, finding the optimal distribution in terms of compliance for two materials with different stiffness in a given container Ω.By enforcing in the energy a dependence on ε like the one we consider, we attempt to encode the fact that the material properties in the set D and in its complement may be very different; specifically, the one sitting in D becomes increasingly soft as ε vanishes.In addition, finer and finer microstructures may be taken into account by assuming as well a dependence on ε of the shapes, so that D is replaced by Ω 0,ε .Problems of this kind are central, e.g., in the modeling of lattice materials for additive manufacturing.We refer to the monograph [3] and the paper [4] for extensive discussions about the connections between homogenization and topology optimization.
Even though { f 0,ε } and f 1 in (1.3) are assumed to satisfy p-growth conditions from above and below, the fact that the 'soft' energies are evaluated on εu rather than u results in a loss of coercivity.As a result, classical weak and strong L p -topologies are not capable to capture the asymptotic behavior of the problem and one needs to resort to an ad hoc notion of 'convergence in the high-contrast sense', which instead keeps track of the finest features of the microstructure, cf.Definition 2.5.Denoting by U 1 the set of maps in L p (Ω; R N ) such that A u = 0 in Ω, our main result consists in showing that the limiting behavior of the energies in (1.3) is encoded by the functional F : U 1 → R defined as F (u) := α 0 + ˆΩ f hom u(x) dx.
In the formula above, α 0 is a constant determined by f 0 (see (2.8)), f hom is a suitable energy density related to f 1 (see (2.9)), and the condition A u = 0 in Ω has to be interpreted in a distributional sense (see Subsection 2.2).
Our first main result is presented in Theorem 2.6, and, loosely speaking, states the following: Theorem.The functional F is the Γ-limit as ε → 0 of the energies in (1.3) restricted to A -free fields with respect to the high-contrast convergence.
Key techniques to establish the theorem are p-equiintegrability arguments, as well as the notions of two-scale convergence [2] and of periodic unfolding [32,33,81,82].The proof relies essentially on the possibility of decoupling the behavior of the material in the 'soft' and 'stiff' contributions.The Γ-convergence of the 'stiff' portion is a corollary of homogenization results in the setting of A -quasiconvex [46].The study of the 'soft' part is instead more challenging, for the presence of the high-contrast microstructure and its interplay with the differential constraint result in the emergence of a second, hidden scale.This makes both the identification of a lower bound and the proof of its optimality need a delicate combination of two-scale results and weak-L p compactness.
The decoupling of the system into regions having different material features hinges on careful compactness and splitting arguments.The latter are in turn based on two essential properties of the differential constraint A : the fact that A -free maps in the periodicity cell have null average and the existence of a suitable extension operator on perforated domains (see Assumptions 1 and 2).In our general setting, these conditions are quite hard to ensure, in contrast to the well-understood case in which the set Ω is contractible, A is the curl operator, and the A -free fields are gradients.In particular, to the authors' knowledge, even for standard operators, it is still an open question whether extension operators preserving the A -free constraint from a perforated domain to the 'filled' set exist.Similar results are available for gradients both in the Sobolev setting [1] and in the weaker one of antiplane fracture [20].We also mention the recent result in [19] for the case of symmetric gradients in linearized fracture (in the space GSBD [39]).
The analysis of the differential constraints satisfying the aforementioned Assumptions 1 and 2 is the focus of Section 7. A special case in which the first requirement holds is that of operators A admitting 'potentials', namely such that A u = 0 if and only if u = Bw for a suitable differential operator B and a field w.The second main contribution of this paper consists in showing that for a broad class of operators A existence of potentials and existence of extension operators are closely related.Roughly speaking, we prove the following.

Theorem.
Let Ω be a 'nice' set and let A be a linear homogeneous differential operator with constant coefficients and constant rank.If an extension operator from A -free maps on Ω to A -free maps on the whole space exists, then A admits a potential on Ω.Conversely, if A admits a potential on Ω for which a suitable Korn-type inequality holds true, then there exists an extension operator from Ω to the whole space which preserves the property of being A -free.
We refer to Theorems 2.10 and 2.11 for the precise formulation of the result, as well as to Section 7 for a broader discussion on the topic.We point out that assuming the existence of potentials imposes some topological constraints on the set Ω.An example is easily provided by the case A = curl for d = 3: one needs to require at least Ω to be simply connected, cf.Remark 7.2 and Examples 7.17-7.20.Essential tools for the proof of Theorems 2.10 and 2.11 are the theory of Fourier multipliers and the notion of generalized Moore-Penrose inverse.
Before proceeding with the mathematical details of our analysis, we conclude the section with some bibliographical notes.
The very first mathematical analysis of high-contrast materials was developed in the seminal work [8] by G. L. Auriault, who employed formal asymptotic expansions.In the early 2000, V. V. Zhikov [83] introduced a novel approach by extending classical two-scale techniques [2] to the setting of PDEs with rapidly oscillating coefficients.The study of high-contrast problems has been ever since at the center of an extensive scientific effort, with applications ranging from elastodynamics [77] to Maxwell's equations [14,27].A full characterization of the settings of linear elasticity and of conducting materials was provided in [22], while the nonlinear elastic setting was considered in [25,26] (see also [17]).The effects of microstructure on free-discontinuity functionals have been studied when high-contrast behavior appears in bulk terms only [11,12], when it is just in surface terms [75,76], and when both contributions are affected [18].Mumford-Shah energies with high-contrast surface contributions have been recently characterized in [68].The analysis of variational models for layered high-contrast materials was undertaken in [30,31,40].
As for the notion of A -quasiconvexity, its introduction goes back to [36], while an extensive study was carried out in [50] for operators A with constant coefficients and constant rank (see (2.2) and (2.3) below).In [16] and [46], under p-growth assumptions on the energy density, relaxation and homogenization results were obtained for integral functionals evaluated on A -free fields (i.e.fields in the kernel of A ); see also [45] for a related analysis on quasicrystals.Problems featuring simultaneously homogenization and dimension reduction were addressed in [61,62], whereas oscillations and concentrations generated by A -free fields were characterized in [47].The case of non-positive energy densities has been studied in [60].A complete theory under linear growth assumptions on the energy density was established in [5,7,9,64].The characterization of A -quasiconvexity was extended to operators with variable coefficients in [74].We refer to [41,42] for homogenization results in this purview, and to [43] for a corresponding relaxation formula.Applications to the theory of compressible Euler systems, as well as to adaptive image processing and to data-driven finite elasticity were the subject of [28], [44], and [29], respectively.To complete our review on A -quasiconvexity, we mention the works [70,73] on BV A -spaces for elliptic and cancelling operators, [63] for a characterization of associated Young measures, as well as [51,52,71,72] for the corresponding Sobolev analysis, and [55] for a compensated-compactness result.
1.1.Structure of the paper.The plan of the paper is as follows.In Section 2, we describe the setting of the problem and the related assumptions, and we formulate the precise statements of our main results, i.e., Theorems 2.6, 2.10, and 2.11.To lay the ground for the proofs, we collect auxiliary assertions and properties concerning measurable selection criteria, A -free sequences, twoscale convergence, and Fourier analysis in Section 3. In Section 4 we study compactness properties of sequences of fields with equibounded energies, we detail the splitting argument, and present the proof of Theorem 2.6.Sections 5 and 6 contain the Γ-convergence analysis for the 'soft' and 'stiff' energy-contributions. Section 7 is focused on the description of the class of the differential operators that are admissible for our analysis, and deals with the proofs of Theorems 2.10 and 2.11.

Setting and main results
This section is devoted to the set-up of our analysis and to the presentation of the main achievements of this contribution.We will first fix the notation and the hypotheses used throughout the paper.The major results will be summarized in Subsection 2.3.

Energy functionals. Let
be a bounded, connected, open set with Lipschitz boundary.We denote by χ 0,ε and χ 1,ε the characteristic functions respectively of Ω 0,ε and Ω 1,ε in Ω (see (1.1) and (1.2)), i.e. for i = 0, 1, Similarly, we let We assume that { f 0,ε } ε>0 and f 1 in (1.3) fulfill the following set of hypotheses: H3: there exist a, λ, Λ > 0 such that for a.e.x ∈ R d , for all ξ ∈ R N , and all ε > 0, for a fixed p ∈ (1, +∞); H4: there exists µ > 0 such that for a.e.x ∈ Ω, for all ξ, η ∈ R N , and ε > 0, where p is the same as in H3; H5: there exists Remark 2.1 (Degeneracy of the 'soft' component).We draw again attention to the fact that, in spite of the standard coercivity and growth of f 0,ε , the problem we address features some degeneracy, which is expressed by the factor ε multiplying the argument of the 'soft' integrand (see (1.3)).From a modeling perspective, this ε accounts for the high-contrast between the two components, in that it makes the 'soft' component less and less sensitive to external stimuli.This can be easily seen in the simple instance f 0,ε (ξ) = f 1 (ξ) = |ξ| p : formula (1.3) becomes and the ratio 1/ε p between the 'stiffness coefficients' of the two components blows up as ε vanishes.
Owing to the growth conditions prescribed by hypothesis H3, the natural environment for our problem is the space L p .We will often work with Q-periodic functions, i.e., those u : R d → R N such that u(x + z) = u(x) for all x ∈ R d and all z ∈ Z d .We will use the subscript 'per' to denote subspaces of Q-periodic maps; for instance, We endow the previous space with the norm of L p (Q; R N ).With a slight abuse of notation, we will tacitly identify L p (Q; R N ) with L p per (R d ; R N ) by mapping the unit cube into the unit torus in R d and extending the corresponding fields by periodicity.Analogously, we will denote the subspace of L p loc (Ω × R d ; R N ) containing all the functions that are Q-periodic w.r.t.their second argument by L p (Ω; L p per (R d ; R N )) and implicitly identify this space with L p (Ω × Q; R N ).

2.2.
The differential constraint.Let p ∈ (1, +∞) be as in H3 and H4 above.We suppose that the constraint that we couple with the energy in (1.3) falls within the framework of A -quasiconvexity as described by I. Fonseca & S. Müller [50].Namely, for N, M ∈ N \ { 0 }, we let A be a partial differential operator on R d from R N to R M whose action on a function u : R d → R N is given by where A (i) : R N → R M are linear maps for all i = 1, . . ., d.In other words, A is a linear first-order differential operator with constant coefficients.
Our fundamental requirement on A is the constant rank assumption introduced by F. Murat [66], that is, we suppose that there exists r ∈ N such that the symbol A of A satisfies the following: for all ω = (ω 1 , . . ., ω d ) ∈ R d \ { 0 }, the operator (O; R M ).To see this, we introduce the formal adjoint of A , denoted by A * .For v : O → R M , we set where A (i), * is the adjoint (i.e. the transpose) of A (i) .In this way, and for u ∈ L p (O; R N ) we define the pairing where p ′ is the conjugate exponent to p.In what follows, if u ∈ L p (O; R N ), equalities of the form A u = 0 in O are always tacitly understood in the sense of W −1,p , or, in other words, in the sense that When such a relation holds, we say that u is A -free on O. Similarly, if u ∈ L p per (R d ; R N ), we say that it is A -free on the unit torus T d when the equality in (2.4) is satisfied for O = Q and for all v ∈ W 1,p ′ per (R d ; R M ); in particular, if u ∈ L p per (R d ; R N ) is A -free on T d , then it is A -free on Q as well.In addition to the constant rank hypothesis, we need to consider further restrictions on the class of operators A for which our analysis will be performed.We formulate two ad hoc assumptions: Assumption 1 (Null-average).For all u ∈ L p per (R d ; R N ) such that u = 0 on D 1 and A u = 0 on T d , it holds ´Q u(y) dy = 0. Assumption 2 (A -free extension).There exists an ε-independent constant c > 0 and a sequence of operators Thanks to the Dunford-Pettis Theorem, this is equivalent to the fact that { |u ε | p } admits a subsequence that is weakly convergent in L 1 (Ω).We point out that in Assumption 2 it is essential to start with maps u which are A -free in the whole set Ω, for this is related to the existence of 'potentials' for the operator A .We refer to the considerations after Theorem 2.11 in Subsection 2.3 for a discussion on this point, while we collect some comments on Assumption 2 in the next remark .
Remark 2.2 (On Assumption 2).In problems involving perforated media the use of extension techniques is fairly common, and Assumption 2 is akin to others in the literature.For comparison, we mention [47,Definition 1.4].The main differences consist in: 1) the fact that the extension operator here is assumed to exist on a periodically perforated domain, with constants independent of the periodicity parameter ε; 2) the fact that the existence of an extended p-equiintegrable sequence here is only required when starting from a sequence that also had p-equiintegrability properties.We refer to [47] for a general discussion on the connections between the existence of extension operators from an open set to a bigger surrounding domain, and the theory of DiPerna-Majda measures.A notion of A -free extension domain was also considered in [60,Definition 4.4].There, however, no p-equiintegrability conditions are prescribed.As pointed out in [60,Section 4.2], such extension results are, in general, non-trivial.Additionally, they depend both on the topology of the set and on the operator A under consideration.In particular, no such A -free extension is possible when A is the operator associated to the Cauchy-Riemann system, not even in the case of very regular sets Ω.
As far as our analysis is concerned, items (1)-(3) in Assumption 2 are among the main ingredients in the proof of Proposition 4.1, and without them we would not be able to establish high-contrast compactness for sequences with equibounded energy (see Definition 2.5).On the contrary, the preservation of p-equiintegrability is not required at this stage, whereas it is used in the proof of our main Γ-convergence result (see the proofs of Propositions 2.9 and 4.4).Nonetheless, point (4) in Assumption 2 could be dispensed with in some simplified versions of our problem, e.g. when f 0,ε = f 0 for every ε > 0 and Ω is a rectangle.
For the sake of brevity, it is convenient to give a name to the class of operators addressed in our study: Definition 2.3.Let A be a linear first-order differential operator as in (2.2).We say that A is admissible if and only if it is of constant rank and it satisfies Assumptions 1 and 2.
It is well-known that the family of constant rank operators is quite large [50]; as for the subclass of the admissible ones, we prove that it is non-empty in Section 7 where, in particular, we analyze the cases of the curl, of the curl curl operator, of the divergence, as well as the setting associated with higher-order gradients.

Main results.
The first main result of this paper, Theorem 2.6 below, is the characterization of the asymptotic behavior as ε → 0 of the functionals in (1.3) when the family is restricted to A -free fields.In order to deduce also some information on the related energy minimizers, we resort to a variational kind of convergence, namely Γ-convergence (see e.g.[15,38]).Here, we consider a quite abstract version of its definition: indeed, we consider functionals defined on a certain set for which it is only known that some of its sequences are converging, the limit point being declared too; no topological structure is provided, instead.We summarize this situation by saying that such a set is endowed with a notion of convergence.Definition 2.4.Let X be a set endowed with a notion of convergence.We say that the family { F ε } ε>0 of functions on X with values in [−∞, +∞] Γ-converges as ε → 0 to the function F : X → [−∞, +∞] if for any x ∈ X and for any sequence { ε k } k∈N such that ε k → 0 the following holds: (1) for any sequence We have already remarked that the natural domain of the functionals (2.10) is contained in L p (Ω; R N ).However, in our setting, the two kinds of convergence in this space that are most frequently considered, the strong and the weak one, are not well-suited to perform a Γ-convergence analysis.Indeed, they do not match the high-contrast nature of the problem and would therefore not yield useful compactness results for sequences with equibounded energy.This leads us to introduce a peculiar notion of convergence.Definition 2.5.We say that a family { u ε } ⊂ L p (Ω; R N ) converges in the high-contrast sense to u ∈ L p (Ω; R N ) (relatively to the sequence of sets { Ω 1,ε }) if εu ε ⇀ 0 weakly in L p (Ω; R N ) and if there exists a second family { ũε } ⊂ L p (Ω; R N ) such that A ũε = 0 in W −1,p (Ω; R M ), u ε = ũε in Ω 1,ε and ũε ⇀ u weakly in L p (Ω; R N ).Some comments are in order.The reason why the definition above is made up of two requirements is that there is no standard convergence which is able to capture alone the behaviors of both components.On one hand, the convergence of extensions is related to the coercivity of the 'stiff' part, and it is a somehow customary requirement in homogenization (see Subsection 2.4); on the other, the convergence εu ⇀ 0 is needed to keep track of the 'soft' component.Here, in the light of Assumption 1, the 0 vector turns out to be the average of some two-scale limit.
If we select A = curl and we assume that Ω is simply connected, the problem may be recast in W 1,p (Ω).In this setting, as far as the asymptotics of the 'stiff' part is concerned, Rellich-Kondrachov's Theorem allows to switch from weak convergence in W 1,p (Ω) to strong convergence in L p (Ω).Therefore, the use of high-contrast convergence on the matrix leads to the standard homogenization results for perforated media.At the same time, it allows to quantify the change in the effective energy of the perforated material that occurs when 'soft' inclusions are added (cf.α 0 in (2.7)).Indeed, it can be readily shown that ε∇w ε ⇀ 0 if F ε (∇w ε ) ≤ c by well-known two-scale properties of gradients.
As a last remark on Definition 2.5, note that it is obvious that the weak convergence in L p (Ω; R N ) of a sequence { u ε } of A -free maps entails high-contrast convergence (it suffices to set ũε := u ε ).In particular, bounded sequences in L p are pre-compact in the high-contrast sense.As Proposition 4.1 proves, however, a much weaker control is enough (see (4.10)), at least for admissible differential operators.
It will be convenient to have a special notation for spaces of functions satisfying the differential constraint encoded by A .For any open Ω ′ ⊂ Ω, we define and Note that limit points of high-contrast convergent sequences are automatically in U 1 .Hereafter, we use the subscripts x and y to denote that the operator A acts on the variables x and y, respectively.We will prove that the limiting behavior of the ε-dependent energies constrained to A -free fields is described by the functional F : U 1 → R defined as where and We are now ready to state our main convergence result.
Theorem 2.6. (2.10) Let us assume that { f 0,ε } and f 1 fulfill H1-H5, and that A is an admissible partial differential operator in the sense of Definition 2.3.Then, the following properties hold: (1) If a family { u ε } ⊂ L p (Ω; R N ) satisfies F ε (u ε ) ≤ c for all ε > 0 and for some c ≥ 0, then it admits a subsequence that is convergent in the high-contrast sense.(2) For any u ∈ U 1 and for any sequence { u ε } ⊂ L p (Ω; R N ) that converges to u in the highcontrast sense, it holds that (3) For any u ∈ U 1 there exists a sequence { u ε } ⊂ U 1 that converges to u in the high-contrast sense and satisfies lim sup By standard Γ-convergence techniques, we find the following corollary of Theorem 2.6: Corollary 2.7.Under the same assumptions and notation of Theorem 2.6, if then there exists a subsequence of { u ε } that is convergent in the high-contrast sense to a minimum point of F .Moreover, inf Our proof of Theorem 2.6 relies on four main intermediate results: 1) a compactness analysis combined with the characterization of high-contrast limits of A -free fields, cf.Proposition 4.1; 2) a splitting argument that allows to reduce the study of the Γ-limit of { F ε } to the study of two independent problems, concerning the asymptotic behavior of the 'soft' and of the 'stiff' energy contributions, respectively, cf.Lemma 4.2 and Proposition 4.4; 3) the identification of an optimal lower bound for the 'soft' part of the energy, cf.Proposition 2.8; 4) the identification of the limiting description for the 'stiff' part, cf.Proposition 2.9.The proof of Theorem 2.6 is exposed in Section 4.
Let us now present shortly the results that we obtain for the 'soft' and 'stiff' parts of the energy.The splitting procedure in Section 4 yields the functionals where the densities { f 0,ε } and f 1 satisfy H1-H5.We will show, respectively in Sections 5 and 6, that Precisely, we establish the following.
If the hypotheses H1 and H3-H5 are satisfied, and if A is an admissible differential operator as in Definition 2.3, then for all u ∈ L p (Ω; R N ) the following hold: (1) for every p-equiintegrable sequence { u ε } ⊂ L p (Ω; R N ) such that u ε = 0 on Ω 1,ε and that εu ε ⇀ u weakly in L p (Ω; R N ), we have (2) there exists a sequence { u ε } ⊂ U 1 with the properties that u ε = 0 in Ω 1,ε for every ε > 0, εu ε ⇀ u weakly in L p (Ω; R N ), and We point out that the p-equiintegrability condition in the first part of the previous statement does not affect the generality of the result, as Proposition 4.4 will prove.For more details on this point, the reader is referred to the splitting argument in Section 4 below.
For what concerns the functionals F 1,ε , the analysis carried out by I. Fonseca & S. Krömer in [46] yields the following: If the hypotheses H1-H4 are satisfied and if A is an admissible differential operator as in Definition 2.3, then the Γ-limit of { F 1,ε } with respect to the weak The last section is focused on the study of admissible differential operators.Differently from the other parts of the paper, the analysis of Section 7 encompasses all linear, k-th order, homogeneous differential operators with constant coefficients and constant rank.It will be useful to consider those operators A for which there exists a second operator B with the property that u = Bw for some Sobolev function w whenever A u = 0; in this case we say that B is a potential for A .Our second main result proves that, for A to admit a potential on a certain open set Ω, it is sufficient that A -free maps on Ω can be extended to A -free maps on the whole space in such a way that a control on the L p norm is ensured.We dub a set Ω with this property an A -extension domain, see Definition 7.1.As Remark 7.2 shows, an A -extension domain has to comply with some topological requirements; in particular, it cannot have holes, in general.
Theorem 2.10 (Existence of potentials for A -free maps).Let A be a linear, k-th order, homogeneous differential operator with constant coefficients and constant rank.If Ω ⊂ R d is a bounded, connected, open set with Lipschitz boundary which is also an A -extension domain, then there exists ℓ ∈ N, and a differential operator B of order ℓ satisfying the following: for all A -free maps u ∈ L p (Ω; R N ) there is a function w ∈ W ℓ,p (Ω; R M ) such that u = Bw almost everywhere in Ω.
An immediate consequence of this result is that Assumption 1 holds whenever the unit cube Q is an A -extension domain (cf.Corollary 7.3).
Our third and last main result shows that for operators admitting a potential satisfying a suitable Korn-type inequality it is possible to define an extension operator preserving A -free maps.
Theorem 2.11.Let A be a linear, k-th order, homogeneous differential operator with constant coefficients and constant rank.Let also B be a linear, ℓ-th order, homogeneous differential operator with constant coefficients such that where A and B are the symbols of A and B, respectively.We further assume that • for any bounded, connected and open set D ⊂ R d with Lipschitz boundary there exist a projection operator on the subspace of B-free maps Then, there exist a constant c := c(d, p, D 1 ) > 0 independent of ε and and Ω, as well as a sequence of maps It is fundamental to observe that the maps under consideration in the theorem above are already a priori in the kernel of the operator A on the whole set Ω.For this reason, it is meaningful to assume the existence of potentials B. In the case in which, instead, our fields were in the kernel of the operator A only in the perforated domain, its lack of contractibility would prevent the existence of a potential to hold even in the simple setting d = 3 and A = curl.We refer once again to Remark 7.2 for a counterexample.

2.4.
Comparison with other works.Before getting to the heart of the matter, we take the chance to compare our main convergence result, Theorem 2.6, with similar ones in the literature.
When A is the curl operator, our analysis is akin to the one performed by M. Cherdantsev & K. D. Cherednichenko in [25].However, our conclusions differ from theirs, also in the simple situation of a contractible Ω.Indeed, even though under this topological assumption curl-free maps coincide with gradients, the notion of convergence that we use is much weaker than the strong two-scale convergence considered in [25].Another key difference with [25] is that in our setting convergence of minimizers follows directly from Theorem 2.6 (see Corollary 2.7), whereas in [25] it had to be shown a posteriori (see [25,Section 10]).Actually, the introduction of high-contrast convergence is motivated exactly by the fact that sequences with uniformly bounded energy F ε are sequentially precompact in that sense.
This notion of convergence is inspired by the one considered by X. Pellet, L. Scardia, & C. I. Zeppieri in [68] to deal with high-contrast Mumford-Shah energies.Unlike that contribution, aside from the role of the differential constraint A , we require additionally that { εu ε } converges to zero, so as to comply with the delicate two-scale characterization in Proposition 4.1.
We also stress that, in the absence of growth conditions from below on the 'soft' part of the energy, the weak L p -topology alone would not ensure convergence of minimizers.We refer to [25, Section 4] and [17,Section 4] for a discussion on this topic.
For a general differential constraint A , our approach does not correspond to the ones in [17,25].If A does not admit a of suitable 'potential' B satisfying a Korn-type inequality (cf.Theorem 2.10 below), then A -free maps and fields u of the form u = Bw are not interchangeable.Another caveat involves the choice of the 'soft' effective energy.Looking at the results in [25], one could expect that in our case the A -quasiconvex envelope of f 0 is to be retrieved in the Γ-limit.We recall that for a continuous function g : R N → R with p-growth the A -quasiconvex envelope of g is defined as The example below shows that Q A f 0 in not the correct limiting energy density.
For such particular choices, we see that the left-hand side of (2.16) equals 0, while for the right-hand side we find

Preliminaries
We gather in this section some preliminary definitions and results to be used later in the paper.
3.1.Measurable selection arguments.We recall here a measurable selection criterion that we will invoke in the proof of Corollary 7.3.For a thorough discussion on the topic we refer to the book by C. Castaing & M. Valadier [24].
Proposition 3.1 (Theorem III.6 in [24]).Let S be a multifunction defined on the measurable space O and taking values in the collection of nonempty complete subsets of the separable metric space X.
If for all open O ⊂ X the set { ω ∈ O : S(ω) ∩ O = ∅ } is measurable, then S admits a measurable selection, that is, there exists a measurable function s : O → X such that s(x) ∈ S(x) for all x ∈ O.

3.2.
A -free fields and two-scale limits.In this subsection we include some useful tools for the variational analysis of problems under differential constraints.
We recall two results about (asymptotically) A -free fields.The operator A is always assumed to be of the form (2.2) and of constant rank.Lemma 3.2 (A -free decomposition, Lemma 2.15 in [50]).Let Ω ⊂ R d be an open, bounded domain, and let p ∈ (1, +∞).
To perform our analysis, we need to track how the differential constraint behaves along sequences that converge in the sense of Definition 2.5 above.In this respect, the notion of two-scale convergence [67,2] will be crucial: In this case we write u ε For the sake of completeness, we record in the following lines the properties of two-scale convergence to be exploited in this work; we refer to [2,81] for further reading.
It is well-known that two-scale convergence in L p can be related to L p convergence by means of the unfolding operator.Lemma 3.6.For ε > 0, let us define the unfolding operator S ε : where ṽ denotes the extension of v by 0 outside Ω.Then, if { u ε } ⊂ L p (Ω; R N ) is a bounded sequence, the following holds: (1) We refer to [32,33,81,82] for further properties of the unfolding operator.A characterization of weak two-scale limits of A -free sequences was established by I. Fonseca & S. Krömer.
As a last technical tool, we show that the unfolding of an A -free map is in turn A -free w.r.t. the periodicity variable.We preliminarily introduce the set Note that Ẑε ⊂ Z ε , where Z ε is the collection of indices in (1.1). where and it can be regarded as an element of W 1,p ′ 0 (Ω; R M ) by extending it to 0 outside ε Q + ε −1 x .In this step (3.1) needs to be used.The conclusion follows now from Assume further that v = 0 in Ω 1,ε .By the definition of the unfolding operator, S ε v(x, y) = 0 for all (x, y) ∈ Ωε × D 1 , so that for any for almost every x ∈ Ωε .

Fourier analysis.
The study of the class of admissible operators in Section 7 is grounded on the theory of Fourier multipliers.For a comprehensive treatment of the matter, we refer to the monographs [78,53]; here we limit ourselves to a short recollection of useful properties.We let S denote the Schwartz space of rapidly decreasing functions and for u ∈ S (R d ; R N ), we let be its Fourier transform.We also denote by F −1 the inverse transform and by Lin(R N ; R N ) the space of linear maps from R N to R N .We recall that a measurable function m : R d → Lin(R N ; R N ) is said to be an L p -multiplier if the linear operator T m defined as ) can be extended to a bounded operator from L p to L p .Given a sufficiently smooth m, S.G.Mikhlin's multipliers theorem provides a condition for it to be a multiplier in terms of the decay of its derivatives.We firstly introduce some notation: when i is a d-dimensional multi-index, i.e. i := (i 1 , . . ., i d ) ∈ N d , we set |i| := j i j and In the scalar case, the criterion reads: We then say that a distribution u belongs to W k,p (R d ) if and we endow the space with the norm It turns out that this definition of W k,p is equivalent to the one given in terms of weak derivatives and that the norms are comparable too.The same approach shows that the dual space W −k,p ′ (R d ) coincides with the space of distributions for which it holds Here, we recall that p ′ := p/(p − 1) and (I − ∆) −k/2 is the pseudo-differential operator defined by the symbol

Proof of Theorem 2.6
This section is devoted to the proof of Theorem 2.6, which stands as our principal result about the asymptotics of the energy functionals F ε in (2.10).For the moment being, we assume that the limiting behaviors of the 'soft' and 'stiff' contributions are known, i.e., we suppose that Propositions 2.8 and 2.9 hold true.Their proofs are dealt with in Sections 5 and 6 below.Our main task in the current section is then to show that sequences with equibounded energies are precompact in a suitable sense and that the asymptotics of the global energy F ε is determined by that of the functionals accounting for the 'soft' and 'stiff' parts.In this respect, it is useful to regard the total energy F ε (u) of u ∈ L p (Ω; R N ) as the sum of F 0,ε (u) and F 1,ε (u), which were defined in (2.11) and (2.12).
The section is organized as follows.We first address the compactness result in Proposition 4.1, second we show how to reduce our problem to the sub-problems regarding the 'soft' and the 'stiff' components, then we prove the Γ-convergence statement in Theorem 2.6.

4.1.
Compactness.The next proposition shows that sequences which are equibounded in energy have high-contrast converging subsequences.This kind of convergence was introduced in Definition 2.5.We remark that, owing to position (2.10),only A -free fields give rise to finite energy configurations.We prove here that high-contrast limits of A -free sequences inherit differential constraints.Proposition 4.1 (High-contrast limits of A -free sequences).Let A be a constant rank differential operator of the form (2.2).Let also { u ε } ⊂ L p (Ω; R N ) be such that there exists c ≥ −aλL d (Ω) for which sup ε>0 F ε (u ε ) ≤ c.Then, (1) There exist a (non-relabeled) subsequence, as well as two maps u 0 ∈ L p (Ω; L p per (R d ; R N )) and u 1 ∈ L p (Ω; R N ), such that { εχ 0,ε u ε } converges to u 0 weakly two-scale and that { χ 1,ε u ε } converges to u 1 weakly in L p (Ω; R N ).Moreover, up to subsequences, { εu ε } and { χ 1,ε u ε } weakly two-scale converge in L p , respectively, to u 0 and to a map The following differential constraints are fulfilled: (3) If A satisfies Assumption 2, setting ũε := E ε A u ε , there exists ũ ∈ L p (Ω; L p per (R d ; R N )) such that, up to subsequences, { ũε } weakly two-scale converges to ũ in L p , and there holds v 1 (x, y) = χ D1 (y)ũ(x, y) almost everywhere in Ω × Q, (4.5) (4) If additionally A satisfies Assumption 1, then and ũε ⇀ u 1 weakly in L p (Ω; R N ).In particular, up to subsequences, { u ε } converges to u 1 in the high-contrast sense. Proof.
12)).The growth assumptions in H3 yield (1) By Lemma 3.5, since bounded sequences in L p are weakly two-scale precompact, there exist u, u 0 , v 1 ∈ L p (Ω; L p per (R d ; R N )) such that, up to a (non-relabeled) subsequence, For what concerns (4.1), we observe that, thanks to the relation between weak two-scale convergence and weak L p -convergence, we have Further, it holds that whence, by the first two convergences in (4.10), we find u 0 (x, y) = χ 0 (y)u(x, y) almost everywhere in Ω × Q.
In particular, also the first equality in (4.1) is satisfied.
(2) Since A u ε = 0 in Ω for all ε > 0, Proposition 3.7 yields immediately (4.2) and (4.3).As for (4.4), given any ψ ∈ W 1,p ′ 0 (D 1 ; R M ), we extend it to the whole R d by setting ψ = 0 in D 0 and ψ(x where the latter equality is due to the choice of the support of ψ.Therefore, the third convergence in (4.10) yields which in turn implies (4.4).
(4) By (4.7) and Assumption 1, we deduce that ´D0 ũ(x, y) dy = 0 for almost every x ∈ Ω.Then, by (4.5),A u 1 = 0 in Ω. Besides, in view of its weak two-scale convergence, { ũε } converges weakly in where we used (4.5) and (4.1).This also shows that { u ε } admits an extension on the 'soft' part that converges to u 1 weakly in L p .Therefore, to grant that there is a subsequence of { u ε } that converges to u 1 in the high-contrast sense, we are only left to observe that εu ε ⇀ ´Q u 0 ( • , y) dy and that such average vanishes because of (4.1), (4.3), and Assumption 1.The proof is now concluded.
As already anticipated in Remark 2.2, we note that the preservation of p-equiintegrability granted by Assumption 2 is not needed to prove item (3) of the previous proposition.We will instead resort to it to establish Proposition 4.4 below.4.2.Splitting.In the light of the of previous subsection, we know that those sequences which are equibounded in energy are precompact with respect to the high-contrast convergence, and that their cluster points fulfill suitable differential constraints.Following the approach of M. Cherdantsev & K. D. Cherednichenko [25], the next step is to show that the asymptotic behavior of the energy along such sequences coincides with the sum of those of the energies of two decoupled systems, one sitting on the 'soft' inclusions, the other on the 'stiff' matrix.

Lemma 4.2 (Splitting).
Let A be a constant rank differential operator of the form (2.2).Let also u ∈ L p (Ω; R N ) be the high-contrast limit of a family { u ε } ⊂ L p (Ω; R N ).Explicitly, assume that εu ε ⇀ 0 weakly in L p (Ω; R N ) and that there is a family { ũε } ⊂ L p (Ω; R N ) with the properties that for some c ≥ −aλ 0 L d (Ω), and if v ε := u ε − ũε , then the following hold: ) Proof.As a consequence of the fact that A ũε = 0 in Ω for all ε, also the weak limit of { ũε }, i.e. u, must be A -free.As immediate consequences of the definition, we also find that v ε = 0 in Ω 1,ε and that A v ε = 0 in Ω. Being { ũε } bounded in L p (Ω; R N ), we also deduce Only the estimates involving the semilimits are now left to prove.It suffices to show that lim ε→0 E ε (u ε ) = 0, with E ε as in (4.14).To this aim, we observe that in view of H4, for almost every x we have , and the conclusion is achieved by exploiting again the boundedness of { ũε } in L p (Ω; R N ).
We conclude this section by showing that, thanks to the A -free decomposition procedure in Lemma 3.2, we can always reduce to the case in which the sequence { v ε } on the left-hand side of (4.15) is p-equiintegrable.This motivates the Γ-liminif inequality contained in Proposition 5.3.We premise a lemma.
Lemma 4.3.Let { f 0,ε } and f 1 satisfy hypotheses H1, H3, and H4.Let also { u ε } , { v ε } ⊂ L p (Ω; R N ) be bounded sequences such that u ε − v ε → 0 in measure, and that { v ε } is p-equiintegrable.Then, ) Proof.Despite the dependence of the energy density f 1 on the oscillating variable, up to a different notational realization, the 'stiff' case is completely analogous to the 'soft' one.For this reason, we detail the proof of (4.16)only We first note that (4.16) is left unchanged if we replace f 0,ε with f 0,ε − f 0,ε (0), and hence we may assume that f 0,ε (0) = 0. Upon extraction of a (non-relabeled) subsequence, we may also suppose that the left-hand side in (4.16) is a limit.In view of [48,Lemma 8.13], for every ε > 0, we can decompose the elements of such subsequence as equiintegrable and converges to zero in measure.Thanks to Vitali's convergence theorem (see e.g.[48,Theorem 2.24]), it follows that u o ε − v ε → 0 strongly L p (Ω; R N ), and, in view of H4, we deduce By employing the definition of R ε , H3, and H4, we find the estimate Since { u o ε } and { u c ε } are bounded in L p (Ω; R N ) and { u o ε } is p-equiintegrable, we infer that the last term tends to zero as ε → 0. Therefore, we see that where the second equality is a consequence of (4.17), the second to last inequality is due to H3, and the last one follows from The improved variant of Lemma 4.2 for the liminf inequality in (4.15) reads as follows.
Proposition 4.4 (Refined splitting for the liminf).Let A be an admissible differential operator.Let u ∈ L p (Ω; R N ) and consider two sequences , then there exists { ṽε } ⊂ L p (Ω; R N ) such that the following hold: Proof.The equiboundedness in energy entails that By Lemma 3.2, there exist a (not relabeled) subsequence of { u ε } and a sequence { w ε } ⊂ L p (Ω; R N ) such that (1) { εw ε } is bounded, A -free, and p-equiintegrable; Note that the last property is not mentioned explicitly in the statement of Lemma 3.2, but it is a byproduct of the construction in its proof (see [50,Lemma 2.15]).
To complete the proof of the corollary, it suffices now show that for q ∈ [1, p), where v ε := u ε − ũε .Indeed, once this is proven, (4.19) is deduced from Lemma 4.2 and Lemma 4.3.
To prove (4.22), we notice that, by the definitions of v ε and ṽε , as well as by Hölder's inequality, for every q ∈ [1, p) it holds Then, in view of (2), of Assumption 2, and of (3), we deduce (4.22).

Γ-convergence.
In this short subsection we tackle the proofs of Theorem 2.6 and Corollary 2.7.
Proof of Theorem 2.6.Statement (1), concerned with the high-contrast compactness of families with equibounded energy, follows from Proposition 4.1.
Let now u ∈ U 1 be fixed and assume that there are two sequences { u ε } , { ũε } ⊂ L p (Ω; R N ) with the property that εu ε ⇀ 0 weakly in L p (Ω; R N ), A ũε = 0 in Ω, ũε ⇀ u weakly in L p (Ω; R N ), and u ε = ũε in Ω 1,ε .If the lower limit of { F ε (u ε ) } is not finite, the estimate of point (2) holds trivially.Otherwise, we define v ε := u ε − ũε and, owing to Proposition 4.4, by applying Proposition 2.8 to { v ε }, as well as Proposition 2.9 to { ũε }, we deduce Finally, we turn to (3).For u ∈ U 1 , again in the light of Propositions 2.8 and 2.9, we can find two sequences Then, setting u ε := v ε + ũε , by Lemma 4.2 we deduce the desired limsup inequality from the ones satisfied by At this stage, the convergence of infima and of minimizers easily follows.
Proof of Corollary 2.7.The equicoercivity of the energies with respect to the high-contrast convergence and the convergence result in Theorem 2.6 yield the conclusion by standard Γ-convergence arguments.

Asymptotics for the soft component
This section is devoted to the proof of Proposition 2.8.We address the asymptotic analysis of the energy stored in the 'soft' component of the system, that is, { F 0,ε } in (2.11), and we prove that the limiting behavior is encoded by the functional F 0 in (2.13).We recall that F 0 is finite only when u = 0 and in that case the value of the functional equals the constant α 0 given by α 0 := sup where the supremum is meant to run over all open sets that are compactly contained in Ω, and, for any open Ω ′ ⊂ Ω, U 0 (Ω ′ ) is as in (2.5).
Remark 5.1 (On the constant α 0 ).By the definition of U 0 (Ω ′ ) in (2.5), we see that Since λ(−a + |ξ| p ) ≤ f 0 (ξ) for every ξ ∈ R N , from (5.1) we deduce that and, more generally, that when In particular, the supremum in the definition of α 0 is not a maximum, in general.Indeed, suppose that Ω ′ 1 ⋐ Ω achieves such supremum.Then, we would get a contradiction whenever there exists Ω ′ 2 ⋐ Ω containing Ω ′ 1 such that the inequality in (5.2) is strict.We separate the discussion of the Γ-liminf and of the Γ-limsup inequalities.
5.1.Γ-liminf inequality.The proof of the Γ-liminf inequality relies on the compactness and splitting results of Section 4, on unfolding techniques, as well as on the following intermediate statement.
Step 1: a simplified setting.We start by establishing an inequality for the case in which the densities on the right-hand side do not depend on ε (cf.[25,Lemma 20]), that is, we first prove that According to the definition of α 0 and to Remark 5.1, for an arbitrary δ > 0, there is an open set In view of (5.3) and (5.4), w ε ∈ U 0 (Ω ′ δ ) for ε > 0 sufficiently small, whence Thanks to the p-equiintegrability of { w ε }, the conclusion (5.6) now follows by letting δ vanish.
Step 2: the general case.We adapt the argument in the proof of [38,Thm. 5.14], relying on the almost everywhere convergence of { f 0,ε } to f 0 , as well as on the p-Lipschitz continuity of these energy densities (see H4 and H5).
In view of (5.6) we deduce where the first inequality is due to (5.7), and the second one to (5.11).The arbitrariness of δ > 0 yields the conclusion.
We are now ready to tackle the Γ-liminf inequality for the 'soft' contribution; the argument is comparable to the one in [25,Lemma 21].
Proposition 5.3.Let { f 0,ε } ε satisfy assumptions H1 and H3-H5, and let u ∈ L p (Ω; R N ).Then, for every p-equiintegrable sequence for every ε > 0, and εu ε ⇀ u weakly in L p (Ω; R N ), we have Proof.If the lower limit of F 0,ε (u ε ) is not finite, then there is nothing to prove.Otherwise, without loss of generality, we focus on the case in which the lower limit of F 0,ε (u ε ) is finite, and we assume that it is a limit.Then, it follows by Proposition 4.1 and Assumption 1 that necessarily u = 0. Therefore, we are left to prove that where α 0 is given by (2.8) and εu ε ⇀ 0 weakly in L p (Ω; R N ).
By the definition of Ω 0,ε (see (1.1)), we have Being z an integer-valued vector, it holds ûε (εz, y) = εu ε ε(y + z) for every z ∈ Z ε and y ∈ D 0 .Hence, we obtain because for all x ∈ ε(Q + z), ⌊x/ε⌋ = z.We can rewrite the last estimate as follows: From this, owing to (5.12) and (5.13), we can invoke Proposition 5.2 to infer In view of the p-equiintegrability of { ûε }, the desired liminf inequality is now obtained by taking the limit as ε → 0.

Γ-limsup inequality.
We now turn to the Γ-limsup inequality for the energy functional associated to the 'soft' portion of the material.The optimality of the lower bound identified in Proposition 5.3 hinges upon the next proposition.
We postpone the proof of Proposition 5.4 to the end of the section, and we discuss next how the Γ-limsup inequality is deduced from it.
We now address the proof of Proposition 5.4.
Proof of Proposition 5.4.We construct a family { ũε } that strongly two-scale converges to w ′ (x, y) := χ Ω ′ (x)w(x, y) in L p , so that { u ε } will be given by u ε := ũε /ε.However, because of measurability issues, we cannot deal directly with a general w ∈ U 0 (Ω ′ ).The argument is thus subdivided in three parts: firstly, we define { ũε } under some extra regularity assumption on w; then, we prove the limsup inequality when w is regular; lastly, we recover the general statement by means of an approximation argument.
Step 1: construction of { ũε } when w is regular.In this step we assume that w possesses some extra regularity in the x variable, namely we take w ∈ U 0 (Ω ′ ) ∩ L p per (R d ; C( Ω; R N )).We construct a sequence { ũε } of A -free maps such that ũε = 0 on Ω 1,ε for all ε > 0 and that ũε 2 → w ′ strongly two scale in L p .Below, when we write w(x, y), we consider the second entry y to be the periodic variable.
We firstly check that { ũε } converges strongly two-scale in L p to w ′ (x, y) := χ Ω ′ (x)w(x, y).To prove the claim, we start by showing that { ũε } is bounded in L p (Ω; R N ).The properties of { w ε } and a change of variables grant that the following identities hold: From Jensen's inequality we then deduce As for the convergence of { ũε }, in view of Lemma 3.5, (5.16) yields the existence of a map ⇀ w weakly two-scale in L p .To identify the limit w, we consider φ ∈ C( Ω; C per (R d ; R N )) and, by similar computations to the ones above, we find where φε (x, y) := φ(ε(y + z), y) if x ∈ ε(Q + z) with z ∈ Ẑ′ ε .By the dominated convergence theorem, we infer that is, ũε 2 ⇀ w ′ weakly two-scale in L p (recall that w(x, y) = 0 if y ∈ D 1 ).In turn, the weak two-scale convergence implies which, combined with (5.16), ensures that In view of the definition of strong two-scale convergence, cf.Definition 3.4, we conclude that ũε Finally, we show that { ũε } is A -free on Ω.To this aim, we fix ψ ∈ W 1,p ′ 0 (Ω; R M ) and, by repeating the steps already seen above, we obtain ) is a cut-off function which equals 1 on D 0 , thanks to the fact that w(x, • ) vanishes on D 1 and is A -free on T d for almost every x ∈ Ω′ ε , we conclude Step 2: limsup inequality when w is regular.We have so far shown that, if and { ũε } is the sequence of Step 1, then where w ′ (x, y) := χ Ω ′ (x)w(x, y).Moreover, u ε = on Ω 0,ε and A u ε = 0 on Ω for all ε > 0. Here, we prove the inequality We preliminarily notice that, by construction, ũε vanishes outside the set Ω′ ε .Therefore, we have the identities Since z ∈ Z d , we have that ũε (ε(y + z)) = S ε ũε (εz, y), S ε being the unfolding operator.Therefore, by exploiting the properties of S ε , the energy of ũε is rewritten as follows: where the last identity is due to the fact that S ε ũε vanishes for x / ∈ Ω′ ε .In the light of the previous equalities, we have Keeping in mind Lemma 3.5, we observe that S ε ũε → w strongly in L p (Ω ′ ×D 0 ; R N ), and consequently, owing to H4 and H5, the limits as ε → 0 of the first and of the second term on the right-hand side equal 0. We thereby conclude that (5.17) holds.
Step 3: recovering the general statement.Let w ∈ U 0 (Ω ′ ).We first extend it to the whole space setting w = 0 for x ∈ R d \ Ω, and then we mollify it with respect to the variable x.This procedure ensures that w, the regularization of w, belongs to . Now, Steps 1-2 yield a recovery sequence for w, and, by means of a diagonal argument and of the continuity of the right-hand side of (5.17) in L p , a recovery sequence for w satisfying all the desired requirements can be constructed.Remark 5.6 (On recovery sequences).Observe that the construction we propose does not yield, in general, a recovery sequence with respect to the weak L p -convergence, that is, a sequence { u ε } ⊂ L p (Ω; R N ) fulfilling Corollary 5.5 with (2) replaced by (2') u ε ⇀ u in L p (Ω; R N ).Indeed, to this aim, letting w ′ be as in Proposition 5.4, we would need to exhibit a sequence { ũε } such that 1 ε ũε 2 → w ′ strongly two-scale in L p , and we would then again set u ε := ũε /ε.
On the whole, the results above form a proof of the Γ-convergence of the energies { F 0.ε }.
Proof of Proposition 2.8.The fact that F 0 is a lower bound for the asymptotic behavior of the sequence { F 0,ε } is established in Proposition 5.3.The optimality of such lower bound is proven in Corollary 5.5.

Asymptotics for the stiff component
In this section, we characterize the limiting behavior of the family { F 1,ε } in (2.12), which accounts for the energy of the 'stiff' component of the system, and we prove that { F 1,ε } Γ-converges to the functional F 1 in (2.14)We rely on the contribution by I. Fonseca & S. Krömer about homogenization of multiple integrals under differential constraints: Theorem 6.1 (Theorem 1.1 in [46]).Let g : R d × R N → [0, +∞) be a Carathéodory function which is Q-periodic with respect to its first argument.Assume that there exist c ≥ 0, Λ > 0, and p > 1 satisfying −c ≤ g(y, ξ) ≤ Λ(1 + |ξ| p ) for a.e.y ∈ R d and all ξ ∈ R N .
Then, for all u ∈ L p (Ω; R N ) such that A u = 0, the functional Γ-converges with respect to the weak L p -convergence to where, for all ξ ∈ R N , As a corollary of the theorem above, we derive the asymptotics of { F 1,ε }.
Proof of Proposition 2.9.For ε > 0, we introduce the set and we observe that Ω1,ε ⊂ Ω 1,ε (recall (1.2)).We let where a and λ are the constants in H3, and χ D1+z is the characteristic function of the set D 1 + z.When u is A -free we can hence write We observe that Theorem 6.1 applies to F1,ε , yielding ,ε (u) in the weak L p -topology.As for the difference between F 1,ε and F1,ε , by H3 one has Notice that the term L d (Ω 1,ε \ Ω1,ε ) is at most of order ε.Thus, the lower bound for { F 1,ε } follows from (6.2) and Theorem 6.1: for all u ∈ L p (Ω; R N ) such that A u = 0 and all A -free sequences { u ε } ⊂ L p (Ω; R N ) that converge to u weakly.For what concerns the upper bound, let us fix u ∈ L p (Ω; R N ) in the kernel of A and consider a recovery sequence { u ε } for { F1,ε } given by Theorem 6.1.Thanks to Lemma 3.2, we get a p-equiintegrable family { v ε } ⊂ L p (Ω; R N ) such that { u ε − v ε } converges to 0 strongly in L q (Ω; R N ) for all q ∈ [1, p).Therefore, ,εn (v εn ).By Lemma 4.3, we obtain the desired upper limit inequality:

Admissible differential constraints
In what follows we deepen our analysis of constant rank operators.We investigate the existence of Sobolev potentials for A -free maps and of extension operators on perforated domains.In particular, we prove Theorems 2.10 and 2.11.
In contrast to the previous sections, here we take into account also differential operators of order greater than 1: we treat the broader class of linear, k-th order, homogeneous differential operators with constant coefficients.To fix the notation, given k, N, M ∈ N \ { 0 }, we consider linear maps A (i) : R N → R M for all d-dimensional multi-indices i with |i| = k and we focus on the operators whose action on a function u : R N → R M is given by We keep in place the constant rank condition, which currently reads: there exists r ∈ N such that the rank of A[ω] equals r for all ω ∈ R d \ { 0 }, where is the symbol of A and ω i = Π d j=1 ω ij j if i = (i 1 , . . ., i d ) is a multi-index.Note that when k = 1 we recover the setting of Subsection 2.2.As before, we say that a map u is A -free in Ω if A u = 0 in W −k,p (Ω; R M ).
7.1.Existence of potentials.This subsection is devoted to the proof of Theorem 2.10.Loosely speaking, our goal is to prove that, given an operator A of constant rank, there exists a second differential operator B such that any A -free map u ∈ L p satisfies u = Bw for a suitable Sobolev function w.When such a relation holds, we will say for brevity that w is a potential for u and that B is a potential for A .We are able to prove existence of potentials for those functions that are A -free on a certain class of subsets of R d , which we introduce in the next definition.
Analogous definitions concerning extensions of sequences that are (asimptotically) A -free are found in [47,Definition 1.4] and [46,Lemma 2.8] (i.e.Lemma 3.3 above), while the reader is referred to Section 7.2 below for a more detailed discussion on extension problems.We also refer to [57] for a related study in the stochastic setting.It is important to notice that the property of being an A -extension domain carries some topological consequences.Remark 7.2 (On the topology of A -extension domains).The requirements of Definition 7.1 may act as implicit restrictions on the topology of the set Ω, according to the specific choice of A .Let us consider, for instance, the case in which d = 3, A =curl, and Ω = R 3 \ Re 3 , { e 1 , e 2 , e 3 } being the canonical basis in R 3 .We assume by contradiction that Ω is an A -extension domain.Then, if for every , 0 , we find that curl u = 0 almost everywhere in Ω, whence curl (E A u) = 0 in R 3 , if E A is the extension operator in Definition 7.1 for A = curl.This fact, in turn, yields a potential w such that E A u = ∇w in R 3 and, from the requirement that E A u = u on Ω, we conclude that u = ∇w on Ω.This leads to a contradiction, for the field u is not conservative.
For the proof of Theorem 2.10 we rely heavily on some recent contributions by B. Raiţă and coauthors, who established existence of potentials for A -free smooth maps and Korn-type inequalities for constant rank operators.Further related bibliographical comments are provided when we discuss Propositions 7.4 and 7.5 below.Before dealing with the proof, we pinpoint here that Theorem 2.10 constitutes a sufficient condition for Assumption 1 to hold: Corollary 7.3.Assume that A is a first order operator satisfying the assumptions of Theorem 2.10 and that the unit cube Q is an A -extension domain.Then, for any open set Ω ′ ⊂ Ω, the space U 0 (Ω ′ ) in (2.5) is characterized as follows: A y u = 0 in W −1,p (T d ; R M ) for a.e.x ∈ Ω ′ , and ˆQ u(x, y) dy = 0 for a.e.x ∈ Ω ′ .(7.1) Proof.For the desired equality to hold, we just need to prove that A y u = 0 in W −1,p (T d ; R M ) for a.e.x ∈ Ω ′ , and ˆQ u(x, y) dy = 0 for a.e.x ∈ Ω ′ .
Let us fix u ∈ U 0 (Ω ′ ).Theorem 2.10 grants that for a.e.x ∈ Ω ′ there exists w x ∈ W ℓ,p per (R d ; R M ) such that u(x, •) = B y w x a.e. in Q.To the aim of defining a measurable selection of the maps { w x } x∈Ω ′ , we exploit Proposition 3.1.
We first observe that for every u ∈ L p (Ω; L p per (R d ; R N )) and x ∈ Ω ′ the set W (x) := w ∈ W ℓ,p per (R d ; R M ) : B y w(y) = u(x, y) for a.e.y ∈ R d is either empty or a closed (and hence complete) subspace of W ℓ,p per (R d ; R M ).Besides, being x → u(x, is measurable as well.It follows that the multifunction x → W (x) admits a measurable selection, which we denote by w(x, • ).
Summing up, we recovered a measurable function w from Ω ′ to W ℓ,p per (R d ; R M ) with the property that u = B y w for almost every x ∈ Ω ′ and y ∈ R d .We conclude that ˆQ u(x, y) dy = |i|=ℓ B (i) ˆQ ∂ y i w(x, y) dy = 0, where the latter inequality follows from the periodicity of w in its second variable.
We devote the remainder of the section to the proof of Theorem 2.10 and to the pertaining tools.We first highlight that B. Raiţă [70] has recently pointed out that the kernel of the symbol of any constant rank operator coincides with the image of the symbol of a suitable operator B, and that the latter is actually a potential for A when tools from Fourier analysis are applicable.Precisely, we have: Proposition 7.4 (Theorem 1 and Lemma 2 in [70]).Let A be a linear homogeneous differential operator on R d with constant coefficients.Then, A is of constant rank if and only if there exists a linear homogeneous differential operator B on R d with constant coefficients such that where A and B are the symbols of A and B, respectively.
Moreover, suppose that A and B are, respectively, operators from R N to R M and from R M to R N .If (7.2) holds, then, for all u ∈ S (R d ; R N ) such that A u = 0, there exists w ∈ S (R d ; R M ) such that u = Bw.
Observe that if the operator B satisfies (7.2), then it is in turn of constant rank; precisely, rank B[ω] = N − r if rank A[ω] = r for all ω ∈ R d \ { 0 }.Proposition 7.4 builds a unified framework that encompasses some well-studied special cases, such as the one of the curl and of the divergence.Results in the same spirit had been previously obtained by J. Van Schaftingen [79, Proposition 4.2] for elliptic operators, which correspond to the subclass of operators whose symbol is injective.
A second relevant finding related to constant rank operators was obtained by A. Guerra & B. Raiţă [54], who showed that this class is exactly the one in which a sort of Korn inequality holds.To state their result, we need to introduce the projection Π A associated with the operator A .For u ∈ S (R d ; R N ), it is defined as where F and F −1 are the Fourier transform and its inverse (see (3.4)), and the map Here and in the rest of the paper, Lin(V ; W ) is the set of linear operators from the vector space V to the vector space W .We have: Proposition 7.5 (Korn-type inequality for constant rank operators [54]).Let p ∈ (1, ∞) and let A be a linear, k-th order, homogeneous differential operator with constant coefficients.Then, A is of constant rank if and only if there exists c := c(d, p) such that For future use, we remark that the conclusion remains valid when φ ∈ S (R d ; R N ), as the Fourier analysis on which the proof is based is still viable in this case, or even when φ ∈ W k,p (R d ; R N ), by approximation.The sufficiency of the constant rank condition had been already observed in the literature, see for instance the bibliography in [54] or the paper by D. Gustafson [56], where a proof for first order operators is given.
To the purpose of constructing Sobolev potentials for A -free fields, we first need to extend the projection operator Π A in (7.3) to nonsmooth maps.We invoke the following result by T. Kato [58], which we present in a form due to F. Murat: Lemma 7.6 (Lemma 3.7 in [66]).Let A be a linear, k-th order, homogeneous differential operator with constant coefficients and constant rank.Then, the operator Π A in (7.3) can be extended to a bounded linear operator from L In the proof of the lemma the constant rank assumption is fundamental.Indeed, it ensures that the map ω → P A [ω] introduced above is analytic on R d \ { 0 } (see [66]); this, in combination with the 0-homogeneity of P A , allows the use of S.G.Mikhlin's multipliers theorem (cf.Corollary 3.10).
Remark 7.7 (Self-adjointness of the projection operator).As a consequence of Parseval's formula and of the self-adjointness of P A , by density there holds We next clarify why Π A is entitled to be named projection: Lemma 7.9 below shows that for an L p -function u one has that A (Π A u) = 0, and that Π A u = u if A u = 0. We premise the instrumental notion of Moore-Penrose generalized inverse Λ † of a linear map Λ. Lemma 7.8 (Properties of the generalized inverse [54,23]).Given the map Λ ∈ Lin(R N ; R M ), we let Λ † ∈ Lin(R M ; R N ) be defined as where (ker Λ) ⊥ = im Λ * is the orthogonal complement of the kernel of Λ and P im Λ is the orthogonal projection on the image of Λ.Then, the following hold: (1) Λ † is the unique element in Lin(R M ; R N ) such that Λ † • Λ = P im Λ * and Λ • Λ † = P im Λ .
( We can now characterize the properties of the projection Π A .Note that they are comparable to the ones presented by I. Fonseca & S. Müller [50, Lemma 2.14] for the periodic setting.We also refer to [62,Subsection 2.8] for an alternative projection operator on the unit torus for which no null-average conditions are imposed.Lemma 7.9.Let A be a linear, k-th order, homogeneous differential operator with constant coefficients and constant rank.For every u ∈ L p (R d ; R N ), there holds: In particular, when A u = 0 in R d , then Π A u = u a.e.
Proof.When u ∈ S (R d ; R N ), the facts that A (Π A u) = 0 and Π A u = u if A u = 0 are simple consequences of (7.3).To extend these properties to the case of a nonsmooth u, we regard ψ ∈ C 1 c (Ω; R M ) as a Schwartz function on the whole space and we find and statement (1) holds.
We now turn to point (2).We argue for h = 0 as the case h > 0 is analogous.We consider at first u ∈ S (R d ; R N ) and we compute the Fourier transform of u − Π A u. Setting û = F u, we find where in the latter inequality we exploited statement (1) in Lemma 7.8.
Since A is k-homogeneous and A † is (−k)-homogeneous, we infer by Corollary 3.10 that A † A is an L p -Fourier multiplier.Therefore, the operator T defined for u ∈ S (R d ; R N ) as Tu := F −1 (û − P A û) can be extended to a bounded linear operator from L p (R d ; R N ) to L p (R d ; R N ), which we still denote by T.
For every u ∈ S (R d ; R N ) we write whence, by the (−k)-homogeneity of A † , Mikhlin's multipliers theorem yields From the definition of T and the characterization of W −k,p recalled in Subsection 3.3, (7.5) follows for u ∈ S (R d ; R N ).The general assertion is then obtained by density.
Eventually, we are able to establish the existence of potentials in a nonsmooth setting and prove the first main result of this section.We will use the following variant of Poincaré-Wirtinger's inequality, which can be derived as a corollary of Rellich-Kondrachov's Theorem.
where Π ∇ ℓ is the projection on the kernel of the operator ∇ ℓ .
Proof of Theorem 2.10.Let us fix u ∈ L p (Ω; R N ) satisfying A u = 0 in W −k,p (Ω; R M ).Since we postulate that Ω is an A -extension domain, there exists an operator E A : L p (Ω; R N ) → L p (R d ; R N ) as in Definition 7.1.In particular, if we let ũ := E A u, then ũ is A -free on R d .As a first step, we approximate ũ by maps in the image of B. By the definition of the projection Π A , there exist a sequence By construction, the functions ũk belong to S (R d ; R N ) and, in view of Proposition 7.4, we recover a sequence { w k } ⊂ S (R d ; R N ) such that ũk = Bw k .Therefore, recalling that Π A ũ = ũ because ũ is A -free, we deduce Next, we proceed by applying Proposition 7.5: since B has constant rank, where ℓ is the order of B. Note that the right-hand side is uniformly bounded in k because { ũk } is convergent in L p (Ω; R N ) by (7.7).As a consequence, inequality (7.6) yields a function w ∈ W ℓ,p (Ω; R M ) such that (up to subsequences) We will comment on the relationships between the previous theorem and the theory developed above at the end of this section, see Remark 7.16.For the moment being, let us just highlight that the hypothesis concerning existence of potentials for A -free fields on D enables us to recast the problem in terms of extension of Sobolev maps.For the latter, an adaptation of well-established arguments yields the following: Proof.The proof follows the same lines of [1,Lemma 2.6].Note that, in order to recover item (3) when ℓ > 1, Poincaré's inequality has to be replaced with (7.6).
We are now in a position to prove the first main result of this section.
Proof of Theorem 7.12.Let us fix u ∈ L p (D; R N ) such that A u = 0.The current assumptions grant that there is a Sobolev potential w ∈ W ℓ,p (D; R M ) for u, i.e. u = Bw, and that it satisfies If E is the extension operator in Lemma 7.14, we set O by the definition of A -free maps and (7.2).
To conclude, we need to show that E A u L p (O;R N ) ≤ c u L p (D;R N ) .By the definition of E A u, we have c being a constant depending on B (and hence on A ). Thanks to Lemma 7.14, we obtain a bound in terms of the potential of u: for some c := c(d, p, A , D, O).The conclusion is achieved by combining the above inequality with (7.10).
The proposition may be proved by adapting the strategy in the seminal work by E. Acerbi & Al.[1].Their result addresses only the case ℓ = 1, and the analogue of (4) is not mentioned; we omit nonetheless the proof in the case ℓ > 1, which is a natural adaptation of [1, Proof of Theorem 2.1].As for point (4), it is a mere consequence of the construction of E ε : it suffices to note that reflections, dilations, and patching by partitions of unity preserve p-equiintegrability.
Thanks to the previous result, we obtain that Assumption 2 is fulfilled by an operator A as soon as it admits a potential for which a Korn-type inequality holds.Note that it is fundamental that we start from fields u which are A -free in the whole set Ω: in principle, the existence of a potential would be false if we worked with maps that are A -free just on the perforated set Ω 1,ε , cf.Remark 7.2.
Proof of Theorem 2.11.Let u ∈ L p (Ω; R N ) be an A -free map and let w ∈ W ℓ,p (Ω; R M ) be its potential.In the same spirit of Theorem 7.12, we set , where E ε is the extension operator in Proposition 7.15.As in the proof of Theorem 7.12, it easy to check that (1) and (3) in Assumption 2 hold.We now turn to item (2), that is, for all A -free u ∈ L p (Ω; R N ).
By the definition of Ω 0,ε in (1.1), there exists an open set Ω ′ ⊂ Ω with Lipschitz boundary such that Ω 0,ε ⊂ Ω ′ and that δ := dist(∂Ω ′ , ∂Ω) > 0. In particular, recalling (3.1), Ω ′ ⊂ Ωε if √ dε < δ, and { Ωε , Ω \ Ω′ } is an open cover of Ω.We observe that E ε w = w a.e. in Ω \ Ω′ ⊂ Ω 1,ε , whence, by the definition on E ε A , E ε A u L p (Ω\Ω ′ ;R N ) = u L p (Ω\Ω ′ ;R N ) ≤ u L p (Ω1,ε;R N ) (7.11)For what concerns the contribution on Ωε , if we let we have . The second inequality is a consequence of the construction of E ε , which is obtained by patching together the extension operators from each 'stiff' unit ε(D 1 + z) to the cell ε(Q + z).Thanks to the definition of Ω1,ε we can further bound the last quantity from above by invoking Korn's inequality ∇ ℓ w − Π B w L p (D1;R M ×d ℓ ) ≤ c Bw L p (D1;R N ) , which holds by assumption.We obtain E ε A u L p ( Ωε;R N ) ≤ c u L p ( Ω1,ε;R N ) ≤ c u L p (Ω1,ε;R N ) .On the whole, the last estimate and (7.11) yield Eventually, we turn to (4).For any ε > 0, let w ε be the potential of u ε .We fix η > 0 arbitrarily and we let E ⊂ Ω be a Lebesgue measurable set.As a first step, we prove that there exists m > 0 such that L d (E) < m implies Since { u ε } is p-equiintegrable, there exists m > 0 with the property that u ε L p (F ;R N ) < η whenever L d (F ) < 2 m.Thanks to the outer regularity of the Lebesgue measure, we can select a finite union of open rectangles U ⊃ E such that L d (U ) < L d (E) + m.Thus, if we set m := m and we assume that L d (E) < m, thanks to (2.15) we deduce where the last inequality is a consequence of the p-equiintegrability of { u ε }.
In conclusion, we infer the p-equiintegrability of { E ε A u ε } from the one of { ∇ ℓ (w ε − Π B w ε ) }.

According to the definition of E ε
A u, we have whence, owing to point (4) in Proposition 7.15, is sufficiently small, as desired.
In light of our analysis, existence of potentials and of extension operators turn out to be almost equivalent; a key role in this respect is played by (7.9).We elaborate on this point in the next remark.All in all, we see that, given a constant rank operator A and a bounded, connected, open set with Lipschitz boundary D, the existence of Sobolev potentials for A -free maps on D and the existence of an A -free extension operator from D to R d are nearly equivalent.More specifically, they would actually be equivalent as soon as we knew that the generalised Korn inequality (7.9) holds when D is a 'nice' open set and B is a constant rank operator.In conclusion, we believe that investigations about the validity of (7.9) constitute a very interesting line of research.
We conclude with a parade of examples.The symbol of curl is then It is easy to check that Assumption 1 holds.Besides, when Ω is simply connected, in view of Proposition 7.15 all the conditions in Assumption 2 are fulfilled too.
Example 7.18 (Operators associated with higher-order gradients).Let Ω be simply connected.For any k ∈ N \ { 0 }, a constant rank differential operator A can be constructed such that A u = 0 if and only if u = ∇ k w for a suitable w [50].Then, similarly to the previous example, for such operator A Assumptions 1 and 2 are consequences, respectively, of a simple check and of Proposition 7.15.
Example 7.19 (The curl curl operator).Let d = M = N = 3 and, for all i, j = 1, 2, 3, let E (i,j) be the matrix whose entries all 0, but for the one in position (i, j), which equals 1.We consider the operator A = curl curl, defined as curl curl u := A (i,j) ∂ 2 u ∂x j ∂x i , where A (i,i) = −E (i+1,i+1) − E (i+2,i+2) (the indices are computed modulo 3) and A (i,j) = E (i,j) .The symbol of this operator is It is interesting to notice that the authors of [59] derive item (2) from (1), that is, they prove an implication of the form (7.12).
It is proved in [6, Subsection 4.1] that Div satisfies (2.15).Hence, Theorem 7.12 holds for A = div and Assumption 2 is fulfilled too.

Figure 1 . 0 D 1 Figure 2 .
Figure 1.The subdivision of the unit cube Q into the 'soft' region D 0 (white) and the 'stiff' region D 1 (grey).

3 )
has rank equal to r.We observe that for every open set O ⊂ R d , A can be regarded as an operator from L p (O; R N ) to W −1,p (O; R M ), where W −1,p (O; R M ) denotes the dual of the Sobolev space W 1,p/(p−1) 0

Proposition 3 . 9 (
Theorem 2 in the Appendix of[65])).Let m : R d \ { 0 } → R be a function of class C k with k > d/2.If there exists c ≥ 0 such that|∂ i m(x)| ≤ c |x| |i| whenever |i| ≤ k, then m is an L p -multiplier.A helpful consequence of the previous result is: Corollary 3.10.Let m : R d \{ 0 } → R be a function of class C k with k > d/2.If m is 0-homogeneous, then it is an L p -multiplier for all p ∈ (1, +∞).Multipliers stand as basics examples of pseudo-differential operators.Their theory, in turn, can be used to characterize Sobolev spaces as follows.Given k ∈ N, we introduce for ω ∈ R d the symbol S[ω] := (1 + |ω| 2 ) k/2 and the associated pseudo-differential operator (I − ∆) k/2 :

Definition 7 . 1 .
Let A be a homogeneous operator of order k.An open set Ω ⊂ R d is an A -extension domain if there exist E A : L p (Ω; R N ) → L p (R d ; R N ) and c > 0 such that the following holds: for all u ) Let A : O → Lin(R N ; R M ) be a smooth map on the open set O ⊂ R d .If rank A[ω] is constant for all ω ∈ O, then the map O ∋ ω → (A[ω]) † is locally bounded and smooth.(3) If A is a linear, k-th order, homogeneous differential operator with constant coefficients, then the map ω → (A[ω]) † is (−k)-homogeneous.

Lemma 7 . 10 .
Let Ω ⊂ R d be a bounded, connected, open set with Lipschitz boundary.There exists a constant c > 0 such that for every u Since the functions w k are smooth, the equality Bw k = B wk holds pointwise and we deduce from (7.7) that, for all φ ∈ C ∞ c (Ω; R N ), ˆΩ u(y) • φ(y) dy = lim k→+∞ ˆΩ wk • B * φ(y) dy = ˆΩ w(y) • B * φ(y) dy, where B * is the adjoint of B. The conclusion is then achieved.

Lemma 7 .
14 (cf.Lemma 2.6 in [1]).Let D, O ⊂ R d be open sets.If D is connected, O is bounded, and ∂D ∩ Ō is a Lipschitz boundary, there exist a bounded linear map E : W ℓ,p (D; R N ) → W ℓ,p (O; R N ) and a constant c := c(d, p, D, O) such that (1) Eu = u a.e. in D ∩ O, (2)

Remark 7 . 16 (
Relations between existence of potentials and of extension operators).Let us compare the main results of the current section.We let the operator A be as above, notably we assume that it has constant rank.Thanks to Theorem 2.10, we know that if D ⊂ R d is a bounded, connected, open set with Lipschitz boundary which is also an A -extension domain in the sense of Definition 7.1, then A -free maps on D admit potentials.In short, for the class of operators under consideration and for sufficiently 'nice' open sets D, it holds D is an A -extension domain =⇒ existence of potentials for A -free fields on D. (7.12) Conversely, if D ⊂ R d is an open bounded set with Lipschitz boundary such that all A -free fields on D admit Sobolev potentials through the operator B and if for the latter (7.9) holds, then D is an A -extension domain.Indeed, by a slight adaptation of the proof, a variant of Theorem 7.12 for the case O = R d can be established.Schematically, again for sufficiently 'nice' open sets D, we have existence of potentials for A -free fields on D Korn-type inequality for B on D =⇒ D is an A -extension domain.

2 Proposition 7 . 21 (( 1 )
and rank A[ω] = 2 for all ω ∈ R d \ { 0 }.When Ω ⊂ R 3 is a bounded and simply connected domain with Lipschitz boundary, we have that A u = 0 if and only if u = Bw for a suitable potential w, where B is the symmetric gradient Bw := (∇w + (∇w) t )/2.As a corollary of a recent result by F.Cagnetti & al. [19, Theorem  1.1], every such Ω is a curl curl-extension domain, whence Assumption 1 is satisfied.Classical Korn's inequality grants also that Assumption 2 holds.Example 7.20 (Divergence).Let us choose A = div, div being the standard divergence operator on R d .Then, N = d, M = 1, and div u =where, for i = 1, . . ., d, e i is the i-th element of the canonical basis of R d and e t i is its transpose.The symbol of div isA[ω] = d i=1 ω i e t i , thus rank A[ω] = 1 for all ω ∈ R d \ { 0 }.For what concerns Assumption 1 and the existence of extension operators, we resort to a result by T. Kato & al., which we present in a simplified setting: Extensions and potentials for divergence-free vector fields[59]).Let Ω ⊂ R d be a bounded and simply connected set with Lipschitz boundary.Then, the following holds: Ω is a div-extension domain in the sense of Definition 7.1.(2)For every u ∈ L p (Ω; R d ) such that div u = 0 in Ω, there exists w ∈ W 1,p (Ω; Antisym(d × d)) satisfying u = Div w := i,j ∂w i,j ∂x j e i in Ω, where Antisym(d × d) is the space of d × d antisymmetric matrices.Besides, w can be selected in such a way that the map u → w from L p (Ω; R d ) to W 1,p (Ω; R d ) is linear and bounded.