A Global Method for Relaxation for Multi-levelled Structured Deformations

We prove an integral representation result for a class of variational functionals appearing in the framework of hierarchical systems of structured deformations via a global method for relaxation. Some applications to specific relaxation problems are also provided.


Introduction
Our purpose in this paper is to establish a global method for relaxation applicable in the context of multilevelled structured deformations.The aim is to provide an integral representation for a class of functionals, defined in the set of (L + 1)-level (first-order) structured deformations (see Definition 2.2), via the study of a related local Dirichlet-type problem, and to identify the corresponding relaxed energy densities, under quite general assumptions (we refer to [7,8] for the introduction of the method in the BV and SBV p contexts).
First-order structured deformations were introduced by Del Piero and Owen [12] in order to provide a mathematical framework that captures the effects at the macroscopic level of smooth deformations and of non-smooth deformations (disarrangements) at one sub-macroscopic level.In the classical theory of mechanics, the deformation of the body is characterised exclusively by the macroscopic deformation field g and its gradient ∇g.In the framework of structured deformations, an additional geometrical field G is introduced with the intent to capture the contribution at the macroscopic scale of smooth sub-macroscopic changes, while the difference ∇g − G captures the contribution at the macroscopic scale of non-smooth submacroscopic changes, such as slips and separations (referred to as disarrangements [13]).The field G is called the deformation without disarrangements, and, heuristically, the disarrangement tensor M := ∇g − G is an indication of how "non-classical" a structured deformation is.This broad theory is rich enough to address mechanical phenomena such as elasticity, plasticity and the behaviour of crystals with defects.
The variational formulation for first-order structured deformations in the SBV setting was first addressed by Choksi and Fonseca [11] where a (first-order) structured deformation is defined to be a pair (g, G) ∈ compactness theorem (see [2,Theorems 4.7 and 4.8]) cannot be applied, due to the lack of a uniform control on the length of the singular set of the approximant bounded energy sequences (cf. the coercivity condition (5)).This is, in fact, the reason why in the structured deformation setting there is a distinction between the part of the deformation arising as the limit of the entire approximants and the part emerging as the limit of only their smooth parts (see the introductory comments in [11] for further details).
We also provide several applications, with the aim of showing that our main integral representation theorem, Theorem 3.2, already covers, and improves, in a unified way, several results available in the literature in the first-order structured deformation context.
As a first application of our general theorem, we are able to extend the integral representation for firstorder structured deformations proved in [11, Theorems 2.16 and 2.17], and later generalized in [20,Theorem 5.1] to allow for an explicit dependence on the spatial variable, to the case of Carathéodory bulk energy densities.This latter setting is more realistic, for instance, it allows for the modelling of materials with a very different behaviour from one point to another, as multigrain-type materials, or other types of mixtures, which also appear in the optimal design context (cf.for instance [21]).On the other hand, the assumptions on the surface energy density can be weakened, compared with [11], although, in the inhomogeneous setting, the continuity with respect to the spatial variable is still needed due to the fact that, in this case, it is meaningless to consider Lebesgue measurability for elliptic integrands defined on N − 1 dimensional sets.See Theorem 4.1 for the precise statement.In particular, under the same set of hypotheses considered therein, we recover the formulae in [11] (see also [3] and [20] for the inhomogeneous setting).We further show that, in the case p > 1, the cell formulae for the interfacial relaxed energy density in [11] and [20], still hold when the bulk energy densities are Carathéodory (see Theorems 4.1 and 4.2).
We stress that a standard relaxation approach, mimicking the arguments in [11], could also be achieved in the measurable bulk energy setting, but this would require the proof of many auxiliary steps, while a global method approach is more direct.
We point out that each step of the recursive relaxation procedure for multi-levelled hierarchies of structured deformations, presented in [4], whose densities, at each stage, satisfy hypotheses (1)- (8), also fits into the scope of Theorem 3.2.We refer to Subsection 4. 4 for more details and also for a different energetic formulation relying on Theorem 3.2 in its full generality.
Another natural application of our abstract result is to homogenization problems, such as the one considered in [1].In that paper, only the case p > 1 was treated under uniform continuity hypotheses on the densities, while here we can extend the result to include also p = 1, Carathéodory bulk densities and weaker assumptions on the elliptic integrands.In this setting, we depart from an energy of the form where ε → 0 + .Besides a periodicity condition, the densities W and ψ satisfy other hypotheses (cf.Theorem 4.3) which ensure that the relaxed functional (4.13) can be placed in the setting of our main theorem.As a further application, we recover the integral representation for one of the relaxed energies in [3].In this paper, an integral representation for the relaxation of an energy arising in the context of secondorder structured deformations is obtained.A simple argument allows for the decomposition of the relaxed functional I as the sum of two terms, I = I 1 + I 2 .Although I 1 does not fit the scope of our global method result, due to the topology which is considered in its definition, we will show that Theorem 3.2 applies to I 2 (in order to avoid conflicting notation to that introduced in (4.2), in Section 4 I 2 will be denoted by J).As in some of the previously mentioned applications, our global method for relaxation applies under milder assumptions than those considered in [3,Theorem 5.7], recovering for our densities the same expressions that were deduced in [3] when those hypotheses are considered.
A classical approach to a relaxation result for hierarchical systems of first-order structured deformations with an arbitrary number of levels L, and the comparison both with the method implied in [4] and with this abstract formulation, will be the subject of a forthcoming work [5].We emphasize that the method in [5], although it is expected to provide more explicit formulae, requires a direct proof for each choice of the level L ∈ N in a iterated way, while the global method does not require any iterative process as outlined in Subsection 4.4.
The paper is organized as follows: in Section 2 we set the notation which will be used in the sequel and recall the notion of a multi-levelled structured deformation, as well as the approximation theorem for these deformations.In Section 3 we state and prove our main theorem (see Theorem 3.2), whereas Section 4 is devoted to the aforementioned applications of our abstract result.

Notation. We will use the following notations
• N denotes the set of natural numbers without the zero element; • Ω ⊂ R N is a bounded, connected open set with Lipschitz boundary; , where S u is the jump set of u, [u] denotes the jump of u on S u , and ν u is the unit normal vector to S u ; finally, ⊗ denotes the dyadic product; • C represents a generic positive constant that may change from line to line.
A detailed exposition on BV functions is presented in [2].
The following result, whose proof may be found in [18], will be used in the proof of Theorem 3.2.
Definition 2.2.For L ∈ N, p 1 and Ω ⊂ R N a bounded connected open set, we define the set of (L + 1)-level (first-order) structured deformations on Ω.
In the case L = 1 and p = 1, this space was introduced and studied in [11], where it was denoted by SD(Ω).In particular, the following approximation result was shown (see [11,Theorem 2.12]).
Theorem 2.3 (Approximation Theorem in SD(Ω)).For every (g, G) ∈ SD(Ω), there exists a sequence u n in SBV (Ω; R d ) which converges to (g, G) in the sense that We now present the definition of convergence of a sequence of SBV functions to an (L+1)-level structured deformation (g, G 1 , . . ., G L ) belonging to either HSD L (Ω) or HSD p L (Ω).
Indeed, for ε > 0 small enough, let To conclude the result it suffices to let ε → 0 + .
Given (g, G 1 , . . ., G L ) ∈ HSD p L (Ω) and O ∈ O ∞ (Ω), we introduce the space of test functions and we let m : Following the ideas of the global method of relaxation introduced in [7], our aim in this section is to prove the theorem below. where ) , and where 0 is the zero matrix in R d×N and Remark 3.3.It follows immediately from the definitions given in (3.3) and in (3.4), and from Theorem 3.6, that if F is translation invariant in the first variable, i.e. if and for every a ∈ R d , then the function f in (3.3) does not depend on a and the function Φ in (3.4) does not depend on λ and θ but only on the difference λ − θ.Indeed, in this case we conclude that With an abuse of notation we write The proof of Theorem 3.2 is based on several auxiliary results and follows the reasoning presented in [7,Theorem 3.7] and [17,Theorem 4.6].For this reason we don't provide the arguments in full detail but point out only the main differences that arise in our setting.We start by proving the following lemma which is used to obtain Theorem 3.6.Lemma 3.4.Assume that (H1) and (H4) hold.For any where Q ν (x 0 , r) is any cube centred at x 0 with side length r, two faces orthogonal to ν and contained in Ω; where Proof.Suppose first that p > 1.Without loss of generality we can assume that , and let δ < α(δ) < 2δ be such that lim where the constant C depends only on the space dimension N and is, therefore, independent of δ.

By (3.6) and the fact that lim
Regarding the second term in (3.8), a similar argument using Hölder's inequality leads to and it suffices to let ε → 0 + to complete the proof in the case p > 1.
When p = 1 the proof is similar and we omit the details.In this case the estimate of the last term in (3.7) is simpler and does not require the use of Hölder's inequality.Also, in this case, more general sets other than cubes may be considered as there is no need to use inequality (3.6) (see also [17]).Following [7,8], for a fixed (u, U 1 , . . ., U L ) ∈ HSD p L (Ω), we set µ := L N ⌊Ω + |D s u| and we define and, for O ∈ O(Ω) and δ > 0, we let Adapting the reasoning given in [17, Lemma 4.2 and Theorem 4.3], with an even easier argument due to our hypothesis (H2) and to the fact that our fields u have bounded variation, we obtain the two results below.
We now present the proof of our main result of this section.
Proof of Theorem 3.2.We begin by proving that, for L N -a.e.x 0 ∈ Ω, Let x 0 be a fixed point in Ω satisfying the following properties where we are denoting by v a the function defined in Ω by v a (x) := u(x 0 ) + ∇u(x 0 )(x − x 0 ).It is well known that the above properties hold for L N -a.e. point x 0 in Ω, taking also in consideration Theorem 3.6 in (3.13) and (3.14).
Having fixed x 0 as above, let δ ∈ (0, 1) and let ε > 0 be small enough so that Q(x 0 , ε) ⊂ Ω.Given the definition of the density f in (3.3), due to (3.13) and (3.14), we want to show that Then, as u = v a on ∂Q(x 0 , ε), we have and, for every i ∈ {1, . . ., L}, let Recall that . Hence, by Remark 3.1, (H4) and (3.16), we have We observe that for every i ∈ {1, . . ., L} we have where in the last line we have used the fact that x 0 is a Lebesgue point for U i , see (3.12).
Using (3.11) and [2, (5.79)] yields lim sup On the other hand, by (3.17) and a change of variables, we can apply [7,Lemma 2.3] to conclude that lim sup Interchanging the roles of (u, U 1 , . . ., U L ) and (v a , U 1 (x 0 ), . . ., U L (x 0 )), the reverse inequality is proved in a similar fashion.This completes the proof of (3.9).
Theorem 3.2 is thus proved.

Applications
In this section we present some applications of the global method for relaxation obtained in Theorem 3.2.
4.1.2-level (first-order) structured deformations.The first application concerns the case of a twolevel structured deformation, that is, we take L = 1 in Definition 2.2.In this setting, given a deformation u ∈ SBV (Ω; R d ), and two non-negative functions W : Ω×R d×N → [0, +∞) and ψ : Ω×R d ×S N −1 → [0, +∞), we consider the initial energy of u defined by which is determined by the bulk and surface energy densities W and ψ, respectively.Then, as justified by the Approximation Theorem 2.5, we assign an energy to a structured deformation (g, G) ∈ HSD p 1 (Ω), which is equivalent to saying that (g, G) ∈ SD(Ω) and G ∈ L p (Ω; R d×N ), via To simplify notation, here and in what follows, we write Notice that this notion of convergence coincides, in the case L = 1, with the one given in Definition 2.4.
Under our coercivity hypothesis (3) below, the definition of I p coincides with the one considered in [11], see [11,Remark 2.15].
The functional in (4.2) was studied in [11], in the homogeneous case, and later in [20], in the case of a uniformly continuous x dependence, where, under certain hypotheses on W and ψ (cf.[20, Theorem 5.1]) it was shown that I p admits an integral representation, that is, that there exist functions In order to present the expressions of the relaxed energy densities H p and h p we start by introducing some notation.
For A, B ∈ R d×N let and for λ ∈ R d and ν ∈ S N −1 let u λ,ν be the function defined by and consider the classes given by for p > 1, and for p = 1, Then, the functions H p and h p appearing in (4.3) are given by (cf.[20, (5.6), (5.7)]) for all x 0 ∈ Ω and A, B ∈ R d×N , and, for all where W ∞ denotes the recession function at infinity of W with respect to the second variable, given by In (4.7), δ 1 (p) = 1 if p = 1 and δ 1 (p) = 0 if p = 1, so that the relaxed surface energy density depends on the recession function of W only in the case p = 1.
In what follows we obtain an integral representation result for I p (g, G), by means of Theorem 3.2, under a similar set of hypotheses on W and ψ as those considered in [11] and [20], but requiring only measurability, rather than uniform continuity, of W in the x variable.
Then, there exist where the relaxed energy densities are given by In the above expressions 0 denotes the zero 2) with L = 1 and F = I p , and C HSD p 1 (g, G; O) is given by (3.1), taking into account that HSD p 1 (Ω) in Definition 2.2 coincides with the set of fields (g, G) ∈ SD(Ω) such that G ∈ L p (Ω; R d×N ).
Proof.Given O ∈ O(Ω) and (g, G) ∈ SD(Ω), with G ∈ L p (Ω; R d×N ), we introduce the localized version of I p (g, G), namely Our goal is to verify that I p (g, G; O) satisfies assumptions (H1)-(H4) of Theorem 3.2 in the case L = 1.We start by proving the following nested subadditivity result and, in addition, and Notice that . Since the distance function to a fixed set is Lipschitz continuous (see [24, Exercise 1.1]), we can apply the change of variables formula [16, Section 3.4.3,Theorem 2], to obtain and, as | det ∇d| is bounded and (4.11) holds, by Fatou's Lemma, it follows that for almost every ρ ∈ [0, δ] we have Fix ρ 0 ∈ [0, δ] such that Gχ O2 (∂O ρ0 ) = 0, Gχ O3\O1 (∂O ρ0 ) = 0 and such that (4.12) holds.For this choice of ρ 0 , we may pass to subsequences of u n and v n (not relabelled) such that the liminf in (4.12) is actually a limit.We observe that O ρ0 is a set with locally Lipschitz boundary since it is a level set of a Lipschitz function (see, e.g., [16]).Hence we can consider u n , v n on ∂O ρ0 in the sense of traces and define Then, by the choice of ρ 0 , w n is admissible for I p (g, G; O 3 ) so, by ( 5), (4.11) and (4.12), we obtain which concludes the proof.
From here, the reasoning in [11,Proposition 2.22], which is still valid with the same proof in the nonhomogeneous case, yields (H1).
To show that (H2) holds, we argue as in [11,Proposition 5.1].Indeed, we can prove lower semicontinuity of (H3) is an immediate consequence of the previous lower semicontinuity property in O, as observed in [7, eq.(2.2)], whereas (H4) follows by standard arguments (as in [11,Lemma 2.18]) from ( 1), ( 2), ( 3) and ( 6) above and by the lower semicontinuity of integral functionals of power type and the total variation along weakly converging sequences.We point out that in order to obtain the lower bound in (H4) we can replace, without loss of generality, W by W + 1 CW .Hence, Theorem 3.2 can be applied to conclude that, for every (g, G) ∈ SD(Ω) × L p (Ω; R d×N ), we have where the relaxed densities f and Φ are given by It is easy to see that the functional I p is invariant under translation in the first variable, that is, Indeed, it suffices to notice that if {u n } is admissible for I p (g, G; O), then the sequence u n + a is admissible for I p (g + a, G; O).Hence, taking into account Remark 3.3 and the abuse of notation stated therein, we obtain (4.8) with f and Φ given by (4.9) and (4.10), respectively.On the other hand, for p > 1, Theorem 3.6 and the fact that F = I p , yield, for every where u λ−θ,ν is given by (4.5), with λ replaced by λ− θ, and we have taken into account the growth condition on W given by (1) and hypothesis (2), and the fact that the latter class of test functions is contained in the initial one.Given that this last expression no longer depends on the initial bulk density W , but only on ψ for which the uniform continuity condition (8) holds, we may apply this condition to replace x by x 0 and obtain We now invoke the periodicity argument used in the first part of the proof of [11,Proposition 4.2] to conclude that which, by a simple change of variables, coincides with so it follows that Φ(x 0 , λ, θ, ν) h p (x 0 , λ − θ, ν).
To prove the reverse inequality, we use the fact that W 0 to obtain Φ(x 0 , λ, θ, ν) = lim sup where the uniform continuity of ψ in the first variable was used in the final equality.We now argue as in [11,Propositions 4.2 and 4.4], in order to replace each weakly converging sequence u n by one which converges strongly to 0 in L p .In this way, we are lead to the conclusion that Φ(x 0 , λ, θ, ν) h p (x 0 , λ − θ, ν).
where W : Ω× R d×N → [0, +∞) is a Carathéodory function and ψ : Ω× R d × S N −1 → [0, +∞) is a continuous function, both Q-periodic in the first variable and such that they satisfy (1)- (5).Let (g, G) ∈ SD(Ω) and let I p,hom be the functional defined by Then, there exist where the limiting energy densities are given by Φ hom (x 0 , λ, θ, ν) := lim sup As in the case of Theorem 4.1, the proof of this theorem amounts to the verification that the functional I p,hom satisfies all of the assumptions of Theorem 3.2, we omit the details.We also refer to [1, Lemma A.1 and Proposition A.2], where the arguments were presented in the case p > 1, but they can be repeated word for word if p = 1.We point out that f hom is actually independent of x 0 and a and Φ hom is independent of x 0 , due to the fact that I p,hom verifies the conditions of [7,Lemma 4.3.3]which in turn can be proven in full analogy with [9,Lemma 3.7].
Notice that, in view of the results in [1], in the case p > 1 and assuming ( 6)-( 9), the densities given by (4.14) and (4.15) coincide with the bulk and surface energy densities H hom and h hom obtained in [1, eq. (1.11) and (1.12), respectively].In particular, when p > 1 (4.15) admits the equivalent representation (see [1, Proposition 3.5]), 4.3.Functionals arising in the analysis of second-order structured deformations.As a further application of our abstract global method for relaxation, we recover the integral representation for one of the energies appearing in [3].In this paper a model for second-order structured deformations is proposed in the space SBV 2 , giving rise to two energies (see [3,Theorem 3.2]).This decomposition relies strictly on hypotheses (I) and (II) below and ( 4)-( 5) from Theorem 4.1, but in the matrix setting.Although the first of these energies does not satisfy the conditions of Theorem 3.2, we will apply our result to the second (to avoid confusion with the notation used in the previous applications we denote it here by J) which is defined on matrix-valued structured deformations, J : SD(Ω; R d×N ) → [0, +∞), and is given by It was proved in [3,Proposition 4.6] that the functional J admits the following alternative characterization, in particular the density of the bulk term does not depend explicitly on the sequence v n , as we can fix the second variable equal to G, but only on the gradient of v n .
In the above expression the density W satisfies the hypotheses (I) (Lipschitz continuity) there exists a constant (II) there exists c W > 0 such that, for every (III) there exists a continuous function ω for every x 0 , x 1 ∈ Ω, A ∈ R d×N , M ∈ R d×N 2 ; (IV) there exists α ∈ (0, 1) and L > 0 such that it was shown in [3, Theorem 4.5] that J is the restriction to the open subsets of Ω of a Radon measure.Standard diagonalization arguments prove that it is sequentially lower semicontinuous in L 1 (O; R d×N ) × M(O; R d×N 2 ), from which locality follows.
Using the characterization of J given in (4.16) it is also easy to see that the growth hypothesis (H4) from Theorem 3.2 holds.
The proof that f (x 0 , A, B, D) = W 2 (x 0 , A, B, D) is similar and we omit the details.Thus our integral representation for J, obtained via the global method given in Theorem 3.2, recovers the one proved in [3, Theorems 3.2 and 5.7], and we emphasize the fact that γ 2 given in (4.19) does not really depend on the variable A. Indeed, as it will be rigorously shown in [5], one can associate to each multi-levelled structured deformation an energy satisfying hypotheses (H1)-(H4) in Section 3. The alternative procedure of assigning this energy very possibly yields an expression lower than the one obtained in [4,Subsection 3.2].
More precisely, referring to the case L = 2 for simplicity of exposition, starting from (4.1) and given (g, G 1 , G 2 , G 3 ) ∈ HSD p 2 (Ω), Theorem 3.2 provides an integral representation for the functional the open unit cube of R N centred at the origin; for any ν ∈ S N −1 , Q ν denotes any open unit cube in R N with two faces orthogonal to ν; • for any x ∈ R N and δ > 0, Q(x, δ) := x + δQ denotes the open cube in R N centred at x with side length δ; likewise Q ν (x, δ) := x + δQ ν ; • O(Ω) is the family of all open subsets of Ω, whereas O ∞ (Ω) is the family of all open subsets of Ω with Lipschitz boundary; • L N and H N −1 denote the N -dimensional Lebesgue measure and the (N − 1)-dimensional Hausdorff measure in R N , respectively; the symbol dx will also be used to denote integration with respect to L N ; • M(Ω; R d×N ) is the set of finite matrix-valued Radon measures on Ω; M + (Ω) is the set of nonnegative finite Radon measures on Ω; given µ ∈ M(Ω; R d×N ), the measure |µ| ∈ M + (Ω) denotes the total variation of µ; • SBV (Ω; R d ) is the set of vector-valued special functions of bounded variation defined on Ω.Given u ∈ SBV (Ω; R d ), its distributional gradient Du admits the decomposition Du

Theorem 4 . 1 .
Let p 1 and let Ω ⊂ R N be a bounded, open set.Consider E given by (4.1) where W

Theorem 4 . 3 .
Let p 1 and Ω ⊂ R N be a bounded, open set.Let ε → 0 + and consider E ε given by

4. 4 .
Multi-levelled structured deformations.The global method for relaxation that we propose in Theorem 3.2, allows us not only to recover the recursive relaxation procedure presented in [4, Subsection 3.2 and Theorem 3.4], since in each step the obtained densities satisfy hypotheses (1)-(9), thus entitling us to apply Theorems 4.1 and 4.2, but also to propose an alternative direct strategy.
∂O 1 ) > ε} and notice that O 2 is covered by the union of the two open sets O 1 and O 2 \ O ε .Thus, by (H1) and (H4) we have