Bloch spectral analysis in the class of non-periodic lami-nates

. In this work, we introduce Bloch waves to study the homogenization process in a class of simple laminates which are obtained as a particular Hashin-Shtrikman microstructure involving translations and dilations in only one direction. This makes this class of microstructures non necessarily periodic in the direction of lamination. We derive explicit formulae for the Bloch wave spectral representation of the homogenized coe ﬃ cients.


Introduction
This paper deals with Bloch wave spectral analysis of a class of non-periodic simple laminates. This work is in the same spirit as our earlier work [1], where we perform the Bloch wave spectral analysis in the class of generalized (non-periodic) Hashin-Shtrikman microstructures. Here we are interested in the Bloch spectral representation of the homogenized coefficients, where the inhomogeneities are governed with both non-uniform scales and transformations in one direction and maintains uniformity with respect to scales and translations in other directions, which concerns both Hashin-Shtrikman and laminates. As a special case, this also includes the case of periodic laminates.
(A n ) is said to converge in the sense of homogenization to a homogenized limit (G-limit) matrix A * ∈ M(α, β; Ω), i.e. A n G − → A * , if for any right hand side f ∈ H −1 (Ω), the sequence u n , solution of the problem where u is the solution of the homogenized equation −div(A * (x)∇u(x)) = f (x) in Ω, u(x) = 0 on ∂Ω.
The homogenized limit A * is locally defined and does not depend on the source term f or the boundary condition on ∂Ω.
Laminated microstructure: Let us now introduce the so-called layers or laminated materials. They are defined as those in which the properties and geometry of the medium vary in only one direction, in the sense that the sequence of matrices A n depends on a single space variable, say A n = A n (x · e r ) (where e r is some standard unit vector in R N ). The corresponding composite is called a laminate, generalizing the one-dimensional settings. In particular, if A n ∈ M(α, β; Ω) satisfies the assumption A n (x) = a n (x · e r )I a.e. x ∈ Ω, the standard G-convergence concept reduces to the usual weak convergence of some combinations of entries of the matrix A n (here, I denotes the identity matrix). Indeed, this yields another type of explicit formula for the homogenized matrix as in the one-dimensional case: A n G-converges to the homogenized matrix A * if and only if the following convergences hold in L ∞ (Ω)−weak* (see [2]): where A * j j is the arithmetic mean of a n (x · e r ) for 1 ≤ j ≤ N, j r and A * rr is the harmonic mean of a n (x · e r ).
Hashin-Shtrikman microstructures: Next, we introduce the Hashin-Shtrikman microstructures and its G-limit. In his book [3, page no. 281], L. Tartar introduces the notion of a homogeneous medium being equivalent to the micro-structured medium. We say that a microstructure A ω (y) = [a ω kl (y)] 1≤k,l≤N ∈ M(α, β; ω) (see (1)) is equivalent to M if after extending A ω by M in R N \ ω, it follows that for any λ ∈ R N there exists Remark 1.1 The function w λ defined in (2) can be equivalently obtained solving the following problem in ω: −div(A ω ∇w λ (y)) = 0 in ω, with the additional boundary condition We remark that, as viewed in this manner on ω, the above system is overdetermined since there exist too many boundary conditions. Indeed, the solution of problem (3) satisfies condition (4) if and only if A ω is equivalent to the matrix M. where the sets K n are finite or countable, such that for any n ∈ N we have (ε p,n ω + y p,n ) ∩ (ε q,n ω + y q,n ) = ∅ ∀p, q ∈ K n , p q, and meas Ω \ ∪ p∈K n (ε p,n ω + y p,n ) = 0.
Moreover, if we define κ n = sup p∈K n ε p,n , we have κ n → 0 as n → ∞. Definition 1.3 Let us consider a sequence of Vitali's coverings of Ω given in Definition 1.2, and a microstructure A ω which we assume to be equivalent to a given matrix M in the sense of Definition 1.1. Using (5)- (6), it is clear that for any n ∈ N and for almost every x ∈ Ω, there exists a unique p ∈ K n (depending on x) such that x ∈ ε p,n ω + y p,n and x−y p,n ε p,n ∈ ω. Then, we define x − y p,n ε p,n , p ∈ K n such that x ∈ ε p,n ω + y p,n , for a.e. x ∈ Ω.
An important convergence property for this kind of materials comes from the following analysis: If, for a given λ ∈ R N , we define v n λ ∈ L 2 (Ω) by v n λ (x) = ε p,n w λ x − y p,n ε p,n + λ · y p,n , p ∈ K n such that x ∈ ε p,n ω + y p,n , for a.e. x ∈ Ω, (8) where w λ is defined in (2), then, one has the following properties (see [3, Page no. 283]): Additionally, as n → ∞, one gets the following convergences: v n λ λ · x weakly in H 1 (Ω), Since the previous results are valid for any λ ∈ R N , by the definition of G-convergence, one can easily check that the entire sequence (A n ω ) satisfies One relevant thing to be noticed is that the G-limit does not depend on the choice of translations y p,n and scales ε p,n , as long as they are imposed to satisfy the Vitali's covering criteria (5) and (6). Let us give two examples of microstructures equivalent to a constant matrix.
and R ∈ (0, 1). We consider the microstructure where α, β ∈ R are known as core and coating conductivities, respectively. Then, A ω is equivalent to γI, where γ satisfies (see [3]): Given ρ 1 , ρ 2 ∈ (−m, ∞) such that ρ 1 < ρ 2 , let us consider ω = E ρ 2 +m 1 ,...,ρ 2 +m N = y : given by Following its construction, the coefficients in Hashin-Shtrikman microstructures are invariant in a certain way of both translations and dilations of the medium. Classical periodic microstructures incorporate uniform translations and uniform dilations with respect to only one scale ε, whereas Hashin-Shtrikman construction incorporates non-uniform translations and dilations with a family of scales {ε p } p . It is a non-periodic and non-commutative class of microstructures, where we would like to develop the Bloch wave spectral analysis. The importance of HS microstructures and their role in homogenization theory is well-known. They are among extreme points of the so-called G-closure set/phase diagram of mixtures of two-phase materials in a prescribed volume proportion. They are also solutions of optimal design problems (see [2,3]).
Let us now define the subclass of non-periodic laminates, contained as a particular case of the generalized Hashin-Shtrikman microstructures, which we are interested in studying. We will refer to them as HS laminates.
We now seek a positive constant m such that a S (y·e r ) is equivalent to m in the e r −direction. To do this, we extend a S (y · e r ) by m for y · e r ∈ R \ [−1, 1] and we study the existence of function w e r ∈ H 1 loc (R) satisfying: From Example 1.1, restricting it to the one-dimensional case (N = 1), we establish the existence of w e r ∈ H 1 loc (R) satisfying (10), insofar m is given by (the harmonic mean of α and β with proportion θ).
In the other directions e j ( j ∈ {1, . . . , r − 1, r + 1, . . . , N}), we define the microstructure by using periodic arrays in the following way: i) We define S ⊥ as the extension of the domain S by periodicity in all directions e j , j r, ii) We use a special sequence of Vitali's coverings (ε p,n , y p,n ) of Ω by reduced and disjoint copies of S ⊥ in the e r -direction such that, for any n ∈ N, we have (ε p,n S ⊥ + y p,n ) ∩ (ε q,n S ⊥ + y q,n ) = ∅ ∀p, q ∈ K n , p q, and meas Ω \ ∪ p∈K n (ε p,n S ⊥ + y p,n ) = 0, with κ n = sup p∈K n ε p,n → 0, for a finite or countable K n . These define the microstructures in A n S as follows: x − y p,n ε p,n · e r I in ε p,n S ⊥ + y p,n a.e. in Ω, p ∈ K n , which makes sense since, for each n, the sets ε p,n S ⊥ + y p,n , p ∈ K, are disjoint. The above construction (13) represents one subclass of non-periodic laminate microstructures in twophase medium. Consequently, one has the following G-convergence of the entire sequence A n S G − → A * S = diag(a S , . . . , a S , . . . , a S ), a S comes at only r-th diagonal entry, where we have denoted by a S and a S the arithmetic mean and harmonic mean of a S , respectively.
Due to the periodicity in the e j -directions ( j r), the homogenized conductivity (A * S ) j j ( j r) is defined by its entries: where, for each j r, χ j ∈ H 1 (S ) solves the following cell problem −div a S (y · e r )(∇χ j (y) + e j ) = 0 in S , y → χ j (y) is 1−periodic in each e j -direction ( j r).
Thus, the limit in (14) is well understood now.

Remark 1.2 As we see, the microstructures governed by (13) are periodic in (N − 1) directions and in one direction it includes one-dimensional Hashin-Shtrikman construction.
We would like now to perform a Bloch spectral analysis for this class of microstructures. Let us briefly explain what is behind Bloch spectral analysis. Following the classical Fourier approach for homogeneous media, one tries to diagonalize the operator and introduces Bloch waves (BW) as its eigenvectors in this approach. One invokes the Floquet approach to solve a parameter-dependent eigenvalue problem; namely, periodic media are multiplicative perturbations of homogeneous media. This gives rise to a generalized periodicity condition for Bloch waves. The homogenized matrix and the oscillatory test functions are obtained as infinitesimal versions of BW and its eigenvalue (energy), which lie at the lower end of the spectrum. It is found that BW (as well as its eigenvalues) carry a discrete (energy) index and a continuous (quasi-momentum) index. The significance lies in the fact that the corresponding eigenvalues represent different energy levels and their discreteness brings some simplification. Another nice feature is the regularity of the lower BW and their energy with respect to the momentum variables. It is no surprise that these two properties play an important role in the homogenization process. As far as we know, the problem of obtaining a BW-type basis is open to arbitrary microstructures, mainly because there might not exist any associated invariance. It is now appropriate to recall that we only need the lower part of the spectrum for homogenization, and even that seems to be an open problem. However, the BW method has already been successfully applied to periodic microstructures. This approach is also known as Bloch wave homogenization (or spectral approach), some references are [4][5][6][7][8]. We must note that the BW method defines a higher order approximation than homogenization. More precisely, exploiting the regularity of the BW fundamental eigenvalue with respect to the momentum variables, it is possible to define a dispersion approximation for periodic and HS media [9][10][11]. In general, the development of the BW approach is open for non-periodic microstructures. It is also used for the non-periodic generalized Hashin-Shtrikman microstructures in paper [1] and for the case of quasi-periodic media in [12]. The present work is a contribution in this direction. More precisely, we develop the BW method for the HS laminates defined above. We take A S (y) = a S (y · e r )I ∈ M(α, β; S ) defined in (9) to consider the following spectral problem parameterized by η ∈ R N : Find µ := µ(η) ∈ C and ϕ S := ϕ S (y; η) (not identically zero) such that where ν = ±e r is the outer normal unit vector on the boundary {y · e r = ±1} and dσ is the surface measure on {y · e r = ±1}.
Here, c is a floating constant depending on the element under consideration. L 2 c,# (S ) and H 1 c,# (Y) are subspace of L 2 (S ) and H 1 (S ) respectively, the second inclusion being proper. They inherit the subspace norm-topology of the parent space.
The motivation for the state space H 1 c,# starting from a S (y · e r ) is equivalent to m in the e rdirection, but independent of other y j variables ( j r). So, in particular, in other directions it is periodic with any period, for convenience we take 2−periodicity (the size of the edges of the reference cell S ).
As a next step, we give the weak formulation of the problem (17) in these function spaces.
We are interested in proving the existence of eigenvalues and their corresponding eigenvectors (µ(η), ϕ S (y; η)) with µ(η) ∈ C and ϕ S (·; η) ∈ H 1 c,# (S ) of where the bilinear forms a S (η)(·, ·) and (·, ·) are defined by a S (η)(u, w) = S a S (y · e r )(y) ∂u ∂y k + iη k u ∂w ∂y k + iη k w dy, Regularity of the ground state: In the next proposition, we establish a regularity result for the ground state on the basis of Kato-Rellich spectral analysis. The proof can be found in [1], where the following notation is used.
Notation 2.1 For integers k, l ∈ {1, . . . , N}, we use the symbols D k , D 2 kl , . . . to denote the derivatives ∂ ∂η k , ∂ 2 ∂η k ∂η l , . . . respectively. Additionally, for a given multi-index = Proposition 2.2 1. Zero is the first eigenvalue of (18) at η = 0 and it is an isolated point of the spectrum with its algebraic multiplicity equals to one.
For k = r, we seek D r ϕ S ,1 (y; 0) = 1 i d dy r ϕ S ,1 (y r ; 0) solving the equation (22) uniquely with the Dirichlet boundary condition (23). In that case, (22) becomes an ordinary differential equation and gets identified with (10) to give 1 i d dy r ϕ S ,1 (y r ; 0) = i|S | −1/2 (w e r (y r ) − y r ) satisfying (25) also. For k r, we notice that D k ϕ S ,1 (y; 0) = 0 is the unique solution of the above system of equations.
Evaluating at η = 0 and using the computation from the previous steps, it simply becomes 1 2 D 2 ii µ 1 (0) = 1 |S | S a S (y · e r )dx − 1 |S | S C S r (w e r (y r ) − y r )dy = m = a S , and 1 2 D 2 j j µ 1 (0) = 1 |S | S a S (y · e r )dx = a S ∀ j r, which are indeed the homogenized coefficients governed with the simple laminates in twophase medium stated in (14). As a next step, one can define the first Bloch transform likewise in [1] and successively perform the limit analysis to obtain the homogenization result as derived in [1]. We briefly mention those results here.
3 Bloch waves at ε p,n −scales, y p,n −translations and Bloch transform Motivated by the HS construction (13), we introduce the operator: x − y p,n ε p,n · e r ∂ ∂x k , x ∈ ε p,n S ⊥ + y p,n , and its shifted version x − y p,n ε p,n · e r ∂ ∂x k + iξ k , x ∈ ε p,n S ⊥ + y p,n .

Homogenization result
We state here our homogenization result. This can be proved by using Bloch waves as introduced at the beginning of this section (for more details, see [1]).
Theorem 3.1 Let us consider Ω be an open bounded subset of R N . We consider the operator A n S defined in (26). Let f ∈ L 2 (Ω) and u n ∈ H 1 0 (Ω) be the unique solution of the boundary value problem A n S u n = f in Ω. Then there exists u ∈ H 1 0 (Ω) such that the sequence u n weakly converges to u in H 1 0 (Ω) with the following convergence of flux σ n S = A n S ∇u n A * S ∇u = σ S weakly in L 2 (Ω) N .
In particular, the limit u satisfies the homogenized equation: where A * S is defined in (14).