Nonlinear Analysis: Real World Applications 18 (2014) 75–85

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Nonlinear Analysis: Real World Applications

journal homepage: www.elsevier.com/locate/nonrwa

A variational approach to the homogenization oflaminate metamaterialsHélia Serrano ∗

Departamento de Matemáticas, E.T.S.I. Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain

a r t i c l e i n f o

Article history:Received 15 March 2012Accepted 13 January 2014

a b s t r a c t

This paper proposes a variational approach to study the asymptotic behaviour of themagnetic induction response of composite materials with a laminate microstructurewhose components have negative magnetic permeability and different positive electricpermittivity. A procedure to compute explicitly the effective electric permittivity ofcomposites made by alternate layers of two single-negative materials is presented.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Homogenization aims at describing mathematically the macroscopic behaviour of microstructures. Microstructures arestructures in a fine length scale between the macroscopic scale and the atomic scale. Many microstructures may be seenin nature, but there are many others which come from industrial processes. Such an example is artificial composites, i.e.materials made by mixing different materials at small scales which happen to be homogeneous materials at macroscopicscale. See [1]. The properties of composites may be described by equations of classical physics at the microscopic lengthscale where the mixture of different components takes place. Nevertheless, their effective properties at macroscopic levelmay be obtained when the small length scales converge to 0. Homogenization comes in at this stage. See [2–7].

In the previous work [8], we have focused on general inhomogeneous anisotropic composites with periodicmicrostructure, whose magnetic properties at microscopic scale are described by the stationary Maxwell equations withperiodic oscillatory coefficients. Precisely, we have obtained the effective magnetic properties at macroscopic scale throughthe homogenization of the vector potential formulation of Maxwell’s equations in magnetism. These composites are calledstatic since their properties do not change with time. See [9].

In this work, we want to go further and study the electromagnetic properties of a special type of composites: laminatemetamaterials. Metamaterials are artificial composites with a periodic or non-periodic structure exhibiting extraordinaryelectromagnetic properties which cannot be found in natural composites. The termmetamaterialwas introduced by RodgerM. Walser, University of Texas at Austin, in 1999 to define initially macroscopic composites having a synthetic, three-dimensional, periodic cellular architecture designed to produce an optimized combination, not available in nature, of two or moreresponses to specific excitation. See [10,11]. Metamaterials may basically be divided into two categories: the electromagnetic(or photonic) crystals, and the effective media. The electromagnetic crystals are structures made of periodic micro- or nano-inclusions whose period is of the same order as the signal wavelength. The effective media are structures whose period ismuch smaller than the signalwavelength so that their electromagnetic properties can be described by using homogenizationtheory. See [12].

The electric permittivity (ε) and the magnetic permeability (µ) are two functions used to characterize, respectively, theelectric and magnetic properties of materials interacting with electromagnetic fields. The majority of materials in nature

∗ Tel.: +34 635789787.E-mail address: heliac.pereira@uclm.es.

1468-1218/$ – see front matter© 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.nonrwa.2014.01.001

76 H. Serrano / Nonlinear Analysis: Real World Applications 18 (2014) 75–85

have positive permittivity and positive permeability so that they are called double-positive (DPS) media. On the other hand,materials with negative permittivity and negative permeability are named double-negative (DNG) media. However, so farthey cannot be found in nature. In the middle range, materials with only one negative parameter are referred to as single-negative (SNG) media, and particularly they can be designated either as epsilon-negative (ENG) or mu-negative (MNG).For instance, cold plasma and silver have negative permittivities at microwave and optical frequencies, respectively, andferromagnetic materials have negative permeability in the VHF and UHF regimes.

Let us consider an inhomogeneous composite with a periodic microstructure, with relative size 1/h (h ∈ N), occupyinga region Ω in R3, whose electric permittivity εh is a strictly positive space-dependent function while the magneticpermeability µh is a strictly negative one. The electromagnetic properties of such a medium at microscopic scale aremodelled by the non-stationary Maxwell equations

∂t D(x, t) − curlH(x, t) = −J(x, t)∂t B(x, t) + curl E(x, t) = 0div B(x, t) = 0divD(x, t) = ρ(x, t)

(1.1)

in Ω × (0, T ), for some T > 0, where D and B stand for, respectively, the electric and magnetic induction, E and H stand forthe electric and magnetic field, respectively, J is the given current density while ρ denotes the charge density. See [13]. Thissystem of four equations with four unknowns may be reduced to a system with only two unknowns if we take into accountthe constitutive relations between the fields, i.e.

D(x, y) = εh(x)E(x, t)H(x, t) = µ−1

h (x)B(x, t),

in Ω × (0, T ), where µ−1h stands for the inverse of µh. Namely, system (1.1) is equivalent to the following one

∂t (εh(x)E(x, t)) − curlµ−1

h (x)B(x, t)

= −J(x, t)∂t B(x, t) + curl E(x, t) = 0div B(x, t) = 0div (εh(x)E(x, t)) = ρ(x, t),

(1.2)

whose unknowns are the electric field E and the magnetic induction B. Notice that, the fourth equation jointly with the firstone lead us to the conservation of electric charge

∂t ρ(x, t) + div J(x, t) = 0 in Ω × (0, T ).

Since we are considering a bounded region Ω with a perfectly conducting boundary ∂Ω , the boundary conditions

E(x, t) × n = 0 and B(x, t) · n = 0 on ∂Ω × (0, T ), (1.3)

where n stands for the unit outward normal vector to ∂Ω , should be added to system (1.2).Once the magnetic induction B is known, we may reconstruct the electric field E from the first equation of system (1.2)

written as

∂t E(x, t) = ε−1h (x) curl

µ−1

h (x)B(x, t)− ε−1

h (x)J(x, t), (1.4)

whenever the electric permittivity εh has an inverse ε−1h and does not depend on time. Precisely,

E(x, t) = E(x, 0) + ε−1h (x)

curl

µ−1

h (x) t

0B(x, τ ) dτ

−

t

0J(x, τ ) dτ

for any (x, t) ∈ Ω × (0, T ). Thus, if we derive in time the second equation of system (1.2) and replace ∂t E by its expressionin (1.4), system (1.2) coupled with the boundary conditions (1.3) reduces to

∂2t B(x, t) + curl

ε−1h (x) curl

µ−1

h (x)B(x, t)

= curlε−1h (x)J(x, t)

in Ω × (0, T )

div B(x, t) = 0 in Ω × (0, T )B(x, t) · n = 0 on ∂Ω × (0, T )

ε−1h curl

µ−1

h B× n = ε−1

h J × n on ∂Ω × (0, T )

(1.5)

where the magnetic induction B is the only unknown, and the given current density J is so that div J = −∂t ρ. Regarding theinitial and final data, we shall assume that

B(x, 0) = B0(x) and ∂t B(x, 0) = ∂t B(x, T ) = 0 in Ω, (1.6)

given a divergence-free field B0 inΩ , sincewewill treat this second-order initial–boundary value problem from a variationalpoint of view.

Recall that, we are specially interested in describing the electromagnetic properties at macroscopic scale of a class ofcomposites called laminates, i.e. materials with amicrostructure formed by, at least, twomaterials placed in alternate layers

H. Serrano / Nonlinear Analysis: Real World Applications 18 (2014) 75–85 77

with a prescribed thickness. So, we consider two materials with different strictly positive electric permittivity, ε1 and ε2,and same negative magnetic permeability, and place them into alternate layers with relative thickness α/h and (1 − α)/h,respectively. The inverse electric permittivity ε−1

h of the composite is given by

ε−1h (x) = ε−1

1 χ(0,α) (hx · e) + ε−12

1 − χ(0,α) (hx · e)

,

where the function χ(0,α) is the characteristic function of interval (0, α) over (0, 1), and e stands for a unit vector in R3,which gives the direction of lamination. The inverse magnetic permeabilityµ−1

h (x) = −β(x) inΩ , for some strictly positivebounded functionβ . Therefore, we are interested in the homogenization (as h goes to∞) of initial–boundary value problemsof type

−∂2t u(x, t) + curl

ε−1h (x) curl (β(x)u(x, t))

= − curl Jh (x, t) in Ω × (0, T )

div u(x, t) = 0 in Ω × (0, T )u(x, t) · n = 0 on ∂Ω × (0, T )ε−1h curlβu

× n = − Jh × n on ∂Ω × (0, T )

u(x, 0) = u0(x) in Ω

∂t u(x, 0) = ∂t u(x, T ) = 0 in Ω,

(1.7)

where the current density is given by

Jh(x, t) = J1(t)χ(0,α) (hx · e) + J2(t)1 − χ(0,α) (hx · e)

,

with J1, J2 ∈ L2(0, T ; R3), and u0 is a given divergence-free function in L2(Ω; R3). The main feature here is the variationalapproach we propose to study this type of second-order differential equations, which depends crucially on the positive signof the electric permittivity εh and the negative sign of the magnetic permeability µh so that β is strictly positive. Precisely,problem (1.7) turns out to be the first-order optimality condition associated with a quadratic functional defined in a Banachspace.

Proposition 1.1. Let β be a strictly positive bounded function. For each h fixed, if uh in V is a minimizer of the energy Ih definedby

Ih(u) =

T

0

Ω

β(x)2

|∂t u|2 +ε−1h (x)2

|curl (βu)|2 + Jh(x, t) · curl (βu)

dx dt, (1.8)

then uh is a weak solution of the initial–boundary value problem (1.7).

The admissible function space V is defined by

V =u ∈ L2(0, T ; X(Ω)) : ∂tu ∈ L2(0, T ; L2(Ω; R3))

.

For a given strictly positive bounded function β : Ω → (0, ∞), the space

X(Ω) =v ∈ L2(Ω; R3) : curl (βv) ∈ L2(Ω; R3), div v = 0 in Ω, v · n = 0 on ∂Ω

is a Hilbert space, with respect to the norm ∥v∥X(Ω) =

∥v∥

2L2(Ω)

+ ∥curl (βv) ∥2L2(Ω)

12, compactly embedded in L2(Ω; R3),

see [14,15]. (See also [16–19].) Therefore, V is a Banach space, with respect to the norm

∥u∥V = ∥u∥L2(0,T ;X(Ω)) + ∥∂t u∥L2(0,T ;L2(Ω;R3))

=

T

0∥u(t)∥2

X(Ω) dt 1

2

+ ∥∂t u∥L2(0,T ;L2(Ω;R3)),

compactly embedded in L2(0, T ; L2(Ω; R3)), see [20,21].In this way, we aim to study the asymptotic behaviour, as h goes to ∞, of the sequence of weak solutions uh of the

family of problems (1.7) from a variational point of view through the study of the asymptotic behaviour of the sequenceof minimizers of the associated family of energies Ih. For such purpose, we use a variational convergence for functionalscalled Γ -convergence, see [22–24].

Notice that, the family of quadratic energies Ih in (1.8) is a special example of amore general class of integral functionalsof type

Ih(u) =

T

0

Ω

W (ah(x, t), ∂t u(x, t), curl (β(x)u(x, t))) dx dt, (1.9)

where the functionW : Rm× R3

× R3→ R is continuous (m ∈ N), and satisfies the conditions:

(i) there exist constants C ≥ c > 0 such that

c|γ |

2+ |ρ|

2− 1

≤ W (λ, γ , ρ) ≤ C

|γ |

2+ |ρ|

2+ 1

, ∀ (λ, γ , ρ) ∈ Rm

× R3× R3

;

78 H. Serrano / Nonlinear Analysis: Real World Applications 18 (2014) 75–85

(ii) there exists a constant c > 0 so that

|W (λ, γ1, ρ1) − W (λ, γ2, ρ2)| ≤ c|γ1 − γ2|

2+ |ρ1 − ρ2|

2 , ∀ λ ∈ Rm, γ1, γ2, ρ1, ρ2 ∈ R3.

We assume here that the sequence ah stands for a sequence of laminates normal to a given unit vector e in R3, i.e. thefunction ah : Ω × (0, T ) → Rm is defined by

ah(x, t) = a1(t)χ(0,α) (hx · e) + a2(t)1 − χ(0,α) (hx · e)

with a1, a2 ∈ L∞(0, T ; Rm). Our main achievement in this work is the following result on the Γ -convergence of sequencesof functionals of type (1.9). Particularly, the Γ -limit density is characterized through a finite-dimensional minimizationproblem depending on the densityW and the values of the sequence ah.

Theorem 1.2. The sequence of functionals Ih defined in (1.9) is Γ -convergent to the functional I defined by

I(u) =

T

0

Ω

W (t, ∂tu(x, t), curl (β(x)u(x, t))) dx dt,

where the homogenized density W : (0, T ) × R3× R3

→ R is given by

W (t, Λ, Θ) = infAi∈R3

i∈1,2,3,4

αCW (a1(t), A1, A2) + (1 − α)CW (a2(t), A3, A4) :

Λ = αA1 + (1 − α)A3(A1 − A3) · e = 0Θ = αA2 + (1 − α)A4(A2 − A4) · e = 0

.

The function CW (λ, ·, ·) stands for the convex hull of W (λ, ·, ·) in R3× R3, for every λ ∈ Rm.

The proof of this main result is based on the properties of Young measures associated with relevant sequences offunctions. Namely, we achieve the previous representation of the Γ -limit density applying the slicing decomposition ofthe joint Young measure (see [25–27]) associated with sequences of type (ah, ∂t uh, curl (βuh)) to prove in two steps alower limit estimate and the existence of a recovering sequence. These ideas were introduced and developed in previousworks [8,28–30]. From this result we may deduce the Γ -convergence of the sequence of quadratic energies in (1.8) as wellas the asymptotic behaviour of the sequence of its minimizers.

In Section 5, the particular case of a laminate composite material formed by alternate layers of two single-negativematerials with positive electric permittivity, ε1 = 7 and ε2 = 3.4, respectively, and constant negative permeabilityµ ≡ −1,is studied. Precisely, the following result is a corollary of Proposition 5.1 jointly with Theorem 3.1.

Proposition 1.3. Let ε1 = 7, ε2 = 3.4, µ(x) = −1, α = 1/3 and J(x, t) = (1, 1, 1) for every (x, t) ∈ Ω × (0, T ). If Bh is thesolution of the initial–boundary value problem (1.5), coupled with the initial and final values (1.6), then the sequence of solutionsBh is such that curl Bh and ∂t Bh weakly converge, as h goes to ∞, to curl B and ∂t B, respectively, in L2(Ω × (0, T ); R3),where B is the solution of the homogenized system

∂2t B(x, t) − curl

ε−1hom curl B(x, t)

= 0 in Ω × (0, T )

div B(x, t) = 0 in Ω × (0, T )B(x, t) · n = 0 on ∂Ω × (0, T )

(−ε−1hom curl B) × n = ε−1

homJ × n on ∂Ω × (0, T )B(x, 0) = B0(x) in Ω

∂t B(x, 0) = ∂t B(x, T ) = 0 in Ω,

(1.10)

where the effective coefficient ε−1hom is a 3 × 3-matrix defined by

ε−1hom =

29119

0 0

0523

0

0 0523

.

Therefore, wemay conclude that, when the number of layers converges to infinity, this particular composite behaves likean anisotropic homogeneous material with negative magnetic permeability µ constant equal to −1, and effective electricpermittivity

εhom =

11929

0 0

0235

0

0 0235

.

H. Serrano / Nonlinear Analysis: Real World Applications 18 (2014) 75–85 79

This work is divided into five sections as follows. In Sections 2 and 3we present an overview on themain notions, resultsand techniques to be used in later sections, in order to be a self-containedwork. Section 2 is dedicated to the Youngmeasuretheory while Section 3 is a brief resume on the notion of Γ -convergence. Proposition 1.1 and Theorem 1.2 are proved inSections 3 and 4, respectively.

2. Laminates and div-Young measures

The main tool used here to study the Γ -convergence of integral functionals is the notion of Young measure (see [27]),provided it is a useful tool to treat minimum problems in the Calculus of Variations. Young measures are families ofprobability measures which, often associated with oscillating sequences, describe their oscillatory behaviour and givea representation of the limits of the composition with non-linear quantities. The following result is a corollary of theFundamental Theorem for Young measures, see [31].

Theorem 2.1. Let p > 0, and vh : D ⊂ Rn→ Rd be measurable functions such that

suph∈N

D|vh(y)|p dy < ∞.

Then, there exist a subsequence vhk and a family νyy∈D of probability measures on Rd such that, for any Carathéodory functionf : D × Rd

→ R bounded from below, it holds

limk→∞

Df (y, vhk(y)) dy =

D

Rd

f (y, λ) dνy(λ) dy

if and only if the sequence f (·, vhk(·)) is equi-integrable.

The family of probability measures νyy∈D is called the Young measure associated with the sequence vhk. As stated inthe previous theorem, whenever the sequence f (·, vhk(·)) is equi-integrable, we may represent its weak limit in L1(D) bymeans of the Young measure associated with the sequence vhk. Moreover, if the sequence vh is bounded in Lp(D; Rd),and f : Rd

→ R is a continuous function such that |f (λ)| ≤ c(|λ|q+ 1), for some constant c > 0 and 0 < q < p, then

the sequence f (vhk(·)) converges weakly to the function f in Lp/q(D) defined by f (y) =

Rd f (λ) dνy(λ). See [25,26,32].Eventhough we do not have equi-integrability, we still have the following lower limit inequality.

Proposition 2.2. If the sequence vh generates the Young measure νyy∈D, then

lim infh→∞

Ef (y, vh(y)) dy ≥

E

Rd

f (y, λ) dνy(λ) dy,

for every Carathéodory function f : E × Rd→ R, and every measurable subset E ⊂ D.

In this work, we are particularly interested in Young measures associated with sequences of pairs and their slicingdecomposition.

Lemma 2.3. Let E ⊂ Rm and F ⊂ Rd be open sets, ν be a positive Radon measure on E × F , and σ be its projection onto E,which is also a Radon measure. Then, for σ -a.e. λ ∈ E, there exists a probability measure µλ on F such that, for every boundedcontinuous function f on E × F ,

(1) the map λ →F f (λ, ρ) dµλ(ρ) is σ -measurable;

(2) it holdsE×F f (λ, ρ) dν(λ, ρ) =

E

F f (λ, ρ) dµλ(ρ)

dσ(λ).

The strong convergence of a sequence of functions in a Lebesgue space is a necessary and sufficient condition for itsassociated Young measure be a Dirac measure concentrated on the strong limit. When we have strong convergence only forsome components of the sequence it is possible to characterize its associated Young measure as follows.

Proposition 2.4. If the sequence wh generates the Young measure σyy∈D, and the sequence zh converges strongly inLp(D; Rd) to z, then the sequence of pairs (wh, zh) generates the Young measure νyy∈D defined by νy = δz(y) ⊗ σy, for a.e.y ∈ D.

Moreover, we are interested in a special type of Young measures, called div-Young measures, generated by sequences ofdivergence-free functions. A family of probabilitymeasures νxx∈Ω supported onRd is said to be a div-Youngmeasure if it isgenerated by a weak convergent sequence Uh in Lp(Ω; Rd) such that divUh = 0 inΩ ⊂ Rn, for every h ∈ N. The followingresult gives us a characterization of div-Young measures based on the first moment and the moment of order p, and it isa particular case of a general result on the characterization of Young measures associated with sequences of solutions ofpartial differential equations, see [33, Theorem 4.1].

80 H. Serrano / Nonlinear Analysis: Real World Applications 18 (2014) 75–85

Theorem 2.5. Let 1 ≤ p < ∞, and νxx∈Ω be a weakly measurable family of probability measures on Rd. There exists a p-equi-integrable sequence Uh in Lp(Ω; Rd) such that divUh = 0 in Ω , and generates the Young measure νxx∈Ω , if and only if

(1) there exists U ∈ Lp(Ω; Rd) such that divU = 0 in Ω , and U(x) =

Rd λ dνx(λ) for a.e. x ∈ Ω;(2)

Ω

Rd |λ|

p dνx(λ) dx < ∞.

If we consider a sequence of functions uh =u1h, u

2h, u

3h

in L2(Ω; R3) such that curl uh converges weakly to curl u in

L2(Ω; R3), where the curl operator is defined by

curl uh =

∂u3

h

∂x2−

∂u2h

∂x3,∂u1

h

∂x3−

∂u3h

∂x1,∂u2

h

∂x1−

∂u1h

∂x2

,

then the sequence curl uh generates a div-Youngmeasure because div (curl uh) = 0 inΩ , for every h ∈ N. Another exampleof a div-Young measure is the Young measure generated by a laminate. Consider the sequence of functions Uh : Ω → Rn

defined by putting

Uh(x) = A1χ(0,α)(hx · e) + A2(1 − χ(0,α)(hx · e)),

where χ(0,α)(y) is the characteristic function of the interval (0, α) over (0, 1), extended by periodicity to R, and e is a unitvector in Rn, such that it generates the homogeneous Young measure ν = αδA1 + (1 − α)δA2 supported on A1, A2. Noticethat div Uh = div

(A1 − A2)χ(0,α)

= 0 in Ω if (A1 − A2) · e = 0.

To conclude this section, we recall a useful result on the approximation of sequences of divergence-free functions, whichis a particular case of [33, Lemma 2.15] (see also [34, Proposition 2.3]).

Proposition 2.6. Let 1 < p < ∞. Let Uh be a bounded sequence in Lp(Ω; Rd) such that divUh converges strongly to 0 inW−1,p(Ω), Uh converges weakly to U in Lp(Ω; Rd), and Uh generates the Young measure νxx∈Ω . Then, there exists a p-equi-integrable sequence Zh ⊂ Lp(Ω; Rd) such that

div Zh = 0 in Ω,

Ω

Zh(x) dx =

Ω

U(x) dx, ∥Zh − Uh∥Lq(Ω;Rd) → 0, for all 1 ≤ q < q,

and Zh also generates the Young measure νxx∈Ω .

3. Γ -convergence of functionals

We aim to study the homogenization of a class of second-order initial–boundary value problems through the Γ -convergence of the associated sequence of quadratic energies. Γ -convergence is a variational convergence for sequences offunctionals based on the analysis of the asymptotic behaviour of minima problems. See [22–24].

Definition 3.1. Let (V , d) be a metric space, and Ih be a sequence of functionals Ih : V → [−∞, +∞]. We say that thesequence Ih Γ (d)-converges to I in V if, for any u ∈ V , it holds

(1) for every sequence uh ⊂ V such that d(uh, u) → 0, then

I(u) ≤ lim infh→∞

Ih(uh);

(2) there exists a sequence uh ⊂ V , such that d(uh, u) → 0, for which

I(u) = limh→∞

Ih(uh).

In order to prove that a sequence of functionals Ih Γ -converges to a functional I , we proceed to prove, from a non-standard framework, that the previous conditions (1) and (2) are fulfilled. Namely, we apply the Young measure theoryto achieve the explicit representation of the Γ -limit density by means of the Young measure associated with relevantsequences of functions. This procedure was introduced in [28] to treat the Γ -convergence of sequences of (non necessarilyperiodic) functionals defined for curl-free functions, and expanded in [29] to study the Γ -convergence on divergence-freefields, see also [8,30]. Recall that Γ -convergence on divergence-free fields was firstly treated in [35] from the standardframework of Γ -convergence.

Once we have studied and achieved the representation of the Γ -limit I of the sequence Ih, we may deduce theasymptotic behaviour of the sequence of minimizers uh of Ih in V , applying the next important result. See [22].

Theorem 3.1. If Ih is a sequence of equicoercive functionals in V such that it Γ (d)-converges to I in V , then the functional Ihas a minimum in V , and

minu∈V

I(u) = limh→∞

infu∈V

Ih(u).

Moreover, if the sequence uh converges and limh→∞ Ih(uh) = limh→∞ infu∈V Ih(u), then the limit of uh is a minimum of I.

H. Serrano / Nonlinear Analysis: Real World Applications 18 (2014) 75–85 81

Let us present here briefly the proof of Proposition 1.1, which is the starting point to study the homogenization ofinitial–boundary value problems of type (1.7) through the Γ -convergence of the sequence of associated energies.

Proof of Proposition 1.1. Let uh ∈ V be such that

Ih(uh) = infu∈V

Ih(u).

Thus, for any ϕ ∈ V , we have

∂

∂sIh(uh + sϕ)|s=0 = 0,

i.e. T

0

Ω

β(x) ∂t uh · ∂t ϕ + ε−1

h (x) curl (β(x)uh) · curl (β(x)ϕ) + Jh(x, t) · curl (β(x)ϕ)dx dt = 0.

Applying the Green formula, it follows thatΩ

β(x) (∂t uh(x, T ) · ϕ(x, T ) − ∂t uh(x, 0) · ϕ(x, 0)) dx −

T

0

Ω

β(x) ∂2t uh · ϕ dx dt

+

T

0

∂Ω

ε−1h (x)curl (β(x)uh)

× n · (β(x)ϕ) dx dt

+

T

0

Ω

curlε−1h (x) curl (β(x)uh)

· (β(x)ϕ) dx dt

= −

T

0

∂Ω

Jh(x, t) × n · (β(x)ϕ) dx dt −

T

0

Ω

curl Jh(x, t) · β(x)ϕ dx dt.

Notice that, for any function v ∈ L2(Ω; R3) such that curl v ∈ L2(Ω; R3), there exists the tangential trace v × n inH−1/2(∂Ω; R3) so that

Ω

v(x) · curlϕ(x) dx =

∂Ω

(v(x) × n) · ϕ(x) dx +

Ω

curl v(x) · ϕ(x) dx.

If we assume that ∂t uh(x, 0) = ∂t uh(x, T ) = 0 in Ω , andε−1h (x)curl (β(x)uh)

× n = −Jh(x, t) × n on ∂Ω × (0, T ),

then, from the arbitrariness of ϕ in V , the minimizer uh satisfies the system

−∂2t uh(x, t) + curl

ε−1h (x) curl (β(x)uh(x, t))

= − curl Jh (x, t) in Ω × (0, T )

div uh(x, t) = 0 in Ω × (0, T )uh(x, t) · n = 0 on ∂Ω × (0, T )ε−1h curlβuh

× n = − Jh × n on ∂Ω × (0, T )

uh(x, 0) = u0(x) in Ω

∂t uh(x, 0) = ∂t uh(x, T ) = 0 in Ω.

4. Proof of Theorem 1.2

The proof is basically divided into two parts. Firstly, we will prove the lower bound estimate. Then, we will prove theexistence of a recovering sequence for which the lower bound obtained previously is attained. The main ideas behind thisproof lie on the characterization of the Young measure associated with sequences of divergence-free fields.

Proof. 1st part. Let us consider any field u in V . Take any bounded sequence uh in V such that curl (βuh) and ∂t uh

converge weakly to curl (βu) and ∂t u, respectively, in L2(Ω × (0, T ); R3). Let ν =νx,tx∈Ω,t∈(0,T )

be the Young measureassociated with the sequence of triples (ah, ∂t uh, curl (βuh)). Notice that the sequence ah generates the Young measureσ = σtt∈(0,T ) given by

σt(λ) = α δa1(t)(λ) + (1 − α) δa2(t)(λ), for a.e. t ∈ (0, T ).

Therefore, for a.e. (x, t) ∈ Ω × (0, T ), we may project the measure νx,t onto R3× R3 so that there exists, for σt-a.e.

λ ∈ Rm, a probability measure µλ,x,t supported on R3× R3 for which the probability measure νx,t may be decomposed as

νx,t(λ, γ , ρ) = µλ,x,t(γ , ρ) ⊗ σt(λ),

i.e.

νx,t(λ, γ , ρ) = α µa1(t),x(γ , ρ) ⊗ δa1(t)(λ) + (1 − α) µa2(t),x(γ , ρ) ⊗ δa2(t)(λ).

82 H. Serrano / Nonlinear Analysis: Real World Applications 18 (2014) 75–85

Thus, it follows from Proposition 2.2 (see also [26, Theorem 6.11]),

lim infh→∞

T

0

Ω

W (ah(x, t), ∂tuh(x, t), curl uh(x, t)) dx dt

≥

T

0

Ω

R3×R3

αW (a1(t), γ , ρ) dµa1(t),x(γ , ρ) +

R3×R3

(1 − α)W (a2(t), γ , ρ) dµa2(t),x(γ , ρ)

dx dt

≥

T

0

Ω

R3×R3

αCW (a1(t), γ , ρ) dµa1(t),x(γ , ρ) +

R3×R3

(1 − α)CW (a2(t), γ , ρ) dµa2(t),x(γ , ρ)

dxdt,

where CW (λ, ·, ·) stands for the convex hull of W (λ, ·, ·) in R3× R3, for every λ ∈ Rm. Since the function CW (λ, ·, ·) is

convex in R3× R3, we may apply the Jensen inequality so that T

0

Ω

R3×R3

αCW (a1(t), γ , ρ) dµa1(t),x(γ , ρ) +

R3×R3

(1 − α)CW (a2(t), γ , ρ) dµa2(t),x(γ , ρ)

dx dt

≥

T

0

Ω

(αCW (a1(t), ϕ1(x, t), φ1(x, t)) + (1 − α)CW (a2(t), ϕ2(x, t), φ2(x, t))) dx dt, (4.1)

where we have defined ϕi : Ω × (0, T ) → R3 and φi : Ω × (0, T ) → R3 by putting

ϕi(x, t) =

R3×R3

γ dµai(t),x(γ , ρ), φi(x, t) =

R3×R3

ρ dµai(t),x(γ , ρ),

respectively, for i = 1, 2.Notice that, the Young measure θ = θx,tx∈Ω,t∈(0,T ) supported on R3 associated with the sequence ∂t uh may be

characterized as

θx,t(γ ) = α µa1(t),x(γ , ρ) ⊗ δa1(t)(λ) + (1 − α) µa2(t),x(γ , ρ) ⊗ δa2(t)(λ),

for a.e. (x, t) ∈ Ω × (0, T ), such that the weak limit ∂t u may be written as

∂t u(x, t) = α ϕ1(x, t) + (1 − α) ϕ2(x, t).

Since we are assuming that div u = 0 in Ω , it follows that θ is a div-Young measure.Moreover, if η = ηx,tx∈Ω,t∈(0,T ) stands for the Young measure supported on R3 associated with the sequence

curl (βuh), then it is also a div-Young measure and may be defined by

ηx,t(ρ) = α µa1(t),x(γ , ρ) ⊗ δa1(t)(λ) + (1 − α) µa2(t),x(γ , ρ) ⊗ δa2(t)(λ),

so that the weak limit curl (βu) may be written as

curl (β(x)u(x, t)) = α φ1(x, t) + (1 − α) φ2(x, t)

for a.e. (x, t) ∈ Ω × (0, T ).Taking into account the above constraints and the arbitrariness of the sequence uh, we may obtain a lower bound for

the integral in (4.1) minimizing the density in ϕi and φi, with i = 1, 2. In this way, wemay establish the following inequality

lim infh→∞

T

0

Ω

W (ah(x, t), ∂t uh(x, t), curl (β(x)uh(x, t))) dx dt

≥

T

0

Ω

W (t, ∂t u(x, t), curl (β(x)u(x, t))) dx dt,

where the homogenized densityW : (0, T ) × R3× R3

→ R is defined by

W (t, Λ, Θ) = infAi∈R3

i∈1,...,4

αCW (a1(t), A1, A2) + (1 − α)CW (a2(t), A3, A4) :

Λ = αA1 + (1 − α)A3(A1 − A3) · e = 0Θ = αA2 + (1 − α)A4(A2 − A4) · e = 0

.

Notice that, the second (respectively fourth) constraint on vectors A1 and A3 (respectively A2 and A4) is a necessary conditionin order to build a recovering sequence of divergence-free fieldswhich alternate their values betweenA1 andA3 (respectivelyA2 and A4) in layers normal to the unit vector e.

2nd part. Let us consider any field u in V . Applying [26, Lemma 7.9], for each j ∈ N, there exist a set of points x(j)k ⊂ Ω\N ,

with |N| = 0, and positive numbers r (j)k such that

x(j)k + r (j)

k Ω

kis a family of pairwise disjoint sets,

Ω =

k

x(j)k + r (j)

k Ω

N,

H. Serrano / Nonlinear Analysis: Real World Applications 18 (2014) 75–85 83

and T

0

Ω

W (t, ∂t u(x, t), curl (β(x)u(x, t))) dx dt

= limj→∞

k

T

0Wt, ∂t u(x

(j)k , t), curl

β(x(j)

k )u(x(j)k , t)

x(j)k + r (j)

k Ω

dt.For each k, j ∈ N and t ∈ (0, T ), let Ai

k,ji∈1,...,4 be the optimal solution of the previous minimization problem so that

Wt, ∂t u(x

(j)k , t), curl

β(x(j)

k )u(x(j)k , t)

= α CW (a1(t), A1

k,j, A2k,j) + (1 − α) CW (a2(t), A3

k,j, A4k,j),

and

∂t u(x(j)k , t) = αA1

k,j + (1 − α)A3k,j, (A1

k,j − A3k,j) · e = 0,

curlβ(x(j)

k )u(x(j)k , t)

= αA2

k,j + (1 − α)A4k,j, (A2

k,j − A4k,j) · e = 0.

(4.2)

Since CW (λ, ·, ·) is the convexification of W (λ, ·, ·) in R3× R3, it follows from the Carathéodory theorem that there exist

probability measures µ1k,j,t and µ2

k,j,t supported on R3× R3 for which

CW (a1(t), A1k,j, A

2k,j) =

R3×R3

W (a1(t), γ , ρ) dµ1k,j,t(γ , ρ)

(A1k,j, A

2k,j) =

R3×R3

γ dµ1k,j,t(γ , ρ),

R3×R3

ρ dµ1k,j,t(γ , ρ)

,

and

CW (a2(t), A3k,j, A

4k,j) =

R3×R3

W (a2(t), γ , ρ) dµ2k,j,t(γ , ρ)

(A3k,j, A

4k,j) =

R3×R3

γ dµ2k,j,t(γ , ρ),

R3×R3

ρ dµ2k,j,t(γ , ρ)

.

Therefore, T

0

Ω

W (t, ∂t u(x, t), curl (β(x)u(x, t))) dx dt = limj→∞

k

T

0

R3×R3

αW (a1(t), γ , ρ) dµ1k,j,t(γ , ρ)

+

R3×R3

(1 − α)W (a2(t), γ , ρ) dµ2k,j,t(γ , ρ)

x(j)k + r (j)

k Ω

dt= lim

j→∞

k

T

0

Rm×R3×R3

W (λ, γ , ρ) dνk,jt (λ, γ , ρ)

x(j)k + r (j)

k Ω

dt,where we have defined the probability measure ν

k,jt by

νk,jt (λ, γ , ρ) = α µ1

k,j,t(γ , ρ) ⊗ δa1(t)(λ) + (1 − α) µ2k,j,t(γ , ρ) ⊗ δa2(t)(λ).

Moreover, for each k, j ∈ N, there exists a sequence of bounded functions w(k,j)h defined in x(j)

k + r (j)k Ω such that the

sequence of triples (ah, ∂t w(k,j)h , curl (βw

(k,j)h )) generates the Young measure ν

k,jt t∈(0,T ), with barycentre

αa1(t) + (1 − α)a2(t), αA1k,j + (1 − α)A3

k,j, αA2k,j + (1 − α)A4

k,j

,

and

limh→∞

x(j)k +r(j)k Ω

W (ah(y, t), ∂t w(k,j)h (y, t), curl (β(y)w(k,j)

h (y, t))) dy

=

Rm×R3×R3

W (λ, γ , ρ) dνk,jt (λ, γ , ρ)

x(j)k + r (j)

k Ω

.In this way, we have that T

0

Ω

W (t, ∂t u(x, t), curl (β(x)u(x, t))) dx dt

= limh→∞

limj→∞

k

T

0

x(j)k +r(j)k Ω

W (ah(y, t), ∂t w(k,j)h (y), curl (β(y)w(k,j)

h (y))) dy dt.

84 H. Serrano / Nonlinear Analysis: Real World Applications 18 (2014) 75–85

Using the previous sequence and a sequence of cut-off functions, we are able to build a sequence of divergence-free fieldsvh in V such that T

0

Ω

W (t, ∂t u(x, t), curl (β(x)u(x, t))) dx dt

= limh→∞

T

0

Ω

W (ah(x, t), ∂t vh(x, t), curl (β(x)vh(x, t))) dx dt.

5. Application of Theorem 1.2

Let us start by considering two artificial, homogeneous, isotropic, single-negative materials, say 1 and 2, with relativeelectric permittivity ε1 = 7 and ε2 = 3.4, respectively, and same relative magnetic permeability µ = −1. Both materialsare opaque to electromagnetic radiation. Notice that material 1 has the same permittivity as the rubber while material 2 asthe polyimide. Consider a laminate structure occupying the unit cube Ω = (0, 1)3 formed by h/2 layers of material 1, withrelative thickness 1/3h, each one alternated with h/2 layers of material 2, with relative thickness 2/3h, normal to the unitvector e = (1, 0, 0). The composite material response depends on the number h of layers we are considering and is fullycharacterized by the permittivity

εh(x) = 7χ0, 13

(hx1) + 3.41 − χ

0, 13

(hx1)

,

and the permeability µh(x) ≡ −1, for every x = (x1, x2, x3) ∈ (0, 1)3. Our aim here is to deduce the asymptotic behaviourof the permittivity and the permeability of this type of composite materials, applying a constant electric current densityJ(x, t) = (1, 1, 1), when the number of layers converges to infinity, i.e. h → ∞. In this case, the quadratic functional Ihassociated with system (1.5) coupled with initial conditions (1.6) reads as

Ih(B) =

T

0

Ω

12|∂t B|2 +

ε−1h (x)2

|curl B|2 + ε−1h (x)(1, 1, 1) · curl B

dx dt. (5.1)

This sequence of functionals Ih Γ -converges to the functional I defined in the following proposition, which turns out to bea direct application of Theorem 1.2.

Proposition 5.1. The sequence Ih defined in (5.1) Γ -converges to

I(B) =

T

0

Ω

12|∂t B(x, t)|2 + (curl B(x, t))T

ε−1hom

2curl B(x, t) + ε−1

homJ(x, t) · curl B(x, t) −72

2737

dxdt,

where ε−1hom is the 3 × 3-matrix defined as

ε−1hom =

29119

0 0

0523

0

0 0523

.

Proof. It follows from Theorem 1.2 that the Γ -limit density W is a function which, for each fixed (Λ, Θ) ∈ R3× R3, gives

the minimum value of the function Ψ : R3× R3

× R3× R3

→ R defined by

Ψ (A1, A2, A3, A4) =16|A1|

2+

142

|A2|2+

121

(1, 1, 1) · A2 +13|A3|

2+

551

|A4|2+

1051

(1, 1, 1) · A4

subject to the constraints

Λ =13A1 +

23A3, (A1 − A3) · e = 0,

Θ =13A2 +

23A4, (A2 − A4) · e = 0.

Taking into account that e = (1, 0, 0), and after some calculations, we conclude that, for each (Λ, Θ) ∈ R3× R3, the

minimum value of Ψ under the previous constraints is

W (Λ, Θ) =12|Λ|

2+ ΘT ε−1

hom

2Θ + ε−1

hom(1, 1, 1) · Θ −72

2737

where the effective matrix ε−1hom is defined previously.

H. Serrano / Nonlinear Analysis: Real World Applications 18 (2014) 75–85 85

A direct consequence of the Γ -convergence of the sequence of energies Ih to I is the homogenization of the family ofinitial–boundary value problems

∂2t B(x, t) + curl

ε−1h (x) curl (−B(x, t))

= curl

ε−1h (x)J(x, t)

in Ω × (0, T )

div B(x, t) = 0 in Ω × (0, T )B(x, t) · n = 0 on ∂Ω × (0, T )

ε−1h curl (−B) × n = ε−1

h J × n on ∂Ω × (0, T )B(x, 0) = B0(x) in Ω

∂t B(x, 0) = ∂t B(x, T ) = 0 in Ω,

which is stated in Proposition 1.3. Precisely, for each h ∈ N, if Bh is the solution of the initial–boundary value problem above,then the sequence Bh is such that curl Bh and ∂t Bh weakly converge, as h goes to ∞, to curl B and ∂t B, respectively, inL2(Ω × (0, T ); R3), where B is the solution of the homogenized system

∂2t B(x, t) + curl

ε−1hom curl (−B(x, t))

= 0 in Ω × (0, T )

div B(x, t) = 0 in Ω × (0, T )B(x, t) · n = 0 on ∂Ω × (0, T )

ε−1hom curl (−B) × n = ε−1

homJ × n on ∂Ω × (0, T )B(x, 0) = B0(x) in Ω

∂t B(x, 0) = ∂t B(x, T ) = 0 in Ω.

Acknowledgements

This work was supported by Project MTM201019739 (Ministerio de Educación, Spain) and Grant TC20101856(Universidad de Castilla-La Mancha).

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