Citation: Christian M. Julien, Alain Mauger. In situ Raman analyses of electrode materials for Li-ion batteries[J]. AIMS Materials Science, 2018, 5(4): 650-698. doi: 10.3934/matersci.2018.4.650
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The formation of shear bands in elasticity is described by a degenerate operator of elliptic-hyperbolic type [1, 23, 4]. The shear bands that are mathematically obtained in this model are infinitesimally thin. To overcome this non-physical description, it is customary to penalize the elasticity operator by a fourth-order singular perturbation [18]. Subsequently, it was suggested that the penalization should be followed by a homogenization procedure, which results in different regimes depending on the order of penalization as compared to the length-scale of the periodic heterogeneities. This was first carried out in [6, 18]. Similar problems in the framework of
We will study the simultaneous homogenization and singular perturbation limits of the following operator
Aκ,ε:=κ2Δ2−∇⋅A(xε)∇, | (1.1) |
where
A1.
A2.
A3. The matrix
We mention briefly the function spaces that make an appearance in this problem. The solutions of the cell problem associated with homogenization of (1.1), as well as Bloch eigenfunctions, are sought in the space .
The method of Bloch waves rests on decomposition of a periodic operator in terms of Bloch waves which may be thought of as a periodic analogue of plane waves. As plane waves decompose a linear operator with constant coefficients by means of the Fourier transform, Bloch waves diagonalize a linear operator with periodic coefficients. This decomposition begins with a direct integral decomposition of a periodic operator
A→⨁∫TdA(η)dη. |
The fiber operator
The homogenization limits for a highly oscillating scalar periodic operator are obtained from its first Bloch mode. The rest of the Bloch modes do not contribute to the homogenization limit. This is a consequence of the separation of the first Bloch eigenvalue from the rest of the spectrum. Such an interpretation of homogenization is also called spectral threshold effect [7]. Moreover, the homogenized tensor is obtained from Hessian of the first Bloch eigenvalue. Therefore, the second-order nature of the differential operator is reflected in the quadratic nature of the first Bloch eigenvalue near the bottom of the spectrum. Indeed, homogenization of higher-even-order periodic operators can also be obtained by the Bloch wave method, where the first Bloch eigenvalue behaves like a polynomial of the corresponding order near the bottom of the spectrum [37, 36].
The homogenization result in Theorem 8.1 exhibits three different regimes depending on the ratio of
●
●
●
In the first regime, the subcritical case, that is,
The approach is to treat
While the motivation for the problem (1.1) comes from the theory of elasticity, it is for the sake of simplicity that we only study the scalar operator. However, it must be noted that Bloch wave homogenization of systems carries some unique difficulties, such as the presence of multiplicity at the bottom of the spectrum. Indeed, these challenges have been surmounted by the use of directional analyticity of Bloch eigenvalues in [35, 7, 3]. Further, the assumption of symmetry, while customary in elasticity, is made for a simplified presentation. A Bloch wave analysis of homogenization of non-selfadjoint operators may be found in [19].
In a forthcoming work, we will obtain quantitative estimates for the combined effects of singular perturbation and homogenization through the notion of Bloch approximation, which was introduced in [11]. Higher order estimates in homogenization have been obtained by these methods, particularly for the dispersive wave equation [14, 15, 2, 22].
The plan of the paper is as follows: In Section 2, we obtain Bloch waves for the singularly perturbed operator
In this section, we will prove the existence of Bloch waves for the singular operator given by
Aρ:=ρ2Δ2−∇⋅A(y)∇. | (2.1) |
Recall that
{ρ2Δ2ψ−∇⋅A(y)∇ψ=λψψ(y+2πp)=e2πip⋅ηψ(y),p∈Zd,η∈Rd. |
The above problem is invariant under
{Aρ(η)ϕ:=ρ2(∇+iη)4ϕ−(∇+iη)⋅A(y)(∇+iη)ϕ=λϕϕ(y+2πp)=ϕ(y),p∈Zd,η∈Y′. | (2.2) |
The operator
The bilinear form
aρ[η](u,v):=∫YA(∇+iη)u⋅¯(∇+iη)vdy+ρ2∫Y(∇+iη)2u¯(∇+iη)2vdy, | (2.3) |
is associated to the operator
Lemma 2.1. There exists a positive real number
aρ[η](u,u)+C∗||u||2L2♯(Y)≥ρ26||Δu||2L2♯(Y)+α2||u||H1♯(Q). | (2.4) |
Proof. We have
aρ[η](u,u)=∫YA(∇+iη)u⋅¯(∇+iη)udy⏟I+ρ2∫Y(∇+iη)2u¯(∇+iη)2udy⏟II. | (2.5) |
We shall estimate the two summands separately. For the first summand, observe that
I=∫YA(∇+iη)u⋅¯(∇+iη)udy=∫YA∇u⋅¯∇udy+2Re{∫YAiηu⋅¯∇udy}+∫YAηu⋅η¯udy, | (2.6) |
where
∫YA∇u⋅∇¯udy=∫YA∇u⋅∇¯udy≥α∫Y|∇u|2dy. | (2.7) |
For the second term of
|2Re{∫YAηu⋅∇¯udy}|≤2∫Y|Aηu⋅∇¯u|dy≤C1∫Y|ηu⋅∇¯u|dy≤C1||ηu||L2♯(Y)||∇u||L2♯(Y)≤C1C2||u||2L2♯(Y)+C1C2||∇u||L2♯(Y). | (2.8) |
Finally, the third term of
|∫YAηu⋅η¯udy|≤C3∫Y|ηu⋅η¯u|dy≤C4||u||L2♯(Y). | (2.9) |
Now, we may choose
I≥α2||u||L2♯(Y)+α2||∇u||L2♯(Y)−(α2+C1C2+C4)||u||2L2♯(Y). | (2.10) |
For the second summand, observe that
II=ρ2∫Y|(∇+iη)2u|2dy=ρ2∫Y|Δu|2dy+ρ2∫Y|η|4|u|2dy+4ρ2∫Y|η⋅∇u|2dy+2ρ2iIm{∫Y|η|2u¯Δudy}+4iρ2Re{∫Y(η⋅∇u)Δ¯udy}+4iρ2Re{∫Y|η|2u(η⋅∇¯u)dy}. | (2.11) |
We estimate the last three terms as follows.
ρ2|2iIm{∫Y|η|2u¯Δudy}|≤2ρ2∫Y|η|2|u||Δu|dy≤2ρ2|η|2||u||L2||Δu||L2≤48ρ2|η|4||u||2L2+ρ248||Δu||2L2. | (2.12) |
ρ2|4iRe{∫Y(η⋅∇)uΔ¯udy}|≤4ρ2∫Y|(η⋅∇)u||Δu|dy≤4ρ2||(η⋅∇)u||L2||Δu||L2≤16ρ23||(η⋅∇)u||2L2+3ρ24||Δu||2L2. | (2.13) |
ρ2|4iRe{∫Y|η|2u(η⋅∇)¯udy}|≤4ρ2∫Y|(η⋅∇)u||η|2|u|dy≤4ρ2|η|2||u||L2||(η⋅∇)u||L2≤192ρ2|η|4||u||2L2+ρ248||(η⋅∇)u||2L2. | (2.14) |
The previous threee estimate use Cauchy-Schwarz inequality for the second step and Young's inequality for the third step. Substituting the inequalities (2.12), (2.13) and (2.14) into (2.11), we get
II≥11ρ248||Δu||2L2(Y)−240ρ2|η|4||u||2L2(Y)−65ρ248||(η⋅∇)u||2L2(Y). | (2.15) |
Since
II≥11ρ248||Δu||2L2(Y)−15ρ2||u||2L2(Y)−65ρ2192||∇u||2L2(Y). |
Now, notice that
||∇u||2L2=∫Y|∇u|2dy=−∫YuΔudy≤6548||u||2L2+1265||Δu||2L2. | (2.16) |
Substituting (2.16) in (2.15), we get
II≥ρ26||Δu||2L2(Y)−16ρ2||u||2L2(Y). | (2.17) |
Combining (2.10) and (2.17), we obtain (2.4) with
C∗=(α2+C1C2+C4+16ρ2). | (2.18) |
Remark 2.2. In the Gå rding type inequality for the operator
Now that we have the coercivity estimate (2.4), we can prove the existence of Bloch eigenvalues and eigenfunctions for the operator
Theorem 2.3. For each
Proof. Lemma 2.1 shows that for every
Remark 2.4. We can prove the existence of Bloch eigenvalues and eigenfunctions for the case
Now that we have proved the existence of Bloch eigenvalues and eigenfunctions, we can state the Bloch Decomposition Theorem which offers a partial diagonalization of the operator
Theorem 2.5. Let
Bρmg(η):=∫Rdg(y)e−iy⋅η¯ϕρm(y;η)dy,m∈N,η∈Y′. | (2.19) |
1. The following inverse formula holds
g(y)=∫Y′∞∑m=1Bρmg(η)ϕρm(y;η)eiy⋅ηdη. | (2.20) |
2. Parseval's identity
||g||2L2(Rd)=∞∑m=1∫Y′|Bρmg(η)|2dη. | (2.21) |
3. Plancherel formula For
∫Rdf(y)¯g(y)dy=∞∑m=1∫Y′Bρmf(η)¯Bρmg(η)dη. | (2.22) |
4. Bloch Decomposition in
BρmF(η)=∫Rde−iy⋅η{u0(y)¯ϕρm(y;η)+iN∑j=1ηjuj(y)¯ϕρm(y;η)}dy−∫Rde−iy⋅ηN∑j=1uj(y)∂¯ϕρm∂yj(y;η)dy. | (2.23) |
The definition above is independent of the particular representative of
5. Finally, for
Bρm(Aρg)(η)=λρm(η)Bρmg(η). | (2.24) |
The Bloch wave method of homogenization requires differentiability of the Bloch eigenvalues and eigenfunctions in a neighbourhood of
Theorem 3.1. For every
1. The first Bloch eigenvalue
2. There is a choice of corresponding eigenfunctions
For the proof, we will make use of Kato-Rellich theorem which establishes the existence of a sequence of eigenvalues and eigenfunctions associated with a selfadjoint holomorphic family of type (B). The definition of selfadjoint holomorphic family of type (B) and other related notions may be found in Kato [21]. Nevertheless, they are stated below for completeness. We begin with the definition of a holomorphic family of forms of type (a).
Definition 3.2.
1. The numerical range of a form
Θ(a)={a(u,u):u∈D(a),||u||=1} |
where
2. The form
Θ(a)⊂Sc,θ:={λ∈C:|arg(λ−c)|≤θ)}. |
3. A sectorial form
Definition 3.3. [Kato] A family of forms
1. each
2.
A family of operators is called a holomorphic family of type (B) if it generates a holomorphic family of forms of type (a).
In [21, 31], Kato-Rellich theorem is stated only for a single parameter family. In [5], one can find the proof of Kato-Rellich theorem for multiple parameters with the added assumption of simplicity for the eigenvalue at
Theorem 3.4. (Kato-Rellich) Let
1. There is exactly one point
2. There is an associated eigenfunction
The proof of Theorem 3.1 proceeds by complexifying the shifted operator
Proof. (Proof of Theorem 3.1)
(i) Complexification of
t(˜η)=∫YA(∇+iσ−τ)u⋅(∇−iσ+τ)¯udy+ρ2∫Y|(∇+iσ+τ)2u|2dy |
for
R:={˜η∈CM:˜η=σ+iτ,σ,τ∈RM,|σ|<1/2,|τ|<1/2}. |
(ii) the form
t(˜η)=∫YA(∇+iσ−τ)u⋅(∇−iσ+τ)¯udy+ρ2∫Y|(∇+iσ+τ)2u|2dy=∫YA(∇+iσ)u⋅(∇−iσ)¯udy−∫YA(τu)⋅∇¯udy+∫YA∇u⋅(τ¯u)dy−∫YAτu⋅τ¯udy+i∫YAσu⋅τ¯udy+i∫YAτu⋅σ¯udy+ρ2∫Y(Δ−|σ|2+|τ|2)u(Δ−|σ|2+|τ|2)¯udy+2ρ2∫Y(Δ−|σ|2+|τ|2)u(τ⋅∇−iσ⋅∇−iσ⋅τ)¯udy+2ρ2∫Y(iσ⋅∇−τ⋅∇−iσ⋅τ)u(Δ−|σ|2+|τ|2)¯udy+4ρ2∫Y(iσ⋅∇−τ⋅∇−iσ⋅τ)u(τ⋅∇−iσ⋅∇−iσ⋅τ)¯udy. |
From above, it is easy to write separately the real and imaginary parts of the form
ℜt(˜η)[u]=∫YA(∇+iσ)u⋅(∇−iσ)¯udy−∫YAτu⋅τ¯udy+ρ2∫Y(Δ−|σ|2+|τ|2)u(Δ−|σ|2+|τ|2)¯udy−4ρ2∫Y|(τ⋅∇)u|2dy+4ρ2∫Y|(σ⋅∇)u|2dy−4ρ2∫Y|(τ⋅σ)u|2dy+8ρ2Re{∫Yi(τ⋅∇)u(σ⋅τ)¯udy}+4ρ2Re{∫Yi(σ⋅∇)uΔ¯udy}+4ρ2Re{∫Yi|σ|2u(σ⋅∇)¯udy}. |
ℑt(˜η)[u]=∫YAσu⋅τ¯udy+∫YAτu⋅σ¯udy+2ℑ{∫YA∇u⋅τ¯udy}+8ρ2Im{∫Yi(τ⋅σ)u(σ⋅∇)¯udy}+8ρ2Im{∫Yi(σ⋅∇)u(σ⋅τ)¯udy}+4ρ2Im{∫YΔu(τ⋅∇)¯udy}−4ρ2Im{∫YiΔu(σ⋅τ)¯udy}+4ρ2Im{∫Y(τ⋅∇)u|σ|2¯udy}+4ρ2Im{∫Yi|σ|2u(σ⋅τ)¯udy}+4ρ2Im{∫Y|τ|2u(τ⋅∇)¯udy}−4ρ2Im{∫Yi|τ|2u(σ⋅τ)¯udy}. |
The following coercivity estimate can be easily found for the real part:
ℜt(˜η)[u]+C5||u||2L2♯(Y)≥α2(||u||2L2♯(Y)+||∇u||2L2♯(Y))+ρ26||Δu||2L2♯(Y). | (3.1) |
Let us define the new form
ℜ˜t(˜η)[u]≥α2(||u||2L2♯(Y)+||∇u||2L2♯(Y))+ρ26||Δu||2L2♯(Y)+C6||u||2L2♯(Y). |
Also, the imaginary part of
ℑ˜t(˜η)[u]≤C7||u||2L2♯(Y)+C8||∇u||2L2♯(Y)+C9||Δu||2L2♯(Y)C10=max{2C8α,6C9ρ2}=C6=C10C7C10(C6||u||2L2♯(Y)+α2||∇u||2L2♯(Y)+ρ26||Δu||2L2♯(Y))≤C10(ℜ˜t(˜η)[u]−α2||u||2L2♯(Y)). |
This shows that
(iii) The form
(iv) The form
(v)
(vi)
(vii)
In Section 3, we have proved that the first Bloch eigenvalue and eigenfunction is analytic in a neighbourhood of
We begin by proving that Bloch eigenvalues are Lipschitz continuous in the dual parameter.
Lemma 4.1. For all
Proof. The following form is associated with
aρ[η](u,u)=∫YA(∇+iη)u⋅¯(∇+iη)udy+ρ2∫Y(∇+iη)2u¯(∇+iη)2udy. | (4.1) |
Hence, for
aρ[η]−aρ[η′]=2Re{∫YAi(η−η′)u⋅¯∇udy}+∫YAηu⋅η¯udy+∫YAη′u⋅η′¯udy+ρ2∫Y(|η|2−|η′|2)|u|2dy+4ρ2∫Y|η⋅∇u|2−|η′⋅∇u|2dy+2ρ2iIm{∫Y(|η|2−|η′|2)uΔ¯udy}+4ρ2iRe{∫Y(η−η′)⋅∇uΔ¯udy}+4ρ2iRe{∫Y|η|2u(η⋅∇)¯u−|η′|2u(η′⋅∇)¯udy}≤C|η−η′|||u||2H1♯(Y)+C′|η−η′|ρ2{||Δu||2L2♯(Y)+||u||2L2♯(Y)}, |
where
λρm(η)≤λρm(η′)+C|η−η′|μm+C′ρ2|η−η′|νm, | (4.2) |
where
{−Δum+um=μmuminYumisY−periodic, |
and
{Δ2vm+vm=νmvminYvmisY−periodic. |
By interchanging the role of
|λρm(η)−λρm(η′)|≤C(μm+ρ2νm)|η−η′|. | (4.3) |
Here,
Now, we will prove a spectral gap result, viz. the second Bloch eigenvalue is bounded below.
Lemma 4.2. For all
λρm(η)≥αλN2, | (4.4) |
where
Proof. Notice that
λρ2(η)=infW⊂H2♯(Y)dim(W)=2maxϕ∈Wϕ≠0∫YA∇(eiη⋅yϕ)⋅∇(e−iη⋅y¯ϕ)dy+ρ2∫Y|Δ(eiη⋅yϕ)|2dy∫Y|ϕ|2dy≥infW⊂H2(Y)dim(W)=2maxψ∈Wψ≠0∫YA∇ψ⋅∇¯ψdy+ρ2∫Y|Δψ|2dy∫Y|ψ|2dy≥infW⊂H2(Y)dim(W)=2maxψ∈Wψ≠0∫YA∇ψ⋅∇¯ψdy∫Y|ψ|2dy≥infW⊂H1(Y)dim(W)=2maxψ∈Wψ≠0∫YA∇ψ⋅∇¯ψdy∫Y|ψ|2dy≥αinfW⊂H1(Y)dim(W)=2maxψ∈Wψ≠0∫Y|∇ψ|2dy∫Y|ψ|2dy=αλN2. |
The bound obtained in Lemma 4.2 will be useful for the small
Lemma 4.3. For all
λρm(η)≥Cρ2κ2−C′, | (4.5) |
where
Proof. Recall the following Gå rding type estimate (2.4) for the form
aρ[η](u,u)+C∗||u||2L2♯(Y)≥ρ26||Δu||2L2♯(Y)+α2||u||H1♯(Q). |
The inequality in Lemma 4.3 follows readily from above by applying the minmax characterization.
Remark 4.4. In Lemma 4.2 and Lemma 4.3, we have avoided estimating the second Bloch eigenvalue by using the spectral problem associated with Neumann bilaplacian as it is known to be ill-posed [30]. Moreover, polyharmonic Neumann eigenvalue problems on polygonal domains (such as
We are finally in a position to prove that the neighbourhood of analyticity of the first Bloch eigenvalue does not depend on the parameter
Theorem 4.5. There exists a neighbourhood
Proof. It was proved in Theorem 3.1 that the first Bloch eigenvalue is analytic in a neighbourhood of
|λρ1(η)−λρ2(η)|≥λρ2(η)−|λρ1(η)−λρ1(0)|−|λρ2(η)−λρ2(0)|(4.3)≥λρ2(η)−2(C+ρ2)|η|, | (4.6) |
where
● For sufficiently large
|λρ1(η)−λρ2(η)|≥λρ2(η)−2(C+ρ2)|η|Lemma4.3≥(C′ρ2−C″)−2(C+ρ2)|η|largeρ≥C‴ρ2−2ρ2|η|>0 |
for
● For remaining values of
|λρ1(η)−λρ2(η)|≥λρ2(η)−2(C+ρ2)|η|Lemma4.2≥αλN2−2(C+ρ2)|η|≥αλN2−2C|η|>0 |
for
Remark 4.6. In the papers [33, 34], an additional artificial parameter is introduced in the Bloch eigenvalue problem to facilitate the homogenization method. Unlike (1.1), these papers employ successive limits of the two parameters instead of simultaneous limits. Therefore, the non-dependence of the neighbourhood of analyticity on the second parameter is not required in [33, 34].
In this section, we will consider the classical cell problem associated with (1.1) and the estimates for the corrector field. This section will allow us to characterize the homogenized tensor for (1.1) and the corrector field in terms of Bloch eigenvalues and eigenfunctions.
For
(5.1) |
By a simple application of Lax-Milgram lemma on
ρ||Δχρj||L2♯(Y)+||χρj||H1♯(Y)≤C. | (5.2) |
If we use
ρ2||∇3χρj||L2♯(Y)≤C. | (5.3) |
We also collect below a few estimates which will be required later. Similar estimates have been proved in [26] to which we refer for more details.
Lemma 5.1. Let
||∇χρ1−∇χρ2||L2(Y)≤C|1−(ρ1/ρ2)2|. | (5.4) |
Proof. Define
ρ21Δ2z−divA(y)∇z=(ρ22−ρ21)Δ2χρ2. | (5.5) |
Now, the quoted estimate readily follows by taking
Lemma 5.2. Let
(5.6) |
Then, there is
||∇χ0j−∇χBj||L2(Y)≤Cϰ | (5.7) |
||∇χρ−∇χB||L2(Y)≤C{ρ||χB||H2(Y)+ϰ}, | (5.8) |
where
(5.9) |
Proof. Observe that given any
||A−B||Lq♯(Y)≤ϰ. |
For example, this can be achieved by a standard smoothing by convolution. Now, by regularity theory,
ρ2Δ2z−divB(y)∇z=−ρ2Δ2χBj+div(A−B)∇χρj. | (5.10) |
We test this equation against
ρ2∫Y|Δz|2dy+∫YA(y)∇z⋅∇zdy≤ρ2∫Y|ΔχBj||Δz|dy+∫Y|A−B||∇χρj||∇z|dy. |
This leads to
ρ2||Δz||2L2+α||∇z||2L2≤ρ2||ΔχBj||L2||Δz||L2+||∇z||L2(∫Y|A−B|2|∇χρj|2dy)1/2. |
By Young's inequality,
||∇z||L2≤C{ρ||ΔχBj||L2+(∫Y|A−B|2|∇χρj|2dy)1/2}. |
On the last term, we apply a form of Meyers estimate for the
For every fixed
Aρ,hom:=MY(A+A∇χρ) | (5.11) |
Definition 5.3 (Homogenized Tensor for
Ahom:={MY(A+A∇χθ)for0<θ<∞whereρ=κε→θ,MY(A+A∇χ0)whenρ=κε→0,MY(A)whenρ=κε→∞. | (5.12) |
In this section, we will give a new characterization of the homogenized tensor (see Definition 5.3), and corrector field (5.1) in terms of the first Bloch eigenvalue and eigenfunction. These characterizations are obtained by differentiating the Bloch spectral problem (2.2) with respect to the dual parameter
We recall the Bloch eigenvalue problem for the operator
ρ2(∇+iη)4ϕρ1(y;η)−(∇+iη)⋅A(y)(∇+iη)ϕρ1(y;η)=λρ1(η)ϕρ1(y;η). | (6.1) |
We know that
We shall normalize the average value of the first Bloch eigenfunction
MY(ϕρ1(⋅,η))=(2π)−d/2 | (6.2) |
for all
MY(∂β0ϕρ1)=0 | (6.3) |
for all
∂βηA≡0forall|β|>4, | (6.4) |
since
Aρ(0)=ρ2∇4−∇⋅A(y)∇∂ej0Aρ=4iρ2ej⋅∇∇2−iej⋅A∇−i∇⋅Aej∂ej+ek0Aρ=−4ρ2δjk∇2−8ρ2∂yj∂yk+2ajk∂ej+ek+el0Aρ=−8iρ2(δjk∂yl+δjl∂yk+δkl∂yj)∂ej+ek+el+em0Aρ=8ρ2(δjkδlm+δjlδkm+δjmδkl), | (6.5) |
where
∂β(fg)=∑γ∈Nd∪{0}(βγ)∂γf∂β−γg, | (6.6) |
where
Cell problems for
Aρ(0)∂β0ϕρ1+d∑j=1βj∂ej0Aρ∂β−ej0ϕρ1+∑j,k(βej+ek)∂ej+ek0Aρ∂β−ej−ek0ϕρ1 |
+∑j,k,l(βej+ek+el)∂ej+ek+el0Aρ∂β−ej−ek−el0ϕρ1+∑j,k,l,m(βej+ek+el+em)∂ej+ek+el+em0Aρ∂β−ej−ek−el−em0ϕρ1=∑γ∈Nd∪{0}(βγ)∂γ0λρ1∂β−γ0ϕρ1. | (6.7) |
Substituting (6.5) in (6.7), we obtain
(ρ2∇4−∇⋅A(y)∇)∂β0ϕρ1=d∑j=1βj(−4iρ2ej⋅∇∇2+iej⋅A∇+i∇⋅Aej)∂β−ej0ϕρ1−∑j,k(βej+ek)(−4ρ2δjk∇2−8ρ2∂yj∂yk+2ajk)∂β−ej−ek0ϕρ1+∑j,k,l(βej+ek+el)(8iρ2(δjk∂yl+δjl∂yk+δkl∂yj))∂β−ej−ek−el0ϕρ1−∑j,k,l,m(βej+ek+el+em)(8ρ2(δjkδlm+δjlδkm+δjmδkl))∂β−ej−ek−el−em0ϕρ1+∑γ∈Nd(βγ)∂γ0λρ1∂β−γ0ϕρ1. | (6.8) |
Expression for
Rearranging (6.8), we get
∂β0λρ1ϕρ1(0)=(ρ2∇4−∇⋅A(y)∇)∂β0ϕρ1+d∑j=1βj(−4iρ2ej⋅∇∇2+iej⋅A∇+i∇⋅Aej)∂β−ej0ϕρ1−∑j,k(βej+ek)(−4ρ2δjk∇2−8ρ2∂yj∂yk+2ajk)∂β−ej−ek0ϕρ1+∑j,k,l(βej+ek+el)(8iρ2(δjk∂yl+δjl∂yk+δkl∂yj))∂β−ej−ek−el0ϕρ1−∑j,k,l,m(βej+ek+el+em)(8ρ2(δjkδlm+δjlδkm+δjmδkl))∂β−ej−ek−el−em0ϕρ1+∑γ∈Ndγ≠β(βγ)∂γ0λρ1∂β−γ0ϕρ1. | (6.9) |
Integrating (6.9) over
∂β0λρ1={2∑j,k(βej+ek)MY(ajk∂β−ej−ek0ϕρ1)−id∑j=1βjMY(ej⋅A∇∂β−ej0ϕρ1)when|β|≠4.2∑j,k(βej+ek)MY(ajk∂β−ej−ek0ϕρ1)−id∑j=1βjMY(ej⋅A∇∂β−ej0ϕρ1)−∑j,k,l,m8ρ2(δjkδlm+δjlδkm+δjmδkl)ϕρ1(0)when|β|=4. | (6.10) |
We specialize to
On the other hand, if we set
(−∇⋅A(y)∇+ρ2Δ2)∂ϕρ1∂ηl(0)=∇⋅A(y)eliϕρ1(0). |
Comparing with (5.1), we conclude that
We also specialize to
12∂2λρ1∂ηk∂ηl(0)=1|Y|∫Y(ek⋅Ael+12ek⋅A∇χρl+12el⋅A∇χρk)dy. | (6.11) |
On comparing (6.11) with (5.11), we obtain the following theorem:
Theorem 6.1. The first Bloch eigenvalue and eigenfunction satisfy:
1.
2. The eigenvalue
∂λρ1∂ηl(0)=0,∀l=1,2,…,d. | (6.12) |
3. For
4. The Hessian of the first Bloch eigenvalue at
12∂2λρ1∂ηk∂ηl(0)=ek⋅Aρ,homel. | (6.13) |
Now, we will prove the stability of homogenized tensor in the limits
Lemma 6.2. Let
Proof. The fact that the first Bloch eigenvalues
|λρ1(η)|≤|λρ1(η)−λρ1(0|(4.3)≤C(μ1+ν1ρ2)|η|<C′forallη∈K, |
where
aρ[η](u,u)=∫YA(∇+iη)u⋅¯(∇+iη)udy+ρ2∫Y(∇+iη)2u¯(∇+iη)2udy=aθ[η](u,u)+(ρ2−θ2)∫Y(∇+iη)2u¯(∇+iη)2udy |
we obtain the following inequality:
λρ1(η)−λθ1(η)≤(ρ2−θ2)ϑ1(η), |
where
aρ[η](u,u)≥aθ[η](u,u)forρ≥θ, |
so that an application of minmax characterization yields:
λρ1(η)≥λθ1(η)forρ≥θ. |
Thus, we obtain
0≤λρ1(η)−λθ1(η)≤(ρ2−θ2)ϑ1(η)forρ≥θ. |
As a consequence, for each
Theorem 6.3. Let
Proof.
Case 1.
Aρ,hom=12∇2ηλρ1(0)→12∇2ηλθ1(0)=Ahomasρ↓θ∈[0,∞). |
Case 2.
|Aρ,hom−MY(A)|=|∫YA(y)∇χρ(y)dy|≤C||∇χρ||L2♯(Y)≤Cρ2, |
where the last inequality follows from Poincaré inequality and (5.3). Therefore, as
Aρ,hom→MY(A)=Ahomasρ↑∞. |
Remark 6.4. As a consequence of Lemma 6.2 and the discussion in Case
|λρ1(η)|+C∗≥λρ1(η)+C∗=aρ[η](ϕρ1(η),ϕρ1(η))+C∗≥α2||ϕρ1(η)||2H1♯(Y). |
Recall that the number
We end this section by finding boundedness estimates for higher order derivatives of the first Bloch eigenvalue and eigenfunction in the dual parameter in the regime
Theorem 6.5. For
1.
2.
3.
4.
5.
Proof. The estimates are computed in tandem from the equations (6.8) and (6.10). One begins by proving the estimate on the derivative of the first Bloch eigenfunction at
In this section, we will relate the Bloch spectral problem (2.2) to the Bloch spectral problem at the
\begin{eqnarray} \begin{cases} \mathcal{A}^{\kappa, \varepsilon}(\eta)\phi^{\kappa, \varepsilon}&:= \kappa^2(\nabla+i\xi)^4\phi^{\kappa, \varepsilon}(x)-(\nabla+i\xi)\cdot A\left(\frac{x}{ \varepsilon}\right)(\nabla+i\xi)\phi^{\kappa, \varepsilon}(x)\\ &\qquad = \lambda^{\kappa, \varepsilon}(\xi)\phi^{\kappa, \varepsilon}(x)\\ \phi^{\kappa, \varepsilon}(x+2\pi p \varepsilon)& = \phi^{\kappa, \varepsilon}(x), p\in\mathbb{Z}^d,\xi\in \frac{Y^{'}}{ \varepsilon}. \end{cases} \end{eqnarray} | (7.1) |
Comparing to (2.2), by homothety and
\begin{align} \lambda^{\kappa, \varepsilon}(\xi) = \varepsilon^{-2}\lambda^\rho( \varepsilon\xi){\rm{ and }}\phi^{\kappa, \varepsilon}(x,\xi) = \phi^\rho\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right). \end{align} | (7.2) |
Now, we can state the Bloch decomposition theorem of
Theorem 7.1. Let
\begin{align} \mathcal{B}^{\kappa, \varepsilon}_mg(\xi):=\int_{\mathbb{R}^d}g(x)e^{-ix\cdot\xi}\overline{\phi_m^{\kappa, \varepsilon}(x;\xi)}\,dx,\; m\in\mathbb{N},\; \xi\in \frac{Y^{'}}{ \varepsilon}. \end{align} | (7.3) |
1. The following inverse formula holds
\begin{align} g(x) = \int_{\frac{Y^{'}}{ \varepsilon}}\sum\limits_{m = 1}^{\infty}\mathcal{B}^{\kappa, \varepsilon}_mg(\xi)\phi_m^{\kappa, \varepsilon}(x;\xi)e^{ix\cdot\xi}\,d\xi. \end{align} | (7.4) |
2. Parseval's identity
\begin{align} ||g||^2_{L^2(\mathbb{R}^d)} = \sum\limits_{m = 1}^{\infty}\int_{\frac{Y^{'}}{ \varepsilon}}|\mathcal{B}^{\kappa, \varepsilon}_mg(\xi)|^2\,d\xi. \end{align} | (7.5) |
3. Plancherel formula For
\begin{align} \int_{\mathbb{R}^d}f(x)\overline{g(x)}\,dx = \sum\limits_{m = 1}^{\infty}\int_{\frac{Y^{'}}{ \varepsilon}}\mathcal{B}^{\kappa, \varepsilon}_mf(\xi)\overline{\mathcal{B}^{\kappa, \varepsilon}_mg(\xi)}\,d\xi. \end{align} | (7.6) |
4. Bloch Decomposition in
\begin{align} \mathcal{B}^{\kappa, \varepsilon}_mF(\xi) = \int_{\mathbb{R}^d}e^{-ix\cdot\xi}\left\{u_0(x)\overline{\phi^{\kappa, \varepsilon}_m(x;\xi)}+i\sum\limits_{j = 1}^N\xi_ju_j(x)\overline{\phi^{\kappa, \varepsilon}_m(x;\xi)}\right\}\,dx\\-\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\sum\limits_{j = 1}^Nu_j(x)\frac{\partial\overline{\phi^{\kappa, \varepsilon}_m}}{\partial x_j}(x;\xi)\,dx. \end{align} | (7.7) |
The definition above is independent of the particular representative of
5. Finally, for
\begin{align} \mathcal{B}^{\kappa, \varepsilon}_m(\mathcal{A}^{\kappa, \varepsilon}g)(\xi) = \lambda^{\kappa, \varepsilon}_m(\xi)\mathcal{B}^{\kappa, \varepsilon}_mg(\xi). \end{align} | (7.8) |
In order to compute the homogenization limit, we need to know the limit of Bloch Transform of a sequence of functions. The following theorem proves that for a sequence of functions convergent in a suitable way, the first Bloch transform converges to the Fourier transform of the limit.
Theorem 7.2. Let
in denotes the characteristic function of the set
Proof. In Theorem 4.5, the existence of the set
\begin{align*} \mathcal{B}_1^{\kappa, \varepsilon} g^ \varepsilon(\xi)& = \int_{\mathbb{R}^d} g(x)e^{-ix\cdot\xi}\overline{{\phi}_1^{\kappa, \varepsilon}}(x;0)\,dx\\ &\qquad+\int_{\mathbb{R}^d} g(x)e^{-ix\cdot\xi}\left(\overline{\phi_1^\rho}\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right) -\overline{\phi^\rho}\left(\frac{x}{ \varepsilon};0\right) \right)dx. \end{align*} |
Now, we need to distinguish between the regimes:
Case 1.
Case 2.
In this section, we will prove the qualitative homogenization result for the singularly perturbed homogenization problem. There are three regimes according to convergence of
Theorem 8.1. Let
\begin{align} \mathcal{A}^{\kappa, \varepsilon} u^ \varepsilon = f\,\;\mathit{in}\;\,\; \Omega, \end{align} | (8.1) |
where
1. For all
\begin{align} A\left(\frac{x}{ \varepsilon}\right)\nabla u^ \varepsilon(x)\rightharpoonup A^{hom}\nabla u^*(x) \;\mathit{in}\; (L^2(\Omega))^d\;\mathit{-weak}\;. \end{align} | (8.2) |
2. The limit
\begin{align} \mathcal{A}^{hom}u^* = -\nabla\cdot A^{hom}\nabla u^* = f\,\;\mathit{in}\;\,\Omega. \end{align} | (8.3) |
Remark 8.2. In the spirit of H-convergence [24], we do not impose any boundary condition on the equation. The H-convergence compactness theorem concerns convergence of sequences on which certain differential constraints have been imposed. In homogenization, the weak convergence of solutions is a consequence of uniform bounds on them, which follow from boundary conditions imposed on the equation. In the theorem quoted above, the uniform boundedness on
The proof of Theorem 8.1 is divided into the following steps. We begin by localizing the equation (8.1) which is posed on
Let
\begin{align} \mathcal{A}^{\kappa, \varepsilon}(\psi_0 u^ \varepsilon)(x) = \psi_0f(x)+g^ \varepsilon(x)+h^{ \varepsilon}(x)+\sum\limits_{m = 1}^4l^{\kappa, \varepsilon}_m(x)\,\;{\rm{ in }}\;\,\mathbb{R}^d, \end{align} | (8.4) |
where
\begin{align} g^ \varepsilon(x)&:=-\frac{\partial \psi_0}{\partial x_k}(x)a^ \varepsilon_{kl}(x)\frac{\partial u^ \varepsilon}{\partial x_l}(x), \end{align} | (8.5) |
\begin{align} h^{ \varepsilon}(x)&:=-\frac{\partial}{\partial x_k}\left(\frac{\partial \psi_0}{\partial x_l}(x)a^{ \varepsilon}_{kl}(x)u^ \varepsilon(x)\right), \end{align} | (8.6) |
\begin{align} l^{\kappa, \varepsilon}_1(x)&:=\kappa^2\frac{\partial^4\psi^0}{\partial x_k^4}(x)u^ \varepsilon(x) . \end{align} | (8.7) |
\begin{align} l^{\kappa, \varepsilon}_2(x)&:= 4\kappa^2\frac{\partial^3\psi^0}{\partial x_k^3}(x)\frac{\partial u^ \varepsilon}{\partial x_k}(x) . \end{align} | (8.8) |
\begin{align} l^{\kappa, \varepsilon}_3(x)&:= 2\kappa^2\frac{\partial^2\psi^0}{\partial x_k^2}(x)\frac{\partial^2u^ \varepsilon}{\partial x_k^2}(x) . \end{align} | (8.9) |
\begin{align} l^{\kappa, \varepsilon}_4(x)&:= 4\kappa^2\frac{\partial\psi^0}{\partial x_k}(x)\frac{\partial^3u^ \varepsilon}{\partial x_k^3}(x)+ 4\kappa^2\frac{\partial^2\psi^0}{\partial x_k^2}(x)\frac{\partial^2u^ \varepsilon}{\partial x_k^2}(x) = 4\kappa^2\frac{\partial}{\partial x_k}\left(\frac{\partial\psi_0}{\partial x_k}\frac{\partial^2u^ \varepsilon}{\partial x_k^2}\right) . \end{align} | (8.10) |
While the sequence
\begin{align} \lambda^{\kappa, \varepsilon}_1(\xi)\mathcal{B}^{\kappa, \varepsilon}_1(\psi_0 u^ \varepsilon)(\xi) = \mathcal{B}^{\kappa , \varepsilon}_1(\psi_0 f)(\xi)+\mathcal{B}^{\kappa, \varepsilon}_1g^ \varepsilon(\xi)+\mathcal{B}^{\kappa, \varepsilon}_1h^{ \varepsilon}(\xi)+\sum\limits_{m = 1}^4\mathcal{B}^{\kappa, \varepsilon}_1l^{\kappa, \varepsilon}_m(\xi). \end{align} | (8.11) |
We shall now pass to the limit
We expand the first Bloch eigenvalue about
\begin{align*} \left(\frac{1}{2}\frac{\partial^2\lambda^\rho_1}{\partial\eta_s\partial\eta_t}(0)\xi_s\xi_t + O( \varepsilon^2) \right) \mathcal{B}^{\kappa, \varepsilon}_1(\psi_0 u^ \varepsilon). \end{align*} |
The higher order derivatives of
\begin{align} e_s\cdot A^{hom}e_t{\partial\eta_s\partial\eta_t}(0)\xi_s\xi_t\widehat{\psi_o u^*}(\xi). \end{align} | (8.12) |
An application of Lemma 7.2 yields the convergence of
The sequence
\begin{align} \chi_{ \varepsilon^{-1}U}(\xi)\mathcal{B}^{\kappa, \varepsilon}_1g^ \varepsilon(\xi)\rightharpoonup-\left(\frac{\partial \psi_0}{\partial x_k}(x)\sigma^*_k(x)\right)^{\bf\widehat{}}(\xi). \end{align} | (8.13) |
We have the following weak convergence in
\begin{align} \lim\limits_{ \varepsilon\to 0}\,\chi_{ \varepsilon^{-1}U}(\xi)\mathcal{B}^{\kappa, \varepsilon}_1h^{ \varepsilon}(\xi) = -i\xi_ka^*_{kl}\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right)^{\bf\widehat{}}(\xi) \end{align} | (8.14) |
We shall prove this in the following steps.
Step 1. By the definition of the Bloch transform (7.7) for elements of
\begin{align} \mathcal{B}^{\kappa, \varepsilon}_1h^{ \varepsilon}(\xi) = -i\xi_k\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial \psi_0}{\partial x_l}(x)a^{ \varepsilon}_{kl}(x)u^ \varepsilon(x)\overline{\phi^\rho_1\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right)}\,dx\\+\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial \psi_0}{\partial x_l}(x)a^{ \varepsilon}_{kl}(x)u^ \varepsilon(x)\frac{\partial\overline{\phi^{\rho}_1}}{\partial x_k}\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right)\,dx. \end{align} | (8.15) |
Step 2. The first term on RHS of (8.15) is the Bloch transform of the expression
Step 3. Now, we analyze the second term on RHS of (8.15). To this end, we make use of analyticity of first Bloch eigenfunction with respect to the dual parameter
\begin{align} \phi_1^\rho(y;\eta) = \phi_1^\rho(y;0)+\eta_s\frac{\partial \phi^\rho_1}{\partial \eta_s}(y;0)+\gamma^\rho(y;\eta). \end{align} | (8.16) |
We know that
\begin{align} \phi_1^{\kappa, \varepsilon}(x;\xi) = \phi_1^\rho\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right) = \phi_1^\rho\left(\frac{x}{ \varepsilon};0\right)+ \varepsilon\xi_s\frac{\partial \phi^\rho_1}{\partial \eta_s}\left(\frac{x}{ \varepsilon};0\right)+\gamma^\rho\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right). \end{align} | (8.17) |
Differentiating the last equation with respect to
\begin{align} \frac{\partial}{\partial x_k}\phi_1^\rho\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right) = \xi_s\frac{\partial}{\partial y_k}\frac{\partial \phi^\rho_1}{\partial \eta_s}\left(\frac{x}{ \varepsilon};0\right)+ \varepsilon^{-1}\frac{\partial \gamma^\rho}{\partial y_k}\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right). \end{align} | (8.18) |
For
\begin{align} \frac{\partial \gamma^\rho}{\partial y_k}(\cdot; \varepsilon\xi) = O(| \varepsilon\xi|^2) = \varepsilon^2O(|\xi|^2)\leq CM^2 \varepsilon^2. \end{align} | (8.19) |
As a consequence,
\begin{align} \varepsilon^{-2}\frac{\partial \gamma^\rho}{\partial y_k}(x/ \varepsilon; \varepsilon\xi)\in L^\infty_{\rm{loc}}(\mathbb{R}^d_\xi;L^2_\sharp( \varepsilon Y)). \end{align} | (8.20) |
The second term on the RHS of (8.15) is given by
\begin{align} \chi_{ \varepsilon^{-1}U}(\xi)\int_K e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)a_{kl}\left(\frac{x}{ \varepsilon}\right)u^ \varepsilon(x)\frac{\partial}{\partial x_k}\left(\overline{\phi^{\rho}_1}\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right)\right)\,dx. \end{align} | (8.21) |
Substituting (8.18) in (8.21), we obtain
\begin{align} \chi_{ \varepsilon^{-1}U}(\xi)\int_K e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)a_{kl}\left(\frac{x}{ \varepsilon}\right)u^ \varepsilon(x)\biggl[\xi_s\frac{\partial}{\partial y_k}\frac{\partial \phi^{\rho}_1}{\partial \eta_s}\left(\frac{x}{ \varepsilon};0\right)+ \varepsilon^{-1}\frac{\partial \gamma^\rho}{\partial y_k}\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right)\biggr]\,dx. \end{align} | (8.22) |
In the last expression, the term involving
\begin{align} \mathcal{M}_Y\left(a_{kl}(y)\frac{\partial \chi^{\theta}_s}{\partial y_k}(y)\right)\xi_s\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)u^*(x)\,dx. \end{align} | (8.23) |
To see this, we write the second term as
\begin{align*} \int_K& e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)a_{kl}\left(\frac{x}{ \varepsilon}\right)u^ \varepsilon(x)\xi_s\frac{\partial}{\partial y_k}\frac{\partial \phi^{\rho}_1}{\partial \eta_s}\left(\frac{x}{ \varepsilon};0\right)\,dx\\ & = \int_K e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)a_{kl}\left(\frac{x}{ \varepsilon}\right)u^ \varepsilon(x)\xi_s\frac{\partial \chi^{\rho}_s}{\partial y_k}\left(\frac{x}{ \varepsilon}\right)\,dx\\ & = \int_K e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)a_{kl}\left(\frac{x}{ \varepsilon}\right)u^ \varepsilon(x)\xi_s\left(\frac{\partial \chi^{\theta}_s}{\partial y_k}\left(\frac{x}{ \varepsilon}\right)+\left[\frac{\partial \chi^{\rho}_s}{\partial y_k}\left(\frac{x}{ \varepsilon}\right)-\frac{\partial \chi^{\theta}_s}{\partial y_k}\left(\frac{x}{ \varepsilon}\right)\right]\right)\,dx. \end{align*} |
The first term in parantheses goes to (8.23) due to strong convergence of
Step 4. By Theorem 6.1 and Remark 6.2, it follows that
\begin{align} \mathcal{M}_Y\left(a_{kl}(y)\frac{\partial}{\partial y_k}\left(\frac{\partial\phi^{\theta}_1}{\partial\eta_s}(y;0)\right)\right) = -i(2\pi)^{-d/2}\mathcal{M}_Y\left(a_{kl}(y)\frac{\partial \chi^{\theta}_s}{\partial y_k}(y)\right). \end{align} | (8.24) |
Therefore, we have the following convergence in
\begin{align} \chi_{ \varepsilon^{-1}U}(\xi)&\mathcal{B}^{\kappa, \varepsilon}_1h^{ \varepsilon}(\xi)\\ &\rightharpoonup-i\xi_s\biggl\{\mathcal{M}_Y(a_{kl})+\mathcal{M}_Y\left(a_{kl}(y)\frac{\partial \chi^{\theta}_s}{\partial y_k}(y)\right)\biggr\}\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right)^{\bf\widehat{}}(\xi)\\ & = -i\xi_s a_{kl}^{*} \left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right)^{\bf\widehat{}}(\xi) \end{align} | (8.25) |
We shall prove that
\begin{align} \lim\limits_{ \varepsilon\to 0}\mathcal{B}_1^{\kappa, \varepsilon}l_1^{\kappa, \varepsilon} = 0. \end{align} | (8.26) |
Observe that
\begin{align} \mathcal{B}_1^{\kappa, \varepsilon}l^{\kappa, \varepsilon}_1(\xi) = \kappa^2\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial^4\psi^0}{\partial x_k^4}(x)u^ \varepsilon(x)\overline{\phi^{\kappa, \varepsilon}(x,\xi)}\,dx. \end{align} | (8.27) |
The integral is the Bloch transform of
We shall prove that
\begin{align} \lim\limits_{ \varepsilon\to 0}\mathcal{B}_1^{\kappa, \varepsilon}l_2^{\kappa, \varepsilon} = 0. \end{align} | (8.28) |
Observe that
\begin{align} \mathcal{B}_1^{\kappa, \varepsilon}l^{\kappa, \varepsilon}_2(\xi) = 4\kappa^2\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial^3\psi^0}{\partial x_k^3}(x)\frac{\partial u^ \varepsilon}{\partial x_k}(x)\overline{\phi^{\kappa, \varepsilon}(x,\xi)}\,dx. \end{align} | (8.29) |
The integral is the Bloch transform of
We shall prove that
\begin{align} \lim\limits_{ \varepsilon\to 0}\mathcal{B}_1^{\kappa, \varepsilon}l_3^{\kappa, \varepsilon} = 0. \end{align} | (8.30) |
Observe that
\begin{align} \mathcal{B}_1^{\kappa, \varepsilon}l^{\kappa, \varepsilon}_3(\xi) = 2\kappa\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial^2\psi^0}{\partial x_k^2}(x)\kappa\frac{\partial^2 u^ \varepsilon}{\partial x_k^2}(x)\overline{\phi^{\kappa, \varepsilon}(x,\xi)}\,dx. \end{align} | (8.31) |
The integral is the Bloch transform of
We shall prove that
\begin{align} \lim\limits_{ \varepsilon\to 0}\mathcal{B}_1^{\kappa, \varepsilon}l_4^{\kappa, \varepsilon} = 0. \end{align} | (8.32) |
Observe that
\begin{align*} l^{\kappa, \varepsilon}_4(x) = 4\kappa\frac{\partial}{\partial x_k}\left(\frac{\partial\psi_0}{\partial x_k}\kappa\frac{\partial^2u^ \varepsilon}{\partial x_k^2}\right) \end{align*} |
belongs to
\begin{align} \mathcal{B}^{\kappa, \varepsilon}_1l^{\kappa, \varepsilon}_4(\xi) = -4i\kappa\xi_k\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial \psi_0}{\partial x_l}(x)\kappa\frac{\partial^2 u^ \varepsilon}{\partial x_k}(x)\overline{\phi^\rho_1\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right)}\,dx\\+4\kappa\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial \psi_0}{\partial x_l}(x)\kappa\frac{\partial^2 u^ \varepsilon}{\partial x_k^2}(x)\frac{\partial\overline{\phi^{\rho}_1}}{\partial x_k}\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right)\,dx. \end{align} | (8.33) |
The analysis of the first term is the same as that of
Finally, passing to the limit in (8.11) as
\begin{align} a_{kl}^*\xi_k\xi_l\widehat{\psi_o u^*}(\xi) = \widehat{\psi_0 f}-\left(\frac{\partial \psi_0}{\partial x_k}(x)\sigma^*_k(x)\right)^{\bf\widehat{}}(\xi)-i\xi_ka^*_{kl}\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right)^{\bf\widehat{}}(\xi). \end{align} | (8.34) |
In this regime, the convergence proofs for
We expand the first Bloch eigenvalue about
\begin{align*} \left(\frac{1}{2}\frac{\partial^2\lambda^\rho_1}{\partial\eta_s\partial\eta_t}(0)\xi_s\xi_t + \frac{ \varepsilon^2}{4!}\partial^{e_s+e_t+e_u+e_v}_0\lambda_1^\rho\xi_s\xi_t\xi_u\xi_v + O( \varepsilon^4) \right) \mathcal{B}^{\kappa, \varepsilon}_1(\psi_0 u^ \varepsilon). \end{align*} |
The fourth order derivative is of order
\begin{align*} \left(\frac{1}{2}\frac{\partial^2\lambda^\rho_1}{\partial\eta_s\partial\eta_t}(0)\xi_s\xi_t + O( \varepsilon^2\rho^2) + O( \varepsilon^4) \right) \mathcal{B}^{\kappa, \varepsilon}_1(\psi_0 u^ \varepsilon)\\ = \left(\frac{1}{2}\frac{\partial^2\lambda^\rho_1}{\partial\eta_s\partial\eta_t}(0)\xi_s\xi_t + O(\kappa^2) + O( \varepsilon^4) \right) \mathcal{B}^{\kappa, \varepsilon}_1(\psi_0 u^ \varepsilon) \end{align*} |
Now, we can pass to the limit
\begin{align*} e_s\cdot A^{hom}e_t{\partial\eta_s\partial\eta_t}(0)\xi_s\xi_t\widehat{\psi_o u^*}(\xi). \end{align*} |
\begin{align*} \lim\limits_{ \varepsilon\to 0}\,\chi_{ \varepsilon^{-1}U}(\xi)\mathcal{B}^{\kappa, \varepsilon}_1h^{ \varepsilon}(\xi) = -i\xi_ka^*_{kl}\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right)^{\bf\widehat{}}(\xi) \end{align*} |
We shall prove this in the following steps.
Step 1. As before, we have
\begin{align} \mathcal{B}^{\kappa, \varepsilon}_1h^{ \varepsilon}(\xi) = -i\xi_k\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial \psi_0}{\partial x_l}(x)a^{ \varepsilon}_{kl}(x)u^ \varepsilon(x)\overline{\phi^\rho_1\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right)}\,dx\\+\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial \psi_0}{\partial x_l}(x)a^{ \varepsilon}_{kl}(x)u^ \varepsilon(x)\frac{\partial\overline{\phi^{\rho}_1}}{\partial x_k}\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right)\,dx. \end{align} | (8.35) |
Step 2. As before, the first term on RHS of (8.35) is the Bloch transform of the expression
\begin{align*} -i\xi_k\mathcal{M}_Y(a_{kl})\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right) = -i\xi_k a^*_{kl}\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right) \end{align*} |
where the last equality is due to Definition 5.3.
Step 3. Now we shall prove that the second term on RHS of (8.35) goes to zero. As before, we make use of analyticity of first Bloch eigenfunction with respect to the dual parameter
\begin{align*} \phi_1^\rho(y;\eta) = \phi_1^\rho(y;0)+\eta_s\frac{\partial \phi^\rho_1}{\partial \eta_s}(y;0)+\gamma^\rho(y;\eta). \end{align*} |
We know that
\begin{align*} \phi_1^{\kappa, \varepsilon}(x;\xi) = \phi_1^\rho\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right) = \phi_1^\rho\left(\frac{x}{ \varepsilon};0\right)+ \varepsilon\xi_s\frac{\partial \phi^\rho_1}{\partial \eta_s}\left(\frac{x}{ \varepsilon};0\right)+\gamma^\rho\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right). \end{align*} |
Differentiating the last equation with respect to
\begin{align} \frac{\partial}{\partial x_k}\phi_1^\rho\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right) = \xi_s\frac{\partial}{\partial y_k}\frac{\partial \phi^\rho_1}{\partial \eta_s}\left(\frac{x}{ \varepsilon};0\right)+ \varepsilon^{-1}\frac{\partial \gamma^\rho}{\partial y_k}\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right). \end{align} | (8.36) |
For
\begin{align} \frac{\partial \gamma^\rho}{\partial y_k}(\cdot; \varepsilon\xi) = O(| \varepsilon\xi|^2) = \varepsilon^2O(|\xi|^2)\leq CM^2 \varepsilon^2. \end{align} | (8.37) |
As a consequence,
\begin{align*} \varepsilon^{-2}\frac{\partial \gamma^\rho}{\partial y_k}(x/ \varepsilon; \varepsilon\xi)\in L^\infty_{\rm{loc}}(\mathbb{R}^d_\xi;L^2_\sharp( \varepsilon Y)). \end{align*} |
The second term on the RHS of (8.35) is given by
\begin{align} \chi_{ \varepsilon^{-1}U}(\xi)\int_K e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)a_{kl}\left(\frac{x}{ \varepsilon}\right)u^ \varepsilon(x)\frac{\partial}{\partial x_k}\left(\overline{\phi^{\rho}_1}\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right)\right)\,dx. \end{align} | (8.38) |
Substituting (8.36) in (8.38), we obtain
\begin{align} \chi_{ \varepsilon^{-1}U}(\xi)\int_K e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)a_{kl}\left(\frac{x}{ \varepsilon}\right)u^ \varepsilon(x)\biggl[\xi_s\frac{\partial}{\partial y_k}\frac{\partial \phi^{\rho}_1}{\partial \eta_s}\left(\frac{x}{ \varepsilon};0\right)+ \varepsilon^{-1}\frac{\partial \gamma^\rho}{\partial y_k}\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right)\biggr]\,dx. \end{align} | (8.39) |
In the last expression, the term involving
The other term also goes to zero as
\frac{\partial}{\partial x_k}\left(\frac{\partial\phi^{\rho}_1}{\partial\eta_s}(x/ \varepsilon;0)\right) = O\left(\frac{1}{\rho^2}\right), |
as shown in Theorem 6.5.
The analysis is the same as before, however the uniform-in-
Hence, for the regime
In this regime, all the convergence proofs are the same as in the regime
For this limit, all steps except the third are the same, hence we only explain the part of Step
\begin{align} \chi_{ \varepsilon^{-1}U}(\xi)\int_K e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)a_{kl}\left(\frac{x}{ \varepsilon}\right)u^ \varepsilon(x)\biggl[\xi_s\frac{\partial}{\partial y_k}\frac{\partial \phi^{\rho}_1}{\partial \eta_s}\left(\frac{x}{ \varepsilon};0\right)+ \varepsilon^{-1}\frac{\partial \gamma^\rho}{\partial y_k}\left(\frac{x}{ \varepsilon}; \varepsilon\xi\right)\biggr]\,dx. \end{align} | (8.40) |
In the last expression, the term involving
\begin{align} \mathcal{M}_Y\left(a_{kl}(y)\frac{\partial \chi^0_s}{\partial y_k}(y)\right)\xi_s\int_{\mathbb{R}^d}e^{-ix\cdot\xi}\frac{\partial\psi_0}{\partial x_l}(x)u^*(x)\,dx. \end{align} | (8.41) |
To see this, we write
\begin{align*} \frac{\partial}{\partial y_k}\frac{\partial \phi^{\rho}_1}{\partial \eta_s}\left(\frac{x}{ \varepsilon};0\right)& = \frac{\partial \chi^{\rho}_s}{\partial y_k}\left(\frac{x}{ \varepsilon}\right)\\ & = \underbrace{\frac{\partial \chi^0_s}{\partial y_k}\left(\frac{x}{ \varepsilon}\right)}_{I}+\underbrace{\left[\frac{\partial \chi^{\rho}_s}{\partial y_k}\left(\frac{x}{ \varepsilon}\right)-\frac{\partial \chi^B_s}{\partial y_k}\left(\frac{x}{ \varepsilon}\right)\right]}_{II}+\underbrace{\left[\frac{\partial \chi^0_s}{\partial y_k}\left(\frac{x}{ \varepsilon}\right)-\frac{\partial \chi^B_s}{\partial y_k}\left(\frac{x}{ \varepsilon}\right)\right]}_{III}. \end{align*} |
The first term
For the expressions
This completes the modification required for the regime
\begin{align*} a_{kl}^*\xi_k\xi_l\widehat{\psi_o u^*}(\xi) = \widehat{\psi_0 f}-\left(\frac{\partial \psi_0}{\partial x_k}(x)\sigma^*_k(x)\right)^{\bf\widehat{}}(\xi)-i\xi_ka^*_{kl}\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right)^{\bf\widehat{}}(\xi) + O(\varkappa). \end{align*} |
However, since
Taking the inverse Fourier transform in the equation (8.34), we obtain the following:
\begin{align} (\mathcal{A}^{hom}(\psi_0 u^*)(x)) = \psi_0 f-\frac{\partial \psi_0}{\partial x_k}(x)\sigma^*_k(x)-a^*_{kl}\frac{\partial}{\partial x_k}\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right), \end{align} | (8.42) |
where the operator
\begin{align} (\mathcal{A}^{hom}(\psi_0 u^*)(x)) = (\psi_0(x)\mathcal{A}^{hom}u^*(x))&-a^*_{kl}\frac{\partial}{\partial x_k}\left(\frac{\partial \psi_0}{\partial x_l}(x)u^*(x)\right)\\ &-a_{kl}^*\frac{\partial\psi_0}{\partial x_k}(x)\frac{\partial u^*}{\partial x_l}(x) \end{align} | (8.43) |
Using equations (8.42) and (8.43), we obtain
\begin{align} \psi_0(x)\left(\mathcal{A}^{hom}u^*-f\right)(x) = \frac{\partial\psi_0}{\partial x_k}\left[a_{kl}^*\frac{\partial u^*}{\partial x_l}(x)-\sigma_k^*(x)\right]. \end{align} | (8.44) |
Let
\begin{align} \psi_0(x)\left[a_{kl}^*\frac{\partial u^*}{\partial x_l}(x)-\sigma_k^*(x)\right] = 0. \end{align} | (8.45) |
Let
\begin{align} \;{\rm{ for }}\; k = 1,2,\ldots,d,\; \left[a_{kl}^*\frac{\partial u^*}{\partial x_l}(x)-\sigma_k^*(x)\right] = 0 \end{align} | (8.46) |
However,
\begin{align} \mathcal{A}^{hom}u^* = f{\rm{ and }}\sigma^*_k(x) = a_{kl}^*\frac{\partial u^*}{\partial x_l}(x). \end{align} | (8.47) |
Thus, we have obtained the limit equation in the physical space. This finishes the proof of Theorem 8.1.
The proof of the qualitative homogenization theorem only requires the first Bloch transform. It is not clear whether the higher Bloch modes make any contribution to the homogenization limit. In this section, we show that they do not. We know that Bloch decomposition is the isomorphism
\begin{align*} \mathcal{B}^{\kappa, \varepsilon}_m \mathcal{A}^{\kappa, \varepsilon} u^ \varepsilon(\xi) = \mathcal{B}^{\kappa, \varepsilon}_mf(\xi)\quad\forall m\geq 1,\forall\,\xi\in \varepsilon^{-1}Y^{'}. \end{align*} |
We claim that one can neglect all the equations corresponding to
Proposition 9.1. Let
v^{\kappa, \varepsilon}(x) = \int_{ \varepsilon^{-1}Y^{'}}\sum\limits_{m = 2}^{\infty}\mathcal{B}^{\kappa, \varepsilon}_m u^ \varepsilon(\xi)\phi_m^{\kappa, \varepsilon}(x;\xi)e^{ix\cdot\xi}\,d\xi, |
then
Proof. Due to boundedness of the sequence
\begin{align} \int_{\mathbb{R}^d}\mathcal{A}^{\kappa, \varepsilon} u^ \varepsilon\,\overline{u^ \varepsilon}\leq C. \end{align} | (9.1) |
However, by Plancherel Theorem (7.6), we have
\begin{align*} \int_{\mathbb{R}^d} \mathcal{A}^{\kappa, \varepsilon} u^ \varepsilon\, \overline{u^ \varepsilon} = \sum\limits_{m = 1}^{\infty}\int_{ \varepsilon^{-1}Y^{'}}\left(\mathcal{B}^{\kappa, \varepsilon}_m\mathcal{A}^{\kappa, \varepsilon} u^ \varepsilon\right)(\xi)\,\overline{\mathcal{B}^{\kappa, \varepsilon}_mu^ \varepsilon(\xi)}\,d\xi\leq C \end{align*} |
Using (7.8), we have
\begin{align*} \sum\limits_{m = 1}^{\infty}\int_{ \varepsilon^{-1}Y^{'}}\lambda^{\kappa, \varepsilon}_m(\xi)|{\mathcal{B}^{\kappa, \varepsilon}_mu^ \varepsilon(\xi)}|^2\,d\xi\leq C. \end{align*} |
Now, by Lemma 4.2
\begin{align} \lambda_m^\rho(\eta)\geq\alpha\lambda_2^N > 0\quad\forall\, m\geq 2\quad\forall\,\eta\in Y^{'}, \end{align} | (9.2) |
where
\begin{align*} \sum\limits_{m = 2}^{\infty}\int_{ \varepsilon^{-1}Y^{'}}|{\mathcal{B}^{\kappa, \varepsilon}_mu^ \varepsilon(\xi)}|^2\,d\xi\leq C \varepsilon^2. \end{align*} |
By Parseval's identity (7.5), the LHS equals
The author gratefully acknowledges the careful examination of the editor and the anonymous referees. The author would also like to thank Ayan Roychowdhury for a discussion about formation of shear bands.
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