
a) The perforated strip
Citation: Stefano Almi, Giuliano Lazzaroni, Ilaria Lucardesi. Crack growth by vanishing viscosity in planar elasticity[J]. Mathematics in Engineering, 2020, 2(1): 141-173. doi: 10.3934/mine.2020008
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In this paper we consider a spectral problem for the Laplace operator in an unbounded strip
a) The perforated strip
We provide asymptotic formulas for the endpoints of the spectral bands and show that these bands collapse asymptotically at the points of the spectrum of the Dirichlet problem in a rectangle obtained by gluing the lateral sides of the periodicity cell. These formulas show that the spectrum has spectral bands of length
It should be emphasized that waveguides with periodically perturbed boundaries have been the subject of research in the last decade: let us mention e.g. [34], [21], [22], [2] and [3] and the references therein. However the type of singular perturbation that we study in our paper has never been addressed. We consider a waveguide perforated by a periodic perforation string, which implies using a combination of homogenization methods and spectral perturbation theory.
As usual in waveguide theory, we first apply the Gelfand transform (cf. [6], [30], [33], [26], [11] and (11)) to convert the original problem, cf. (7), into a family of spectral problems depending on the Floquet-parameter
The above mentioned homogenization spectral problems have different boundary conditions from those considered in the literature (cf. [5], [14] and [16] for an extensive bibliography). Obtaining convergence for their spectra, correcting terms and precise bounds for discrepancies (cf. (10)), as
Let
Π={x=(x1,x2):x1∈R,x2∈(0,H)} | (1) |
be a strip of width
¯ω=ω∪∂ω⊂ϖ0, | (2) |
where
ϖ0=(−1/2,1/2)×(0,H)⊂Π. | (3) |
We also introduce the strip
ωε(j,k)={x:ε−1(x1−j,x2−εkH)∈ω}with j∈Z,k∈{0,…,N−1}, | (4) |
where
Πε=Π∖⋃j∈ZN−1⋃k=0¯ωε(j,k) | (5) |
is made equal to 1 by rescaling, and similarly, the period is made equal to
ϖε=ϖ0∖N−1⋃k=0¯ωε(0,k), |
(see b) in Figure 1). For brevity, we shall denote by
ωε=⋃j∈ZN−1⋃k=0ωε(j,k), | (6) |
while
In the domain (5) we consider the Dirichlet spectral problem
{−Δuε(x)=λεuε(x),x∈Πε,uε(x)=0,x∈∂Πε. | (7) |
The variational formulation of problem (7) refers to the integral identity
(∇uε,∇v)Πε=λε(uε,v)Πε∀v∈H10(Πε), | (8) |
where
Problem (7) gets a positive cutoff value
σε=⋃n∈NBεn, | (9) |
where
In Section 2 we address the setting of the Floquet parametric family of problems (13)-(16), obtained by applying the Gelfand transform (11) to the original problem (7). They are homogenization spectral problems in a perforated domain, the periodicity cell
Λεi(η)→Λ0i as ε→0,∀η∈[−π,π],i=1,2,⋯. |
However, this result does not give information on the spectral gaps.
Using the method of matched asymptotic expansions for the eigenfunctions of the homogenization problems (cf. Section 4) we are led to the unit cell boundary value problem (31)-(33), the so-called local problem, that is, a problem to describe the boundary layer phenomenon. Section 3 is devoted to the study of this stationary problem for the Laplace operator, which is independent of
|Λεi(η)−Λ0i−εΛ1i(η)|≤ciε3/2, | (10) |
with some
As a consequence, we deduce that the bands
[Λ0i+εBi−−ciε3/2,Λ0i+εBi++ciε3/2], |
of length
Dealing with the precise length of the band, we note that the results rely on the fact that the elements of the antidiagonal of the polarization matrix do not vanish (cf. (70)-(75)), but this is a generic property for many geometries of the unit hole
Summarizing, Section 2 addresses some asymptotics for the spectrum of the Floquet-parameter family of spectral problems; Section 3 considers the unit cell problem; Section 4 deals with the asymptotic expansions; in Section 5.1, we formulate the main asymptotic results of the paper, while the proofs are performed in Section 5.2.
In this section, we deal with the setting of the Floquet-parameter dependent spectral problems and the limit behavior of their spectra, cf. Sections 2.1 and 2.2, respectively.
The Floquet-Bloch-Gelfand transform (FBG-transform, in short)
uε(x)→Uε(x;η)=1√2π∑n∈Ze−inηuε(x1+n,x2), | (11) |
see [6] and, e.g., [30], [33], [11], [26] and [4], converts problem (7) into a
ϖε={x∈Πε:|x1|<1/2} | (12) |
see Figure 1, b. Note that
−ΔUε(x;η)=Λε(η)Uε(x;η),x∈ϖε, | (13) |
Uε(x;η)=0,x∈Γε, | (14) |
Uε(1/2,x2;η)=eiηUε(−1/2,x2;η),x2∈(0,H), | (15) |
∂Uε∂x1(12,x2;η)=eiη∂Uε∂x1(−12,x2;η),x2∈(0,H), | (16) |
where
The variational formulation of the spectral problem (13)-(16) reads:
(∇Uε,∇V)ϖε=Λε(Uε,V)ϖεV∈H1,ηper(ϖε;Γε), | (17) |
where
0<Λε1(η)≤Λε2(η)≤⋯≤Λεm(η)≤⋯→∞, | (18) |
which are repeated according to their multiplicities (see Ch. 10 in [1] and Ch. 13 in [30]). The eigenfunctions are assumed to form an orthonormal basis in
The function
η∈[−π,π]↦Λεm(η) | (19) |
is continuous and
Bεm={Λεm(η):η∈[−π,π]} | (20) |
are closed, connected and bounded intervals of the real positive axis
A first approach to the asymptotics for eigenpairs of (13)-(16) is given by the following convergence result, that we show adapting standard techniques in homogenization and spectral perturbation theory: see, e.g., Ch. 3 in [27] for a general framework and [14] for its application to spectral problems in perforated domains with different boundary conditions. Let us recall
ωε(0,0),…,ωε(0,N−1)⊂ϖ0. | (21) |
Theorem 2.1. Let the spectral problem (13)-(16) and the sequence of eigenvalues (18). Then, for any
Λεm(η)→Λ0m,as ε→0, | (22) |
where
0<Λ01<Λ02≤⋯≤Λ0m≤⋯→∞,as m→∞, | (23) |
are the eigenvalues, repeated according to their multiplicities, of the Dirichlet problem
−ΔU0(x)=Λ0U0(x),x∈υ,υ≡(0,1)×(0,H)U0(x)=0,x∈∂υ. | (24) |
Proof. First, for each fixed
0<C≤Λεm(η)≤Cm∀η∈[−π,π]. | (25) |
To obtain the lower bound in (25), it suffices to consider (17) for the eigenpair
Λεm(η)=minEεm⊂H1,ηper(ϖε;Γε)maxV∈Eεm,V≠0(∇V,∇V)ϖε(V,V)ϖε, |
where the minimum is computed over the set of subspaces
Hence, for each
Λεm(η)→Λ0m(η),Uεm(⋅;η)⇀U0m(⋅;η) in H1(ϖ0)−weak, as ε→0, | (26) |
for a certain positive
‖U;L2(ϖ0∖¯ω)‖≤C‖∇U;L2(ϖ0∖¯ω)‖∀U∈H1(ϖ0∖¯ω),U=0 on ∂ω, |
and we deduce
ε−1‖Uεm(⋅;η);L2({|x1|≤ε/2}∩ϖ0)‖2≤Cε‖∇Uεm(⋅;η);L2({|x1|≤ε/2}∩ϖ0)‖2. |
Now, taking limits as
−ΔU0m(x;η)=Λ0m(η)U0m(x;η),x1∈{(−1/2,0)∪(0,1/2)},x2∈(0,H),U0m(x;η)=0 for x2∈{0,H},x1∈(−1/2,1/2) and x1=0,x2∈(0,H),U0m(1/2,x2;η)=eiηU0m(−1/2,x2;η),x2∈(0,H),∂U0m∂x1(1/2,x2;η)=eiη∂U0m∂x1(−1/2,x2;η),x2∈(0,H), | (27) |
where the differential equation has been obtained by taking limits in the variational formulation (17) for
Now, from the orthonormality of
In addition, extending by
u0m(x;η)={U0m(x;η),x1∈(0,1/2),eiηU0m(x1−1,x2;η),x1∈(1/2,1), | (28) |
we obtain a smooth function in the rectangle
Remark 1. Note that the eigenpairs of (24) can be computed explicitly
Λ0np=π2(n2+p2H2),U0np(x)=2√Hsin(nπx1)sin(pπx2/H),p,n∈N. | (29) |
The eigenvalues
In this section, we study the properties of certain solutions of the boundary value problem in the unbounded strip
In order to obtain a corrector for the approach to the eigenpairs of (13)-(16) given by Theorem 2.1, we introduce the stretched coordinates
ξ=(ξ1,ξ2)=ε−1(x1,x2−εkH). | (30) |
which transforms each opening of the string
−ΔξW(ξ)=0,ξ∈Ξ, | (31) |
with the periodicity conditions
W(ξ1,H)=W(ξ1,0),∂W∂ξ2(ξ1,H)=∂W∂ξ2(ξ1,0),ξ1∈R, | (32) |
and the Dirichlet condition on the boundary of the hole
W(ξ)=0,ξ∈∂ω. | (33) |
Regarding (31)-(33), it should be noted that, for any
Δx+Λε=ε−2(Δξ+ε2Λε), |
and
w(x2)W(ε−1x), | (34) |
for the boundary layer. Here,
It is worth recalling that, according to the general theory of elliptic problems in domains with cylindrical outlets to infinity, cf., e.g., Ch. 5 in [26], problem (31)-(33) has just two solutions with a linear polynomial growth as
¯ω⊂(−R,R)×(0,H) | (35) |
and define the cut-off functions
χ±(y)={1, for ±y>2R,0, for ±y<R, | (36) |
where the subindex
Proposition 3.1. There are two normalized solutions of (31)-(33) in the form
W±(ξ)=±χ±(ξ1)ξ1+∑τ=±χτ(ξ1)pτ±+˜W±(ξ),ξ∈Ξ, | (37) |
where the remainder
p(Ξ)=(p++(Ξ)p+−(Ξ)p−+(Ξ)p–(Ξ)). | (38) |
Proof. The existence of two linearly independent normalized solutions
Let us consider the functions
ˆW±(ξ)=W±(ξ)∓χ±(ξ1)ξ1, | (39) |
which, obviously, satisfy (32), (33) and
−ΔξˆW±(ξ)=F±(ξ),ξ∈Ξ, | (40) |
with
Let
‖W,H‖=‖∇yW;L2(Ξ)‖. |
The variational formulation of (40), (32) and (33) reads: to find
(∇yˆW±,∇yV)Ξ=(F±,V)Ξ∀V∈H. | (41) |
Since
In addition, since for each
ˆWτ(ξ)=cτ±+O(e−(±ξ1)2π/H) as ±ξ1→+∞, |
where the constants
cτ±=limT→∞1H∫H0ˆWτ(±T,ξ2)dξ2=limT→∞1H∫H0(Wτ(±T,ξ2)−τδτ,±T)dξ2. | (42) |
Obviously,
In this section, we detect certain properties of the matrix
Proposition 3.2. The matrix
Proof. We represent (37) in the form
W±(ξ)=W±0(ξ)+{±ξ1−R,±ξ1>R,0,±ξ1<R. | (43) |
The function
[W±0]±(ξ2)=0,[∂W±0∂|ξ1|]±(ξ2)=−1,ξ2∈(0,H), |
where
In what follows, we write the equations for
∫H0Wτ0(±R,ξ2)dξ2=−∫H0Wτ0(±R,ξ2)[∂W±0∂|ξ1|]±(ξ2)dξ2=−(∇ξWτ0,∇ξW±0)Ξ. | (44) |
On the other hand, on account of (43) and the definition of
Wτ0(±R,ξ2)=Wτ(±R,ξ2) and [∂Wτ∂|ξ1|]±(ξ2)=0. |
Consequently, we can write
∫H0Wτ0(±R,ξ2)dξ2=−∫H0Wτ(±R,ξ2)[∂W±0∂|ξ1|]±(ξ2)dξ2=∫H0(Wτ(±R,ξ2)[∂W±0∂|ξ1|]±(ξ2)−W±0(±R,ξ2)[∂Wτ∂|ξ1|]±(ξ2))dξ2, |
and using again the Green formula for
∫H0Wτ0(±R,ξ2)dξ2=+limT→∞∫H0(Wτ(τT,ξ2)∂W±0∂|ξ1|(τT,ξ2)−W±0(τT,ξ2)∂Wτ∂|ξ1|(τT,ξ2))dξ2=−H(pτ±(Ξ)+δτ,±R). | (45) |
Here, we have used the following facts:
W±0(ξ)=χ±(ξ1)(p±±+R)+χ∓(ξ1)p∓±+˜W±(ξ). |
Considering (44) and (45) we have shown the equality for the Gram matrix
(∇ξWτ0,∇ξW±0)Ξ=H(pτ±(Ξ)+δτ,±R), |
which gives the symmetry and the positiveness of the matrix
Let us note that our results above apply for Lipschitz domains or even cracks as it was pointed out in Section 2.1. Now, we get the following results in Propositions 3.3 and 3.4 depending on whether
Proposition 3.3. Let
H(2p+−−p++−p–)>mes2(ω). |
Proof. We consider the linear combination
W0(ξ)=W+(ξ)−W−(ξ)−ξ1=χ+(ξ1)(p++−p+−)−χ−(ξ1)(p–−p−+)+˜W0(ξ). |
It satisfies
−ΔξW0(ξ)=0,ξ∈Ξ,W0(ξ)=−ξ1,ξ∈∂ω, |
with the periodicity conditions in the strip, and
0<‖∇W0;L2(Ξ)‖2+mes2(ω)=−∫∂ωξ1∂ν(ξ1)dν+∫∂ωW0∂ν(W0(ξ))dν=−∫∂ωξ1∂ν(ξ1+W0(ξ))dν=∫∂ω(∂νξ1(ξ1+W0(ξ))−ξ1∂ν(ξ1+W0(ξ)))dν=−limT→∞∑±±∫H0W0(±T,ξ2)dξ2=−H(p+++p–−p+−−p−+). |
Remark 2. observe that for a hole
p++=p–. | (46) |
Indeed, this is due to the fact that each one of the two normalized solutions in (37) are related with each other by symmetry. Also, we note that, on account of Proposition 3.2, the symmetry
Proposition 3.4. Let
p+−=p−+>0. | (47) |
In addition,
Proof. First, let us note that due to the symmetry
W−(ξ1,ξ2)={−ξ1+W∗(−ξ1,ξ2),ξ1<0,W∗(ξ1,ξ2),ξ1>0. | (48) |
where
−ΔξW∗(ξ)=0, for ξ∈Π+,W∗(0,ξ2)=0, for ξ2∈(h,H−h),−∂ξ1W∗(0,ξ2)=1/2, for ξ2∈(0,h)∪(H−h,H). | (49) |
Indeed, denoting by
[˜W∗](0,ξ2)=0,[∂˜W∗∂ξ1](0,ξ2)=−1, |
and hence, the function on the right hand side of (48) is a harmonic function in
Now, considering (49), integrating by parts on
∫Υ(0)W∗(0,ξ2)dξ2=limT→∞H∫0W∗(T,ξ2)dξ2=Hp−+(Ξ). |
Similarly, from (49), we get
0=−∫Π+W∗(ξ)ΔξW∗(ξ)dξ=∫Π+|∇ξW∗(ξ)|2dξ−12∫Υ(0)W∗(0,ξ2)dξ2. |
Therefore, we deduce
H2p−+(Ξ)=∫Π+|∇ξW∗(ξ)|2dξ>0 | (50) |
and from the symmetry of
Also, from the definition (48), we have
From Proposition 3.4, note that when
¯ω={ξ:δ−2ξ21+(ξ2−H/2)2≤τ2},τ=H/2−h, | (51) |
(47) holds true, for a small
In this section we construct asymptotic expansions for the eigenpairs (
Let
Λεm=Λ0m+εΛ1m(η)+⋯. | (52) |
To construct asymptotics of the corresponding eigenfunctions
Uεm(x;η)=U0m(x;η)+εU1m(x;η)+⋯ | (53) |
as the outer expansion, and
Uεm(x;η)=ε∑±wm±(x2;η)W±(xε)+⋯ | (54) |
as the inner expansion near the perforation string, cf. (4) and (21).
Above,
U0m(x;η)={U0m(x),x1∈(0,1/2),e−iηU0m(x1+1,x2),x1∈(−1/2,0), | (55) |
First, let us notice that
Uεm(x;η)=0+x1∂U0m∂x1(0,x2)+εU1m(+0,x2;η)+⋯,x1>0,Uεm(x;η)=0+x1e−iη∂U0m∂x1(1,x2)+εU1m(−0,x2;η)+⋯,x1<0, | (56) |
where, for second formula (56), we have used (55).
The inner expansion (54) is processed by means of decompositions (37). We have
Uεm(x;η)=εwm+(x2;η)(ξ1+p++)+εwm−(x2;η)p−++⋯,ξ1>0,Uεm(x;η)=εwm−(x2;η)(−ξ1+p–)+εwm+(x2;η)p+−+⋯,ξ1<0. | (57) |
Recalling relationship between
wm+(x2;η)=∂U0m∂x1(0,x2),wm−(x2;η)=−e−iη∂U0m∂x1(1,x2), | (58) |
and also obtain the equalities
U1m(+0,x2;η)=∑τ=±wmτ(x2;η)pτ+,U1m(−0,x2;η)=∑τ=±wmτ(x2;η)pτ−. | (59) |
Formulas (58) define coefficients of the linear combination (54) while formulas (59) are the boundary conditions for the correction term in (53). Moreover, inserting ans
{−ΔxU1m(x;η)−Λ0mU1m(x;η)=Λ1m(η)U0m(x;η),x∈ϖ0,x1≠0,U1m(x1,H;η)=U1m(x1,0;η)=0,x1∈(−1/2,0)∪(0,1/2), | (60) |
and the quasi-periodic conditions with
Since
∫ϖ0Λ1m(η)U0m(x;η)¯U0m(x;η)dx=H∫0U1m(−0,x2;η)¯∂U0m∂x1(−0,x2;η)dx2−H∫0U1m(+0,x2;η)¯∂U0m∂x1(+0,x2;η)dx2. |
Thus, by (55) and (59), the only compatibility condition in (60) (recall that
Λ1m(η)=−H∫0¯Bm(x2;η)⋅p(Ξ)Bm(x2;η)dx2 | (61) |
where
Bm(x2;η)=(∂U0m∂x1(0,x2),−e−iη∂U0m∂x1(1,x2))T∈C2, | (62) |
and it determines uniquely the second term of the ansatz (52). Here and in what follows, the top index
Also, from (53), (54) and (57) the composite expansion approaching
Uεm(x;η)≈U0m(x;η)+εU1m(x;η)+ε∑τ=±wmτ(x2;η)Wτ(xε)−(εwm±(x2;η)(ε−1|x1|+p±±)+εwm∓(x2;η)p∓±),±x1≥0. | (63) |
We address the case where
Λεm+l(η)→Λ0m+l as ε→0, for l=0,⋯,κm−1. | (64) |
Let
Following Section 4.1, for each
Λεm+l=Λ0m+εΛ1m+l(η)+⋯, | (65) |
the outer expansion for
Uεm+l(x;η)=U0m+l(x;η)+εU1m+l(x;η)+⋯, | (66) |
and the inner expansion
Uεm(x;η)=ε∑±wm+l±(x2;η)W±(xε)+⋯, | (67) |
where the terms
By repeating the reasoning in Section 4.2, we obtain formulas for the above mentioned terms in (65), (66) and (67) by replacing index
−ΔxU1m+l(x;η)−Λ0mU1m+l(x;η)=Λ1m+l(η)U0m+l(x;η),x∈ϖ0,x1≠0, |
and integrating by parts, we obtain
∫ϖ0Λ1m+l(η)U0m+l(x;η)¯U0m+l′(x;η)dx=−H∫0(∂U0m+l′∂x1(0,x2),−eiη∂U0m+l′∂x1(1,x2))⋅p(Ξ)×(∂U0m+l∂x1(0,x2),−e−iη∂U0m+l∂x1(1,x2))Tdx2. |
Since the eigenfunctions
H∫0∂U0m+l′∂x1(x∗1,x2)∂U0m+l∂x1(x∗1,x2)dx2=0, with x∗1∈{0,1},l≠l′, |
and, hence, for each
Λ1m+l(η)=−H∫0¯Bm+l(x2;η)⋅p(Ξ)Bm+l(x2;η)dx2, | (68) |
where
Bm+l(x2;η)=(∂U0m+l∂x1(0,x2),−e−iη∂U0m+l∂x1(1,x2))T. | (69) |
Therefore we have determined completely all the terms in the asymptotic ans
In this section, we justify the results obtained by means of matched asymptotic expasions in Section 4. Since the case in which all the eigenvalues of the Dirichlet problem (24) are simple can be a generic property, we first consider this case, cf. Theorem 5.1 and Corollary 5.1, and then the case in which these eigenvalues have a multiplicity greater than
Theorem 5.1. Let
|Λεm(η)−Λ0m−εΛ1m(η)|≤cmε3/2 | (70) |
and there are no other different eigenvalues in the sequence (18) satisfying (70).
Theorem 5.1 shows that
In order to detect the gaps between consecutive spectral bands (20) it is worthy writing formulas
Λ1m(η)=B0(m)+B1(m)cos(η), with B0(m)=H∫0(p++|∂U0m∂x1(0,x2)|2+p–|∂U0m∂x1(1,x2)|2)dx2,B1(m)=2p+−H∫0∂U0m∂x1(0,x2)∂U0m∂x1(1,x2)dx2, | (71) |
which are obtained from (61) and (62). Formula (29) demonstrates that
B0(m)=(p+++p–)H∫0|∂U0m∂x1(0,x2)|2dx2, |
and that the integral in
Remark 3. Let us consider that the eigenvalue
B0(m)=2(p+++p–)n2π2,B1(m)=(−1)n4p+−n2π2, |
and, consequently,
Λ1m(η)=2(p+++p–)n2π2+(−1)n4p+−n2π2cos(η). | (72) |
Corollary 5.1. Under the hypothesis of Theorem 5.1, the endpoints
|Bε±(m)−Λ0m−ε(B0(m)±|B1(m)|)|≤cmε3/2. | (73) |
Hence, the length of the band
Note that for the holes such that the polarization matrix (38) satisfies
Theorem 5.2. Let
|Λεm+l0(η)−Λ0m−εΛ1m+l(η)|≤cmε3/2. | (74) |
In addition, when
Corollary 5.2. Under the hypothesis in Theorem 5.2, the spectral bands
[Λ0m+εmin0≤l≤κm−1η∈[−π,π]Λ1m+l(η)−cmε3/2,Λ0m+εmax0≤l≤κm−1η∈[−π,π]Λ1m+l(η)+cmε3/2]. | (75) |
Hence, the length of the the bands
Remark 4. Under the hypothesis of Theorem 5.2, it may happen that, for
Remark 5. Notice that the positive cutoff value
In this section we prove the results of Theorems 5.1 and 5.2 and of their respective corollaries.
Proof of Theorem 5.1. Let us fix
⟨Tε(η)Uε,Vε⟩=(Uε,Vε)ϖε∀Uε,Vε∈H1,ηper(ϖε;Γε). | (76) |
The integral identity (17) for problem (13)-(16) can be rewritten as the abstract equation
Tε(η)Uε(⋅;η)=τε(η)Uε(⋅;η)in H1,ηper(ϖε;Γε), |
with the new spectral parameter
τε(η)=(1+Λε(η))−1. | (77) |
Since
{τεm(η)=(1+Λεm(η))−1}m∈N. |
For the point
tεm(η)=(1+Λ0m+εΛ1m(η))−1, | (78) |
cf. (52) and (61), we construct a function
‖Uεm;H1,ηper(ϖε;Γε)‖≥cm, | (79) |
‖Tε(η)Uεm−tεm(η)Uεm;H1,ηper(ϖε;Γε)‖≤Cmε3/2, | (80) |
where
‖(Tε(η)−tεm(η))−1;H1,ηper(ϖε;Γε)→H1,ηper(ϖε;Γε)‖≥c−1mε−3/2, |
with
\operatorname{dist}(t^{ \varepsilon}_{m}(\eta), \sigma(T^ \varepsilon(\eta)) = \|(T^ \varepsilon(\eta) -t^{ \varepsilon}_{m}(\eta))^{-1}; H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)\to H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon) \|^{-1} |
supported by the spectral decomposition of the resolvent (cf., e.g., Section V.5 in [9] and Ch. 6 in [1]), we deduce that the closed segment
[t^{ \varepsilon}_{m}(\eta)- \mathbf{c}_m \varepsilon^{3/2}, t^{ \varepsilon}_{m}(\eta) + \mathbf{c}_m \varepsilon^{3/2}] |
contains at least one eigenvalue
\begin{equation} \left|(1+\Lambda^{ \varepsilon}_{p}(\eta))^{-1}- (1+ \Lambda^{0}_{m}+ \varepsilon\Lambda^{1}_{m}(\eta))^{-1}\right| \leq \mathbf{c}_m \varepsilon^{3/2}. \end{equation} | (81) |
Then, simple algebraic calculations (cf. (81) and (25)) show that, for a
\begin{equation} \left| \Lambda^{ \varepsilon}_{p}(\eta)- \Lambda^{0}_{m}- \varepsilon\Lambda^{1 }_{m}(\eta)\right| \leq \mathcal{C}_m \varepsilon^{3/2} \end{equation} | (82) |
is satisfied with a constant
To conclude with the proof of Theorem 5.1, there remains to present a function
\begin{equation} \mathcal{V}^{ \varepsilon m}_{out}(x;\eta) = U^{0}_{m}(x;\eta)+ \varepsilon U^{1}_{m}(x;\eta), \end{equation} | (83) |
where
\begin{equation} \mathcal{V}^{ \varepsilon m}_{in}(x;\eta) = \varepsilon \sum\limits_{\pm}w^m_\pm(x_2;\eta)W^\pm( \varepsilon^{-1}x), \end{equation} | (84) |
and
\begin{equation} \mathcal{V}^{ \varepsilon m}_{mat}(x;\eta) = \varepsilon w^m_\pm(x_2;\eta)( \varepsilon^{-1}|x_1|+p_{\pm\pm})+ \varepsilon w^m_\mp (x_2;\eta) p_{\mp\pm}, \quad \pm x_1 > 0, \end{equation} | (85) |
with
\begin{equation} \mathcal{U}^{ \varepsilon}_{m}(x;\eta) = X^ \varepsilon(x_1) \mathcal{V}^{ \varepsilon m}_{out}(x;\eta) + \mathcal{X}(x_1) \mathcal{V}^{ \varepsilon m}_{in}(x;\eta) -X^ \varepsilon(x_1) \mathcal{X}(x_1) \mathcal{V}^{ \varepsilon m}_{mat}(x;\eta), \end{equation} | (86) |
where
\begin{equation} X^ \varepsilon(x_1) = \left\{ \begin{array}{ll} 1, &\hbox{ for } |x_1| > 2R \varepsilon, \\ 0, &\hbox{ for } |x_1| < R \varepsilon, \end{array} \right. \mbox{ and } \quad \mathcal{X}(x_1) = \left\{ \begin{array}{ll} 1, &\hbox{ for } |x_1| < 1/6, \\ 0, &\hbox{ for } |x_1| > 1/3. \end{array} \right. \end{equation} | (87) |
Note that (85) takes into account components in both expressions (83) and (84), but the last subtrahend in
First, let us show that
First of all, we recall (83) and (87) to derive
\begin{multline*} \| \mathcal{U}^{ \varepsilon}_{m} ; H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)\| \\\geq \left\| \mathcal{U}^{ \varepsilon}_{m} ; L^2((1/3, 1/2)\times (0, H))\right\| = \| \mathcal{V}^{ \varepsilon m}_{out} ; L^2((1/3, 1/2)\times (0, H))\| \\ \geq \left\|U^{0}_{m} ; L^2((1/3, 1/2)\times (0, H))\right\| - \varepsilon\|U^{1}_{m} ; L^2((1/3, 1/2)\times (0, H))\| \geq c > 0, \end{multline*} |
for a small
Using (76) and (78), we have
\begin{multline} \|T^ \varepsilon(\eta) \mathcal{U}^{ \varepsilon}_{m} -t^{ \varepsilon}_{m}(\eta) \mathcal{U}^{ \varepsilon}_{m}; H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)\| = \sup\left|\langle T^ \varepsilon(\eta) \mathcal{U}^{ \varepsilon}_{m} -t^{ \varepsilon}_{m}(\eta) \mathcal{U}^{ \varepsilon}_{m}, \mathcal{W}^{ \varepsilon}\rangle\right|\\ = (1+ \Lambda^{0}_{m}+ \varepsilon\Lambda^{1}_{m}(\eta))^{-1}\sup \left|(\nabla \mathcal{U}^{ \varepsilon}_{m}, \nabla \mathcal{W}^{ \varepsilon})_{\varpi^ \varepsilon} -(\Lambda^{0}_{m}+ \varepsilon\Lambda^{1}_{m}(\eta))( \mathcal{U}^{ \varepsilon}_{m}, \mathcal{W}^{ \varepsilon})_{\varpi^ \varepsilon}\right|, \end{multline} | (88) |
where the supremum is computed over all
\| \mathcal{W}^{ \varepsilon}; H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)\|\leq 1. |
Taking into account the Dirichlet conditions on
\int_{ \Pi_T\setminus\overline{\omega} } \vert U \vert^2\, dy \leq C_{T}\int_{ \Pi_T\setminus\overline{\omega} } \vert \nabla_y U \vert^2 dy \quad \forall U \in H^1( \Pi_T\setminus\overline{\omega} ) , \, U = 0 \mbox{ on } \partial \omega, |
where
\int_0^\infty \frac{1}{t^2} z(t)^2\, dt\leq 4 \int_0^\infty \left\vert \frac{dz}{dt}(t)\right\vert^2\, dt \quad \forall z\in C^1[0, \infty), \, z(0) = 0. |
Then, we have
\begin{equation} \|( \varepsilon+|x_1|)^{-1} \mathcal{W}^{ \varepsilon}; L^2(\varpi^ \varepsilon)\|\leq c \|\nabla \mathcal{W}^{ \varepsilon}; L^2(\varpi^ \varepsilon)\|\leq c. \end{equation} | (89) |
Clearly, from (71),
\begin{multline*} \left|(\nabla \mathcal{U}^{ \varepsilon}_{m}, \nabla \mathcal{W}^{ \varepsilon})_{\varpi^ \varepsilon} -(\Lambda^{0}_{m}+ \varepsilon\Lambda^{1}_{m}(\eta))( \mathcal{U}^{ \varepsilon}_{m}, \mathcal{W}^{ \varepsilon})_{\varpi^ \varepsilon}\right|\\ = \left|(\Delta \mathcal{U}^{ \varepsilon}_{m}+(\Lambda^{0}_{m}+ \varepsilon\Lambda^{1}_{m}(\eta)) \mathcal{U}^{ \varepsilon}_{m}, \mathcal{W}^{ \varepsilon})_{\varpi^ \varepsilon}\right|. \end{multline*} |
On the basis of (83)-(86) we write
\begin{multline} \Delta \mathcal{U}^{ \varepsilon}_{m} + (\Lambda^{0}_{m}+ \varepsilon\Lambda^{1}_{m}(\eta)) \mathcal{U}^{ \varepsilon}_{m}\\ = X^ \varepsilon \left(\Delta U^{0}_{m}+\Lambda^{0}_{m} U^{0}_{m} + \varepsilon(\Delta U^{1}_{m}+\Lambda^{0}_{m} U^{1}_{m} +\Lambda^{1}_{m} U^{0}_{m})+ \varepsilon^2 \Lambda^{1}_{m} U^{1}_{m}\right)\\ \quad \quad\quad + [\Delta, X^ \varepsilon]( \mathcal{V}^{ \varepsilon m}_{out}- \mathcal{V}^{ \varepsilon m}_{mat})+ \mathcal{X}(\Delta \mathcal{V}^{ \varepsilon m}_{in}- X^ \varepsilon\Delta \mathcal{V}^{ \varepsilon m}_{mat})+[\Delta, \mathcal{X}]( \mathcal{V}^{ \varepsilon m}_{in}- \mathcal{V}^{ \varepsilon m}_{mat})\\ + (\Lambda^{0}_{m} + \varepsilon\Lambda^{1}_{m}) \mathcal{X}( \mathcal{V}^{ \varepsilon m}_{in}-X^ \varepsilon \mathcal{V}^{ \varepsilon m}_{mat}) = : S_1^ \varepsilon + S^ \varepsilon_2 + S_3^ \varepsilon + S^ \varepsilon_4 + S_5^ \varepsilon. \quad \quad\, \, \end{multline} | (90) |
Here,
Considering
\left|(S^ \varepsilon_1, \mathcal{W}^ \varepsilon)_{\varpi^ \varepsilon}\right|\leq \varepsilon^2 \Lambda^{1}_{m}(\eta) \|U^{1}_{m} ; L^2(\varpi^ \varepsilon)\|\| \mathcal{W}^{ \varepsilon}; L^2(\varpi^ \varepsilon)\|\leq C_m \varepsilon^2. |
As regards
\begin{eqnarray*} \left| \mathcal{V}^{ \varepsilon m}_{out}(x;\eta)- \mathcal{V}^{ \varepsilon m}_{mat}(x;\eta)\right|&\leq & c(|x_1|^2+ \varepsilon |x_1|), \\ \left|\frac{\partial \mathcal{V}^{ \varepsilon m}_{out }}{\partial x_1}(x;\eta)-\frac{\partial \mathcal{V}^{ \varepsilon m}_{mat }}{\partial x_1}(x;\eta)\right|&\leq &c(|x_1|+ \varepsilon), \quad\pm x_1\in [ \varepsilon R, 2 \varepsilon R]. \end{eqnarray*} |
Above, we have also used the smoothness of the function
\begin{eqnarray*} \left|(S^ \varepsilon_2, \mathcal{W}^ \varepsilon)_{\varpi^ \varepsilon}\right| \leq & \|S^ \varepsilon_2 ; L^2(\varpi^ \varepsilon_{ \varepsilon R})\|\| \mathcal{W}^{ \varepsilon}; L^2(\varpi^ \varepsilon_{ \varepsilon R})\|\leq c \varepsilon \|( \varepsilon+|x_1|)^{-1} \mathcal{W}^{ \varepsilon}; L^2(\varpi^ \varepsilon)\|\\ & \times \left(\int\limits^H_0\int\limits^{2 \varepsilon R}_{ \varepsilon R}\left(\frac{1}{ \varepsilon^2}\left|\frac{ \mathcal{V}^{ \varepsilon m}_{out }}{\partial x_1}-\frac{ \mathcal{V}^{ \varepsilon m}_{mat }}{\partial x_1}\right|^2+\frac{1}{ \varepsilon^4}\left| \mathcal{V}^{ \varepsilon m}_{out}- \mathcal{V}^{ \varepsilon m}_{mat}\right|^2\right)d|x_1|dx_2\right)^\frac 12 \\ \leq & c\left(\frac{1}{ \varepsilon^2} \varepsilon^2+\frac{1}{ \varepsilon^4} \varepsilon^4\right)^\frac 12\left(mes_2 \varpi^ \varepsilon_{ \varepsilon R} \right)^\frac 12 \varepsilon \|( \varepsilon+|x_1|)^{-1} \mathcal{W}^{ \varepsilon}; L^2(\varpi^ \varepsilon)\|\leq c \varepsilon^\frac{3}{2}. \end{eqnarray*} |
Dealing with
\begin{multline*} \Delta \mathcal{V}^{ \varepsilon m}_{in}(x;\eta)- X^ \varepsilon(x_1)\Delta \mathcal{V}^{ \varepsilon m}_{mat}(x;\eta)\\ = 2\sum\limits_{\pm} \frac{\partial w^m_\pm}{\partial x_2}(x_2;\eta)\frac{\partial W^\pm}{\partial \xi_2}(y) + \varepsilon \sum\limits_{\pm} \frac{\partial^2 w^m_\pm}{\partial x_2^2}(x_2;\eta)\widetilde{W}^\pm(y), \end{multline*} |
when
\begin{multline*} \left|(S^ \varepsilon_3, \mathcal{W}^ \varepsilon)_{\varpi^ \varepsilon}\right| \leq \\ c\left( \left\|( \varepsilon+|x_1|)\frac{\partial W^\pm}{\partial \xi_2}; L^2(\varpi^ \varepsilon)\right\| + \varepsilon \left\|( \varepsilon+|x_1|)\widetilde{W}^\pm; L^2(\varpi^ \varepsilon)\right\| \right) \left\|\frac{1}{ \varepsilon+|x_1|} \mathcal{W}^{ \varepsilon}; L^2(\varpi^ \varepsilon)\right\| \\ \leq c\left( \int\limits_0^{1/2} ( \varepsilon + t)^2e^{-2\delta t/ \varepsilon} dt\right)^\frac 12 \|\nabla \mathcal{W}^{ \varepsilon}; L^2(\varpi^ \varepsilon)\|\leq c \varepsilon^\frac{3}{2}. \end{multline*} |
Above, obviously, we take the positive constant
\left|(S^ \varepsilon_5, \mathcal{W}^ \varepsilon)_{\varpi^ \varepsilon}\right|\leq c \varepsilon^\frac{3}{2}. |
Moreover, the supports of the coefficients
\left|(S^ \varepsilon_4, \mathcal{W}^ \varepsilon)_{\varpi^ \varepsilon}\right|\leq c{e^{-2\delta/(3 \varepsilon) }}. |
Revisiting the obtained estimates we find the worst bound
The fact that the constants
\begin{equation} -\big(\frac{\partial}{\partial y_1} + {\rm i} \eta \big) \big(\frac{\partial}{\partial y_1} + {\rm i}\eta \big)- \frac{\partial^2}{\partial y_2^2}, \end{equation} | (91) |
and therefore, performing this change in (60), gives the solution
Proof of Corollary 5.1. Due to the continuity of the function (19), the maximum and minimum of
In order to show (73) for the maximum
\Lambda^{0}_{m}+ \varepsilon B_0(m) + \varepsilon |B_1(m)|-c_m \varepsilon^{3/2} \leq \Lambda_m^{ \varepsilon}(\eta^+) \leq \Lambda^{0}_{m}+ \varepsilon B_0(m) + \varepsilon |B_1(m)| + c_m \varepsilon^{3/2} |
and
\Lambda^{0}_{m} + \varepsilon \Lambda^1_m(\eta^+_{ \varepsilon, m}) -c_m \varepsilon^{3/2} \leq\Lambda_m^{ \varepsilon}(\eta^+_{ \varepsilon, m}) \leq \Lambda^{0}_{m}+ \varepsilon \Lambda^1_m(\eta^+_{ \varepsilon, m}) + c_m \varepsilon^{3/2}. |
Consequently, from (71), we derive
\begin{multline*} \Lambda^{0}_{m}+ \varepsilon B_0(m) + \varepsilon |B_1(m)|-c_m \varepsilon^{3/2} \leq \Lambda_m^{ \varepsilon}(\eta^+ ) \leq \Lambda_m^{ \varepsilon}(\eta^+_{ \varepsilon, m}) \\ \leq \Lambda^{0}_{m}+ \varepsilon B_0(m) + \varepsilon |B_1(m)| + c_m \varepsilon^{3/2}, \end{multline*} |
which gives (73) for
We proceed in a similar way for the minimum
\left [\Lambda^{0}_{m}+ \varepsilon B_0(m) - \varepsilon |B_1(m)|-c_m \varepsilon^{3/2} \, , \, \Lambda^{0}_{m}+ \varepsilon B_0(m) + \varepsilon |B_1(m)| + c_m \varepsilon^{3/2}\right] |
Therefore, the whole band
Proof of Theorem 5.2. This proof holds exactly the same scheme of Theorem 5.1. Indeed, for each
\begin{equation} \left| \Lambda^{ \varepsilon}_{p}(\eta)- \Lambda^{0}_{m}- \varepsilon\Lambda^{1 }_{m+l}(\eta)\right| \leq \mathcal{C}_{m, l} \varepsilon^{3/2} \end{equation} | (92) |
is satisfied for a certain natural
We use a classical argument of contradiction (cf. [15] and [25]). We consider the set of functions
\begin{equation} \| \mathcal{U}^{ \varepsilon}_{m+l} ; H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)\|\geq \widetilde c_m \quad \mbox{ and }\quad \left \vert \langle \mathcal{U}^{ \varepsilon}_{m+l} \, , \, \mathcal{U}^{ \varepsilon}_{m+l'} \rangle \right \vert \leq \widetilde C_m \varepsilon^{1/2}, \mbox{ with } l\not = l'\, , \end{equation} | (93) |
for certain constants
Then, we define
\begin{equation} \left|(1+\Lambda^{ \varepsilon}_{p(l)}(\eta))^{-1}- (1+ \Lambda^{0}_{m}+ \varepsilon\Lambda^{1}_{m+l}(\eta))^{-1}\right| \leq \widetilde{\mathbf{c }}_m \varepsilon^{3/2}, \end{equation} | (94) |
for a constant
\begin{equation} \left \| \, \widetilde{\mathcal{W }}^ \varepsilon_{m+l} -\widetilde{\mathcal{U }}^ \varepsilon_{m+l} ; H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon) \right \|\leq \widetilde{\mathcal{C }}_m , \end{equation} | (95) |
where
\left \| \, \widetilde{\mathcal{U }}^ \varepsilon_{m+l} - \mathcal{W}^ \varepsilon_{m+l} ; H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon) \right \|\leq \widetilde{\mathbf{c }}_m^{-1} \max\limits_{0\leq l\leq \kappa_m-1} \mathcal{C}_{l, m} , |
and some straightforward computation (cf., eg., Lemma 1 in Ch. 3 of [27]). Now, from (93) and (95) and straightforward computations we obtain
\begin{equation} \left \vert \langle\widetilde{\mathcal{W }}^{ \varepsilon}_{m+l} \, , \, \widetilde{\mathcal{W }}^{ \varepsilon}_{m+l'} \rangle \right \vert \leq 5\widetilde{\mathcal{C }}_m \, \mbox{ with } \, l\not = l'\, , \end{equation} | (96) |
and this allows us to assert that set
\sum\limits_{l = 0}^{\kappa_m-1} \alpha_l^ \varepsilon \widetilde{\mathcal{W }}^ \varepsilon_{m+l} = 0. |
Let us consider
\langle \widetilde{\mathcal{W }}^ \varepsilon_{m}, \widetilde{\mathcal{W }}^ \varepsilon_{m} \rangle \leq \sum\limits_{l = 1}^{\kappa_m-1} \left\vert \frac{\alpha_l^ \varepsilon}{\alpha_0^ \varepsilon} \right\vert \left\vert \langle \widetilde{\mathcal{W }}^ \varepsilon_{m+l}, \widetilde{\mathcal{W }}^ \varepsilon_{m} \rangle \right \vert \leq (\kappa_m-1) 5\widetilde{\mathcal{C }}_m . |
Now, setting
Thus,
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