Citation: Yanfeng Shi, Shuo Qiu, Jiqiang Liu, Tinghuai Ma. Novel efficient lattice-based IBE schemes with CPK for fog computing[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 8105-8122. doi: 10.3934/mbe.2020411
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Cloaking using transformation optics (changes of variables) was introduced by Pendry, Schurig and Smith [30] for the Maxwell system and by Leonhardt [16] in the geometric optics setting. These authors used a singular change of variables, which blows up a point into a cloaked region. The same transformation had been used to establish (singular) non-uniqueness in Calderon's problem in [10]. To avoid using the singular structure, various regularized schemes have been proposed. One of them was suggested by Kohn, Shen, Vogelius and Weinstein [11], where instead of a point, a small ball of radius ε is blown up to the cloaked region. Approximate cloaking for acoustic waves has been studied in the quasistatic regime [11,26], the time harmonic regime [12,19,27,20], and the time regime [28,29], and approximate cloaking for electromagnetic waves has been studied in the time harmonic regime [4,14,24], see also the references therein. Finite energy solutions for the singular scheme have been studied extensively [9,32,33]. There are also other ways to achieve cloaking effects, such as the use of plasmonic coating [2], active exterior sources [31], complementary media [13,22], or via localized resonance [23] (see also [17,21]).
The goal of this paper is to investigate approximate cloaking for the heat equation using transformation optics. Thermal cloaking via transformation optics was initiated by Guenneau, Amra and Venante [8]. Craster, Guenneau, Hutridurga and Pavliotis [6] investigate the approximate cloaking for the heat equation using the approximate scheme in the spirit of [11]. They show that for the time large enough, the largeness depends on ε, the degree of visibility is of the order εd (d=2,3) for sources that are independent of time. Their analysis is first based on the fact that as time goes to infinity, the solutions converge to the stationary states and then uses known results on approximate cloaking in the quasistatic regime [11,26].
In this paper, we show that approximate cloaking is achieved at any positive time and established the degree of invisibility of order ε in three dimensions and |lnε|−1 in two dimensions. Our results hold for a general source that depends on both time and space variables, and our estimates depend only on the range of the materials inside the cloaked region. The degree of visibility obtained herein is optimal due to the fact that a finite time interval is considered (compare with [6]). The analysis in this paper is of frequency type via Fourier transform with respect to time. This approach is robust and can be used in different context. A technical issue is on the blow up of the fundamental solution of the Helmholtz type equations in two dimensions in the low frequency regime. We emphasize that even though our setting is in a bounded domain, we employs Fourier transform in time instead of eigenmodes decomposition. This has the advantage that one can put the non-perturbed system and the cloaking system in the same context.
We next describe the problem in more detail and state the main result. Our starting point is the regularization scheme [11] in which a transformation blows up a small ball Bε (0<ε<1/2) instead of a point into the cloaked region B1 in Rd (d=2,3). Here and in what follows, for r>0, Br denotes the ball centered at the origin and of radius r in Rd. Our assumption on the geometry of the cloaked region is mainly to simplify the notations. Concerning the transformation, we consider the map Fε:Rd→Rd defined by
Fε(x)={x in Rd∖B2,(2−2ε2−ε+|x|2−ε)x|x| in B2∖Bε,xε in Bε. | (1.1) |
In what follows, we use the standard notations
F∗A(y)=∇F(x)A(x)∇FT(x)|det∇F(x)|,F∗ρ(y)=ρ(x)|det∇F(x)|,x=F−1(y), | (1.2) |
for the "pushforward" of a symmetric, matrix-valued function A, and a scalar function ρ, by the diffeomorphism F, and I denotes the identity matrix. The cloaking device in the region B2∖B1 constructed from the transformation technique is given by
(Fε∗I,Fε∗1) in B2∖B1, | (1.3) |
a pair of a matrix-valued function and a function that characterize the material properties in B2∖B1. Physically, this is the pair of the thermal diffusivity and the mass density of the material.
Let Ω with B2⋐Ω⊂Rd (d=2,3)* be a bounded region for which the heat flow is considered. Suppose that the medium outside B2 (the cloaking device and the cloaked region) is homogeneous so that it is characterized by the pair (I,1), and the cloaked region B1 is characterized by a pair (aO,ρO) where aO is a matrix-valued function and ρO is a real function, both defined in B1. The medium in Ω is then given by
*The notation D⋐Ω means that the closure of D is a subset of Ω.
(Ac,ρc)={(I,1) in Ω∖B2,(Fε∗I,Fε∗1) in B2∖B1,(aO,ρO) in B1. | (1.4) |
In what follows, we make the usual assumption that aO is symmetric and uniformly elliptic and ρO is a positive function bounded above and below by positive constants, i.e., for a.e. x∈B1,
Λ−1|ξ|2≤⟨aO(x)ξ,ξ⟩≤Λ|ξ|2 for all ξ∈Rd, | (1.5) |
and
Λ−1≤ρO(x)≤Λ, | (1.6) |
for some Λ≥1. Given a function f∈L1((0,+∞),L2(Ω)) and an initial condition u0∈L2(Ω), in the medium characterzied by (Ac,ρc), one obtains a unique weak solution uc∈L2((0,∞);H1(Ω)) ∩C([0,+∞);L2(Ω)) of the system
{∂t(ρcuc)−div(Ac∇uc)=f in (0,+∞)×Ω,uc=0 on (0,+∞)×∂Ω,uc(t=0,⋅)=u0 in Ω, | (1.7) |
and in the homogeneneous medium characterized by (I,1), one gets a unique weak solution u∈L2((0,∞);H1(Ω))∩C([0,+∞);L2(Ω)) of the system
{∂tu−Δu=f in (0,+∞)×Ω,uc=0 on (0,+∞)×∂Ω,uc(t=0,⋅)=u0 in Ω. | (1.8) |
The approximate cloaking meaning of the scheme (1.4) is given in the following result:
Theorem 1.1. Let u0∈L2(Ω) and f∈L1((0,+∞);L2(Ω)) be such that suppu0,suppf(t,⋅)⊂Ω∖B2 for t>0. Assume that uc and u are the solution of (1.7) and (1.8) respectively. Then, for 0<ε<1/2,
‖uc(t,⋅)−u(t,⋅)‖H1(Ω∖B2)≤Ce(ε,d)(‖f‖L1((0,+∞);L2(Ω))+‖u0‖L2(Ω)), |
for some positive constant C depending on Λ but independent of f, u0, and ε, where
e(ε,d)={εif d=3,|lnε|−1if d=2. |
As a consequence of Theorem 1.1, limε→0uc(t,⋅)=u(t,⋅) in (0,+∞)×(Ω∖B2) for all f with compact support outside (0,+∞)×B2 and for all u0 with compact support outside B2. One therefore cannot detect the difference between (Ac,ρc) and (I,1) as ε→0 by observation of uc outside B2: Cloaking is achieved for observers outside B2 in the limit as ε→0.
We now briefly describe the idea of the proof. The starting point of the analysis is the invariance of the heat equations under a change of variables which we now state.
Lemma 1.1. Let d≥2, T>0, Ω be a bounded open subset of Rd of class C1, and let A be an elliptic symmetric matrix-valued function, and ρ be a bounded, measurable function defined on Ω bounded above and below by positive constants. Let F:Ω↦Ω be bijective such that F and F−1 are Lipschitz, det∇F>c for a.e. x∈Ω for some c>0, and F(x)=x near ∂Ω. Let f∈L1((0,T);L2(Ω)) and u0∈L2(Ω). Then u∈L2((0,T);H10(Ω))∩C([0,T);L2(Ω)) is the weak solution of
{∂t(ρu)−div(A∇u)=fin ΩT,u=0on (0,T)×∂Ω,u(0,⋅)=u0in Ω, | (1.9) |
if and only if v(t,⋅):=u(t,⋅)∘F−1∈L2((0,T);H10(Ω))∩C([0,T);L2(Ω)) is the weak solution of
{∂t(F∗ρv)−div(F∗A∇v)=F∗fin ΩT,u=0on (0,T)×∂Ω,v(0,⋅)=u0∘F−1in Ω. | (1.10) |
Recall that F∗ is defined in (1.2). In this paper, we use the following standard definition of weak solutions:
Definition 1.1. Let d≥2 and T>0. We say a function
u∈L2((0,T);H10(Ω))∩C([0,T);L2(Ω)) |
is a weak solution to (1.9) if u(0,⋅)=u0 in Ω and u satisfies
ddt∫Ωρu(t,⋅)φ+∫ΩA∇u(t,⋅)∇φ=∫Ωf(t,⋅)φ in (0,T), | (1.11) |
in the distributional sense for all φ∈H10(Ω).
The existence and uniqueness of weak solutions are standard, see, e.g., [1] (in fact, in [1], f is assumed in L2((0,T);L2(Ω)), however, the conclusion holds also for f∈L1((0,T);L2(Ω)) with a similar proof, see, e.g., [25]). The proof of Lemma 1.1 is similar to that of the Helmholtz equation, see, e.g., [12] (see also [6] for a parabolic version).
We now return to the idea of the proof of Theorem 1.1. Set
uε(t,⋅)=uc(t,⋅)∘F−1ε for t∈(0,+∞). |
Then uε is the unique solution of the system
{∂t(ρεuε)−div(Aε∇uε)=f in (0,+∞)×Ω,uε=0 on (0,+∞)×∂Ω,uε(t=0,⋅)=u0 in Ω, | (1.12) |
where
(Aε,ρε)={(I,1) in Ω∖Bε,(ε2−daO(⋅/ε),ε−dρO(⋅/ε)) in Bε. | (1.13) |
Moreover,
uc−u=uε−u in (0,+∞)×(Ω∖B2). |
In comparing the coefficients of the systems verified by u and uε, the analysis can be derived from the study of the effect of a small inclusion Bε. The case in which finite isotropic materials contain inside the small inclusion was investigated in [3] (see also [5] for a related context). The analysis in [3] partly involved the polarization tensor information and took the advantage of the fact that the coefficients inside the small inclusion are finite. In the cloaking context, Craster et al. [6] derived an estimate of the order εd for a time larger than a threshold one. Their analysis is based on long time behavior of solutions to parabolic equations and estimates for the degree of visibility of the conducting problem, see [11,26], hence the threshold time goes to infinity as ε→0.
In this paper, to overcome the blow up of the coefficients inside the small inclusion and to achieve the cloaking effect at any positive time, we follow the approach of Nguyen and Vogelius in [28]. The idea is to derive appropriate estimates for the effect of small inclusions in the time domain from the ones in the frequency domain using the Fourier transform with respect to time. Due to the dissipative nature of the heat equation, the problem in the frequency for the heat equation is more stable than the one corresponding to the acoustic waves, see, e.g., [27,28], and the analysis is somehow easier to handle in the high frequency regime. After using a standard blow-up argument, a technical point in the analysis is to obtain an estimate for the solutions of the equation Δv+iωε2v=0 in Rd∖B1 (ω>0) at the distance of the order 1/ε in which the dependence on ε and ω are explicit (see Lemma 2.2). Due to the blow up of the fundamental solution in two dimensions, the analysis requires new ideas. We emphasize that even though our setting is in a bounded domain with zero Dirichlet boundary condition, we employs Fourier transform in time instead of eigenmodes decomposition as in [6] to put both systems of uε and u in the same context.
To implement the analysis in the frequency domain, let us introduce the Fourier transform with respect to time t:
ˆφ(k,x)=∫Rφ(t,x)eiktdt for k∈R, | (2.1) |
for φ∈L2((−∞,+∞),L2(Rd)). Extending u,uc, uρ and f by 0 for t<0, and considering the Fourier with respect to time at the frequency ω>0, we obtain
Δˆu+iωˆu=−(ˆf+u0) in Ω, |
and
div(Aε∇ˆuε)+iωρεˆuε=−(ˆf+u0) in Ω, |
where
(Aε,ρε)={(I,1) in Ω∖Bε,(ε2−daO(⋅/ε),ε−dρO(⋅/ε)) in Bε. |
The main ingredient in the proof of Theorem 1.1 is the following:
Proposition 2.1. Let ω>0, 0<ε<1/2, and let g∈L2(Ω) with suppg⊂Ω∖B2. Assume that v,vε∈H1(Ω) are respectively the unique solution of the systems
{Δv+iωv=gin Ω,v=0on ∂Ω, |
and
{div(Aε∇vε)+iωρεvε=gin Ω,vε=0on ∂Ω. |
We have
‖vε−v‖H1(Ω∖B2)≤Ce(ε,ω,d)(1+ω−1/2)‖g‖L2(Ω), | (2.2) |
for some positive constant C independent of ε, ω and g. Here
e(ε,ω,3)=εe−ω1/2/4, | (2.3) |
and
e(ε,ω,2)={e−ω1/2/4/|lnε|if ω≥1/2,lnω/ln(ωε)if 0<ω<1/2. | (2.4) |
The rest of this section is divided into three subsections. In the first subsection, we present several lemmas used in the proof of Proposition 2.1. The proofs of Proposition 2.1 and Theorem 1.1 are then given in the second and the third subsections, respectively.
In this subsection, we state and prove several useful lemmas used in the proof of Proposition 2.1. Throughout, D⊂B1 denotes a smooth, bounded, open subset of Rd such that Rd∖D is connected, and ν denotes the unit normal vector field on ∂D, directed into Rd∖D.
The first result is the following simple one:
Lemma 2.1. Let d=2,3, k>0, and let v∈H1(Rd∖D) be such that Δv+ikv=0 in Rd∖D. We have, for R>2,
‖v‖H1(BR∖D)≤CR(1+k)‖v‖H1/2(∂D), | (2.5) |
for some positive constants CR independent of k and v.
Proof. Multiplying the equation by ˉv (the conjugate of v) and integrating by parts, we have
∫Rd∖D|∇v|2−ik∫Rd∖D|v|2=∫∂D∂νvˉv. |
This implies
∫Rd∖D|∇v|2+k∫Rd∖D|v|2≤C‖∂νv‖H−1/2(∂D)‖v‖H1/2(∂D). | (2.6) |
Here and in what follows, C denotes a positive constant independent of v and k. Since Δv=−ikv in B2∖D, by the trace theory, see, e.g., [7,Theorem 2.5], we have
‖∂νv‖H−1/2(∂D)≤C(‖∇v‖L2(B2∖D)+‖Δv‖L2(B2∖D))≤C(‖∇v‖L2(B2∖D)+k‖v‖L2(B2∖D)). | (2.7) |
Combining (2.6) and (2.7) yields
∫Rd∖D|∇v|2+k∫Rd∖D|v|2≤C(1+k)‖v‖2H1/2(∂D). | (2.8) |
The conclusion follows when k≥1.
Next, consider the case 0<k<1. In the case where d=3, the conclusion is a direct consequence of (2.8) and the Hardy inequality (see, e.g., [18,Lemma 2.5.7]):
∫R3∖D|v|2|x|2≤C∫R3∖D|∇v|2. | (2.9) |
We next consider the case where d=2. One just needs to show
∫BR∖D|v|2≤C‖v‖2H1/2(∂D). | (2.10) |
By the Hardy inequality (see, e.g., [18,Lemma 2.5.7]),
∫R2∖D|v|2|x|2ln(2+|x|)2≤C(∫R2∖D|∇v|2+∫B2∖D|v|2), | (2.11) |
it suffices to prove (2.10) for R=2 by contradiction. Suppose that there exists a sequence (kn)→0 and a sequence (vn)∈H1(R2∖D) such that
Δvn+iknvn=0 in R2∖D,‖vn‖L2(B2∖D)=1, and limn→+∞‖vn‖H1/2(∂D)=0. |
Denote
W1(R2∖D)={u∈L1loc(R2∖D);u(x)ln(2+|x|)√1+|x|2∈L2(R2∖D) and ∇u∈L2(R2∖D)}. |
By (2.8) and (2.11), one might assume that vn converges to v weakly in H1loc(R2∖D) and strongly in L2(B2∖D). Moreover, v∈W1(R2∖D) and v satisfies
Δv=0 in R2∖D,v=0 on ∂D, | (2.12) |
and
‖v‖L2(B2∖D)=1. | (2.13) |
From (2.12), we have v=0 in R2∖D (see, e.g., [18]) which contradicts (2.13). The proof is complete.
We also have
Lemma 2.2. Let d=2,3, ω>0, 0<ε<1/2, and let v∈H1(Rd∖D) be a solution of Δv+iωε2v=0 in Rd∖D. We have, for 3/2<|x|<R,
|v(x/ε)|≤Ce(ε,ω,d)‖v‖H1/2(∂D), | (2.14) |
for some positive constant C=CR independent of ε, ω and v.
Recall that e(ε,ω,d) is given in (2.3) and (2.4).
Proof. By the trace theory and the regularity theory of elliptic equations, we have
‖v‖L2(∂B2)+‖∇v‖L2(∂B2)≤C‖v‖H2(B5/2∖B3/2)≤C(1+ω1/2ε)‖v‖H1(B3∖B1). | (2.15) |
It follows from Lemma 2.1 that
‖v‖L2(∂B2)+‖∇v‖L2(∂B2)≤C(1+ω3/2)‖v‖H1/2(∂D). | (2.16) |
Here and in what follows in this proof, C denotes a positive constant depending only on R and D.
The representation formula gives
v(x)=∫∂B2(Gℓ(x,y)∂rv(y)−∂ryGℓ(x,y)v(y))dy for x∈Rd∖ˉB2, | (2.17) |
where ℓ=eiπ/4εω1/2, and, for x≠y,
Gℓ(x,y)=eiℓ|x−y|4π|x−y| if d=3 and Gℓ(x,y)=i4H(1)0(ℓ|x−y|) if d=2. |
Here H(1)0 is the Hankel function of the first kind of order 0. Recall, see, e.g., [15,Chapter 5], that
H(1)0(z)=2iπln|z|2+1+2iγπ+O(|z|2log|z|) as z→0,z∉(−∞,0], | (2.18) |
and
H(1)0(z)=√2πzei(z+π4)(1+O(|z|−1))z→∞,z∉(−∞,0]. | (2.19) |
We now consider the case d=3. We have, for 3/2<|x|<R and y∈∂B2,
|eiℓ|x/ε−y||≤e−√22ω1/2|x−εy|≤e−ω1/2|x|/3. |
It follows that, for 3/2<|x|<R and y∈∂B2,
|Gℓ(x/ε,y)|≤Cεe−3ω1/2/10. | (2.20) |
Similarly, one has, for 3/2<|x|<R and y∈∂B2,
|∂ryGℓ(x/ε,y)|≤C(ε2ω1/2|x|+ε2|x|2)e−ω1/2|x|/3≤Cεe−3ω1/2/10. | (2.21) |
Combining (2.17), (2.20) and (2.21) yields
|v(x/ε)|≤Cεe−3ω1/2/10(‖v‖L2(∂B2)+‖∇v‖L2(∂B2)) for 3/2<|x|<R. |
We derive from (2.16) that
|v(x/ε)|≤Cεe−ω1/2/4‖v‖H1/2(∂D) for 3/2<|x|<R; |
which is the conclusion in the case d=3.
We next deal with the case where d=2 and ω>ε−2/4, which is equivalent to |ℓ|>1/2. From (2.19), we derive that, for 3/2<|x|<R and y∈∂B2,
|Gℓ(x/ε,y)|≤Cω−1/4e−3ω1/2/10 and |∂ryGℓ(x/ε,y)|≤Cεω1/4e−3ω1/2/10. | (2.22) |
Using (2.16) and combining (2.17) and (2.22), we obtain, since ω>ε−2/4,
|v(x/ε)|≤Cεe−ω1/2/4‖v‖H1/2(∂D) for 3/2<|x|<R, |
which gives the conclusion in this case.
We finally deal with the case where d=2 and 0<ω<ε−2/4, which is equivalent to |ℓ|<1/2. From (2.17), we obtain, for x∈∂B4,
v(x)=∫∂B2([Gℓ(x,y)−Gℓ(x,0)]∂rv(y)−∂ryGℓ(x,y)v(y))dy+∫∂B2Gℓ(x,0)∂rv(y)dy. | (2.23) |
Since d=2, we have
‖v‖L∞(B5∖B3)≤C‖v‖H2(B5∖B3)≤C‖v‖H2(B5∖B2)≤C(1+ω1/2)‖v‖H1(B6∖B1). |
It follows from Lemma 2.1 and the trace theory that
‖v‖L∞(B5∖B3)+‖v‖L2(∂B2)+‖∇v‖L2(∂B2)≤C(1+ω3/2)‖v‖H1/2(∂D). | (2.24) |
Since, by (2.18),
|∇yGℓ(x,y)|≤C for x∈∂B4 and y∈∂B2 |
and
|Gℓ(x,0)|≥C|ln|ℓ|| for x∈∂B4, |
we derive from (2.23) and (2.24) that
|∫∂B2∂rv(y)dy|≤C(1+ω3/2)|ln|ℓ||‖v‖H1/2(∂D). | (2.25) |
Again using (2.17), we get, for 3/2<|x|<R,
v(x/ε)=∫∂B2([Gℓ(x/ε,y)−Gℓ(x/ε,0)]∂rv(y)−∂ryGℓ(x/ε,y)v(y))dy+∫∂B2Gℓ(x/ε,0)∂rv(y)dy. | (2.26) |
Since, by (2.18), for 0<ω<1/2,
|Gℓ(x/ε,0)|≤C|lnω| and |∇yGℓ(x/ε,y)|≤Cε for 3/2<|x|<R,y∈∂B2, |
and, by (2.19), for 1/2<ω<ε−2/4,
|Gℓ(x/ε,0)|≤Cω−1/4e−3ω1/2/10 and |∇yGℓ(x/ε,y)|≤Cεω1/4e−3ω1/2/10 for 3/2<|x|<R,y∈∂B2, |
we derive from (2.24), (2.25) and (2.26) that, for 3/2<|x|<R,
|v(x/ε)|≤{C|lnω||ln|ℓ||‖v‖H1/2(∂D) if 0<ω<1/2,Cω3/2e−3ω1/2/10|ln|ℓ||‖v‖H1/2(∂D) if 1/2<ω<ε−2/4, |
which yields the conclusion in the case 0<ω<ε−2/4. The proof is complete.
In this proof, C denotes a positive constant depending only on Ω and Λ. Multiplying the equation of vε by ˉvε and integrating in Ω, we derive that
∫Ω⟨Aε∇vε,∇vε⟩+ω∫Ωρε|vε|2≤C‖g‖2L2(Ω). | (2.27) |
Here we used Poincaré's inequality
‖vε‖L2(Ω)≤C‖∇vε‖L2(Ω). |
It follows from (2.27) that
‖vε(ε⋅)‖2H1/2(∂B1)≤C‖vε(ε⋅)‖2H1(B1)≤C∫Bε1εd−2|∇vε|2+1εd|vε|2≤C(1+ω−1)‖g‖2L2(Ω). | (2.28) |
Similarly, using the equation for v and Poincaré's inequality, we get
‖v‖H1(Ω)≤C‖g‖L2(Ω). | (2.29) |
Since Δv+iωv=0 in B2, using Caccioppolli's inequality, we have
‖v‖H3(B1)≤C‖v‖H2(B3/2)≤C‖v‖H1(B2)≤C‖g‖L2(Ω). | (2.30) |
By Sobolev embedding, as d≤3,
‖v‖W1,∞(B1)≤C‖v‖H3(B1). | (2.31) |
It follows that
‖v(ε⋅)‖H1/2(∂B1)≤C‖v(ε⋅)‖H1(B1)≤C‖v‖W1,∞(B1)≤C‖g‖L2(Ω). | (2.32) |
Set
wε=vε−v in Ω∖Bε. |
Then wε∈H1(Ω∖Bε) and satisfies
{Δwε+iωwε=0 in Ω∖Bε,wε=vε−v on ∂Bε,wε=0 on ∂Ω. | (2.33) |
Let ˜wε∈H1(Rd∖Bε) be the unique solution of the system
{Δ˜wε+iω˜wε=0 in Rd∖Bε,˜wε=wε on ∂Bε, | (2.34) |
and set
˜Wε=˜wε(ε⋅) in Rd∖B1. |
Then \widetilde W_{\varepsilon} \in H^1(\mathbb{R}^d \setminus B_{1}) is the unique solution of the system
\begin{equation} \left\{ \begin{array}{cl} \Delta \widetilde W_{\varepsilon} + i \omega \varepsilon^2 \widetilde W_{\varepsilon} = 0 & \mbox{ in } \mathbb{R}^d \setminus B_{1}, \\[6pt] \widetilde W_{\varepsilon} = w_\varepsilon (\varepsilon \, \cdot \, ) & \mbox{ on } \partial B_{1}. \end{array}\right. \end{equation} | (2.35) |
Fix r_0 > 2 such that \Omega \subset B_{r_0} . By Lemma 2.2, we have, for 1 \le |x| < r_0 , that
\begin{equation*} | \widetilde W_{\varepsilon} (x / \varepsilon)| \le C e(\varepsilon, \omega, d) \|w_\varepsilon (\varepsilon \, \cdot \, ) \|_{H^{1/2}(\partial B_1)}, \end{equation*} |
which yields, for x \in B_{r_0} \setminus B_{1} , that
\begin{equation*} | \widetilde w_{\varepsilon}(x)| \le C e(\varepsilon, \omega, d) \|w_\varepsilon (\varepsilon \, \cdot \, ) \|_{H^{1/2}(\partial B_1)}. \end{equation*} |
Since \Delta \widetilde w_{\varepsilon} + i \omega \widetilde w_{\varepsilon} = 0 in B_{r_0} \setminus B_{1} , it follows from Caccioppoli's inequality that
\begin{equation} \| \widetilde w_{\varepsilon} \|_{H^1(B_2 \setminus B_{3/2})} \le C e(\varepsilon, \omega, d) \|w_\varepsilon (\varepsilon \, \cdot \, ) \|_{H^{1/2}(\partial B_1)}. \end{equation} | (2.36) |
Fix \varphi \in C^2(\mathbb{R}^d) such that \varphi = 1 in B_{3/2} and \varphi = 0 in \mathbb{R}^d \setminus B_2 , and set
\chi_{\varepsilon} = w_{\varepsilon} - \varphi \widetilde w_{\varepsilon} \mbox{ in } \Omega \setminus B_\varepsilon. |
Then \chi_{\varepsilon}\in H^1_0(\Omega \setminus B_\varepsilon) and satisfies
\begin{equation*} \Delta \chi_{\varepsilon} +i\omega \chi_{\varepsilon} = -\Delta \varphi \widetilde w_{\varepsilon} - 2\nabla \varphi \cdot \nabla \widetilde w_{\varepsilon} \mbox{ in } \Omega\setminus B_\varepsilon. \end{equation*} |
Multiplying the equation of \chi_{\varepsilon} by \bar \chi_{\varepsilon} and integrating by parts, we obtain, by Poincaré's inequality,
\begin{equation} \| \chi_{\varepsilon} \|_{H^1(\Omega \setminus B_\varepsilon)}\le C \| \widetilde w_{\varepsilon} \|_{H^1(B_2 \setminus B_{3/2})}. \end{equation} | (2.37) |
Combining (2.36) and (2.37) yields
\begin{equation} \| w_{\varepsilon} \|_{H^1(\Omega \setminus B_2)} \le C e(\varepsilon, \omega, d) \|w_\varepsilon (\varepsilon \, \cdot \, ) \|_{H^{1/2}(\partial B_1)}. \end{equation} | (2.38) |
The conclusion now follows from (2.28) and (2.32).
Let v_\varepsilon = u_\varepsilon - u . Using the fact that v_\varepsilon is real, by the inversion theorem and Minkowski's inequality, we have, for t > 0 ,
\begin{align} \|v_\varepsilon(t, \cdot) \|_{L^2(\Omega \setminus B_2)} \le C \int_0^\infty \|\hat{v}_\varepsilon (\omega, \cdot) \|_{L^2(\Omega \setminus B_2) } \, d \omega. \end{align} | (2.39) |
Using Proposition 2.1, we get
\begin{align*} \int_0^\infty \|\hat{v}_\varepsilon (\omega, \cdot) \|_{L^2(\Omega \setminus B_2) } \, d \omega &\le C \int_0^\infty (1+\omega^{-1/2})e(\varepsilon, \omega, d)\| \hat{f}(\omega)+u_0\|_{L^2(\Omega \setminus B_2) } \, d \omega \\[6pt] & \le C \mbox{esssup}_{\omega \gt 0} \|\hat{f}(\omega)+u_0\|_{L^2(\Omega \setminus B_2) } \int_0^\infty (1+\omega^{-1/2})e(\varepsilon, \omega, d) \, d \omega \\[6pt] & \le Ce(\varepsilon, d) \big(\|f\|_{L^1\big( (0, + \infty); L^2(\Omega) \big)} +\|u_0\|_{L^2(\Omega)} \big). \end{align*} |
It follows from (2.39) that, for t > 0 ,
\begin{align*} \|v_\varepsilon(t, \cdot) \|_{L^2(\Omega \setminus B_2)} \le Ce(\varepsilon, d) \big(\|f\|_{L^1\big( (0, + \infty); L^2(\Omega) \big)} +\|u_0\|_{L^2(\Omega)} \big). \end{align*} |
Similarly, we have, for t > 0 ,
\begin{equation*} \|\nabla v_\varepsilon(t, \cdot) \|_{L^2(\Omega \setminus B_2)} \le Ce (\varepsilon, d) \big(\|f\|_{L^1\big( (0, + \infty); L^2(\Omega) \big)} +\|u_0\|_{L^2(\Omega)} \big). \end{equation*} |
The conclusion follows.
The second author is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2015.21.
The authors declare no conflict of interest in this paper.
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