On the Calder\`on problem in periodic cylindrical domain with partial Dirichlet and Neumann data

0.31 MB·

by  Mourad Choulli

ON THE CALDERÒN PROBLEM IN PERIODIC CYLINDRICAL DOMAIN WITH
PARTIAL DIRICHLET AND NEUMANN DATA
MOURADCHOULLI,YAVARKIAN,ERICSOCCORSI
6
1
0 Abstract. We consider the Calderòn problem in an infinite cylindrical domain, whose cross section is a
2 bounded domain of the plane. We prove log-log stability in the determination of the isotropic periodic
conductivity coefficient from partial Dirichlet data and partial Neumann boundary observations of the
n
solution.
a
J

0
2
1. Introduction

P] Let ω be a bounded domain of R2 which contains the origin, with a C2 boundary. Set Ω:=R×ω and
A denote anypointx∈Ωby x=(x1,x′),where x1 ∈R andx′ :=(x2,x3)∈ω. GivenV ∈L∞(Ω), real-valued
and 1-periodic with respect to x , i.e.
. 1
h
t V(x1+1,x′)=V(x1,x′), x′ ∈ω, x1 ∈R, (1.1)
a
m we consider the boundary value problem (BVP) with non-homogeneous Dirichlet data f,
[ (−∆+V)u = 0, in Ω,
(1.2)
u = f, on Γ:=∂Ω=R×∂ω.
1
(cid:26)
v Next we fix ξ ∈S1 :={y ∈R2; |y|=1} and define the ξ -shadowed (resp., ξ -illuminated) face of ∂ω as
8 0 0 0
5 ∂ω+ :={x′ ∈∂ω; ν′(x′)·ξ >0} (resp. ∂ω− :={x′ ∈∂ω; ν′(x′)·ξ 60}), (1.3)
3 ξ0 0 ξ0 0
5 where ν′ is the outgoing unit normal vector to ∂ω. Here and henceforth the symbol · denotes the Euclidian
0 scalar product in Rk, k >2, and |y|:=(y·y)21 for all y ∈Rk.
1. ThenforanyclosedneighborhoodG′ of∂ω− in∂ω,weknowfrom[CKS2,Theorem1.1]thatknowledge
ξ0
0 of the partial Dirichlet-to-Neumann (DN) map restricted to G:=R×G′,
6
1 ΛV :f 7→∂νu|G, (1.4)
:
v uniquely and logarithmic-stably determines V. Here we used the usual notation ∂ u := ∇u·ν, where ∇
ν
i denotes the gradient operator with respect to x∈Ω, u is the solution to (1.2), and
X
r ν(x1,x′):=(0,ν′(x′)), x=(x1,x′)∈Γ,
a
is the outward unit vector normal to Γ. Otherwise stated, the unknown potential V appearing in the first
line of (1.2) can be stably recovered from boundary observation of the current flowing through G, upon
probing the system (1.2) with non-homogeneous Dirichlet data.
Notice that in the above mentioned result, only the output, i.e. the measurementof the current flowing
acrossΓ, is local (or partial)in the sense that it is performed onG and not onthe whole boundary Γ, while
the input, i.e. the Dirichlet data, remains global as it is possibly supported everywhere on Γ. Therefore,
[CKS2,Theorem1.1]claimslogarithmicstabilityintheinverseproblemofdeterminingtheelectricpotential
V in the first line of (1.2) from the knowledge of partial Neumann, and full Dirichlet, data. In the present
paper, we aim for the same type of result under the additional constraint that not only the Neumann data,
but also the Dirichlet data, be partial. Namely, given an arbitrary closed neighborhood F′ of ∂ω+ in ∂ω,
ξ
such that
F′∩G′ 6=∅ and F′∪G′ =∂ω, (1.5)
1
2 MOURADCHOULLI,YAVARKIAN,ERICSOCCORSI
weseekstableidentificationofV bytheinput-restrictedDNmap(1.4)toDirichletdatafunctionsf supported
in F :=R×F′.
1.1. State of the art. Since the seminal paper [Ca] by Calderón, the electrical impedance tomography
problem, or Calderón problem, of retrieving the conductivity from the knowledge of the DN map on the
boundary of a bounded domain, has attracted many attention. If the conductivity coefficient is scalar, then
the Liouville transform allows us to rewrite the Calderón problem into the inverse problem of determining
the electric potential in Laplace operator, from boundary measurements. There is an extensive literature
on the Calderón problem. For isotropic conductivities, a great deal of work has been spent to weaken the
regularityassumptiononthe conductivity, inthe study ofthe uniqueness issue,seee.g. [BT,HT]. Inallthe
above mentioned papers, the full DN map are needed, i.e. lateral observations are performed on the whole
boundary. ThefirstuniquenessresultfrompartialdatafortheCalderónproblem,wasobtainedindimension
3orgreater,byBukhgeimandUhlmannin[BU]. Theirresult,whichrequiresthatDirichletdatabeimposed
on the whole boundary, and that Neumann data boundary be observed on slightly more than half of the
boundary, was improved by Kenig, Sjöstrand and Uhlmann in [KSU], where both input and ouput data are
measured on subsets of the boundary. In the two-dimensional case, Imanuvilov, Uhlmann and Yamamoto
proved in [IUY1, IUY2] that the partial DN map uniquely determines the conductivity. We also mention
that the special case of the Calderón problem in a bounded cylindrical domain of R3, was treated in [IY1].
ThestabilityissuefortheCalderónproblemwasaddressedbyAlessandriniin[Al]. Heprovedalog-type
stability estimate with respect to the full DN map. Such a result, which is known to be optimal, see [Ma],
degenerates to log-log stability with partial Neumann data, see [HW, CKS1]. In [CDR1, CDR2], Caro, Dos
Santos Ferreira and Ruiz proved stability results of log-log type, corresponding to the uniqueness results
of [KSU] in dimension 3 or greater. We refer to [BIY, NSa, Sa] for stability estimates associated with the
two-dimensionalCalderónproblem, and we point out that both the electric and the magnetic potentials are
stably determined by the partial DN map in [T].
Notice that all the above mentioned papers are concerned with the Calderónproblem in a bounded do-
main. Itturnsoutthatthereisonlyasmallnumberofmathematicalpapersdealingwithinversecoefficients
problems in an unbounded domain. Several authors considered the problem of recovering coefficients in an
unbounded domain from boundary measurements. Object identification in an infinite slab, was proved in
[Ik, SW]. Unique determination of a compactly supported electric potential of the Laplace equation in an
infinite slab, by partial DN map, is established in [LU]. This result was extended to the magnetic case in
[KLU], and to bi-harmonic operators in [Y].
Thestabilityissueininversecoefficientsproblemsstatedinaninfinitecylindricalwaveguideisaddressed
in[BKS,CKS,CKS1,CS,KKS,Ki,KPS1,KPS2],andalog-logtypestabilityestimatebypartialNeumann
data for the periodic electric potential of the Laplace equation can be found in [CKS2]. In the present
paper, we are aiming for the same result as in [CKS2], where the full Dirichlet data is replaced by partial
voltage. The proof the corresponding stability estimate relies on two different types of complex geometric
optics(CGO)solutionstothequasi-periodicLaplaceequationin(0,1)×ω,whicharesupportedinF. These
functions are built in Section 3 by means of a suitable Carleman estimate. This technique is inspired by
[KSU], but, in contrast to [CDR1, CDR2], due to the quasi-periodic boundary conditions imposed on the
CGO solution, we cannot apply the Carleman estimate of [KSU], here.
1.2. Settings and main result. We stick with the notations of [CKS2] and denote by C the square root
ω
of the first eigenvalue of the Dirichlet Laplacian in L2(ω), that is the largest of those positive constants c,
such that the Poincaré inequality
k∇′uk >ckuk , u∈H1(ω), (1.6)
L2(ω) L2(ω) 0
holds true. Here ∇′ := (∂ ,∂ ) stands for the gradient with respect to x′ = (x ,x ). This can be
x2 x3 2 3
equivalently reformulated as
C :=sup{c>0 satisfying (1.6)}. (1.7)
ω
ON THE CALDERÒN PROBLEM IN PERIODIC CYLINDRICAL DOMAIN WITH PARTIAL DATA 3
Next,forM ∈(0,C )andM ∈[M ,+∞),weintroducethesetV (M )ofadmissibleunknownpotentials
− ω + − ω ±
in the same way as in [CKS2, Sect. 1.2]:
Vω(M±):={V ∈L∞(Ω;R) satisfying (1.1), kVkL∞(Ω) 6M+ and kmax(0,−V)kL∞(Ω) 6M−}. (1.8)
Before stating the main result of this paper we need to define the DN map associated with the BVP
(1.2) and V ∈ V (M ). To this end, we introduce the Hilbert space H (Ω) := {u ∈ L2(Ω); ∆u ∈ L2(Ω)},
ω ± ∆
endowed with the norm
kuk2 :=kuk2 +k∆uk2 ,
H∆(Ω) L2(Ω) L2(Ω)
and refer to [CKS2, Lemma 2.2] in order to extend the mapping
T u:=u (resp.,T u:=∂ u ), u∈C∞(Ω),
0 |Γ 1 ν |Γ 0
into a continuous function T0 : H∆(Ω) → H−2(R;H−12(∂ω)) (resp., T1 : H∆(Ω) → H−2(R;H−32(∂ω))).
Since T is one-to-one from B :={u∈L2(Ω); ∆u=0} onto
0
H(Γ):=T H (Ω)={T u; u∈H (Ω)},
0 ∆ 0 ∆
by [CKS2, Lemma 2.3], we put
kfk := T−1f = T−1f , (1.9)
H(Γ) 0 H∆(Ω) 0 L2(Ω)
where T−1 denotes the operator inverse to T(cid:13) : B(cid:13)→ H(Γ).(cid:13)Throu(cid:13)ghout this text, we consider Dirichlet
0 (cid:13)0 (cid:13) (cid:13) (cid:13)
data in H(Γ) which are supported in F, i.e. input functions belonging to
H (F):={f ∈H(Γ); suppf ⊂F}.
c
To any f ∈ H (F), we associate the unique solution u ∈ H (Ω) to (1.2), given by [CKS2, Proposition 1.1
c ∆
(i)], and define the partial DN map associated with (1.2), as
Λ :f ∈H (F)7→T u . (1.10)
V c 1 |G
Upon denoting by B(X ,X ), where X , j = 1,2, are two arbitrary Banach spaces, the class of bounded
1 2 j
operators T :X →X , we recall from [CKS2, Proposition 1.1 (ii)-(iii)] that
1 2
ΛV ∈B(Hc(F),H−2(R,H−23(G′))) and ΛV −ΛW ∈B(Hc(F),L2(G)), V, W ∈Vω(M±). (1.11)
The main result of this article, which claims that unknown potentials of V (M ) are stably determined
ω ±
in the elementary cell Ωˇ :=(0,1)×ω, by the partial DN map, is stated as follows.
Theorem 1.1. Let V ∈ V (M ), j = 1,2, where M ∈ [M ,+∞), M ∈ (0,C ), and C is defined by
j ω ± + − − ω ω
(1.7). Then, there exist two constants C > 0 and γ > 0, both of them depending only on ω, M , F′, and
∗ ±
G′, such that the estimate
kV −V k 6CΦ(kΛ −Λ k), (1.12)
1 2 H−1(Ωˇ) V1 V2
holds with
γ if γ >γ∗,
Φ(γ):= (ln|lnγ|)−1 if γ ∈(0,γ∗), (1.13)

0 if γ =0.

Here k·k denotes the usual norm in B(H (F),L2(G)).
c
The statement of Theorem 1.1 remains valid for any periodic potential V ∈ L∞(Ω), provided 0 lies in
the resolventsetofA ,the self-adjointrealizationinL2(Ω)ofthe DirichletLaplacian−∆+V. Inthis case,
V
the multiplicative constants C and γ , appearing in (1.12)-(1.13), depend on (the inverse of) the distance
∗
d > 0, between 0 and the spectrum of A . In the particular case where V ∈ V (M ), with M ∈ (0,C ),
V ω ± − ω
we have d>C −M , and the implicit condition d>0 imposed on V, can be replaced by the explicit one
ω −
on the negative part of the potential, i.e. kmax(0,−V)kL∞(Ω) 6M−.
4 MOURADCHOULLI,YAVARKIAN,ERICSOCCORSI
1.3. Application to the Calderón Problem. The inverse problem addressedin Subsection 1.2 is closely
related to the periodic Calderón problem in Ω, i.e. the inverse problem of determining the conductivity
coefficient a, obeying
a(x +1,x′)=a(x ,x′), x′ ∈ω, x ∈R, (1.14)
1 1 1
from partial boundary data of the BVP in the divergence form
−div(a∇u) = 0, in Ω,
(1.15)
u = f, on Γ.
(cid:26)
LetT denotethetraceoperatoru7→u onH1(Ω). WeequipthespaceK (Γ):=T (H1(Ω))withthenorm
0 |Γ 0
kfk :=inf{kuk ; T u=f},
K(Γ) H1(Ω) 0
and recall for any a∈C1(Ω) satisfying the ellipticity condition
a(x)>a >0, x∈Ω, (1.16)
∗
for some fixed positive constant a , that the BVP (1.15) admits a unique solution u ∈ H1(Ω) for each
∗
f ∈K (Γ). Moreover,the full DN map associated with (1.15), defined by f 7→aT u, where T u:=∂ u , is
1 1 ν |Γ
a bounded operator from K (Γ) to H−1(R;H−21(∂ω)). Here, we rather consider the partial DN map,
Σa :f ∈K (Γ)∩a−12(Hc(F))7→aT1u|G, (1.17)
where a−21(Hc(F)):={a−21f; f ∈Hc(F)}.
Further, since the BVP (1.15) is brought by the Liouville transform into the form (1.2), with V :=
a
a−12∆a12, then, with reference to (1.8), we impose that Va be bounded in Ω and satisfies the following
conditions
kVakL∞(Ω) 6M+ and kmax(0,−Va)kL∞(Ω) 6M−, (1.18)
where M ∈(0,C )andM ∈[M ,+∞)area priori arbitrarilyfixedconstants. Namely,we introduce the
− ω + −
set of admissible conductivities, as
Aω(a∗,M±):= a∈C1(Ω;R) satisfying ∆a∈L∞(Ω), kakW1,∞(Ω) 6M+,(1.14),(1.16),and (1.18) .
(1.19)
(cid:8) (cid:9)
We check by standard computations that the condition (1.18) is automatically verified, provided the con-
ductivity a∈Aω(a∗,M±) is taken so small that kak2W1,∞(Ω)+2a∗k∆akL∞(Ω) 64M−a2∗, or even that
4M
kakW2,∞(Ω) 6 (4M−+1−)21 +1a∗,
in the particular case where a∈W2,∞(Ω).
The mainresultofthis sectionclaims stabledeterminationofsuchadmissibleconductivities a,fromthe
knowledge of Σ . It is stated as follows.
a
Corollary 1.2. Fix a >0, and let M be as in Theorem 1.1. Pick a ∈A (a ,M ), for j =1,2, obeying
∗ ± j ω ∗ ±
a (x)=a (x), x∈∂Ω (1.20)
1 2
and
∂ a (x)=∂ a (x), x∈F ∩G. (1.21)
ν 1 ν 2
Then Σ −Σ is extendable to a bounded operator from a−12(H (F)) into L2(G). Moreover, there exists
a1 a2 1 c
two constant C >0 and γ >0, both of them depending only on ω, M , a , F′, and G′, such that we have
∗ ± ∗
ka −a k 6CΦ a−12 kΣ −Σ k , (1.22)
1 2 H1(Ωˇ) ∗ a1 a2
(cid:16) (cid:17)
whereΦisthesameas inTheorem 1.1. Herek·kdenotes theusualoperatornorminB(a−21(H (F)),L2(G)).
1 c
ON THE CALDERÒN PROBLEM IN PERIODIC CYLINDRICAL DOMAIN WITH PARTIAL DATA 5
1.4. Floquet decomposition. Inthis subsection,wereformulatethe inverseproblempresentedinSubsec-
tion 1.2 into a family of inverse coefficients problems associated with the BVP
(−∆+V)v = 0, in Ωˇ :=(0,1)×ω,
v = g, on Γˇ :=(0,1)×∂ω,
 (1.23)
 ∂ v(1,v·()1−,·)e−iθ∂eiθvv((00,,··)) == 00,, iinn ωω,,
x1 x1
for θ ∈ [0,2π), and suitable Dirichlet data g. This is by means of the Floquet-Bloch-Gel’fand (FBG)
transformintroduced in [CKS2, Section 3.1]. We stick with the notations of [CKS2, Section 3.1],and, for Y
beingeitherω of∂ω,wedenotebyU theFBGtransformfromL2(R×Y)onto ⊕ L2((0,1)×Y)dθ. That
(0,2π) 2π
is to say, the FBG transform U maps L2(Ω) onto (⊕0,2π)L2(Ωˇ)2dπθ if Y = ω, anRd L2(Γ) onto (⊕0,2π)L2(Γˇ)2dπθ
when Y = ∂ω. We recall that the operator U is unitary in both cases. We start by introducing several
R R
functionalspacesandtrace operatorsthatareneeded by theanalysisofthe inverseproblemassociatedwith
(1.23).
1.4.1. Functional spaces and traceoperators. Fixθ ∈[0,2π). Withreferenceto[CKS,Section6.1]or[CKS2,
Section 3.1], we set for each n∈N∪{∞},
Cn([0,1]×ω):= u∈Cn([0,1]×ω); ∂j u(1,·)−eiθ∂j u(0,·)=0 in ω, j 6n ,
θ x1 x1
and for Y being either ω or ∂ω(cid:8), we put (cid:9)
1 1
Hs((0,1)×Y):= u∈Hs((0,1)×Y); ∂j u(1,·)−eiθ∂j u(0,·)=0 in ω, j <s− if s> ,
θ x1 x1 2 2
(cid:26) (cid:27)
and
1
Hs((0,1)×Y):=Hs((0,1)×Y) if s∈ 0, .
θ 2
(cid:20) (cid:21)
Further, we recall from [CKS2, Eq. (3.29)] that UH (Ω)= ⊕ H (Γˇ)dθ, where
∆ (0,2π) ∆,θ 2π
H (Ωˇ):={u∈L2(Ωˇ); ∆u∈L2(Ωˇ) and u(1,·)−eiθu(0,·)R=∂ u(1,·)−eiθ∂ u(0,·)=0 in ω}.
∆,θ x1 x1
Moreover,the space C∞([0,2π]×ω) is dense in H (Ωˇ), and we have UT U−1 = ⊕ T dθ for j =0,1,
θ ∆,θ j (0,2π) j,θ2π
where the linear bounded operator
R
Tj,θ :H∆,θ(Ωˇ)→Hθ−2(0,1,H−2j2+1(∂ω)),
fulfills T u=u if j =0, and T u=∂ u if j =1, provided u∈C∞([0,1]×ω). Therefore, putting
0,θ |Γˇ 1,θ ν |Γˇ θ
H (Γˇ):={T u; u∈H (Ωˇ)}, and H (Fˇ):={f ∈H (Γˇ), supp f ⊂Fˇ},
θ 0,θ ∆,θ c,θ θ
we get that UH(Γ) = ⊕ H (Γˇ)dθ and UH (F) = ⊕ H (Fˇ)dθ. As in [CKS2, Eq. (3.30)], the
(0,2π) θ 2π c (0,2π) c,θ 2π
space H (Γˇ) is endowed with the norm kgk :=kv k , where v denotes the unique L2(Ωˇ)-solution
θ R Hθ(Γˇ) g LR2(Ωˇ) g
to (1.23) with V =0, given by [CKS2, Proposition 3.2 (i)].
1.4.2. Inverse fibered problems. Let V ∈V (M ), where M are as in Theorem 1.1. Then, for any f be in
ω ± ±
H (F), u is the H (Ω)-solution to (1.2), if and only if, for almost every θ ∈ [0,2π), (Uu) is the H (Ωˇ)-
c ∆ θ ∆
solution to (1.23), associated with g = (Uf) ∈ H (Fˇ). The corresponding partial DN map, defined by
θ c,θ
Λ : g ∈ H (Fˇ) 7→ T v , where v is the unique H (Ωˇ)-solution to (1.23), is a bounded operator from
V,θ c,θ 1,θ |Gˇ ∆
Hc,θ(Fˇ) into Hθ−2(0,1;H−23(G′)), and we have
⊕ dθ
UΛ U−1 = Λ , (1.24)
V V,θ
2π
Z(0,2π)
6 MOURADCHOULLI,YAVARKIAN,ERICSOCCORSI
according to [CKS2, Proposition 7.1]. Further, if V and V are two potentials lying in V (M ), then
1 2 ω ±
Λ −Λ ∈ B(H (Fˇ),L2(Gˇ)), for each θ ∈ [0,2π), by (1.11) and (1.24). Moreover, Λ −Λ being
V1,θ V2,θ c,θ V1 V2
unitarily equivalent to the family of partial DN maps {Λ −Λ , θ ∈[0,2π)} , it holds true that
V1,θ V2,θ
kΛV1 −ΛV2kB(Hc(F),L2(G)) =θ∈s[u0,p2π)kΛV1,θ−ΛV2,θkB(Hc,θ(Fˇ),L2(Gˇ)). (1.25)
Therefore, it is clear from (1.25) that Theorem 1.1 is a byproduct of the following statement.
Theorem 1.3. Let M and V , j = 1,2, be as in Theorem 1.1. Fix θ ∈ [0,2π). Then, there exist two
± j
constants C >0 and γ >0, both of them depending only on ω, M , F′, and G′, such that we have
θ θ,∗ ±
kV −V k 6C Φ (kΛ −Λ k). (1.26)
1 2 H−1(Ωˇ) θ θ V1,θ V2,θ
Here, Φ is the function defined in Theorem 1.1, upon substituting γ for γ in (1.13), and k·k denotes the
θ θ,∗
usual norm in B(H (Fˇ),L2(Gˇ)).
c,θ
We notice that the constants C and γ of Theorem 1.3, may possibly depend on θ. Nevertheless,
θ θ,∗
we infer from (1.25) that this is no longer the case for C and γ, appearing in the stability estimate (1.12)
of Theorem 1.1, as we can choose C = C and γ = γ for any arbitrary θ ∈ [0,2π). Therefore, we may
θ θ,∗
completely leave aside the question of how C and γ depend on θ. For this reason, we shall not specify
θ θ,∗
the possible dependence with respect to θ of the various constants appearing in the remaining part of this
text. Finally, we stress out that the function Φ does actually depend on θ through the constant γ , as it is
θ θ
obtained by substituting γ for θ in the definition (1.13).
θ
1.5. Outline. The remaining part of this text is organized as follows. In Sections 2 and 3, we build the
two different types of (CGO) solutions to the BVP (1.23), for θ ∈ [0,2π), needed for the proof of Theorem
1.3, which is presented in Section 4. Section 5 contains the proof of Corollary 1.2. Finally, a suitable
characterizationof the space H , θ ∈[0,2π), used in Section 3, is derived in Section A in appendix.
∆,θ
2. Complex geometric optics solutions
In this section we build CGO solutions to the system
(−∆+V)u = 0, in Ωˇ,
u(1,·) = eiθu(0,·), on ω, (2.27)

∂ u(1,·) = eiθ∂ u(0,·), on ω,
 x1 x1
where θ ∈ [0,2π) and the real valued potential V ∈ L∞(Ωˇ) are arbitrarily fixed. More precisely, we seek a
sufficiently rich set of solutions u to (2.27), parametrized by
ζ
ζ ∈Z :={ζ ∈i(θ+2πZ)×C2; |Rζ|=|Iζ| and Rζ·Iζ =0}, (2.28)
θ
of the form
u (x)=(1+v (x))eζ·x, x∈Ωˇ, (2.29)
ζ ζ
where the behavior of the function v with respect to ζ, is prescribed in a sense we shall specify further.
ζ
Notice from definition (2.28) that for all ζ ∈Z , we have
θ
∆eζ·x =0, x∈Ω. (2.30)
Actually, the analysis carried out in this text requires two different types of CGO solutions to (2.27),
denoted generically by u and u , with different features we shall make precise below. The corresponding
ζ1 ζ2
parameters ζ and ζ are the same as in [CKS2, section 4]. Namely, given k ∈Z and η ∈R2\{0}, we pick
1 2
ξ ∈S1 such that ξ·η =0, and we set
(θ+2π([r]+1)) 1,−2πk η , if k is even,
ℓ=ℓ(k,η,r,θ):= |η|2 (2.31)
 θ+2π [r]+ 3 (cid:16)1,−2πk η(cid:17) , if k is odd,
 2 |η|2
(cid:0) (cid:0) (cid:1)(cid:1)(cid:16) (cid:17)

ON THE CALDERÒN PROBLEM IN PERIODIC CYLINDRICAL DOMAIN WITH PARTIAL DATA 7
foreachr >0,insucha waythatℓ·(2πk,η)=ℓ′·ξ =0. Here[r] standsforthe integerpartofr, thatis the
unique integer fulfilling [r]6r<[r]+1, and we used the notationℓ=(ℓ ,ℓ′)∈R×R2. Next, we introduce
1
|η|2
τ =τ(k,η,r,θ):= +π2k2+|ℓ|2, (2.32)
s 4
and notice that
|(2πk,η)| |2πk|
2πr<τ 6 +4π(r+1) 1+ . (2.33)
2 |η|
(cid:18) (cid:19)
Thus, putting
η η
ζ := iπk,−τξ+i +iℓ and ζ := −iπk,τξ−i +iℓ, (2.34)
1 2
2 2
(cid:16) (cid:17) (cid:16) (cid:17)
it is easy to see that
ζ ,ζ ∈Z satisfy the equation ζ +ζ =i(2πk,η). (2.35)
1 2 θ 1 2
2.1. Second order smooth CGO solutions. The first type of CGO solutions u we need are smooth
ζ1
CGO functions, in the sense that the remainder term v appearing in (2.29) is taken in
ζ1
H2 (Ωˇ):={f ∈H2(Ωˇ); f(1,·)=f(0,·) and ∂ f(1,·)=∂ f(0,·) in ω}.
per x1 x1
The existence of such functions was already established in [CKS2, Lemma 4.1], as follows.
Lemma 2.1. Let V ∈ V (M ). Pick θ ∈ [0,2π), k ∈ Z, and η ∈ R2 \{0}, and choose ξ ∈ S1 such that
ω ±
ξ·η = 0. Let ζ be given by (2.34), as a function of τ, defined in (2.31)-(2.32). Then, one can find τ > 0
1 1
so that for all τ > τ , there exists v ∈ H2 (Ωˇ) such that the function u , defined by (2.29) with ζ = ζ ,
1 ζ1 per ζ1 1
is solution to (2.27). Moreover, for every s∈[0,2], the estimate
kv k 6C τs−1, (2.36)
ζ1 Hs(Ωˇ) s
holds uniformly in τ >τ , for some constant C >0, depending only on s, ω, and M .
1 s ±
The second type of test functions needed for the proof of Theorem 1.1 are CGO solutions to (2.27),
vanishing on a suitable subset of the boundary (0,1)×∂ω. They are described in Subsection 2.2.
2.2. CGO solutions vanishing on a sub-part of the boundary. For ξ ∈S1 and ε>0, we set
∂ω+ :={x′ ∈∂ω; ξ·ν′(x′)>ε} and ∂ω− :={x′ ∈∂ω; ξ·ν′(x′)6ε}, (2.37)
ε,ξ ε,ξ
and we write Γˇ± instead of (0,1)×∂ω± , in the sequel.
ε,ξ ε,ξ
Bearing in mind that F′ is a closed neighborhood in ∂ω, of the subset ∂ω+ defined by (1.3), we pick
ξ0
ε>0 so small that
∀ξ ∈S1, (|ξ−ξ |6ε)=⇒ ∂ω− ⊂F′ . (2.38)
0 ε,−ξ
(cid:16) (cid:17)
In this subsection, we aim for building solutions u ∈ H (Ωˇ) of the form (2.29) with ζ = ζ , where ζ is
∆,θ 2 2
given by (2.31)-(2.32) and (2.34), to the BVP
(−∆+V)u = 0, in Ωˇ,
u(1,·) = eiθu(0,·), on ω,
 ∂ u(1,·) = eiθ∂ u(0,·), on ω, (2.39)
 x1 u = 0, x1 on Γˇ+ .
ε,−ξ
2
The result we have in mindis as follows.
8 MOURADCHOULLI,YAVARKIAN,ERICSOCCORSI
Proposition 2.2. Let V, θ, k, η, ξ, and τ, be the same as in Lemma 2.1, and let ζ be defined by (2.34).
2
Then, one can find τ > 0, so that for each τ > τ , there exists v ∈ H (Ωˇ) such that the function u ,
2 2 ζ2 ∆,0 ζ2
defined by (2.29) with ζ =ζ , is a H (Ωˇ)-solution to (2.39). Moreover, v satisfies
2 ∆,θ ζ2
kvζ2kL2(Ωˇ) 6Cτ−21, τ >τ2, (2.40)
for some constant C >0 depending only on ω, M and F′.
+
The proof of Proposition 2.2 is postponed to Section 3.
We notice from the last line in (2.39), that the solution u given by Proposition 2.2 for τ >τ , verifies
ζ2 2
T u =0 on Γˇ\Fˇ, as we have Γˇ\Fˇ ⊂Γˇ\Γˇ− ⊂Γˇ+ . Therefore, it holds true that
0,θ ζ2 ε,−ξ ε,−ξ
2
T u ∈H (Fˇ), τ >τ . (2.41)
0,θ ζ2 c,θ 2
Further, we have
kT u k 6C ku k +kVu k 6C(1+M )ku k , (2.42)
0,θ ζ2 Hθ(Γˇ) ζ2 L2(Ωˇ) ζ2 L2(Ωˇ) + ζ2 L2(Ωˇ)
(cid:16) (cid:17)
for some positive constant C = C(ω), by the continuity of T : H (Ωˇ) → H (Γˇ) and the first line of
0,θ ∆,θ θ
(2.39). As
ζ ·x−τξ·x′ ∈iR, x=(x ,x′)∈R×R2, (2.43)
2 1
by (2.31)-(2.32) and (2.34), we infer from the identity uζ2 =eζ2·x(1+vζ2), that
ku k 6keζ2·xk +keζ2·xv k 6keτξ·x′k +keτξ·x′v k ,
ζ2 L2(Ωˇ) L2(Ωˇ) ζ2 L2(Ωˇ) L2(Ωˇ) ζ2 L2(Ωˇ)
and hence that
kuζ2kL2(Ωˇ) 6ecωτ meas(ω)+kvζ2kL2(Ωˇ) ,
where c :=sup{|x′|, x′ ∈ω} and meas(ω) denote(cid:16)s the two-dimensionalL(cid:17)ebesgue measure of ω. From this,
ω
(2.40) and (2.42), it then follows that
kT u k 6Cecωτ, τ >τ , (2.44)
0,θ ζ2 Hθ(Γˇ) 2
where the constant C >0, depends only on ω, M , and F′.
+
3. Proof of Proposition 2.2
The proof of Proposition2.2 is by means of two technical results, given in Subsection 3.1. The first one
is a Carleman estimate for the quasi-periodic Laplace operator −∆+V in L2(Ωˇ), which was inspired by
[BU, Lemma 2.1]. The second one is an existence result for the BVP (1.23) with non zero source term.
3.1. Two useful tools. We start by recalling the following Carleman inequality, borrowed from [CKS2,
Corollary 5.2].
Lemma 3.1. For M > 0 arbitrarily fixed, let V ∈ L∞(Ω) satisfy (1.1) and kVkL∞(Ω) 6 M. Then, there
exist two constants C > 0 and τ > 0, both of them depending only on ω and M, such that for all ξ ∈ S1,
0
all θ ∈[0,2π), and all w ∈C2([0,1]×ω) obeying w =0, we have
θ |Γˇ
Cke−τξ·x′wk2L2(Ωˇ)+τke−τξ·x′(ξ·ν′)21∂νwk2L2(Γˇ+)
ξ
6 ke−τξ·x′(−∆+V)wk2L2(Ωˇ)+τke−τξ·x′|ξ·ν′|21∂νwk2L2(Γˇ−), (3.45)
ξ
provided τ >τ .
0
Here and henceforth, we write Γˇ± instead of (0,1)×∂ω±.
ξ ξ
Armed with Lemma 3.1, we turn now to establishing the following existence result for the BVP (1.23)
with non zero source term.
ON THE CALDERÒN PROBLEM IN PERIODIC CYLINDRICAL DOMAIN WITH PARTIAL DATA 9
Lemma 3.2. For M > 0, let V ∈ L∞(Ωˇ) be real valued and such that kVk 6 M, and let θ ∈ [0,2π),
L∞(Ωˇ)
ξ ∈ S1, f ∈L2(Ωˇ), and g ∈L2(Γˇ−ξ \Γˇ+ξ; |ξ·ν′|−12dx). Then, for every τ ∈ [τ0,+∞), where τ0 is the same
as in Lemma 3.1, there exists v ∈H (Ωˇ) fulfilling
∆,θ
(−∆+V)v = f, in Ωˇ,
(3.46)
v = g, on Γˇ−\Γˇ+.
( ξ ξ
Moreover, v satisfies the estimate
ke−τξ·x′vkL2(Ωˇ) 6C τ−1ke−τξ·x′fkL2(Ωˇ)+τ−12ke−τξ·x′|ξ·ν′|−12gkL2(Γˇ−\Γˇ+) , (3.47)
ξ ξ
(cid:16) (cid:17)
for some constant C >0, depending only on ω and M.
Proof. We denote by C2 ([0,1]×ω) the set of C2([0,1]×ω)-functions vanishing on the boundary Γˇ, i.e.
0,θ θ
C2 ([0,1]×ω):= w ∈C2([0,1]×ω); w =0 .
0,θ θ |Γˇ
n o
By substituting (−ξ) for ξ in the Carleman estimate (3.45), we get for all w ∈C2 ([0,1]×ω), that
0,θ
τ eτξ·x′w +τ21 eτξ·x′|ξ·ν′|21∂νw
L2(Ωˇ) L2(Γˇ−)
(cid:13) (cid:13) (cid:13) (cid:13) ξ
(cid:13) (cid:13) (cid:13) (cid:13)
6 C(cid:13) eτξ·x′(cid:13)(−∆+V)w(cid:13) + eτξ·x′(τξ(cid:13)·ν′)12∂νw . (3.48)
L2(Ωˇ) L2(Γˇ+)
(cid:18)(cid:13) (cid:13) (cid:13) (cid:13) ξ (cid:19)
(cid:13) (cid:13) (cid:13) (cid:13)
Put M := ̺ C2 ([0,1]×ω(cid:13)) = ̺(w);w ∈(cid:13)C2 ([0,1(cid:13)]×ω) , where ̺(w(cid:13)) := (−∆+V)w,∂ w .
θ 0,θ 0,θ ν |Γˇ+
ξ
Since ̺ is one-t(cid:16)o-one from C2 ((cid:17)[0,1]n×ω) onto M , according too (3.48), then the an(cid:16)tilinear form (cid:17)
0,θ θ
Υ:̺(w)7→hf,wiL2(Ωˇ)−hg,∂νwiL2(Γˇ−\Γˇ+), (3.49)
ξ ξ
is well defined on Mθ, regarded as a subspace of Lτ,1 := L2(Ωˇ;eτξ·x′dx)×L2(Γˇ+ξ;eτξ·x′|τξ ·ν′(x′)|12dx).
2
Moreover,for each w ∈C2 ([0,1]×ω), we have
0,θ
|Υ(̺(w))| 6 e−τξ·x′f eτξ·x′w + e−τξ·x′|ξ·ν′|−21g eτξ·x′|ξ·ν′|21∂νw
L2(Ωˇ) L2(Ωˇ) L2(Γˇ−\Γˇ+) L2(Γˇ−)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) ξ ξ (cid:13) (cid:13) ξ
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
6 (cid:13)τ−1 e−(cid:13)τξ·x′f (cid:13) +τ(cid:13)−12 e−τ(cid:13)ξ·x′|ξ·ν′|−21g (cid:13) (cid:13) (cid:13)
L2(Ωˇ) L2(Γˇ−\Γˇ+)
(cid:18) (cid:13) (cid:13) (cid:13) (cid:13) ξ ξ (cid:19)
(cid:13) (cid:13) (cid:13) (cid:13)
× τ (cid:13)eτξ·x′w (cid:13) +τ21 eτ(cid:13)ξ·x′|ξ·ν′|21∂νw (cid:13)
L2(Ωˇ) L2(Γˇ−)
(cid:18) (cid:13) (cid:13) (cid:13) (cid:13) ξ (cid:19)
(cid:13) (cid:13) (cid:13) (cid:13)
6 C τ−(cid:13)1 e−τξ·(cid:13)x′f +(cid:13)τ−12 e−τξ·x′|ξ·ν′(cid:13)|−21g
L2(Ωˇ) L2(Γˇ−\Γˇ+)
(cid:18) (cid:13) (cid:13) (cid:13) (cid:13) ξ ξ (cid:19)
(cid:13) (cid:13) (cid:13) (cid:13)
× eτξ(cid:13)·x′(−∆+(cid:13)V)w +(cid:13)eτξ·x′|τξ·ν′|21∂νw(cid:13) ,
L2(Ωˇ) L2(Γˇ+)
(cid:18)(cid:13) (cid:13) (cid:13) (cid:13) ξ (cid:19)
by (3.48), and hence (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
|Υ(̺(w))|6C τ−1 e−τξ·x′f +τ−21 e−τξ·x′|ξ·ν′|−21g k̺(w)kL ,
(cid:18) (cid:13) (cid:13)L2(Ωˇ) (cid:13) (cid:13)L2(Γˇ−ξ\Γˇ+ξ)(cid:19) τ,12
where C is the same constant a(cid:13)s in (3.4(cid:13)8). Thus, by(cid:13)Hahn Banach’s th(cid:13)eorem, Υ extends to an antilinear
(cid:13) (cid:13) (cid:13) (cid:13)
form on L , still denoted by Υ, satisfying
τ,1
2
kΥk6C τ−1 e−τξ·x′f +τ−12 e−τξ·x′|ξ·ν′|−21g . (3.50)
L2(Ωˇ) L2(Γˇ−\Γˇ+)
(cid:18) (cid:13) (cid:13) (cid:13) (cid:13) ξ ξ (cid:19)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
10 MOURADCHOULLI,YAVARKIAN,ERICSOCCORSI
Therefore,thereexists(v,g˜)inL−τ,−1 :=L2(Ωˇ;e−τξ·x′dx)×L2(Γˇ+ξ;e−τξ·x′|τξ·ν′(x′)|−12dx),thedualspace
2
to L with pivot space L2(Ωˇ)×L2(Γˇ+), such that
τ,1 ξ
2
Υ(̺(w))=h(v,g˜),̺(w)i =hv,(−∆+V)wi +hg˜,∂ wi , w ∈C2 ([0,1]×ω). (3.51)
L2(Ωˇ)×L2(Γˇ+) L2(Ωˇ) ν L2(Γˇ+) 0,θ
ξ ξ
This and (3.49) yield that
hv,(−∆+V)wiL2(Ωˇ)+hg˜,∂νwiL2(Γˇ+) =hf,wiL2(Ωˇ)−hg,∂νwiL2(Γˇ−\Γˇ+), w∈C02,θ([0,1]×ω). (3.52)
ξ ξ ξ
Taking w∈C∞(Ωˇ)⊂C2 ([0,1]×ω) in (3.52), we getthat (−∆+V)v =f inthe distributional sense onΩˇ.
0 0,θ
Moreover,since f ∈L2(Ωˇ;e−τξ·x′dx) and L2(Ωˇ;e−τξ·x′dx)⊂L2(Ωˇ), then the identity (−∆+V)v =f holds
in L2(Ωˇ). This entails that v ∈H (Ωˇ).
∆
We turn now to proving that v ∈ H (Ωˇ). With reference to Lemma A.1 in Appendix, it suffices to
∆,θ
show that
(∆′−(θ+2kπ)2)vˆ 2 <∞, (3.53)
k,θ L2(ω)
k∈Z
X(cid:13) (cid:13)
where (cid:13) (cid:13)
vˆ (x′):=hv(·,x′),ϕ i , x′ ∈ω, k ∈Z, (3.54)
k,θ k,θ L2(0,1)
and
ϕ (x ):=ei(θ+2kπ)x1, x ∈(0,1), k ∈Z. (3.55)
k,θ 1 1
To do that, we fix k ∈Z, pick χ∈C∞(ω), and apply (3.52) with w(x)=ϕ (x )χ(x′), getting
0 k,θ 1
hv,ϕ −∆′+(θ+2kπ)2 χi =hh,ϕ χi , (3.56)
k,θ L2(Ωˇ) k,θ L2(Ωˇ)
with h:=f −Vv. Next, by Fubini(cid:0)’s theorem, we hav(cid:1)e
1
hv,ϕ −∆′+(θ+2kπ)2 χi = v(x ,x′)ϕ (x )dx −∆′+(θ+2kπ)2 χ(x′)dx′
k,θ L2(Ωˇ) 1 k,θ 1 1
Zω(cid:18)Z0 (cid:19)
(cid:0) (cid:1) = hvˆ , −∆′+(θ+2kπ)2 χi (cid:0) (cid:1)
k,θ L2(ω)
= h −∆(cid:0)′+(θ+2kπ)2 vˆk,θ(cid:1),χi(C∞)′(ω),C∞(ω), (3.57)
0 0
and similarly (cid:0) (cid:1)
hh,ϕk,θχiL2(Ωˇ) =hhˆk,θ,χi(C0∞)′(ω),C0∞(ω). (3.58)
Putting (3.56)–(3.58) together, we find that
−∆′+(θ+2kπ)2 vˆ =hˆ , k ∈Z. (3.59)
k,θ k,θ
Since {ϕ , k ∈Z} is an Hibertian(cid:0) basis of L2(0,1) (cid:1)and L2(Ωˇ)=L2(0,1;L2(ω)), then w 7→{wˆ , k ∈Z}
k,θ k,θ
is a unitary transform of L2(Ωˇ) onto L2(ω). From this, (3.59), and the fact that h ∈ L2(Ωˇ), it then
k∈Z
follows that
−∆′+(θ+L2kπ)2 vˆ 2 = khˆ k2 =khk2 ,
k,θ L2(ω) k,θ L2(ω) L2(Ωˇ)
k∈Z k∈Z
X(cid:13)(cid:0) (cid:1) (cid:13) X
which entails (3.53). (cid:13) (cid:13)
Further, as v ∈H (Ωˇ), we infer from the Green formula that
∆,θ
hv,(−∆+V)wi =h(−∆+V)v,wi −hv,∂ wi , w∈C2 ([0,1]×ω). (3.60)
L2(Ωˇ) L2(Ωˇ) ν L2(Γˇ) 0,θ
Bearing in mind hat (−∆+V)v =f, it follows from (3.52) and (3.60) that
hv,∂νwiL2(Γˇ) =hg,∂νwiL2(Γˇ−\Γˇ+)+hg˜,∂νwiL2(Γˇ+), w ∈C02,θ([0,1]×ω). (3.61)
ξ ξ ξ
Since w is arbitrary in C2 ([0,1]×ω), and Γˇ = Γˇ−\Γˇ+ ∪Γˇ+, we deduce from (3.61) that v = g on
0,θ ξ ξ ξ
Γˇ−\Γˇ+, and v =g˜ on Γˇ+. (cid:16) (cid:17)
ξ ξ ξ