Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary

0.27 MB·English

by  Michel Cristofol

Determining the waveguide conductivity in a hyperbolic
equation from a single measurement on the lateral boundary
1Michel Cristofol, 2Shumin Li, 3Eric Soccorsi
Abstract
5 We consider the multidimensional inverse problem of determining the conductivity
1 coefficientofahyperbolicequationinaninfinitecylindricaldomain,fromasingleboundary
0 observationofthesolution. We proveHo¨lderstabilitywiththe aidofaCarlemanestimate
2
specifically designed for hyperbolic waveguides.

n
a
J AMS 2010 Mathematics Subject Classification: 35R30.

7
Keywords: Inverse problem, hyperbolic equation, conductivity, Carleman estimate, infi-
]
P nite cylindrical domain.
A
.
h
at 1 Statement of the problem and results
m
[ 1.1 Introduction

1
Thepresentpaperdealswiththeinverseproblemofdeterminingthetime-independentisotropic
v
4 conductivity coefficient c : Ω → R appearing in the hyperbolic partial differential equation
8 ∂2 −∇·c∇ = 0, where Ω := ω ×R is an infinite cylindrical domain whose cross section ω is
t
3 a bounded open subset of Rn−1, n ≥ 2. Namely, ℓ > 0 being arbitrarily fixed, we seek H¨older
1
0 stability in the identification of c in Ω := ω×(−ℓ,ℓ) from the observation of u on the lateral
ℓ
1. boundary ΓL := ∂ω×(−L,L) over the course of time (0,T), for L > ℓ and T > 0 sufficiently
0 large.
5
Several stability results in the inverse problem of determining one or several unknown co-
1
: efficients of a hyperbolic equation from a finite number of measurements of the solution are
v
available in the mathematics literature [1, 5, 6, 7, 17, 18, 26, 29]. Their derivation relies on
i
X
Bukhgeim-Klibanov’s method [9], which is by means of a Carleman inequality specifically de-
r
signed for hyperbolic systems. More precisely, [17, 29] study the determination of the zero-th
a
order term p : Ω → R appearing in ∂ −∆+p = 0, while [1, 7] deal with the identification of
t
the speed c : Ω → R in the hyperbolic equation ∂ −cA = 0 where A = A(x,D ) is a second
t x
order differential operator. The case of a principal matrix term in the divergence form, arising
from anisotropic media, was treated by Bellassoued, Jellali and Yamamoto in [5], using the
full data (i.e. the measurements are performed on the whole boundary). Using the FBI trans-
form Bellassoued and Yamamoto claimed logarithmic stability in [6] from arbitrarily small
1Aix-Marseille Universit´e, CNRS,I2M UMR 7373, E´cole Centrale. E-mail: [email protected].
2UniversityofScienceandTechnologyofChina,SchoolofMathematicalSciences,P.O.Box4,Hefei,Anhui
230026, People’s Republic of China. E-mail: [email protected].
3Aix-MarseilleUniversit´e,CNRS,CPTUMR7332, 13288Marseille, France&Universit´edeToulon,CNRS,
CPT UMR 7332, 83957 La Garde, France. E-mail: [email protected].
1
boundary observations. Imanuvilov and Yamamoto derived stability results in [18] by means
of H−1 Carleman inequality, from data observation on subdomainsfulfilling specific geometric
assumptions. In [26] Klibanov and Yamamoto employed a different approach inspired by [25].
Similarly, numerous authors have usedthe Dirichlet-to-Neumann operator to claim stability
in the determination of unknown coefficients of a hyperbolic equation. We refer to [4, 19, 28]
for a non-exhaustive list of such references.
In all the above mentioned papers, the domain was bounded. Recently, the Bukhgeim-
Klibanov method was adapted to the framework of infinite quantum cylindrical domains in
[10, 23, 24]. Inverse boundary value problems stated in unbounded waveguides were also
studied in [11, 12, 13] with the help of the Dirichlet-to-Neumann operator. In all the six
previous articles, the observation is taken on the infinitely extended lateral boundary of the
waveguide. The approach developed in this paper is rather different in the sense that we
aim to retrieve any arbitrary bounded subpart of the non-compactly supported conductivity c
from one data taken on a compact subset of the lateral boundary. This requires that suitable
smoothness properties of the solution to (1.1) be preliminarily established in the context of
the unbounded domain Ω.
The paper is organized as follows. Section 2 is devoted to the analysis of the direct problem
associated with the hyperbolic system under study. In Section 3 we prove a global Carleman
estimate specifically designed for hyperbolic systems in the cylindrical domain Ω. Finally
Section 4 contains the analysis of the inverse problem and the proof of the main result.
1.2 Settings
1.2.1 Notations
Throughout this text we write x = (x′,x ) ∈ Ω for every x′ := (x ,...,x ) ∈ ω and x ∈ R.
n 1 n−1 n
1/2
Further, we denote by |y| := m y2 the Euclidian norm of y = (y ,...,y ) ∈ Rm,
i=1 j 1 m
m ∈ N∗, and we put Sn−1 := x(cid:16)′ = (x ,.(cid:17)..,x ) ∈ Rn−1, |x′|= 1 .
P 1 n−1
For the sake of shortness we write ∂ for ∂/∂x , j ∈ N∗ := {m ∈ N∗, m ≤ n+1}. For
(cid:8) j j n+1 (cid:9)
convenience the time variable t is sometimes denoted by x so that ∂ = ∂/∂t = ∂ . We
n+1 t n+1
set ∇ := (∂1,...,∂n)T, ∇x′ := (∂1,...,∂n−1)T and ∇x,t = (∂1,...,∂n,∂t)T.
For any open subset D of Rm, m ∈ N∗, we denote by Hp(D) the p-th order Sobolev space
on D for every p ∈ N, where H0(D) stands for L2(D). We write h·,·i (resp., k·k ) for the
p,D p,D
usual scalar product (resp., norm) in Hp(D) and we denote by H1(D) the closure of C∞(D)
0 0
in the topology of H1(D)
Finally, for d > 0 we put Ω := ω × (−d,d), Q := Ω ×(0,T), Γ := ∂ω ×(−d,d) and
d d d d
Σ := ∂ω×(−d,d)×(0,T).
d
1.2.2 Statement of the problem
We examine the following initial boundary value problem (IBVP in short)
∂2u−∇·c∇u= 0 in Q := Ω×(0,T),
t
u(·,0) = θ , ∂ u(·,0) = θ in Ω, (1.1)
0 t 1

u= 0 on Σ := Γ×(0,T),


2
with initial conditions (θ ,θ ), where c is the unknown conductivity coefficient we aim to
0 1
retrieve. This is by means of the Bukhgeim-Klibanov method imposing that the solution u to
(1.1) be sufficiently smooth and appropriately bounded.
Throughout the entire text we shall suppose that c fulfills the ellipticity condition
c≥ c in Ω, (1.2)
m
for some positive constant c . Notice that we may assume, and this will be systematically the
m
case in the sequel, without restricting the generality of the foregoing, that c ∈ (0,1).
m
Let us now say a few words on the solution to (1.1). In order to exhibit sufficient conditions
on the initial conditions (θ ,θ ) (together with the cross section ω and the conductivity c)
0 1
ensuring that the solution to (1.1) is within an appropriate functional class we shall make
precise further, we need to introduce the self-adjoint operator A = A , associated with c,
c
generated in L2(Ω) by the closed sesquilinear form
q [u] := kc1/2uk2 = c(x)|u(x)|2dx, u∈ D(q ):= H1(Ω).
A 0,Ω A 0
ZΩ
Evidently, Aacts on its domainas −∇·c∇. SinceAis positivein L2(Ω), by (1.2), theoperator
A1/2 is well defined from the spectral theorem, and D(A1/2) = D(q ) = H1(Ω). For the sake
A 0
of definiteness, we set A0 := I and D(A0) := L2(Ω), where I denotes the identity operator in
L2(Ω), and we put
Am/2v := A(m−1)/2(A1/2v), v ∈ D(Am/2) := {v ∈ D(A(m−1)/2), A1/2v ∈ D(A(m−1)/2)},
for each m ∈ N∗. It turns out that the linear space D(Am/2) endowed with the scalar product
m
hv,wi := hAj/2v,Aj/2wi ,
D(Am/2) 0,Ω
j=0
X
is Hilbertian, and it is established in Proposition 2.6 that
D(Am) = {v ∈ H2m(Ω); v,Av,...,Ap−1v ∈H1(Ω)}, m ∈ {p−1/2,p}, p ∈ N∗, (1.3)
0
provided ∂ω is C2m and c∈ W2m−1,∞(Ω). As a matter of fact we know from Corollary 2.7 for
any natural number m, that the system (1.1) admits a unique solution
m+1
u∈ Ck([0,T];Hm+1−k(Ω)), (1.4)
k=0
\
provided the boundary ∂ω is Cm+1, the conductivity c ∈ Wm,∞(Ω;R) fulfills (1.2) and
(θ0,θ1) ∈ D(A(m+1)/2)×D(Am/2). Moreover, if kckWm,∞(Ω) ≤ cM for some constant cM > 0,
then the solution u to (1.1) satisfies the estimate
m+1
kuk ≤ C(kθ k +kθ k ), (1.5)
Ck([0,T];Hm+1−k(Ω)) 0 m+1,Ω 1 m,Ω
k=0
X
where C > 0 depends only on T, ω and c .
M
3
1.2.3 Admissible conductivity coefficients and initial data
In order to solve the inverse problem associated with (1.1) we seek solutions belonging to
∩4 Ck([0,T];H5−k(Ω)). Hence we chose m = 4 in (1.4) andimpose on c to bein W4,∞(Ω;R)
k=3
and satisfy (1.2). In what follows we note c a positive constant fulfilling
M
kckW4,∞(Ω) ≤ cM. (1.6)
Since our strategy is based on a Carleman estimate for the hyperbolic system (1.1), it is also
required that the condition
a′·∇x′c≥ a0 in Ω, (1.7)
hold for some a′ = (a ,...,a ) ∈ Sn−1 and a > 0. Hence, given O , a neighborhood of Γ
1 n−1 0 ∗
in Rn−1, and c ∈ W4,∞(O ∩Ω;R) satisfying
∗ ∗
c∗ ≥ cm and a′·∇x′c∗ ≥ a0 in O∗∩Ω, (1.8)
we introduce the set Λ = Λ (a′,a ,c ,c ,c ) of admissible conductivity coefficients as
O∗ O∗ 0 ∗ m M
Λ := {c ∈ W4,∞(Ω;R) obeying (1.2) and (1.6)−(1.7); c= c in O ∩Ω}. (1.9)
O∗ ∗ ∗
Notice that the above choice of m = 4 dictates that (θ ,θ ) be taken in D(A5/2)×D(A2),
0 1
which is embedded in H5(Ω)×H4(Ω) according to (1.3). Furthermore, it is required by the
analysis of the inverse problem carried out in this article that θ be in W3,∞(Ω) and satisfy
0
−a′·∇x′θ0 ≥ η0 in ω∗, (1.10)
for some η > 0 and some open subset ω in Rn−1, with C2 boundary, satisfying
0 ∗
ω\O ⊂ ω ⊂ ω. (1.11)
∗ ∗
Thus, for M > 0 such that
0
1
kθ0kW3,∞(Ω)+ kθjk5−j,Ω ≤ M0, (1.12)
j=0
X
we define the set Θ = Θ (a′,M ,η ) of admissible initial conditions (θ ,θ ) as
ω∗ ω∗ 0 0 0 1
Θ := (θ ,θ ) ∈ D(A5/2)∩W3,∞(Ω) ×D(A2), fulfilling (1.10) and (1.12) . (1.13)
ω∗ 0 1
n (cid:16) (cid:17) o
Having introduced all these notations we may now state the main result of this paper.
1.2.4 Main result
The following result claims H¨older stability in the inverse problem of determining c in Ω ,
ℓ
where ℓ > 0 is arbitrary, from the knowledge of one boundary measurement of the solution to
(1.1), performed on Σ for L > ℓ sufficiently large. The corresponding observation is viewed
L
as a vector of the Hilbert space
H (Σ ) := H4(0,T;L2(Γ ))∩H3(0,T;H1(Γ )),
L L L
endowed with the norm,
kuk2 := kuk2 +kuk2 , u ∈H (Σ ).
H(ΣL) H4(0,T;L2(ΓL)) H3(0,T;H1(ΓL)) L
4
Theorem 1.1. Assume that ∂ω is C5 and let O be a neighborhood of Γ in Rn−1. For
∗
a′ = (a ,...,a ) ∈ Sn−1, a > 0, c ∈ (0,1), c > c and c ∈ W4,∞(O ∩Ω;R) fulfilling
1 n−1 0 m M m ∗ ∗
(1.8), pick c , j = 1,2, in Λ (a′,a ,c ,c ,c ), defined by (1.9). Further, given M > 0,
j O∗ 0 ∗ m M 0
η > 0 and an open subset ω ⊂ Rn−1 obeying (1.11), chose θ ∈ Θ (a′,M ,η ), defined in
0 ∗ 0 ω∗ 0 0
(1.13), and θ = 0.
1
Then for any ℓ > 0 we may find L > ℓ and T > 0, such that the 5 Ck([0,T],H5−k(Ω))-
k=0
solution u , j = 1,2, to (1.1) associated with (θ ,θ ), where c is substituted for c, satisfies
j 0 1 j
T
kc −c k ≤ Cku −u kκ .
1 2 H1(Ωℓ) 1 2 H(ΣL)
Here C > 0 and κ ∈ (0,1) are two constants depending only on ω, ℓ, M , η , a′, a , c , c
0 0 0 ⋆ m
and c .
M
We stress out that the measurement of the observation data is performed on Γ and not on
L
the whole boundary ∂Ω .
L
2 Analysis of the direct problem
In this section we establish existence and uniqueness results as well as regularity properties,
for the solution to hyperbolic (1.1)-like IBVP systems. The corresponding results are similar
to the ones obtained for hyperbolic equations in bounded domains (see e. g. [15, Sect. 7.2,
Theorem 6]) but since Ω is unbounded here, they cannot be derived from them.
2.1 Existence and uniqueness result
With reference to (1.1) we consider the boundary value problem
∂2v+Av = f in Q
t (2.1)
v(·,0) = g, ∂ v(·,0) = h in Ω,
t
(cid:26)
wheref,g andharesuitabledata, andwerecallfrom4[27, Sect. 3, Theorem8.2] thefollowing
existence and uniqueness result.
Proposition 2.1. Assume that f ∈ C0([0,T];D(A0)), g ∈ D(A1/2) and h ∈ D(A0). Then
there exists a unique solution v to (2.1) such that
∂kv ∈ C0([0,T];D(A(1−k)/2)), k = 0,1. (2.2)
t
Moreover we have the estimate
1
sup k∂tkv(·,t)kD(A(1−k)/2) ≤ C kfkC0([0,T];D(A0))+kgkD(A1/2)+khkD(A0) . (2.3)
t∈[0,T]
Xk=0 (cid:16) (cid:17)
4Upon taking H := D(A0) = L2(Ω), V := D(A1/2) = H1(Ω) and a(t;u,v) := hA1/2u,A1/2vi =
0 0,Ω
R c(x)∇u(x)·∇v(x)dx for u,v∈V and all t∈[0,T].
Ω
5
2.2 Improved regularity
Proposition 2.2. Assume that f ∈ C1([0,T];D(A0)), g ∈ D(A) and h ∈ D(A1/2). Then the
solution v to (2.1) satisfies
∂kv ∈ C0([0,T];D(A(2−k)/2)), k = 0,1,2, (2.4)
t
and we have the estimate
2
sup k∂tkv(·,t)kD(A(2−k)/2) ≤ C kfkC1([0,T];D(A0)) +kgkD(A)+khkD(A1/2) . (2.5)
t∈[0,T]
Xk=0 (cid:16) (cid:17)
Proof. By differentiating (2.1) with respect to t, we check that w := ∂ v obeys
t
∂2w+Aw = ∂ f in Q
t t (2.6)
w(·,0) = h, ∂ w(·,0) = f(·,0)−Ag in Ω.
t
(cid:26)
Since f(·,0)−Ag is lying in D(A0) then we have ∂k+1v = ∂kw ∈ C0([0,T];D(A(1−k)/2)) for
t t
k = 0,1, with the estimate
1
sup k∂tk+1v(·,t)kD(A(1−k)/2) ≤ C k∂tfkC0([0,T];D(A0)) +khkD(A1/2)+kf(·,0)−AgkD(A0)
t∈[0,T]
Xk=0 (cid:16) (cid:17)
≤ C kfkC1([0,T];D(A0))+kgkD(A)+khkD(A1/2) (2.7)
(cid:16) (cid:17)
by Proposition 2.1. Further, as Av = f − ∂2v from the first line in (2.1), we get that v ∈
t
C0([0,T];D(A)), and that kv(·,t)kD(A) is majorized by the right hand side of (2.7), uniformly
in t ∈ [0,T]. This and (2.7) yield the desired result.
2.3 Higher regularity
Theorem 2.3. Let m be a nonnegative integer. We assume that g ∈ D(A(m+1)/2), h ∈
D(Am/2), and
∂kf ∈ C0([0,T];D(A(m−k)/2)), k = 0,...,m.
t
Then there exists a unique solution v to (2.1), such that
∂kv ∈C0([0,T];D(A(m+1−k)/2)), k = 0,1,...,m+1. (2.8)
t
Moreover we have the estimate
m+1
sup k∂kv(·,t)k
t D(A(m+1−k)/2)
t∈[0,T]
k=0
X
m
≤ C k∂kfk +kgk +khk . (2.9)
t C0([0,T];D(A(m−k)/2)) D(A(m+1)/2) D(Am/2)
!
k=0
X
6
Proof. a) The proof is by an induction on m, the case m = 0 following from Proposition 2.1.
b) We assume that the theorem is valid for some m ∈ N and suppose that
g ∈ D(A(m+2)/2), h ∈ D(A(m+1)/2),
(2.10)
∂kf ∈ C0([0,T];D(A(m+1−k)/2)), k = 0,...,m+1.
(cid:26) t
We use the same strategy as in the proof of Proposotion 2.2. That is we differentiate (2.1)
with respect to t and get that w := ∂ v is solution to (2.6). Next, using that h ∈ D(A(m+1)/2),
t
f(0)−Ag ∈ D(Am/2) and
∂k(∂ f)= ∂k+1f ∈C0([0,T];D(A(m−k)/2)), k = 0,...,m,
t t t
from (2.10), we get that ∂k+1v = ∂kw ∈ C0([0,T];D(A(m+1−k)/2)) for k = 0,1,...,m+1, and
t t
the estimate:
m+1
sup k∂k+1v(·,t)k
t D(A(m+1−k)/2)
t∈[0,T]
k=0
X
m
≤ C k∂k+1fk +khk +kf(0)−Agk .
t C0([0,T];D(A(m−k)/2)) D(A(m+1)/2) D(Am/2)
!
k=0
X
This entails ∂kv ∈ C0([0,T];D(A(m+2−k)/2)) for k = 1,...,m+2, and
t
m+2
sup k∂kv(·,t)k
t D(A(m+2−k)/2)
t∈[0,T]
k=1
X
m+1
≤ C k∂kfk +khk +kgk . (2.11)
t C0([0,T];D(A(m+1−k)/2)) D(A(m+1)/2) D(A(m+2)/2)
!
k=0
X
Further, as Av = f −∂2v from the first line in (2.1), we find out that
t
kv(·,t)kD(A(m+2)/2) ≤ C kAv(·,t)kD(Am/2)+kvkD(A0)
(cid:16) (cid:17)
≤ C kf(·,t)kD(Am/2)+k∂t2v(·,t)kD(Am/2)+kvkD(A0) . (2.12)
(cid:16) (cid:17)
Hereweusedtheidentity kv(·,t)k2 = kAv(·,t)k2 + 1 kAk/2v(·,t)k2 and
D(A(m+2)/2) D(Am/2) k=0 D(A0)
theestimate kA1/2v(·,t)kD(A0) ≤ 1k=0kAkv(·,t)kD(A0). SincekvkPD(A0) andk∂t2v(·,t)kD(Am/2)
are majorized by the right hand side of (2.11), uniformly in t ∈ [0,T], (2.11)-(2.12) yield the
P
assertion of the theorem for m+1.
Remark 2.4. The result of Theorem 2.3 is similar to the one of [15, Sect. 7.2, Theorem 6],
which holds for a bounded domain. This can be seen from the characterization of the D(Ak/2)
for k = 0,...,m, displayed in Subsection 2.4. Namely, it is worth noticing that the mth-order
compatibility conditions [15, Sect. 7.2, Eq. (62)] imposed on f, g and h, are actually hidden
in the operatorial formulation of Theorem 2.3.
7
2.4 Characterizing the domain of Am/2 for m ∈ N∗
2.4.1 Elliptic boundary regularity
In this subsection we extend the classical elliptic boundary regularity result for the operator
∇·c∇ , which is well known in any sufficiently smooth bounded subdomain of Rn (see e.g.
[15, Sect. 6.3, Theorem 5]), to the case of the infinite waveguide Ω under study. The proof
of this result boils down to [24, Lemma 2.4] which claims elliptic boundary regularity for the
Dirichlet Laplacian in Ω.
Lemma 2.5. Let r be a nonnegative integer. We assume that ∂ω is Cr+2 and that c ∈
Wr+1,∞(Ω) obeys (1.2). Then, for any ϕ ∈ Hr(Ω), there exists a unique solution v ∈ Hr+2(Ω)
to the boundary problem
−∇·c∇v = ϕ in Ω
(2.13)
v = 0 on ∂Ω.
(cid:26)
Moreover we have the estimate
kvk ≤ C kϕk , (2.14)
r+2,Ω r r,Ω
where C is a positive constant depending only on r, ω, the constant c appearing in (1.2) and
r m
kckWr+1,∞(Ω).
Proof. The proof is by induction on r.
a) We firstconsider the case r = 0. Due to (1.2) there is a uniquesolution v ∈H1(Ω) to (2.13)
0
by the Lax-Milgram theorem. Moreover v satisfies the energy estimate
kvk ≤ Ckϕk , (2.15)
1,Ω 0,Ω
where the constant C > 0 depends only on ω and c . Here we used (1.2) and the Poincar´e
m
inequality,whichholdstrueinΩsinceωisbounded. Furthermore,vissolutiontotheboundary
value problem
−∆v = f in Ω
(2.16)
v = 0 on ∂Ω,
(cid:26)
where
f := c−1(ϕ+∇c·∇v). (2.17)
Since f ∈ L2(Ω) then v ∈ H2(Ω) by [24, Lemma 2.4], and kvk is upper bounded, up to
2,Ω
some multiplicative constant depending only on ω, by kfk . As a consequence we have
0,Ω
kvk ≤ C′(kϕk +kvk ),
2,Ω 0,Ω 1,Ω
from (2.17), the constant C′ > 0 depending only on ω, cm and kckW1,∞(Ω). This and (2.15)
yield (2.14) with r = 0.
b) Suppose that the statement of the lemma is true for r ∈ N fixed, and assume that ∂ω is
Cr+3, c ∈ Wr+2,∞(Ω) and ϕ ∈ Hr+1(Ω). Hence the solution v to (2.16) belongs to Hr+2(Ω)
and satisfies the estimate (2.14), by induction assumption, and we have f ∈ Hr+1(Ω) in virtue
of (2.17). Further,v beingsolutionto(2.16)wheretheboundary∂ω isCr+3 thenv ∈ Hr+3(Ω)
8
by [24, Lemma 2.4]. Moreover kvk is upper bounded (up to some multiplicative constant
r+3,Ω
depending only on r and ω) by kfk . From this and (2.17) then follows that
r+1,Ω
kvk ≤ C′′(kϕk +kvk ),
r+3,Ω r+1,Ω r+2,Ω
where the constant C′′ > 0 depends only on r, ω, cm and kckWr+2,∞(Ω). Putting this together
with (2.14), we obtain (2.14) where r is replaced by r+1, proving that the statement of the
lemma remains valid upon substituting r+1 for r.
2.4.2 The domain of Am/2 for m ∈ N∗
In this subsection we characterize the domain of Am/2 for m ∈ N∗.
Proposition 2.6. Let m ∈ N∗ and let k be either m−1/2 or m. Assume that ∂ω is C2k and
that c ∈W2k−1,∞(Ω) fullfills (1.2). Then we have
D(Ak)= {u ∈ H2k(Ω), u,Au, ...,Am−1u ∈H1(Ω)}.
0
Moreover, the norm associated with D(Ak) is equivalent to the usual one in H2k(Ω): we
may find a constant c(k) > 1, depending only on k, ω, the constant c defined in (1.2) and
m
kck , such that we have
W2k−1,∞(Ω)
c(k)−1kukD(Ak) ≤ kuk2k,Ω ≤ c(k)kukD(Ak), u∈ D(Ak).
Proof. It suffices to show that
D(Ak) ⊂ {u∈ H2k(Ω), u,Au, ...,Am−1u∈ H1(Ω)}, (2.18)
0
and
kuk2k,Ω ≤ c(k)kukD(Ak), u∈ D(Ak). (2.19)
The proof is by induction on m.
a) We start with m = 1 and notice from the very definition of A1/2 that D(A1/2) = D(q ) =
A
H1(Ω). Moreover we have
0
kA1/2uk2 = q [u] ≥ c k∇uk2 , u∈ D(A1/2),
0,Ω A m 0,Ω
−1/2
in virtue of (1.2). Bearing in mind that c ∈ (0,1), we obtain that kuk ≤ c kuk
m 1,Ω m D(A1/2)
for every u ∈ D(A1/2). This establishes (2.19) for k =1/2.
Similarly, bearing in mind that D(A) = {u ∈ H1(Ω), Au ∈ L2(Ω)}, we apply Lemma 2.5
0
with r = 0 and ϕ = Au, where u ∈ D(A) is arbitrary. We find that u ∈ H2(Ω) satisfies
kuk ≤ C kAuk , which entails (2.18)-(2.19) for k = 1.
2,Ω 0 0,Ω
b) Let us now suppose that the statement of the lemma is true for some m ∈ N∗ fixed. Pick
k ∈ {m −1/2,m} and assume that ∂ω is C2(k+1) and that c ∈ W2k+1,∞(Ω) satisfies (1.2).
As D(Ak+1) = {u ∈ D(Ak), Au ∈ D(Ak)}, we deduce from the induction assumption that we
have
D(Ak+1)= {u ∈ H2k(Ω), Au∈ H2k(Ω) and u,...,Amu ∈ H1(Ω)},
0
9
with kAjuk ≤ c(k)kAjuk for j = 0,1. Thus, applying Lemma 2.5, with r = 2k and
2k,Ω D(Ak)
ϕ = Au, for u ∈ D(Ak+1), we get that u ∈ H2(k+1)(Ω), proving (2.18) where (k+1,m+1) is
substituted for (k,m). Moreover, it holds true that
kuk ≤ C kAuk ,
2(k+1),Ω 2k 2k,Ω
and sinceAu ∈D(Ak), theinduction assumption yields kAuk2k,Ω ≤ c(k)kAukD(Ak). Therefore
kuk is majorized, up to a multiplicative constant depending only on ω, c and m, by
2(k+1),Ω
kukD(Ak+1), which is (2.19) where k+1 is substituted for k.
In view of Theorem 2.3 and Proposition 2.6 we obtain the following result.
Corollary 2.7. Let m be a natural number. Assume that ∂ω is Cm+1, that c ∈ Wm,∞(Ω)
fulfills (1.2) and that (θ ,θ ) ∈ D(A(m+1)/2) × D(Am/2). Then the initial boundary value
0 1
problem (1.1) admits a unique solution
m+1
u∈ Ck([0,T];Hm+1−k(Ω)).
k=0
\
Moreover, we have
m+1
kuk ≤ C(kθ k +kθ k ), (2.20)
Ck([0,T];Hm+1−k(Ω)) 0 m+1,Ω 1 m,Ω
k=0
X
for some constant C > 0 depending only on T, ω and kckWm,∞(Ω).
3 Global Carleman estimate for hyperbolic equations in cylin-
drical domains
In this section we establish a global Carleman estimate for the system (1.1). To this purpose
we start by time-symmetrizing the solution u of (1.1). Namely, we put
u(x,t) := u(x,−t), x ∈ Ω, t ∈(−T,0). (3.1)
Under the conditions of Theorem 1.1, and since θ = 0, it is not hard to check that
1
4
u∈ Ck([−T,T];H5−k(Ω)).
k=3
\
With a slight abuse of notations we put Q := Ω × (−T,T), Σ := Γ × (−T,T) and Q :=
L
Ω ×(−T,T), Σ := ∂ω×(−L,L)×(−T,T) for any L > 0, in the remaining part of this text.
L L
10