Carleman estimates for the Zaremba Boundary Condition and Stabilization of Waves

0.37 MB·English

by  Pierre Cornilleau

Carleman estimates for the Zaremba Boundary Condition and
Stabilization of Waves.
Pierre Cornilleau∗& Luc Robbiano†
August 1, 2012
2
1
0
2
Abstract
l
u
Inthispaper,weshall proveaCarleman estimate fortheso-called Zarembaproblem. Using
J
sometechniquesofinterpolationandspectralestimates,wededucearesultofstabilizationforthe
1
waveequation bymeans of alinear Neumannfeedback on theboundary. This extendsprevious
3
resultsfromtheliterature: indeed,ourlogarithmicdecayresultisobtainedwhilethepartwhere

thefeedbackisappliedcontactstheboundaryzonedrivenbyanhomogeneousDirichletcondition.
]
P WealsoderiveacontrollabilityresultfortheheatequationwiththeZarembaboundarycondition.
A
.
h
t Keywords
a
m
Carlemanestimates,StabilizationofWaves,Zarembaproblem,pseudo–differentialcalculus,con-
[ trollability.

2
v
Contents
4
6
1 1 Introduction 2
5
1.1 General background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
.
0 1.2 Stabilization of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1
1.3 Carleman estimates for the Zaremba Boundary Condition . . . . . . . . . . . . . . . . 4
1
1.3.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1
: 1.3.2 Carleman estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
v
i
X 2 Proof of Theorem 2 7
2.1 Estimates in zone µ<0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
r
a 2.2 Estimates in zone µ> (∂ ϕ)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
− xn
2.2.1 Estimate of (D op(ρ χ ))u . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
xn − 1 1
2.2.2 Estimates of v and v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
0 1
2.2.3 Estimate of u in H1(x >0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
n
2.3 End of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Proof of Theorem 1 19
3.1 Different Carleman estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Proof of Proposition 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 End of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 Preliminary settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.2 Proofs of Proposition 1.1 and Theorem 1 . . . . . . . . . . . . . . . . . . . . . 26
∗Teacher atLyc´eeduParcdesLoges,1,boulevarddesChamps-E´lys´ees,91012E´vry,France.
e-mail: [email protected].
†Laboratoire de Math´ematiques de Versailles, Universit´e de Versailles St Quentin, CNRS, 45, Avenue des Etats-
Unis,78035Versailles,France. e-mail: [email protected]
1
4 Comments and further applications 28
A Symbolic Calculus 29
B Symbol reduction and roots properties 32
C A norm estimate 35
1 Introduction
1.1 General background
Weareinterestedhereinthestabilizationofthewaveequationonaboundedconnectedregularopen
setofRd. Ourstabilizationwillbe obtainedbymeans ofa feedbackona partofthe boundarywhile
the other part of the boundary is submitted to an homogeneous Dirichlet condition.
Since the worksof Bardos,Lebeau, Rauch(see [2]), the caseof stabilizationfor the waveequationis
well understood(by the so calledGeometric Control Condition) for Dirichlet orNeumann boundary
condition. Indeed, if the part of the boundary driven by the homogeneous Dirichlet condition does
not contact the region where the feedback is applied, Lebeau has given a sharp sufficient condition
for exponentialstabilizationofthe waveequation(see [20,Th´eor`eme3]and[21]). Moreover,Lebeau
and Robbiano (see [23]) have shown that, in the case where the Neumann boundary condition is
applied on the entire boundary, a weak conditionon the feedback (which does not satisfy Geometric
Control Condition) provides logarithmic decay of regular solutions.
On the other hand, multiplier techniques (see [15, 8]) give some results of exponential stabilization
(even if the part of the boundary driven by the homogeneous Dirichlet condition touches the region
where the feedback is applied) but under very strong assumptions on the form of the boundary
conditions.
Our goal here is to obtain some stabilization of logarithmic type under weak assumptions for the
boundary conditions. More precisely, we will see that, for solutions driven by an homogeneous
Dirichlet boundary condition on a part of the boundary and submitted to a feeback of the form
∂ u= a(x)∂ u
ν t
−
ontheotherpartoftheboundary,whereaissomenon-trivialnon-negativefunction,theirenergywith
initial data in the domain of k (denoting the infinitesimal generator of our evolution equation)
A A
decays like ln(t) k when t goes to infinity.
−
To this end, we willneedsome Carlemanestimates forthe so-calledZarembaBoundaryProblem
∆ u=f in X,
X
u=f on ∂X ,
0 D

∂ u=f on ∂X ,
 ν 1 N
where X is some regular manifold with boundary ∂X splitted into ∂X and ∂X and normal
D N
vectorfield ν. However, we will mainly tackle some local problem and the following model case (in
Rn with the flat metric)
+
∆u=f in x >0 ,
n
{ }
u=f on x =0,x >0 ,
0 n 1
 { }
∂ u=f on x >0,x >0 ,
 xn 1 { n 1 }
should help the reader to understand the main difficulties of this problem.

The Zaremba problem lies in the large class of boundary pseudodifferential operators, studied by
many authors. The first one was probably Eskin (see the monograph [9] where pseudodifferential
elliptic boundary problems are studied) but then Boutet de Monvel - in [5] - raisedthe fundamental
transmission condition. It was shown to play a key role in the resolution of such problems (see the
booksofGrubb[12]and[13, Chapter10]wherethe algebraofpseudodifferentialproblemsisstudied
in details).
2
Unfortunately,theZarembaproblemcannotbesolvedbythispseudodifferentialcalculus. Indeed,its
resolutioninvolvesapseudodifferentialoperatoronthe boundary thatdoes notsatisfythe transmis-
sion condition (see [16]). It lies in the general class of operators introduced by Rempel and Schulze
in [25] which allow to construct a parametrix for mixed elliptic problems - including the Zaremba
problem (see [16] and, more specifically, Section 4.1). However, up our knowledge, a Carleman esti-
mate for the Zaremba problem could not be obtained so far.
Carleman estimates have many applications ranging from the quantification of unique continuation
problems, inverse problems, to stabilization issues and control theory (see the survey paper [17] for
a general presentation of these topics). This last application was the motivation for the proof of
a suitable Carleman estimate (in the papers of either Lebeau and Robbiano [22] or Fursikov and
Imanuvilov [10]) and is still animating nowadays a large developpement of Carleman estimates (see
e.g. [19, 18] were controllability of parabolic systems with non-smooth coefficients is studied). Fi-
nally, we use the approach developped in [21, 23, 7] (also used by other authors - see, e.g., [3]) to
deduce our stabilization result. We shall also address a controllability result for the heat equation
with the Zaremba boundary condition (based on the approach developped in [22]).
Acknowledgements. TheauthorsthankJ´erˆomeLeRousseauandNicolasLernerforinteresting
discussions related to this work.
1.2 Stabilization of Waves
Let Ω be a bounded connected open set of Rd with C boundary ∂Ω. Let also Γ a smooth hy-
∞
persurface of ∂Ω which splits the boundary into the two non-empty open sets ∂Ω , ∂Ω so that
D N
∂Ω=∂Ω ∂Ω Γ (see Figure 1).
D N
⊔ ⊔
We study the decay of the solution of the following problem
(∂2 ∆)u=0 in Ω R+,
t − ×
u=0 on ∂Ω R+,
 D× (1)
∂ u+a(x)∂ u=0 on ∂Ω R+,
 (uν,∂ u) =t (u ,u ) in Ω, N ×
t t=0 0 1
|
where (u ,u ) H1(Ω) L2(Ω) is such that u = 0 in ∂Ω and a is, for some ρ (0,1), a
0 1 0 D
non-negative fun∈ction of C×ρ(∂Ω ), the space of Ho¨lder continuous functions on ∂Ω . ∈
N N
Forthe sakeofsimplicity, weherefocus onthe classicalLaplacian∆butallthe resultsdescribed
below remain true with the Laplacian associated to a smooth metric (see Section 4).
∂Ω
N
Γ
Ω
∂Ω
D
Figure 1: A configuration example.
We denote H = u H1(Ω);u =0 in ∂Ω L2(Ω) and define
0 0 D
{ ∈ }×
0 I
=
A ∆ 0
(cid:18) (cid:19)
3
with domain
( )= (u ,u ) H;∆u L2(Ω),u H1(Ω),u =0 on ∂Ω and ∂ u +a(x)u =0 on ∂Ω .
0 1 0 1 0 D ν 0 1 N
D A { ∈ ∈ ∈ }
For any solution u of (1), we define its energy by
1
E(u,t)= ∂ u(x,t)2+ ∂ u(x,t)2dx
t x
2 | | | |
ZΩ
where ∂ =(∂ ,...,∂ ).
x x1 xd
Denoting the resolvent set of by
A
ρ( )= µ C; µI : ( ) H is an isomorphism ,
A { ∈ A− D A → }
we will establish the following spectral estimate:
Proposition 1.1. Let ρ>1/2 and a Cρ(∂Ω ) a non-negative function.
N
∈
If a=0 and a(x) 0, then one has iR ρ( ) and there exists C >0 such that
6 −x−−→Γ ⊂ A
→
λ R, ( iλI)−1 H H 6CeC|λ|.
∀ ∈ k A− k →
Hence, using an useful result of Burq (see [7, Theorem 3]), we get our logarithmic decay result:
Theorem 1. Let ρ>1/2 and a Cρ(∂Ω ) a non-negative function.
N
∈
If a(x) 0 and a = 0 then, for every k 1, there exists C > 0 such that, for every (u u )
k 0, 1
−x−−→Γ 6 ≥ ∈
( k) th→e corresponding solution u of (1) satisfies
D A
C
t>0, E(u,t)1/2 6 k (u ,u ) .
∀ log(2+t)k k 0 1 k ( k)
D A
These results are completely analogous to the ones obtained by Lebeau and Robbiano in [23].
The outline of the proof is also quite similar to the one proposed there except that the situation is
a bit different here because of the mixed character of the boundary value problem.
The key point is also to establish some Carleman estimate in a neighborhood of Γ and to obtain
some interpolation inequality (see [23, Th´eor`eme 3]). This last result concerns an abstract problem
derived from the spectral problem.
DefiningX =( 1,1) Ω,∂X =( 1,1) ∂Ω ,∂X =( 1,1) ∂Ω ,weconsiderthecorresponding
N N D D
− × − × − ×
problem:
∆ v =v in X,
X 0
(∂ +ia(x)∂ )v =v on ∂X , (2)
 ν x0 1 N
v =0 on ∂X ,
 D
for some data v0 L2(X) andv1 L2(∂XN).
∈ ∈
If Y = ( 1/2,1/2) Ω and ∂Xδ = ( 1,1) x ∂Ω ;a(x) > δ , we will prove the following
− × N − ×{ ∈ N }
interpolation result.
Proposition 1.2. Let ρ>1/2 and a Cρ(∂Ω ) a non-negative function.
N
∈
If a(x) 0 and a=0, there exists δ >0, C >0 and τ (0,1) such that for any τ [0,τ ] and
0 0
−x−−→Γ 6 ∈ ∈
for any fu→nction v solution of (2), the following inequality holds
τ
kvkH1(Y) 6C kv0kL2(X)+kv1kL2(∂XN)+kvkL2(∂XNδ)+k∂x0vkL2(∂XNδ) kvk1H−1τ(X).
(cid:16) (cid:17)
1.3 Carleman estimates for the Zaremba Boundary Condition
We will now present our Carlemanestimates and establish first some useful notations. Let n 2 be
≥
the dimension of the connected manifold X.
4
1.3.1 Notations
Pseudodifferential operators We use the notation introduced in [22].
First, we shall use in the sequel the notations hξi:=(1+|ξ|2)12 and Dxj = hi∂xj for 1≤j ≤n.
Let us now introduce semi-classical ψDOs. We denote by Sm(Rn Rn), Sm for short, the space
×
of smooth functions a(x,ξ,h), defined for h (0,h ] for some h > 0, that satisfy the following
0 0
∈
property: for all α, β multi-indices, there exists C 0, such that
α,β
≥
∂α∂βa(x,ξ,h) C ξ m β , x Rn, ξ Rn, h (0,h ].
x ξ ≤ α,βh i −| | ∈ ∈ ∈ 0
(cid:12) (cid:12)
Then,forallsequen(cid:12)cesa Sm(cid:12) j,j N,thereexistsasymbola Sm suchthata hja ,
(cid:12) m−j ∈ (cid:12)− ∈ ∈ ∼ j m−j
in the sense that a hja hNSm N (see for instance [24, Proposition 2.3.2] or [14,
Proposition 18.1.3]),−withj<aN as pmr−injci∈pal symb−ol. We define Ψm as the space of ψDOsP=Op(a),
m
for a Sm, formally dePfined by A
∈
1
u(x)= eihx−t,ξi/ha(x,ξ,h)u(t) dt dξ, u S′(Rn).
A (2πh)n ∈
ZZ
We now introduce tangential symbols and associated operators.
We setx=(x,x ), x =(x ,...,x )andξ =(ξ ,...,ξ )accordingly. We denote by Sm(Rn
′ n ′ 1 n 1 ′ 1 n 1
Rn 1),Sm forshort,thespaceofsm−oothfunctionsb(x,ξ ,h−),definedforh (0,h ]forsomeTh >×0,
− ′ 0 0
that satisTfy the following property: for all α, β multi-indices, there exists C∈ 0, such that
α,β
≥
∂α∂βb(x,ξ ,h) C ξ m β , x Rn, ξ Rn 1, h (0,h ].
x ξ′ ′ ≤ α,βh ′i −| | ∈ ′ ∈ − ∈ 0
(cid:12) (cid:12)
As above, for al(cid:12)(cid:12)l sequences bm(cid:12)(cid:12)j Sm−j, j N, there exists a symbol b Sm such that b
Ψmjhajsbmth−ej,sipnatcheeosfentasnegtehnattiabl−−ψDO∈j<sNBThj=bmo−pj(∈b∈)h(NobSsTmer−vNe,thweithnobtmatiaosnpwrinecaipd∈aolpstyTimsbdoiffl.eWrenetdferfionm∼e
aPbTove to avoid confusion), for b PSm, formally defined by
∈ T
1
Bu(x)= ei x′ t′,ξ′ /hb(x,ξ ,h)u(t,x ) dt dξ , u S (Rn).
(2πh)n 1 h − i ′ ′ n ′ ′ ∈ ′
− ZZ
We shall also denote the principal symbol b by σ(B).
m
Different norms We use L2 and Hs semi-classical norms on Rn, on x >0 , on x =0 and
sc { n } { n }
on x = 0, x > 0 . We recall that, in this paper, we use the usual semi-classical notations,
n 1
{ ± }
namely D = h∂ , and the symbols are quantified in semi-classical sense. In particular all the
xj i xj
norms depend on h.
To distinguish these different norms, we denote by
u 2 = u(x)2dx, u = Op( ξ s)u
s
k k ZRn| | k k k h i k
and
n
u 2 = u(x)2dx, u 2 = u 2 + D u 2 .
k kL2(xn>0) Z{x∈Rn, xn>0}| | k kHs1c(xn>0) k kL2(xn>0) Xj=1k xj kL2(xn>0)
Finally, on x =0, we use the norms
n
v 2 = v(x′)2dx′, v s = op( ξ′ s)v ,
| | ZRn−1| | | | | h i |
and the space Hs ( x >0) of the restrictions of Hs (Rn 1) functions equipped with the norm
sc ± 1 sc −
v = inf op( ξ s)w.
| |Hssc(±x1>0) w∈Hssc(Rn−1)| h ′i |
w|±x1>0=v
5
In particular for v Hs(Rn 1), we have
−
∈
v v
| |±x1>0|Hssc(±x1>0) ≤| |s
and we write, when there is no ambiguity, v instead of v .
| |Hssc(±x1>0) | |±x1>0|Hssc(±x1>0)
1.3.2 Carleman estimate
We now detail the local Carleman estimate obtained for the Zaremba boundary problem.
Let B = x Rn; x 6κ and P a differential operator whose form is
κ
{ ∈ | | }
1
P = ∂2 +R x, ∂
− xn i x′
(cid:18) (cid:19)
where ∂ =(∂ ,...,∂ ) and the symbol r(x,ξ ) of R is real, homogeneous of degree 2 in ξ and
x′ x1 xn−1 ′ ′
satisfies
c>0; (x,ξ ) B Rn 1, r(x,ξ ) cξ 2,
′ κ − ′ ′
∃ ∀ ∈ × ≥ | |

ξ Rn 1,r(0,ξ )= ξ 2.
 ′ − ′ ′
∀ ∈ | |
As usual in the context of Carleman estimates, we define the conjugate Pϕ =h2eϕ/h P e−ϕ/h for
ϕ any real-valued C function. Since ◦ ◦
∞
2
1 i 1 i
P =h2 ∂ + ∂ ϕ +h2R x, ∂ + ∂ ϕ ,
ϕ i xn h xn i x′ h x′
(cid:18) (cid:19) (cid:18) (cid:19)
the corresponding semi-classical principal symbol satisfies
pϕ(x,ξ)=(ξn+i∂xnϕ(x))2+r(x,ξ′+i∂x′ϕ(x)).
We assume that ϕ is such that, for some κ >0,
0
∂ϕ
x B , (x)=0 (3)
∀ ∈ κ0 ∂x 6
n
andthatHo¨rmanderpseudo-convexityhypothesis(see[14,Paragraph28.2,28.3])holdsforP onB
κ0
(x,ξ) B Rn,p (x,ξ)=0 Rep ,Imp (x,ξ)>0, (4)
∀ ∈ κ0 × ϕ ⇒{ ϕ ϕ}
where the usual Poissonbracket is defined, for p,q smooth functions, by
p,q (x,ξ)=(∂ p.∂ q ∂ p.∂ q)(x,ξ).
ξ x x ξ
{ } −
a
Remark 1. For instance in the model case P = ∆, we can choose ϕ(x ) = x + x2. We have
− n n 2 n
indeed p (x,ξ)=(ξ +i(1+ax ))2+ ξ 2 thus Rep ,Imp (x,ξ)=4a(ξ2+(1+ax )2)>0 if x
ϕ n n | ′| { ϕ ϕ} n n n
is small enough.
In the general case, changing ϕ into eβϕ for β > 0 large enough, hypothesis (4) can be satisfied
(see [14, Proposition 28.3.3] or [22, Proof of Lemma 3, page 352]).
Our local Carleman estimate for the Zaremba Boundary Condition can now be stated in the
following form.
Theorem 2. There exists ε>0 such that if ϕ satisfies
∂ϕ
>0 on x =0 B and ∂ ϕ(0) ε∂ ϕ(0)
∂x { n }∩ κ0 | x′ |≤ xn
(cid:18) n (cid:19)
6
and (3), (4) hold then, there exists κ (0,κ ] and C,h > 0, such that, for any h (0,h ),
0 0 0
∈ ∈
g H1/2(x >0), g H 1/2(x <0) and any g H1(Rn) supported in B which satisfies
0 1 1 − 1 κ
∈ ∈ ∈
g =g if x =0 and x >0,
P(g) L2(Rn) and 0 n 1
∈ (cid:26) ∂xng =g1 if xn =0 and x1 <0,
the following inequality holds:
geϕ/h + geϕ/h + h(∂ g)eϕ/h
k kHs1c(xn>0) | |1/2 | xn |−1/2
C h 1/2 h2P(g)eϕ/h + g eϕ/h + hg eϕ/h .
≤ − k kL2(xn>0) | 0 |Hs1c/2(x1>0) | 1 |Hs−c1/2(x1<0)
(cid:16) (cid:17)
Remark 2. The estimate in the theorem, except for the boundary terms, is the usual Carleman
estimate. Let us also remind that all the norms are semi-classical: in particular geϕ/h is
k kHs1c(xn>0)
equivalenttoh eϕ/h∂ g + eϕ/hg . Forthe othernorms,we referthe readertothe
k x kL2(xn>0) k kL2(xn>0)
definitions in paragraph1.3.1.
Remark 3. The norms . and . on the boundary x = 0 of the left hand side of this
1/2 1/2 n
inequality cannot be repla|ce|dby the|n|−orms . and . ( provided that the data g , g are estimated
1 0 1
|| ||
in the spaces H1(x >0) and L2(x <0)).
1 1
Indeed, in the special case where P = ∆, it is well-known that the variational solution of the
−
boundary value problem
∆u=f in X,
−
u=0 on ∂X ,
D

∂ u=0 on ∂X ,
 ν N
may be, even for smooth data f, suchthat ∂νu / L2(∂X).
∈
We refer the reader to the famous two-dimensionnal conterexample of Shamir (see [26]) where one
consider, in polar coordinates, the sets
X = (r,θ);r (0,1),θ (0,π) , ∂X = (r,π);r (0,1) , ∂X =∂X ∂X .
N D N
{ ∈ ∈ } { ∈ } \
and the function
θ
u(r,θ)=φ(r)r1/2sin
2
(cid:18) (cid:19)
withφ C ([0,1])acut-offfunctionsuchthatφ=1insomeneighborhoodof0andsupp(φ) [0,1).
∞
∈ ⊂
The paper is structured as follows: our proof of the main Carleman estimate (Theorem 2) is
divided into the three subsections of Section 2. This will allow us to deduce the interpolation
inequality of Proposition 1.2 and finally Theorem 1 in Section 3. In Section 4, we conclude by
some comments on the geometry and sketch a proof of controllability of the heat equation with the
Zaremba boundary condition.
2 Proof of Theorem 2
We firstrecallsomewell-knownfactsaboutpseudodifferentialoperators. Wereferthe readerto[24].
For simplicity, we write in all this section instead of . Note that there will
k·kHs(xn>0) k·kHssc(xn>0)
be no confusion as we do not use the classical norm on Hs(x >0).
n
Composition formula. If a Sm, b Sm′ then Op(a) Op(b)=Op(c) for c Sm+m′ given by
∈ ∈ ◦ ∈
(h/i)α
c(x,ξ,h)= | |∂αa∂αb (x,ξ,h)+hN+1R(x,ξ,h)
 α! ξ x 
|αX|6N
 
7
where
N +1 1 1
R(x,ξ,h)= (1 t)N e iz.ζ/h∂αa(x,ξ+ζ,h)∂αb(x+tz,ξ,h)dzdζdt.
(2πh)n Z0 − |α|X=N+1i|α|α!ZR2n − ξ x
We will also use the composition formula for tangential operators,which is completely analogous.
In the sequel, we will also need the following straightforwardresult.
Lemma 2.1. Let a Sm. Then
∈ T
h
[D ,op(a)]= op(∂ a).
xn i xn
Next, we use the same notations as in [23] and put
p (x,ξ)=ξ2 +2i(∂ ϕ)ξ +q (x,ξ )+2iq (x,ξ ) (5)
ϕ n xn n 2 ′ 1 ′
where q (x,ξ )= (∂ ϕ)2+r(x,ξ ) r(x,∂ ϕ(x)) and q (x,ξ )=r˜(x,∂ ϕ(x),ξ ).
2 ′ − xn ′ − x′ 1 ′ x′ ′
Here we denote by r˜(x,.,.) the bilinear form associatedto r(x,.) (i.e. such that r(x,ξ )=r˜(x,ξ ,ξ )
′ ′ ′
for all ξ Rn 1).
′ −
∈
We also define
q (x,ξ )2
1 ′
µ(x,ξ′):=q2(x,ξ′)+ (∂ ϕ(x))2.
xn
The sign of µ is of great importance to localize the roots of p in ξ . We may explain this from the
ϕ n
model case presented in the introduction. In this framework, one has P = ∆ and we may choose
−
ϕ=ϕ(x ) (more precisely of the form ϕ(x )=x +ax2/2 for some a>0) so that
n n n n
p (x,ξ )=(ξ +i∂ ϕ(x))2+ ξ 2,
ϕ ′ n xn | ′|
q2(x,ξ′)=−(∂xnϕ(x))2+|ξ′|2, q1(x,ξ′)=0 and µ(x,ξ′)=|ξ′|2−(∂xnϕ(x))2.
Moreover,the roots of p in ξ are given by
ϕ n
ρ1(x,ξ′)=−i(∂xnϕ(x)−|ξ′|), ρ2(x,ξ′)=−i(∂xnϕ(x)+|ξ′|)
and satisfy
µ(x,ξ )>0 Im(ρ (x,ξ ))>0>Im(ρ (x,ξ ))
′ 1 ′ 2 ′
⇒
whereas
µ(x,ξ )<0 Im(ρ (x,ξ ))<0.
′ 1,2 ′
⇒
In the microlocal zone µ < 0, the operator p is elliptic and since its roots in ξ have negative
ϕ n
imaginary part, one will be able to estimate directly the traces of g in terms of the interior data
P(g). On the contrary, in the microlocal zone µ>0, even is p is elliptic, only one of its root in ξ
ϕ n
has negative imaginary part and elliptic estimates would only get an equation on the traces of g. In
our general framework, we prove several analogous properties presented in Lemma B.1 (which are
very close to the ones of [23, Lemme 3]) and the case µ>0 will in fact be treated in section 2.2.
Our proof of Theorem 2 is consequently divided in two main parts. In the first one, we establish
a microlocal Carleman inequality concentrated where µ < 0 and, in the second one, we focus on
the microlocal region µ> (∂ ϕ)2. We will finally gather the results of these two parts in a short
− xn
concluding section.
Notations: In the sequel, we set, for w a function defined on Rn,
w if x >0,
w = n
0 if x <0.
n
(cid:26)
We also denote, for z C/R and s R,
∈ − ∈
zs =exp(slog(z))
wherelogisdefinedasanholomorphicfunctiononC R . Moreover,weusethenotation√z :=z1/2.
\ −
8
2.1 Estimates in zone µ < 0
We remind that we have denoted v =eϕ/hg. We also define the set
= (x,ξ ) Rn Rn 1, µ(x,ξ ) α(∂ ϕ)2
Eα { ′ ∈ × − ′ ≤− xn }
where α>0 is a sufficiently small parameter to be fixed later.
The proof we give essentially follows that of Lemma 4 in [23] and Proposition 2.2 in [19].
Let χ supported in and satisfying χ = 1 in a neighborhood of . Obviously χ S0
2α 3α
because χ− =0 when ξ Eis large enough. If u−=op(χ )v, one has E − ∈ T
′
− | | −
P u=op(χ )P v+[P ,op(χ )]v =f where
ϕ ϕ ϕ 1
− −
f C P v +Ch v . (6)
k 1kL2(xn>0) ≤ k ϕ kL2(xn>0) k kH1(xn>0)
Denoting δ(j) =(d/dx )jδ , straightforwardcomputation show that we have
n |xn=0
Pϕu=f1−h2γ0(u)δ′−ih(γ1(u)+2i∂xnϕ(x′,0)γ0(u))δ (7)
whereγ (u):=u andγ (u):=D u = ih∂ u arethefirstsemi-classicaltraces.
We now0construc|txna=l0o+cal par1ametrix foxrnP|xn.=0+ − xn |xn=0+
ϕ
Let χ(x,ξ) S0 such that χ = 1 for sufficiently large ξ as well as in a neighborhood of supp(χ )
∈ | | −
with moreover
supp(χ)∩p−ϕ1({0})=∅.
Note that it is indeed possible because the real null set p 1( 0 ) is bounded in ξ and, using Lemma
−ϕ { }
B.1, the roots of p in ξ are not real.
ϕ n
We define
χ(x,ξ)
e (x,ξ)= S 2.
0 −
p (x,ξ) ∈
ϕ
One may find e S 3 such that E =Op(e +he ) satisfies, for some R S 2,
1 − 0 1 2 −
∈ ∈
E P =Op(χ)+h2R .
ϕ 2
◦
Indeed, by symbolic calculus, one may verify that e = χ∂xpϕ.∂ξpϕ. In the sequel, we shall denote
1 p3
ϕ
e:=e +he .
0 1
We set the new quantities
w :=γ (u), w :=γ (u)+2i∂ ϕ(x,0)γ (u) (8)
1 0 0 1 xn ′ 0
and we apply our parametrix E to the equation (7) which may be written now in the form
h
P u=f + w δ h2w δ .
ϕ 1 0 1 ′
i −
One computes the action of E on w and w and finds
0 1
h 1
E w δ (x,x )= ei(x′ y′).ξ′/htˆ(x ,x,ξ )w (y )dy dξ ,
i 0 ′ n (2πh)n 1 − 0 n ′ ′ 0 ′ ′ ′
(cid:18) (cid:19) − ZZ
1
E h2w δ (x,x )= ei(x′ y′).ξ′/htˆ(x ,x,ξ )w (y )dy dξ ,
− 1 ′ ′ n (2πh)n 1 − 1 n ′ ′ 1 ′ ′ ′
− ZZ
(cid:0) (cid:1)
where
1
tˆ0(xn,x′,ξ′)= eixnξn/he(x,ξ)dξn,
2iπ R
Z
1
tˆ1(xn,x′,ξ′)= eixnξn/hξne(x,ξ)dξn.
2iπ R
Z
9
We note that the integraldefining tˆ is absolutely convergingbut that the integraldefining tˆ has to
0 1
be understood is the sense of the oscillatory integrals (see for instance [14, Section 7.8]).
Using the fact that e(x,ξ ,ξ ) is holomorphic for large ξ and actually a rational function with
′ n n
| |
respect to ξ , we can change the contour R into the contour defined by γ =[ C ξ ,C ξ ] ξ
n ′ ′ n
− h i h i ∪{ ∈
C; ξ = C ξ ,Im(ξ ) > 0 oriented counterclockwise where C > 0 is chosen sufficiently large so
n ′ n
| | h i }
that χ=1 if ξ C ξ .
n ′
| |≥ h i
Doing so, we get
u=E(f )+T w +T w +r
1 0 0 1 1 1
where
r =(I Op(χ))u+h2R u (9)
1 2
−
and, if j =0,1 and x >0, the tangential operators T of symbols
n j
1
tˆj(x,ξ′)= 2iπ eixn(ξn/h)e(x′,xn,ξ′,ξn)ξnjdξn, (10)
Zγ
The symbols 1 χ and χ are not in the same symbol class but it is known (see Lebeau-Robbiano
− −
[22] and Le Rousseau-Robbiano [19, Lemma 2.2]) that, since supp(1 χ) suppχ = , we have
− ∩ − ∅
(I Op(χ))op(χ ) hNΨ N.
−
− − ∈
N N
\∈
Consequently, recalling (9), one has the estimate
r 6Ch v =Ch v Ch v . (11)
k 1k2 k k k kL2(xn>0) ≤ k kH1(xn>0)
We now choose χ (x,ξ ) S0 so that supp(χ ) χ =1 , χ is supported in and χ=1 in
1 ′ 1 1 α
a neighborhood of supp(χ ).∈ T − ⊂{ } E
1
We set t =tˆχ for j =0,1 which allows us to get
j j 1
u=E(f )+op(t )w +op(t )w +r +r , (12)
1 0 0 1 1 1 2
where
r =op((1 χ )tˆ)w +op((1 χ )tˆ)w .
2 1 0 0 1 1 1
− −
One now notes that p (x,ξ) >c ξ 2 on supp(χ). Consequently, one obtains
ϕ
| | h i
E(f ) 6C f =C f . (13)
k 1 k1 k 1k k 1kL2(xn>0)
Moreover,using (10), one may obtain
l N,α Nn 1,β Nn 1, ∂l ∂α∂βtˆ Ch l ξ 1+j+l β .
∀ ∈ ∈ − ∈ − | xn x′ ξ′ j|≤ − h ′i− −| |
Consequently,notingthatr doesnotinvolvederivationswithrespecttox andthatsupp(1 χ )
2 n 1
− ∩
supp(χ )= , one obtains
−|xn=0 ∅
r 6Ch( v + D v ) (14)
k 2k1 k kH1(xn>0) | xn |xn=0|−1/2
from the composition of tangential operators and using the following trace formula (see [23, page
486])
ψ 6Ch 1/2 ψ .
| |xn=0| − k kH1(xn>0)
Regarding the two last terms, we use that µ(x,ξ )<0 for (x,ξ ) supp(χ ). Hence, by Lemma
′ ′ 1
∈
B.1, p (x,ξ ,ξ ) 1 is an holomorphic function of ξ on Im(ξ ) 0 for (x,ξ ) supp(χ ).
ϕ ′ n − n n ′ 1
{ ≥ } ∈
Recalling the form of e, one consequently has, for j =0,1,
1 ξj ξj(∂ p .∂ p )(x,ξ ,ξ )
(tj)(x,ξ′) = 2iπχ1(x,ξ′) eixnξn/hp (x,ξn,ξ )dξn+h eixnξn/h n xpϕ3(xξ,ξϕ,ξ ) ′ n dξn
(cid:18)Zγ ϕ ′ n Zγ ϕ ′ n (cid:19)
= 0.
10