Progrecs in Nonlinear Differential Fqiiations and Their Applications
Carleman Estimates
and Applications to Uniqueness and
Control Theory Ferruccio Colombini
Claude Zuily Editors
Birkhäuser
Progress in Nonlinear Differential Equations and Their Applications Volume 46
Editor Haim Brezis Université Pierre et Marie Curie Paris and Rutgers University New Brunswick, N.J.
Editorial Board Antonio Ambrosetti, Scuola Normale Superiore, Pisa A. Bahri, Rutgers University, New Brunswick Felix Browder, Rutgers University, New Brunswick Luis Cafarelli, Institute for Advanced Study, Princeton Lawrence C. Evans, University of California, Berkeley Mariano Giaquinta, University of Pisa David Kinderlehrer, Carnegie-Mellon University, Pittsburgh Sergiu Klainerman, Princeton University Robert Kohn, New York University P. L. Lions, University of Paris IX Jean Mawhin, Universit6 Catholique de Louvain Louis Nirenberg, New York University Lambertus Peletier, University of Leiden Paul Rabinowitz, University of Wisconsin, Madison John Toland, University of Bath
Carleman Estimates and Applications to Uniqueness and Control Theory
Femiccio Colombini Claude Zuily Editors
Birkhäuser Boston • Basel • Berlin
Femiccio Colombini Dipartimento di Matematica Università di Pisa 56127 Pisa Italy
Claude Zuily Laboratoire de Mathématique Université de Paris Sud—Orsay 91405 Orsay France
Library oI Congress CatalOgIDg.ID-PUNICStIOn Data
Cajieman estimates and applications to uniqueness and control theory / Colombini. Claude Zuily. editors. p. cm. - (Progress in nonlinear differential equations and their applications v. 46) Includes bibliographical references. ISBN 0-8176-4230-7 (acid-free paper) - ISBN 3-7643-4230.7 (acid-free paper) I. Continuation methods. 2. Control theory. I. Colombint. F. (Ferroccio) II. Zuily. Claude. 1945- III. Series. QA377.C325 2001 5l5'.353—dc2l
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AMS Subject Classifications: 35B60. 35315, 35Q30, 35310, 35R45, 34135. 35L05, 35L70
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Contents Preface Stabilization for the Wave Equation on Exterior Domains by Lassaad Aloui and Moez Khenissi
vii 1
Carleman Estimate and Decay Rate of the Local Energy for the Neumann Problem of Elasticity by Mourad Bellassoued
15
Microlocal Defect Measures for Systems by Nicolas Burq
37
Strong Uniqueness for Laplace and Bi-Laplace Operators in the Limit Case by Ferruccio Colombini and Cataldo Grammatico
49
Stabilization for the Semilinear Wave Equation in Bounded Domains by Belhassen Dehman
61
Recent Results on Unique Continuation for Second Order Elliptic Equations by Herbert Koch and Daniel Tataru
73
Strong Uniqueness for Fourth Order Elliptic Differential Operators by Philippe Le Borgne
85
Second Microlocalization Methods for Degenerate Cauchy—Riemann Equations by Nicolas Lemer
109
with Boundary A Gdrding Inequality on a by Nicolas Lerner and Xavier Saint Raymond
129
Some Necessary Conditions for Hyperbolic Systems by Tatsuo Nishitani
139
Strong Unique Continuation Properly for First Order Elliptic Systems by Takashi Okaji
149
vi
Contents
Observabiliry of the Schrodinger Equation
byKimDangPhung
16S
Unique Conlinuation from Sets of Positive Measure by Rachid Regbaoui
179
Some Results and Open Problems on the Controllability of Linear and Semilinear Heal Equations by Enrique Zuazua
191
Preface The articles in this volume reflect a subsequent development after a scientific meeting entitled Carleman Estimates and Control Theory, held in Cortona in September 1999. The 14 research-level articles, written by experts, focus on new results on Carleman estimates and their applications to uniqueness and controllability of partial differential equations and systems. The main topics are unique continuation for elliptic PDEs and systems, control theory and inverse problems. New results on strong uniqueness for second or higher order operators are explored in detail in several papers. In the area of control theory, the reader will find applications of Carleman estimates to stabilization, observability and exact control for the wave and the Schrodinger equations. A final paper presents a challenging list of open problems on the topic of controllability of linear and semilinear heat equations. The papers contaIn exhaustive and essentially self-contained proofs directly accessible to mathematicians, physicists, and graduate students with an elementary background in PDES. Contributors are L. Aloui, M. Bellassoued, N. Burq, F. Colombini, B. Dehman,
C. Grammatico, M. Khenissi, H. Koch, P. Le Borgne, N. Lerner, T. Nishitani, T. Okaji. K.D. Phung, R. Regbaoui, X. Saint Raymond, D. Tataru, and E. Zuazua.
Ferruccio Colombini Claude Zuily May2001
Stabilization for the Wave Equation on Exterior Domains L. Aloui and M. Khenissi 1
Introduction
The aim of this article is the study of stabilization for the wave equation on exterior domains, with Dirichiet boundary condition. More precisely, let O be a bounded, smooth domain of R" (n odd); we consider the following wave equation on Q =C0:
onRxQ (E)
=0
I
with the initial data f =
HD x L2, the completion of (11. for the energy norm. It is well known that equation (E) has a unique global solution u in the space C(R, H0) ri C' (IL L2). Moreover, the total energy of the solution is
conserved.
The goal of this work is to study the behaviour of local energy defined by
ER(f) =
IPIIIHR
=
f
(IVfi(x)12 + 1f2(z)12)dx
BR is a ball of radius I? containing the obstacle 0. Many authors have studied this question (see [161, [13]). We particularly mention the where
work of Moraw-etz [12] who established a polynomial decay of this energy for star-shaped obstacles. This result was improved by Lax, Phillips, and Morawetz [5] who showed exponential decay.
In 1967 Lax and Phillips [6) conjectured that this decay is equivalent to the fact that the obstacle is nontrapping. The necessary condition was proved by Ralston [14], and the sufficient condition by Melrose [11] who used, in particular, the Melrose—Sjöstrand theorem [10], on propagation of singularities.
We finally quote the recent work of N. Burq [2) who established the logarithmic decay of the local energy without any geometric condition on the obstacle.
2
L. Aloui, M. Khenissi
We obtain an exponential decay of local energy by adding to the equation a dissipative term of type a microlocal geometric assumption, called "exterior geometric control", which was inspired by the condition introduced by B.L.R. (see [1]). This theorem contains the result of Melrose [11]. The proof is based on Lax—Phillips theory [6], well adapted to the case of the dissipative equation. We also use mlcrolocal analysis techniques, in particular the theorem of propagation of microlocal defect measures of G. Lebeau [8].
2
Preliminaries
First, we recall some results of Lax—Phillips theory for the wave equation. Let us consider the following wave equation in the free space L where H0
lu(O)=fl,Oeu(O)=f2 is the completion
withf=(fj,f2)EH0
of
for the norm
+ 1f2(x)12) dx
111112
It is well known that (L) has a unique global solution u. If we set Uo(t)f = then (Jo(t) form a unitary, strongly continuous group on H0,
generated by the unbounded operator A0 D(Ao) = Following Lax
D÷,0 the space
D...,0
{f
of
and
Phillips,
{f
H0, Aol E H0}.
(2.2)
we denote by
= (fl,f2) E H0
such that Uo(t)f = 0
t,t
in IxI
0},
outgoing data, and
{f
=
E H0 such that Uo(t)f = 0
the space of incoming data. And, for R >
= D+.R = {f E H0 = D,R = {f E H0 D+
0,
=
0 in
such that Uo(t)f
=
0
D...
in ]xj < —t,t
.1
IIm(z2)u12) +
crJ
+
=
=
rJ
+
(e2" +
the previous estimate with Proposition 1.2. we obtain
J
r=R2
fiR2
+J
crJ fiR2
—
Jr=R1 (r21u12 +
+
(2.8)
By the trace formula we get (see Burq [1] (3.6))
J
+ FVuI2)(e2" +
J
fiR2
+ (2.9)
Taking into account AelL =
f
—
+
z2u
in
a neighborhood of 5(0. R1) we can get +
fiR2
J
r=
+
(2.10)
Carleman Estimate and Decay Rate
23
Then we have
+
J
f
r=R1
(r21u12
+
+
Hence by the fact that K
and by (2.12). (2.11) we obtain
f
+ JIm(z2)121u12)
+ r2 f
+ e2
(2.11)
jIm(Bu .
r=R2
crf
(2.12)
0R2
On the other hand we get
f
0R2
i.ii=J 0R2
=
J
r=fl2
z21u12—E(u111).
(2.13)
Im(z2)JttJ2,
(2.14)
Then we have
Irn
f
r=R2
(Bu ii)
=
f Im(f .
—
OR2
combining (2.14) with (2.12) we obtain
{f
1/12
+ r2IIm(z2)121u12 + r2ff
c.rJ
T21u12 + Vu12.
OR2
(2.15) < C1; this is further equivalent to 1m(z) < Now assume that Then the term IIm(z2)12Iu12 can be easily incorporated in the right hand side in (2.16) for large r. Finally, if Im z we get the estimate eC'T
f
1112
cJ (r21u12 + JVuJ2).
OR2
This completes the proof of Theorem 0.2.
(2.16)
24
3
M. Bellassoued
Proof of Carleman estimates
This section is devoted to proving estimates of Carleman type near the boundary for solutions to boundary value problem of the form
JA(x,D)u=f B(z, D)u = g
on
where A(x, D) is a partial differential operator with principal symbol given by — r2Id (3.2) + (.t + = and B(x, D)u is defined by (0.3). Here we remark that the phenomenon of
A(x,
Rayleigh waves is connected to the failure of the Lopatinskii condition, and
our analysis is completely different from the scalar case treated by Burq [1]. D. Tataru [20] was the first to consider the Carleman estimates and the uniform Lopatinskii condition for scalar operators; here we shall use the method developed in [20] for construction of the symmetrizer. To our knowledge, very little literature on the system problem is available, even without additional conditions on the boundary. Indeed, no general method is available to solve such problems, except to multiply the system by the cofactors matrix and then use the machinery of scalar Carleman estimates (see Hörmander [2]) for the determinant. Unfortunately this method needs height regularity conditions on the coefficients, and especially in the case of the boundary problem it increases the multiplicity of real characteristics near the boundary. And hence the Lopatinskii condition is not easily satisfied. D. Tataru [20] gives a rigorous study of the Lopatinskii condition and Carleman estimates. In fact Tataru proved the Carleman estimates in the general case for scalar operators under the Lopatinskii condition. But in the case of elasticity systems the situation is more complicated. Indeed, the operator has a principal symbol matrix 3 x 3, and especially in the case of Neumann boundary condition, the phenomenon of Rayleigh waves is connected to failure of the Lopatinskii condition. In this step our proof diverges completely from the proof of Burq [Bu]. Our approach, consisting of diagonalizing the system near the boundary is the main technical part of this work. 3.1 3.1.1
Reduction of the problems Reduction of the Laplacian
be a bounded smooth domain of R" with boundary 0f10 of class Ccc. In a neighborhood of a given x0 äf?0, we denote by x = the system of normal geodesic coordinates where x' E and E IR are characterized by Let
=
=
> 0};
dist(x',x) = dist(x,Oflo).
Carleman Estimate and Decay Rate
25
In this system of coordinates the principal symbol of the Laplace operator takes the form
= tt(x
+ r(x.
=
(3.3)
is a quadratic form, such that there exist for C >
where r(x.
0,
T(8ft0).
for any x E K.
(3.4)
We set
=
+
(3.5)
then we have
=
= 0.
(3.6)
function with be a lxi 0 and — op(k,)v = on {x. = a) where r) is a tangential symbol of order 1, then for large enough r we have
+ lIvIItT +
+ rlop(xo)vIL.) (3.28)
whenever v E
Carleman Estimate and Decay Rate
Lemma 34. There exist C >
0
such that for any large enough r we have
CT2
+
+
If we assume that
whenever v E
29
—
(3.29) cip(k2)v
for any
such that
=92
= 0)
Ofl
then we
E
have
+ r1g212 C
+
+ (3.30)
whenever v E C00'(K).
Remark 3.5. We have similar lemmas if we assume that a Dirichlet condition in a boundary v = 91 on =0 3.2.2 Estimation for A
By applying Lemma 3.3 and Lemma 3.4 we get the following estimate of A.
Lemma 3.6.
There
exist C >
IIop(A)v112
whenever v E D,2v — op(k)v
=
g on
any Uoi,(A)v112
+
0
such that for any large enough r we have
+
(3.31)
Furthermore, if we assume that = 0} such that
>
0
and
for
then we have
+ IIvII?,. +
+
(3.32)
whenever v
Now we give a simpler estimate which completely neglects the boundary conditions.
Lemma 3.7. There exist C > 0 such that for any large enough r we have Ilop(A)uW +
+
(3.33)
whenever u E
4
End of the proof of Proposition 3.2
The purpose of this section is to prove Proposition 3.2. The essential ingredient in the proof is to estimate the traces of u by the operators A and B.
M. Bellassoued
30
Proposition 4.1. There for any
exists Co large enough r we have
> 0 and C > 0 such that if
>
Co
IIA(x. D. r)ul 12 + rIB(x, D, r)uC?_O,.dB,T
+
+ u
denote
u
7=
and
=
op(A)ii
(4.2)
where op(A) is the differential operator with principal symbol
=
+
(4.3)
It is easy to see that
onxn0 where 7=
(44
[op(A):op(xo)Ju.
Let us reduce the probiem (4.4) to a first order system. Put v = '(< D'. Then the system (4.4) is reduced to the first system r > ii.
45 where the principal symbol of op(A) is given by A=
(
0
(4.6)
>' B1.B0)
8= +
=
with
(4.7)
and
F=t(0.f):
(4.8)
further —
= =
+
e.
r)).
Carleman Estimate and Decay Rate
31
be fixed in Supp(xo). In this case the eigenvalues of A are
Let
± ict,. and 4 = zt E R. Denote = =
with ±Im(4) > 0 and
±
where ro) corresponding to form a basis of the generalized eigenspace of A(xo, eigenvalues with positive or negative imaginary part. Let, for y E {p, +
= tir where is a small circle with the center ± ia.,. Using this projection operator, we put = j = I...., n — 1 and = where a;) as a smooth >= 1. and = (st positively homogeneous function of degree zero and define a pseudodifferential S(x. r) with principal symbol r). Then by the argument in Taylor 120] (see also Yamaznoto [24)) there exists a pseudodifferential operator K(x. r) of order —1 such that the boundary value problem (4.5) is reduced to
f w = (I + F = (I + K)'S'F, 8= BS(I + K)-' and N = diag(W'. N); moreover the eigenvalues of the principal symbol of
N have negative imaginary parts. Denote the boundary operator B of and (4.7) by the subspace generated by (st st); then we have
r) = IKer(zt — A)1 e
= 4.2
—
A)].
Proof of PropositIon 4.1 p> 0; then there exist C> 0 such
Lemma 4.2. Let 1Z = diag(O. that
= diag(0.e(x,(r)) and we have
C
Cdiag(0. Id)
in on
= 0} n suppXo.
Proof of Proposition 4.1 Denote the function =
(4.12)
M. Bellassoued
32
Taking into account (4.10) we have = —21m(op(1Z)op(fl)w, w)
+ 21m(op(R)w. F) +
)w, w).
(4.13)
The integration in the normal direction gives w)o
=2j
Im(ap(R)ap(fl)w,
- 21mj
(op(R)w,1)
-
Then according to Lemma 4.2 and the Gãrding inequality we obtain for
w = (w+,w) and large r Im(op(1Z)op(N)w,w)
(4.14)
and further, for any E
JoI(op(R)w,F)Idxn < rCrIIwfl2 +
(4.15)
Applying Lemma 4.2 ii), we obtain w) + C18w12
(4.16)
Combining (4.18) (4.17) (4.16) with (4.15) we get
CrIIw
112
+
+
(4.17)
This implies the estimate (4.1).
4.3
Proof of Lemma 4.2
r)
First we prove that for any (x.
the restriction 8+ of B in
r) is an isomorphism. The eigenvalues of A are = with multiplicity respectivly (n — = —irp' ±
and Now let X = (X1, X2) E 20; then X satisfies
1) and 1. be an eigenvector of A associated to
0
f(e.r>X2=zoXi
(
1A(zo).Xi=0. (a) Calculus of eigenvector associated to Denote by
}
a n—2
basis of
A(zt)4=0
then we have
fori€{1,...,n_2}
418 ) .
Carleman Estimate and Decay Rate
where A(zfl =
+
Now we set the following
+
+
33
vector in C's:
=
0}. Note bTX to be the bundle of rank dim X, whose sections are the tangent vector fields to OX, bT.X its dual bundle (the Meirose cotangent bundle) and j : T'X bTsx the canomc application. j is defined by Note
= xe).
= Note that Car P { (y, x, ifold of P,
e2
Z = j(Car P),
= r(x, y,
}
(2.9)
the characteristic man-
2=ZU
s2 =
(2.10)
SZ =
The spaces SZ and S2 are locally compact metric spaces. For Q A°, with principal symbol q = c(Q), note that ic(q)(p)
=q(j'(p)).
(2.12)
The main result ensuring the existence of a measure describing the asymptotic polarization of the sequence is the following:
Proposition 2.1. There exist a subsequence of (uk) (still noted (uk)) and an hermitian positive measure p on SZ such that VQ E A°
urn
lim fx k-.4-oo = Remark 2.2. To
prove this result satisfies any boundary condition.
+
Quk (2.13)
we do not need to assume that (Uk)
The proof of this proposition relies (as the proof in the more simple case when the boundary is empty) on Gârding inequalities (see [6]).
N. Burq
40
The propagation theorem
3
In this section we suppose that the sequence (usc) satisfies (2.7).
The Meirose and Sjöstrand flow
3.1
We work near a point
For
E
&'
close
to
close to 0, note that
and
= (Id —
+ eb(e))
b is the principal symbol of the operator B appearing in (2.7). The matrix the hyperbolic reflection describes, for e E fi, = associated to the boundary condition (the relation can be, since ç E fi, computed using geometric optics methods, see (10]). where
In this section we note E a small conical neighborhood of the point
EG in Z=j(CarP).
P0
A ray is a continuous application from an interval I C R to E, s such
'y(s),
that
(i) If
0
0
such that x(-y(s)) >
0
pour
Is — sil <e.
flu (ir'(p1) is a singleton in jr'(O)) and for any I E C°° from p''(O) to R, ir—invariant, if f is the unique continuous
(11) If Pi
application from E to R such that the following diagram R
commutes, then s i—i f[7(s)J is derivable at s = d
-
=
1
—1
and
(P1)].
(3.2)
In the following we suppose that there is no infinite contact between the bicharacteristic of p and the boundary. This hypothesis implies the existence and uniqueness of the ray passing through any point, which gives the definition of the Melrose and Sjöstrand flow on Z. By a suitable change of parameter along this flow, we obtain a flow on SZ. Consider S a hypersurface transverse to the flow. Then locally, SZ = x S where s is the parameter along the flow.
Microlocal Defect Measures for Systems
41
Theorem 3.1. The measure jz is supported in SZ and there exists a function (s, z) E R3 x S '—i M(s, z)
(3.3)
ji-almost everywhere continuous such that the pull back of the measure by M (i.e., the measure Ts/A = MjM defined (for a E C°(SZ)) by
a) =
(hz,
MaM)
(3.4)
0
(3.5)
satisfies
=
(we say that the measure p is invariant along the flow associated to M). Ftzrthermore, the function M is continuous except at point z0) E ii where we have
M(so +0,zo) =
—0,zo)
(3.6)
and along any ray the matrix M is solution to a differential equation (with jumps at 11) whose coefficients can be explicitly computed in terms of the geometry and the different terms in the operators P and B.
Remark 3.2. Roughly speaking, in the result above, the norm of the ma-
trix M describes the damping of the measure p, whereas the rotation component of M describes the way the polarization of the measure (the asymptotic polarization of the sequence
is
turning.
The proof of Theorem 3.1 does not rely on any hard to prove propagation of singularities result. It is direct and uses equations satisfied by the measure in the distributional meaning. It is based on an induction argument on the order of tangency of the points close to which we are working (see
Definition 3.3. If
=(si,zo)ERXS=SZ (3.7)
are two points on the same ray, and e1, e2 CN two directions, we say that and are connected by the propagation flow if
M(s2,zo)ei = M(si,zo)e2.
(3.8)
N. Burq
42
The Lamé system
4 4.1
Transversal and longitudinal waves
Consider the Lamé system in a smooth bounded domain
C Rd; d =
2; 3.
(4.1)
with
t=o= uj E L2
U0 E
U
> 0 et A +
> 0. There is a natural energy
E (u) (t) = j
+ plVuI2 + (A +
div u12.
(4.2)
The following result shows that any solution of (4.1) can be decomposed into two components: the transversal one (UT) and the longitudinal one (uL):
Lemma 4.1. There exists (so,
1. u =
x
E
curl i,1 = UL +UT, UL =
1+3
such
that
UT = curl
2.
3. divl,b=0,curluL=O,divuT=O. 4. There exists C > 0 such that for any interval I CC R
(u),
(4.4)
Ikt'11H2(Jxn)
4.2
(4.3)
Geometry
Consider M =
x fZ. Note that
Car C c
= {(t,r, z,
; (x, t) E M, r2 =
(4.5)
Car T C
= {(t,r, z,
(x, t) E M, r2 =
(4.6)
the two characteristic manifolds of the wave operators. Note that = j(CarC; 7) C bT.M are their projections. In a geodesic coordinate system where
=
(4.7)
Microlocal Defect Measures for Systems
43
with Q, of order i, and with r0 = r
{x = O} = {(t, y, r, with VL,T =
cj4
;
I/LTT + r0
0),
(4.8)
and
T*OM=nLUcLU6L=nTUgTUET,
(4.9)
with
= {(t, y, r,
+ r0 > 0}, + r0 = + ro < 0).
;
gL,T = {(t, y, r, I.. eL,T = {(t, y, r,
Measures According to the results of Section 2, it is possible to associate to any 4.3
sequence (uk) whose energy is bounded, two measures as in the previous
section describing the asymptotic polarization of the sequences (4) and (ut,). Furthermore there exists two borelian sets, AT and AL such that /AT(AL) = O,1LL(AT)
=0 and SZ =
U AL.
From now on we suppose that the measure ItLL is equal to 0. Using the results of the previous section (and making global the constructions), it is possible to show that the measure is invariant along a certain flow defined on SZ x C3. The flow is the flows associated to Dirichiet boundary conditions on flL and to some more complicate boundary condition on
(cTU'Ir)\IIL. The condition div Uk = 0 gives a polarization condition on the measure pr. This condition implies that the measure pr is polarized in directions orthogonal to the direction of propagation (p is equal to flpfl where fl is the projector on the hyperplane normal to Since the flow associated to the Dirichlet boundary condition is the simple reflection = Id), this polarization and the invariance imply that near 7(L, the measure is polarized in a direction orthogonal to both, the incoming and the reflected ray; hence it is polarized along the normal to the boundary and to the ray (which is 1-dimensionnal if the ray is not normal to the boundary and is then called the critical direction). Suppose now that any ray hits the boundary in two points 91;2 E 'Ni. that are not normal to the boundary. Then near the point Q2), the measure is polarized along the critical direction at x2 and also polarized along the direction transported by the flow from the critical direction at x1. If these two directions are not the same, then the measure PT has to be polarized along two directions which are not the same, and hence PT = 0 near By propagation we obtain that is null along the ray passing through In summary if any ray hits the boundary at two points Q1;2 flL which are not normal to the boundary 0) and where the critical directions are not connected by the flow, then 0.
N. Burq
44
Application to the thermoelastic system
5
C R"; d =
Consider a smooth bounded domain of the system of thermoelasticity:
3 and (u, 9) solution
2;
OeO—M+f3divOtu=0,
ulan0,
t=o= U0
U
Glt=oOo and
L2 (Q)3,
Otu It=o= u1
L2(f 1);
its natural energy
E(u, 9)(t) = j
+ (A + ,i)ldiv u[2 +
+
j
=
0.
dx, (5.2)
In [10], C. Lebeau and E. Zuazua show the following:
Theorem 5.1. (Lebeau—Zuazua) For solutions to (5.1) the energy decay is uniform: there exists C,E >0 such that for any (uo,u,,Oo) H0' (ffld x E (u, 9) (t)
(u, 9) (0)
(5.3)
if and only if the two following conditions hold:
(i) any solution
H0'
(11)d
of
div
= =0
in
(5.4)
(5.5)
is equal to 0.
(ii) There existsT>0 andC>0 such that for any (uo,u,)E L2
the
solution of (4.1) satisfies
0. Suppose that there is a ray for PT = 40? —
which encounters the set
\ {IhilI = 0)
{t E [0,Tfl
(5.8)
only at points where the critical directions are connected by the polarization flow (or does not encounter this set). Then there exists a sequence of initial data (i4, = 0) such that the solution of (5.1) satisfies
=1 =1
urn
A classical uniqueness-compacity argument (see [21) shows that conditions (i) et (ii)) are equivalent to conditions (i) et (ii'), with
(ii') There exists T> 0 and C > 0 such that for any (u0, u,) E H0' (Q)d x L2
(ç1)d
the solution of (4.1) satisfies 2
U0 H1(1Z)d + it1
c[frf
2
+
ul2dxdt+
To prove Theorem 5.2, we argue by contradiction: Suppose that condition
(iii) is fulfilled and condition (ii') is not. Then there exists a sequence (UIC) of solutions of (4.1) such that
k2
U0 H1(fl)d
>k
+ Ujk2L2(C1)d
[JT1
Idiv ukl2dxdt+
+
.
(5.11)
Renormalizing the sequence (t4, ?4), we can assume that the initial energy is equal to 1. Then it is possible to apply to this sequence the constructions
46
N. Burq
Since the right-hand side in
above and associate two measures p'p and (5.11) is bounded, we obtain that r
urn
I
k-..+ooJ0
I jdiv
= 0.
(5.12)
0. The geometric hypothesis and the is also propagation result imply as above that the measure
which implies that PL
equal to 0. But this is in contradiction with the fact that the initial energy is equal to 1 (which implies that (pT+pL, ljo,T( = T)). To prove Theorem 5.4, we suppose that there exists a "bad" ray (and a "bad" direction). Then we construct a sequence Uk converging weakly to 0, of energy equal to 1, and such that the measures associated are as follows:
1. The measure PL is null.
2. The measure direction.
is supported by the "bad" ray (and along the "bad"
In fact, we do such a construction for well-prepared initial data, such that these conditions are fulfilled for small time. Then we apply our propagation
result to show that the conditions are fulfilled for any time. It is at this point that the fact that the ray and the direction are "bad" is important. The first condition implies that
L div ul2dxdt +
+
=0
(5.13)
whereas the energy is equal to 1. Hence (ii) is false.
References [11 N. Burq and C. Lebeau, Mesures de défaut do compacité, application au système de lamé, A panzitre aux Annales de L 'Ecole Normale Supérieure, 2001.
[2J C. Bardos, C. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, Siam Journal of Control and Optimization 305 (1992), 1024-1065.
N. Burq, Mesures semi-classiques et mesures de défaut, Séminaire Bourbaki, March 1997.
N. Denker, On the propagation of the polarization set for systems of real principal type. Journal of Functional AnaLysis 46 (1982), 351—373.
C. Gerard, Propagation de la polarisation pour des problèmes aux limites convexes pour les bicaracteristiques, Cornmun. Partial Differ. Equations 10 (1985), 1347-1382.
Microlocal Defect Measures for Systems 16]
47
Gerard, Microlocal defect measures, Communications in Partial Differential Equations 16 (1991), 1761—1794. P.
[7] P. Gerard and E. Leichtnarn, Ergodic properties of eigenfunctions for the dirichlet problem, Duke Mathematical Journal 71 (1993), 559—607. 18]
H. Koch and D. Thtaru, On the spectrum of hyperbolic semigroups, Comm. Partial Differential Equations 20(5-6) (1995), 901—937.
[91
G. Lebeau, Equation des ondes ainorties, In A. Boutet de Monvel and
V. Marchenko, editors, Algebraic arid Geometric Methods in Mathematical Physics, Kiuwer Academic, The Netherlands, 1996, p. 73—109.
[10] C. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Ration. Mech. Anal. 148(3) (1999), 179—231.
[11] L. Miller, Propagation d'ondes semi-classiques It travers une interface et mesures 2-microlocales, Ph.D. thesis, Ecole Polytechnique, 1996.
[121 L. Tartar, H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proceedings of the Royal Society Edinburgh 115-A (1990), 193—230. Université de Paris-Sud Orsay, France
Strong Uniqueness for Laplace and Bi-Laplace Operators in the Limit Case F. Colombini and C. Grammatico 1
Introduction
In this article we study some limiting cases of strong unique continuation for inequalities of the type
XEfL or
+
+
is a neighbourhood of the origin in constants. where
xE
and A. 8, C are positive
which are C°°-flat Let Cr(1l) denote the space of functions in at the origin. We say that the relation (1.1) (respectively, (1.2)) has the property of strong unique continuation at the origin, if the only function u E Cr(i)) (1.1) (respectively, (1.2)) is the zero function. This problem has been studied by several authors, such as Alinhac— Baouendi [1J— [2], Hörmander
Sogge (3J, Regbaoui
Jerison—Kenig [8], Barcelo—Kenig—Ruiz—
Colombini—Grammatico (4], Le Borgne (9] and
many others. In particular, Regbaoui [10] studies the relation (1.1) for general operators P(x,D) = and P(O,D) = & In 1998 Le Borgne [9] studied the strong unique continuation for (1.2), but on the right-hand side he adds third order derivatives with potential IsI —
£>0.
Using methods similar to those in [4), after writing (1.1) (respectively, (1.2)) in polar coordinates, we shall give Carleman estimates to prove our results. We point out that Theorems 2.1 and 2.3, proved in [4] and [5] respectively, give almost optimal results. In this paper we prove strong uniqueness results which are not included in these theorems.
50
F.
Colombini, C. Cranunatico
Finally let us recall that Wolff [11] has shown that, for n 4, there is a not identically zero, such that, for a certain constant C, O
and
Ek±Eh
h.
and (,) denote the norm and scalar product in
Notation: Let Weset, for
(see[6])
0
01 +
il, are suitable vector fields tangent to From now on B(0, R) = {x E In •
for some positive constants C, C', C", then u
Remark 2.11. For n =
4
+
(2.9)
9
0 in Q.
the preceding theorem holds with bounds only
for C', C".
As a consequence of the construction of the function w in Theorem 2.3 when h is even, we have the following
Theorem 2.12. For every C > 9/4 there exists a function w with supp w
R2 satisfying the estimate ID,D,w(x)12 +
for a suitable constant M.
IVw(x)12
x
Cr(R2) (2.10)
Laplace and Bi-Laplace Operators
53
Proofs
3
Let 1(k) = dim Ek, We recall that, with Ek as above, dim Ek = and write I = 1,2,... ,1(k), for an orthonormal base of Ek. Finally, we introduce the coordinates (T,c.i) E R x with T = lg r. For the proofs of the above theorems we make use of Carleman type inequalities.
3.1
Endpoint case for
We give the proof for the radial case, while the tangential case can be found in(4).
Proof of Theorem 2.5. Let V E
C00° ((—cc,
We can write
+oo) x
1(k)
00
=
fk,1
(T)
k=O 1=1
with as above. We note that 00
ff (V
dTdw
=
1(k)
>f
(T)(2dT,
k=O 1=1
where dw is the standard measure on Set Q= and
= e_TTQ (eTTV)
(3.2)
,
with r real parameter. We can write (1Q7V(T, .)112
+ r(r + ii —2) + &)V(T,
= lI($ + (2r + n —
.)112,
so using the relation
=
it is easy to see that
Jf IQ,-VJ2dTdw -2r(r + n -2) > + (2r2 + 2r(n
—
ff 2) + (n — 2)2)
+ T2('r + n - 2)211
ff
iOi'Vi2dTdw
+ JJ I&Vl2dTdw.
F. Colombini, C. Granunatico
54
From (3.3) we deduce 00
ff
+ n — 2)2
1(k)
>21 Ifkj(T)j2dT k=O 00
1(k)
— 2r(r + n — 2)>2 > k(k + n — 2)1 Ifk,l(T)j2dT k=O 1=1
00
1(k)
+ >2 >2k2(k+n_2)2fIfk,,(T)I2dT k=O 1=1
+ 2r2 so
for any
E
Jf
R, kEN and 1= 1,2,... ,l(k), we have
r2 Jf
ff
00
1(k)
+ >2
—
k)2(r + k + n — 2)21
k=O 1=1
From now on, we take for r the positive solution of the equations
r(r+n—2)=m(m+n—2)+m+ n—i 2
mEN.
(3.3)
It is easy to see that m < r < m+i; now we fix r in such a way and we analyse several cases:
• for
k
= m,
n—3 (r—m)2 (r+m+n—2)2 =r 2+(n—2)r+ n—i 2 2 finally
• for k
m + 1,
(i-—(m+1))2(r+(m+i)+n—2)2 = r2-i-(n—2)r+
n—i n—3
Therefore, keeping in mind (3.3), if n 3, we have
JJ
T2
ff
IVI2dTdW
+
+ rJJ IVl2dTdw.
ff (3.4)
Laplace and Bi-Laplace Operators
55
Setting U = eTTV, we obtain from (3.4)
ff
T1'
Qu12 dTdw
i°w12 dTdw,
Jf
(3.5)
Co°°((—oc,+00) x and for any r given by (3.3). Choose now x E C°°(R) equal to 1 in (—cc, T0) with eTh < R and radial increasing in Pri such that c (—oo, lgR). Let
for every U
—
fo if In 1/2
iflxl1.
= j E N, put Let u be any smooth function flat at the origin for which (2.6) holds; For
satisfies (3.5), and passing to the limit the same then the function inequality is satisfied by we have, for r as above, By applying (3.5) to e_21Tf1Qu(T, )ll2dT
j
-Co
+ JT0
e_2TTIIQ(xu)(T, )lI2dT
I
)112 dT
(3.6)
From (3.6), taking into account (2.6), we have (for r as above) +00
I JT0
I1Q (ru) (T, .)112 dT
(r - C)]
,To
e2rT
(T, .)112
-Co
and hence
I
+00
1IQ(xu)(T,)lI2dT (r —C) I
Uu(T,.)lI2dT.
JTo
Letting r go to +00, with r as in (3.3), we deduce that u and therefore 0 in 3.2
0 in
Endpoint cases for &
Before proving the remaining theorems we make some remarks which hold for all cases. As in the proof of Theorem 2.5 we give Carleman estimates using decomposition in harmonic spherical functions. x Let V E be as in (3.1). We set
P= P1.V =
(eTTV)
F. Colombini, C. Crammatico
56
with r real parameter. We note that PT =
with Q,. as in (3.2). So, setting 'f = r — 2 and taking (3.3) into account, it is easy to see that
oo
1(k)
(3.7)
+ k=O 1=1
with
CT,k=(f—k)2(f+k+n—2)2(r—k)2(r+k+n—2)2.
(3.8)
Now, for each case, we choose suitable values of the parameter r. Given that choice, the proof follows the same pattern for each different case. Proof of Theorem 2.6. In this case we take for r the positive solution of the equations
r(r+n—4)=m(m+n—2)--m+
n
2
mEN.
(3.9)
It is easy to see that m < r < m + 1; now we fix r as above and analyse several cases:
• for
k
= m, Crm = [3m(m+n —2) +
• for k = m
—
—
2)2J;
1,
cr,vn_1 =
[3(m_1)(m_1+n_2)+
Therefore, keeping in mind (3.7), we deduce
9ff I&Vt2 ffdw + +
(n
- 2)2
- 2)4JfIVI2dTdw +
JJ
dTdw
Laplace and Bi-Laplace Operators
57
,m—2
where V(T,w)—_ k=O
Thus, setting U = eTTV and U =
we obtain, as in (3.5), a similar and for any r estimate in U and U for every U E +oo) x given by (3.9). Then we can conclude as in the preceding theorem. Proof of Theorem 2.8. Now, we choose r > 0 such that
r(r+n —4)=m(rn+n—2)—m+ with m a positive integer.
We note that m
0
m€N.
In any case, we have
• for any k E N, 9r2k(k + n —
+ i* (k + n — 2) + r3.
2)
(3.13)
Thus, in these cases, from (3.13) it follows that there are no bounds for the constants C, C', C".
2. If n = 4, taking r >
0
such that r2
we obtain Gr,m = (3T2 +
3)2 +
(3.14)
and —
CT,m_l = (3T2
—
(3.15)
Hence, keeping in mind that the principal term in the Carleman estimate is 9r2k(k + 2), it follows from (3.14) and (3.15) that there is a bound for C', C", but not on C.
3. If n 3 it is easy to see that there does not exist r E [m,m+1], such that Crm 9r2m (m + n —2) and CT.m_i
9r2(m—1)(vn—1+n—2)
Thus, in this way we cannot give a Carleman estimate for the term 9
Laplace and Bi-Laplace Operators
59
Proof of Theorem 2.12. It is easy to see that the function w that we have constructed in Theorem 2.3 verifies
0 and some real p 1 satisfying (d
(2.3) —
2)p < d.
Furthermore a(x) is a positive continuous function on Under these conditions, it is well known that problem (2.1) is well posed, i.e.. for every initial data {u°, u1 } E H0' x L2(Q) it has a unique global solution E C°([0,+oo[,H01 (ri))
Stabilization for the Semilinear Wave Equation
63
We attach to such a solution its energy at time t,
E(u)(t) = 1/2
f
dx
x)12)
(lOiti(t, x)12 +
dx + j F(u(t,x))
f(s)ds.
where F(u)
= J0 A simple integration by parts shows that for 0
t1
t2, one has
pt2
E(u)(t2) — E(u)(ti) = —
j
tz
/ a(x)IOcu(t,x)I2dtdx.
JO
the energy is decreasing in time. The system (2.1) is called dissipative. The goal of the present work is precisely to study this dissipation. So
Theorem 2.1. Let
be a bounded open subset of Rd. connected, of class
And let f be a function of C'(R) satisfying (2.2) and (2.3), and a(x)
a continuous positive function on Il such that: (2.4)
where w is an open subset of
neighbourhood of the boundary Ofl. Then
we have local stabilization for (2.1). i.e.. for every real number R > 0, there exist two constants C> 1 and -y > 0 depending on R, such that the inequality
E(u)(t)
t
0
(2.5)
holds for every solution u of system (2.1) if the initial data {u°, u1 } satisfy R.
IIUIIH1(fl) + IIU IILZ(n)
(2.6)
When d = 1 or 2. or when d 3 and the nonlinearity 1(u) is subcritical. p < d/(d—2) in (2.3). we can improve this result by weakening the geometric condition imposed on the open set w. For that, we the couple (w, T) verifies the following a properties: (GC): Geometric control—i.e.. every generalized bicharacteristic ray of and length > T meets the open set (UC): Unique continuation—the unique solution of the system i.e., 1
10w+b(t,x)w=0 on ]0.T[xcz foranytE]0,T[ Lb E
and w
H'(]O.T[xIl)
is the null solution. We can then state the following theorem.
Theorem 2.2. Under the hypotheses of Theorem 2.1, we also assume that d = 1 or 2, or if d 3 then the condition (2.3) is satisfied with 1 p< d/(d — 2) (subcritical case). Moreover, we assume that for T > 0 large enough, the couple (w, T) satisfies (CC) and (UC). Then the statements of Theorem 2.1 still hold true.
64
3
B. Dehman
Comments and remarks 3.1 As announced in the introduction. Theorems 2.1 and 2.2 state a result of local stabilization for the energy.
3.2 The geometric control condition (GC) of Theorem 2.2 is automatically fulfilled when w is a neighbourhood of the boundary (Theorem 2.1). It is almost sufficient and necessary for the control and stabilization of the linear wave equation (Bardos. Lebeau. Rauch [1] and Burq [2]). Recently in [3). by slightly modifying the definition, Burq and Gerard showed it to be a reaV sufficent and necessary condition for boundary control. This justifies its use in Theorems 2.1 and 2.2. 3.3 As for the unique continuation condition (UC). it. needs further investigation. In the analytic framework, it is satisfied by any open set w and any time T > 0, as a consequence of Holmgren's theorem. It is also satisfied if b(t, x) = b(x) is of class for any open set w and T > 0 large enough, due to Robbiano's theorem 18]. Finally for & E it holds in particular for w a neighbourhood of the boundary OC and T large enough, which is the case of Theorem 2.1 (see Ruiz [9) and Tataru [11]). Now, in the general case. the necessity of this hypothesis seems to be an open problem. Let us finally note that we did not attempt. in this work, to use (UC) in its optimal form.
3.4 The proof of Theorem 2.1 rests essentially on microlocal analysis arguments. Besides the inequalities coming from linear" geometric control, we use in a critical way the properties of microlocal defect measures associated to the sequences of energy bounded solutions of (2.1). The proof of Theorem 2.2 is simpler and uses the compactness of the injection H1(cl) L2P(1Z).
4
Proof of Theorems 2.1 and 2.2
It is well known that it suffies to prove the estimate
E(u)(T)
C/
Jo
j a(x)182u(t,x)I2dtdx
for some time T> 0, and for every solution u of (2.1). satisfying (2.6). Here we will take a time T satisfying simultaneously the assumptions (CC) and (UC), which are fulfilled in the case of Theorem 2.1.
Stabilization for the Semilinear Wave Equation
In a first step we write the solution u of(2.1) as u = satisfy respectively
(00=0
and
where
foranyt>0
(4.2)
=
= u°(x) and
I.
65
and
=
—a(x)Otu
on ]O,+oo[xIl
—
(4.3) tb(O.x)
=
0.
Let us now remark that, due to (2.2) and (2.3), we have 1(0) = 0 and (44)
+ IsI")
which implies that IF(u)I
C
using the injection initial energy), we deduce that
+ IttI2P).
L2P(Q) and hypothesis (2.6) (bounded
So
E(u)(0)
C
+
C
(4.5)
+
Following then Zuazua [12]. we obtain by applying the geometric control inequalities (CC) to the linear system (4.2) for any t 0,
E(u)(t)
(JTJ)
E(u)(0)
C (11u011
+ flu' 11L2)
That is
E(u)(t)
CjJ [a(x)lOtu(t,x)12 + 1u12] dtdx + (4.6)
On the other hand, the standard hyperbolic estimate applied to (4.3) gives
cj CJoI
j Ia(x)Otu + f(u)12 dt dx + 1u12 + ttzI2"]dtdx.
B. Dehman
66
Combining this inequality with (4.6), we obtain
E(u)(t)
CI
I [a(x)iOtu(t,x)12 + 1u12 + 1u12"]dtdx Vt
Jo
0.
(4.7)
Jcz
Our goal is now to eliminate successively in this estimate the terms
and ui2. For that, we argue by contradiction and we consider a sequence (un) of solutions of (2.1), satisfying
j
0
Let
+ iunl2ldtdx
J =
the sequence ( )
+
=
1
1
n? 1.
n satisfies then
= 0 on JO, +oo[xQ
+ an = 0 for any t > 0
(4.8)
E(vn)(0)=1.
(4 9
(4.10)
Furthermore, taking in account (2.6) and (4.4), we have
+ lvni") and so
+ if(anvn)I This implies that the sequence (va) verifies an estimation analogous to (4.7), and of course the inequality T
JJ
+
!,
fl
1.
(4.12)
0
so it has a subsequence (still denoted by (va)) weakly convergent in this space. But —. 0 in L2(]0,T[xQ) due to 0 in H1 (JO, T[xfl). (4.12); then Eventually after extracting a new subsequence, we deduce that (va) is bounded in
0
in H'(cl),Vt EJ0,T[.
(4.13)
—i 0 in L2(J0,T[xIl), so the sequence Indeed, = f0 belongs to L1([0, T]) and goes to 0 in this space. It has then a subsequence which we will denote gn, satisfying —' 0 for almost every t E [0, T]. Thus —p 0 in L2(fl) for almost every t [0,71. On theother hand, for each integern, C°([0,TJ, H01 )riC'([0,TJ,L2), and, as well as for the sequence
a simple integration by parts shows
Stabilization for the Semilinear Wave Equation
67
that it is of bounded energy, independent of n. In particular, 3 C > 0 such C for any t [0,1'). So, let to, fixed in [0,1'] and that IIôtvrz(t,.)I1L2(n) 0 (ik) a sequence of]0, T[ converging toto such that lim IIvn(tk, for any k. We have — vn(tk..)flL2(n)
— tkl.
SUJ)
ItO —
tE (0.7']
And the inequality .)IIL2(fl) — tkl + IIvn(tk, .)I1L2(n) allows us to conclude immediately that urn .)I1L2(n) = 0. Let us remark that the central argument we have used is the equicontinuity of the sequence in which is a consequence of (2.6). On the other hand. let E (Q); we have
I
=
—
I
0
for any t
[0. T} due to the previous argument. The proof of t [0.TJ is thus complete. Now, if d = 1 or 2 or if d 3 and the nonlinearity f(u) satisfies p < d/(d — 2) (subcritical case, cf. (2.3)), we obtain by using the compactness of the injection '—p
—. 0 in L2'(Q) for any t
[0,7').
(4.14)
This leads by Lebesgue's theorem to I.
0
JJ 0
n
(4.15)
00.
Q
Then, combining (4.12) and (4.7), we deduce that contradicts (4.10). The estimation
E(u)(t) C I
rT r
I (a(x)IOtu(t,x)12 + 1u12]dtdr Vt Jo Jcz
—
>0
0:
which
(4.16)
is thus proved.
When d 3 and p = d/(d—2), the compactness argument used in (4.14) is false. We extend then (va) to the whole space by i)
f
if x E (t ) = 1,0 otherwise
for any t 0. And we extend the function a(z), by continuity through the boundary Ofl, by a continuous function a(x), compactly supported in such that a(x) ao/2 for any x belonging to a small neighborhood of the boundary OfZ in Rd.
B. Dehman
68
The sequence
satisfies inequality (4.12) and the equation
=
+
—
(4.17)
®
where i9/thi is the normal derivative on the boundary and cial measure. (i,,) is clearly bounded in H'(IO, And
is its superfi-
is bounded in L2(1O,T[xRd) due to (4.11). On the other hand, it is well known that is bounded in L2(1O, T[ x ofZ). Then Ov,,/th# ® is bounded in H_h/2_((1O,T[zRd) for anye > 0. Thus the right-hand member of (4.17) is compactly supported, and compact in After the extraction of a subsequence, and taking in account (4.12), we can sup—s 0 in pose that Now we will make use of the notion of microlocal defect measures. We recall the definition.
Let U be an open set of Rk and (un) a bounded sequence of 0. We denote by
the cosphere bundle of U, i.e., the set
E
=1}. Then we have:
Theorem 4.1. ((41, Theorem 1) There exists a subsequence
and a positive Radon measure p on SU, such that, for any pseudodifferential operator A, defined on U, polyhomogeneous of order 0, properly supported, we have: a(x,
= is the principal symbol of A. p is called a microlocal defect
where a(z, measure of the sequence (un).
Remark 4.2. If the support of p is empty, we see easily, by taking as the pseudodifferential A any truncature function iJ'(x) that 0 in
We also recall the following theorem of microlocal elliptic regularity for these measures. We stay in the previous framework and we consider a differential operator with coefficients on U, P(x, = with principal symbol Then we have: =
Theorem 4.3. ([41. Proposition 2.1 and Corollary 2.2) Let (un) be the previously defined. We aLso assume that (P(x, is compact in (U). If p is a microlocal defect measure associated to (un), then /4 satisfies the algebric relation sequence of
=0 i.e.. supp j4 C
= 0}.
Stabilization for the Semilinear Wave Equation
69
—a 0 in T[ We come back now to our problem. We know that x Rd). Let ji be a ruicrolocal defect measure (m.d.m) associated to (i',,) in in L2(]0, T[XR'). H' (]0. T[ x Rd), i.e., p is an m.d.m associated to Deriving equation (4.17), one verifies easily that
is compact in H2(JO. The theorem of microlocal elliptic regularity for the m.d.m. then implies: the characteristic set of the wave operator. On the C {r2 = other hand, (4.12) gives, in particular:
—p0 in L2(]0,T(xRd),
that is —i 0 in L2(jO, T[xw), or
0 in H'(JO.T[xw). We obtain then
E Sd), which gives 0 (recall that Rigorously speaking, this convergence holds for some but we will continue to denote it by subsequence of in W', contained in So, let w1 be a small compact neighborhood of x w,) for any E > 0. From this, we We have i',, —. 0 in H'([e,T — for any E > 0. Furthermore. deduce that — 0 in H' ([E. T — x n we can write
So p is zero on in
p
j
0
p
j
IVxvnl2dxdt=J 0
p
JWlnfl -J
2p. is bounded. In particuliar. the norm On the other hand, we can write, thanks to the Holder inequality:
jf
dt dx
(critical Sobolev exponent in Rd+1) and a. > 0 where q' = such that aq + i3q' = 2p. Taking into account that 0 in H'(JO.T[xfl). we obtain. modulo a subsequence that. 0. which ensures that
I JRd I Jo
dt dx
—i
0.
Recapitulating the previous arguments. we obtain ,-
/ / Jo Combining this result with (4.12) and (4.7). we establish again the estima-
tion (4.16). End of the proof.
In this section. we eliminate the term 1u12 in estimation (4.16). This proof was developed by Zuazua in [12). but we recall it briefly, to make this article complete. We argue again by contradiction. and consider a sequence (un) of solutions 01(2.1) such that: 10T
n
12 dt dx
—
The first condition implies that h is globally Lipschitz, the second says that W' is slowly varying and the third one implies that W does not stay long near half-integers. The function k, on the other hand, is a perturbation which is small with respect to h, namely + lxiiVki + ixi2iV2kl 0),
in addition, we supC3, C4 are positive constants and C3 < pose that P(x, D) is a differential operator with complex Lipschitz continuous where C1,
coefficients and also that P(x,D) = Qi(x,D)Q2(r,D) where Qi(n,D) and Q2(z, D) are two second order differential elliptic operators such that Qj (0, D) = Q2(0, D) = —A. The proof of the theorem mentioned above uses the classical Carleman method.
1
1.1
Introduction Definitions
Let Q be a connected open subset of (n 2) containing 0. We recall that a function u E L2(1l) has a zero of infinite order at 0 if u satisfies the condition
Iui2dx=O(RN),
VN>0,
R—'O.
If u is smooth, then = 0 for all o N", and it is said that u is fiat at zero. Consider a differential inequality of the form IP(x, D)ui
0, satisfijing the differential inequality
U E
1P(x,D)ul
+c21 lxi
lxi
lxi
lxi (1.3)
and u has a zero of infinite order at 0, then u 1.4
0 in Il.
Remarks
The assumptions of the theorem can be improved appreciably by noticing that: 1. It is enough to show the theorem for the operator A(x, D) defined above, since under the assumption (1.3), the operator R(x, D) is absorbed 'A noticeable improvement of the differential inequality (1.3) was obtained from profitable discussions during the conference; I thank in particular R. ftegbaoui whose remarks really interested me.
P. Le Borgne
90
by a member of the right-hand side of the differential inequality; it would be the same for any operator for order 3 with bounded coefficients. 2. Coefficients of the operator Q2(x, D) can be chosen to be differentiable with Lipschitz derivatives (i.e., with bounded second order derivatives). Let us note that in this case, the operator P(x, D) does not have Lipschitz coefficients for the terms of order lower than 3. 3. The conditions on u can be reduced to:
•
UE
• u is locally integrable and verifies almost everywhere (1.3). The inequality (1.3) then makes sense and the theorems of elliptic regularity involve u E 4. By using the unique continuation result due to P.M. Goorjian 110] and R.N. Pederson [20], it is enough to prove that the solution u of (1.3), which has a zero of infinite order at 0, vanishes in a neighbourhood of 0. By using the fact that is connected, the unique continuation property allows us to affirm that u vanishes identically.
To conclude, our result is illustrated by the relative weakness of the differential inequality:
• it contains the third order derivatives of u;
• it gives a positive answer to the critical problem e = 0, V0(x) for lal
3.
The strong uniqueness problem for an operator P(x, D) such that P(0, D) = is possibly not factorized is still open.
2
Steps of the proof
Theorem 2.1. Let u E then u for all ,
= 0;
= &, where &
(iii)
the Laplace—Beltrami operator on the
is
sphere;
(iv) the adjoint 1,' of the operator
is
written as
is
denoted
= (n —
1)w,
—
(v)
Other notation will be used hereafter:
• for 1 j n, the vector field
the vector field +8
is denoted D,;
• for any a E N's, = •. . •
:= where
a = (a1,..-
• for k = 1,-.. , n, Ak =
in L2(R x sphere
:= belongs to
; we denote and A0 = where a = (ao,". E
product of the form
• the norm
and
• -
every
indicates the norm in the space L2(R", 1x1"dx), i.e., dtdw) we denote the measure of surface on the (this norm was denoted until now .
• the scalar product (-,•)
is
the scalar product in the space L2(R x
The property (v) of the vector fields H4(R x
allows us to obtain for w E
Re(&w, w) = —
(3.3)
and also
&w) = — The inequality (2.1) is a consequence of the theorem which follows.
(3.4)
Fourth Order Differential Operators
95
Carleman's inequality for &
3.2
Theorem 3.1. (Carleman's inequality for &) For any a constant C > 0 such that, for all function v E T E {n + n E JW} sufficiently large we have
f
dx
Cr4J + Cr2
+
> 0, there exists
\ {0}), for all
dx dx
J
f
—
+Cr2
dx
>
f
dx. (3.5)
The proof of this theorem is rather technical; we give a sketch of it. We refer the reader who wishes to obtain more details to P. Le Borgne [17]. For any function v \0), we define w E x S"') by setting = e_Tt&e1t; then we have w= By definition
f
ff
dx
= The computation of the right-hand side of the above equality by using integration by parts can then be carried out starting from the development in polar coordinates of the operator Note that to show Theorem 2.1, one can apply Theorem 3.1 to a function yR of support included in a ball of center 0 and radius R with R since R can be close to 0, r must be able to be selected arbitrarily large. In the polar coordinates (t, w) previously introduced, we have
=
+ (n —
+
(3.6)
Since w is related to class C°° with compact support, from (3.6) we can by using integration by estimate the square in L2 (R x ')of parts. This is easy: on the one hand the differential operator obtained has constant coefficients; on the other hand the properties (3.3) and (3.4) of the vector fields 1Z, make integration by parts very simple. dt we In the calculations using integration by parts of ff isolate on the one hand the terms where the following appears:
ff H2 dt dw,
ff
dt
P. Le Borgne
96
Eff for k =
1,
2, and on the other hand terms where we have
fJIotwI2dtdw, >1J1og11;wl2dtdw,
Jf
dt
Jf
dt
These two sums are respectively noted I(r, w) and J(r, in). The other terms of the development are determined; they have positive coefficients corresponding to the greatest powers of r. The terms with the lower powers of i- will constitute a remainder. By underlining the positive terms obtained, we can thus write
Jf
dt dw
I(r, w) + J(r, w) +
CkT8_2k 2
mension quelconque pour des inégalités différentielles elliptiques, Duke Math. Journal 48 (1981), 49—68. [41
N. Aronszajn, A unique continuation theorem for solutions of elliptic
partial differential equations or inequalities of second order, J. Math. Pures Appi. 36 (1957), 235—249.
Aronszajn, A. Krzywicki, and J. Szarski, A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. for Mat. 4 (1962), 417—453. N.
[6] T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux dérivées partielles a deux variables indépendantes, Ark. for Mat. 26 B. 17 (1938). 1—9. F. Colombini and C. Grammatico, A counterexample to strong unique-
ness for all powers of the Laplace operator, Comm. Partial Differential equations 25(2000), 585—600.
[8] F. Colombini and C. Grammatico, Some remarks on strong unique continuation for the Laplace operator and its powers. Comm. Partial Differential Equations 24(5—6) (1999) 1079—1094.
[9) H.O. Cordes, Uber die bestimmheit des losungen elliptischer differentialgleichungen durch anfangsvorgaben, Nachr. A had. Wiss. Gottingen Math. Phys. KI. ha 11 (1956), 239—258.
[10] P.M. Goorjian, The uniqueness of the Cauchy problem for partial differential equations which may have multiple characteritics, Trans. Amer. Math. Soc. 146 (1969), 493—509.
[11] C. Graminatico, A result on strong unique continuation for the Laplace operator, Comm. Partial Differential Equations 22 (1997), 1475—1491.
108
P. Le Borgne
[12] L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, 1963.
[13] L. Hörmander, On the uniqueness of the Cauchy problem I-Il. Math. Scand. 6 (1958), 213—225; 7 (1959), 177—190.
(14] L. Hörmander, The Analysis of Linear Partial Differential Operators, 3, Springer-Verlag, 1985.
[15] L. Hörmander, The Analysis of Linear Partial Differential Operators, 4, Springer-Verlag, 1985. [16] L. Hörmander, Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations 8 (1983), 2 1—64.
[17] P. Le Borgne, Unicité forte pour le produit de deux opérateurs elliptiques d'ordre 2, prépublication, Département de Mathématiques, Université de Reims, (1998), to appear in Indiana Univ. Math. J. [18] N. Lerner, Résultats d'unicité forte pour des opérateurs elliptiques a coefficients Gevrey, Comm. Partial Differential Equations 6 (1981), 1163-1177.
[191 R.N. Pederson, On the unique continuation theorem for certain second and fourth order elliptic equations, Comm. Pure Appi. Math. 11 (1958), 67—80.
[20] R.N. Pederson, Uniqueness in Cauchy's problem for equations with double characteristics. Ark. Math. 6 (1967), 535—549.
[21] A. Pus, A smooth linear elliptic differential equation without any solution in a sphere, Comm. Pure and Appi. Math. 14 (1961), 599—617. [221 M.H. Protter, Unique continuation for effiptic equations, Trans. Amer. Math. Soc. 95 (1961), 81—91.
[23] R. Regbaoui, Strong unique continuation for second order elliptic differential operators, Journal of Differential Equations 141(2) (1997), 201—217.
(24] T. Shirota, A remark on the unique continuation theorem for certain fourth order elliptic equations, Proc. Jap. Acad. 36 (1960), 571—573.
[25] C. Zuily, Uniqueness and non uniqueness in the Cauchy Problem, Progress in Mathematics 33, Birkäuser, 1983. Université de Reims Département de Mathématiques Moulin de la Housse B.P. 1039. 51687 Reims Cédex 2. France
Second Microlocalization Methods for Degenerate Cauchy—Riemann Equations Nicolas Lerner ABSTRACT We study a class of degenerate Cauchy—Riemann equations and we show that the second microlocalization with respect to a hypersurface is a useful tool to formulate and prove propagation and solvability resuLts.
1
Introduction
We want to study the following class of operators: + iAo(t, x, DX)DZ, + Ro(t, x, The symbols where (t,z) E R x R", D1 = and Ro(t,x,e) belong to the standard class (e.g., are smooth functions homogeneous of degree 0 with respect to the c-variable for 1). The function A0 is
assumed to be nonnegative so that the operator (1.1) satisfies the so-called condition (P). For a pseudo-differential operator with a complex-valued symbol Pi +ip2, where' AL
0,
condition (P) means that p2 does not change sign along the bicharacteristic curves of Pi. Condition (P) is equivalent to local solvability for differential operators of principal type. In fact it was proved by Reals and Fefferman in [1J that (1.1) is actually a microlocal model for locally solvable differential operators.2 Later on, Hörmander proved semi-global existence theorems for operators satisfying condition (P) (see Chapter 26 in [6]) and a propagation-of-singularity result for these operators. A consequence of the latter is the existence of a smooth (local) solution u to the equation
Pu =
/
1L is the Liouville vector field, given in coordinates by E 2However the microlocalization required to get to the model (1.1) is not homogeneous and much finer than a conic localization.
N. Lerner
110
for a smooth right-hand-side I whenever P satisfies condition (P). Moreover, for an operator of order m satisfying condition (P), f in the Sobolev It is not known space H8 and e > 0, there exists a solution u E if one can make e = 0 in the above statement for a semi-global solution, provided that P satisfies a nontrapping condition. Our goal in these notes is to set the stage to proving a two-microlocal optimal propagation result for operators satisfying condition (P). Since after a nonhomogeneous canonical transformation, all the difficulties concentrate on operators of type (1.1), we shall limit our discussion here to these operators. It should be emphasized that the Cauchy problem with respect to {t = T} is not well-posed in the sense of Hadamard for (1.1) unless A0 is identically 0. As a matter of fact, the Cauchy problem is not even well posed for in R2. However we want to study (1.1) the standard CR equation + as an evolution equation and it is of course helpful to look at the ODE
+ A0
=
0, the region
i.e., with, X =
—
0} is a forward3 domain for propagation,
such that
sup
0, IC'I(s,X)Ids.
I
One gets as well a backward region with X = (x,
such that
0,
)C4(s,X)[ds.
sup IT...
These with
inequalities
suggest that the following PDE problem is
well posed,
T_
= = Fju(T_,
I x')
F1 stands for the Fourier transform with respect to x1. In fact, the = 0} will allow us to sepasecond microlocalization with respect to rate the forward propagation region from the backward region. Roughly speaking, we shall be able to the characteristic function of the set where
O}. In the analytic framework, Sjöstrand introduced second microlocalization methods in [14]. Analogous methods were used for proving propagation-ofsingularities of Sobolev type by Lebeau [11], Bony [2] and Delort 151. Some 3Note that inequality (1.3) proves that if $ is "singular" (e.g., infinite) at time t, then it is also singular at later times. On the other hand, "regularity" is propagating in the other direction since (1.3) proves as well that regularity at time T implies regularity
for t 0, we need only to
conjugate the operator d/dt — Q(t). In fact, we have
=
—
so that with v(t) = Iv(t)I
and
Re(Q(t)—p) s
o,
the already proven inequality
Iv(T_)I
+
f
—
(Q(s)
—
p)v(s)t ds
implies e_hA(t_T41u(t)j < Iu(T41 +
J
—
Q(s)u(s)I ds.
0
In fact this lemma can be improved by taking into account the nonnegativity of the operator p — R.eQ(t). 6We can suppose that 0(t) > 0 on 1, replacing a by a + c with £ positive.
Degenerate Cauchy—Riemann Equations
Lemma 2.1'. Under the assumptions of Lemma 2.1, we have, for t E 1/2
+f +
—
f
- Q(s)u(s)Ids.
Proof. As done previously, we need only to prove (2.5) when
(2.5)
=
0.
In this
case (2.3) gives Iu(t)12
—
Iu(T_)!2
+
f
2Re(—.Q(s)u(s),u(s))ds
= j 2Re(f(s),u(s))ds,
which implies pt
pi
+ /
Jr
2Re(—Q(s)u(s).u(s))ds
1, we have
(ird + Proof. With ü standing for the Fourier transform with respect to x1, we have
x') =
f
x1,x')dx1,
and thus Iü(t,
x')12 dJ
x1, x')J2dxi.
Consequently, for N 1,
J =
(1 +
J +J
f +
(1
f
:
+
N. Lerner
122
which proves the lemma. Then we have proved the following:
Theorem 4.2. Let n be a positive integer. There exists an integer ii
(de-
only on the dimension n), such that the following property is satisfied. Let I = [T_,T+] be a compact interval, let L be given by (4.1). There exists a positive constant C0 depending only on ii semi-norms of ao, ro in (4.1), such that if is a positive number and u(t,x) E S(R x RTh) with supp u c { lxii d) such that if pending
1.' C0,
p1/2
Coe"iIi < A,
then
rf sup iu(t)I
tEl
+ p1"2
Li
iu(t)l2dtj
(IYwu(T+)l + where Y,
5
rr
11/2
+ Li +
j
(4.12)
are defined in (3.2) and 1Q10 is given in (4.10).
A two-microlocal propagation result
The previous theorem is enough to prove a semi-global solvability result for the transpose of L, and since L is also of type (4.1), we get a semi-global solvability result for this type of operators. However, the estimates (4.11) and (4.12) are much more precise and indicate that the following evolution problem is well posed:
fLu=f, E_u(T_) = v....
(resp.E_) is the projection on the subspace of L2(R") such that, using the Fourier transform with respect to x1, supp 111 c {ei 0} (reap. 0)). We shall not pursue more formally this direction but instead we will concentrate our attention on a propagation result which is suggested by Theorem 4.2. Let us first define the natural two-microlocal spaces associated to our second microlocalization with respect to the hypersurface {Ej = 0). Let s, s' be real numbers ; we define the Hubert spaces Here
H8.9'(Rv*)
= {u
(D)8(Di)8'u
E L2(Rhl)},
where
(D) = (1 + 1D12)'12,
(D1) = (1 + D12)'/2.
Degenerate Cauchy—Riemann Equations
123
Let m, m' be real numbers. We introduce the class of symbols (a semiclassical version is used in Section 3 and here we slightly abuse notation by using the same letters for the metrics and a different definition for
= {a
x
E
(5.2)
Using Hörmander's notation with metrics, this means
=
dsI2 +
= g).
+
The standard class
S((e)m,
+
(5.3)
= C).
C g
Lemma 5.1. The metrics G,g satisfy the and
=
(5.4)
Moreover the metric g is slowly varying and temperate and weight. For a1 E
= aIa2+
a2 E
with with
=
is
a g-
we have
r E Sml+m2,m'i+m'2_2. (55)
If a E 50,1 is nonnegative, then is bounded from Hm+v,m'+v' to Hm,m'.
a
The proof of the first statement follows the proof of Lemma 3.1 so closely
that we leave it to the reader. The other statements are standard consequences of the properties of g (see Chapter 18 in 16]).
and the Poisson Remark 5.2. Note that a1a2 belongs to bracket {ai,a2} is in The"gain" in the asymptotic expansion occurs only with respect to the smallest of the large parameters < (c)). If ,c is a bounded function of one variable such that K' is compactly supported, then ic(s) E Moreover, if a E 5tm the function The assumption on the nonnegative E 5m,1 since sm C a can be relaxed to a E this is a version of the Fefferman—Phong inequality that we shall not need here. On the other hand, assuming a nonnegative and in S'° (or even St,O with e > 0) does not suffice to get semi-boundedness.
Now we want to define the two-microlocal regularity of a distribution u in V'(Q) where is an open set of Let us quickly review the standard x ]Rlz\{0} and m be one-microlocal regularity. Let will denote the canonical a real number. The mapping 7r1: —. projection. We say that u is in if there exists a conic neighborhood
124
V0 of
N. Lerner
such that for all
E Cr(irj(Vo)), all symbols a E The Htm wave-front set
suppa C V0, aWXU
=
with is defined
E
It is a closed conic subset of
If
?
\jOJ
—
(
.
.
with 0, to say that u E will simply mean that u = 0, we shall say that However if U
(resp.
E
such that for all x E
if there exists a conic neighborhood V0 of (XO, C Vofl
(resp. suppa c V0fl
> 0},
o=*
such that
j + r(1 —
0.
> 0 and
(1.8)
We now define O'O E Q+ by Co
=
max
+(ic,a) —j— lal+r(1
—'co)
r—
which is positive by assumption. Note that F(j, a, (3; 0) + r(1
1 and al
—
'co)
0 if
r by (1.4). Hence 'co + 1. By definition it is obvious that F(j, a, j3; 0) + r(1 — 'co) ao(r — al) if 0 and hence
F(j,a,fl;ao) 0
(2.2)
vanishes in Cl (Theorem 2 in [7)). Unfortunately, their assumption (2.2) that the solution of exponential order at the origin must vanish is too restrictive, at least, in a certain case. Indeed, we can show that if all the eigenvalues of
N(O,0) are equal to a nonreal complex number or its complex conjugate, then the function U E C' that satisfies such systems and vanishes of infinite order at the origin is identically zero. In addition, we can treat a more general class of differential inequalities. We emphasize that there are no regularity assumptions on the eigenvalues of N in our work as well as in
In this section B'"(Cl) denotes the class of functions f defined on Cl satisfying that I is Holder continuous of order X' E
11(X) — f(X')I 1, with a boundary of class C°°. Let T>0 and e E x ]0,T[;R). We say that the function e Let
controls exactly for the wave equation with partially null initial data if for all (cl), there is a boundary control g H'(R,; L2 such E that the solution of problem
I
O(,0)0 inl
Observability of the Schrödinger Equation
satisfies
0
167
in fl x [T, -i-oo[.
We say that the function 9 controls geometrically if any generalized bicharacteristic ray meets the set e 0 on a non—diffractive point. (see [4D•
We propose to establish the following exact control result:
Theorem 2.1. If the function 9 : (x, t)
E (x) 0(t) contn,ls Il exactly for the wave equation with partially null initial data, then for all e > 0,
there exists a control for all initial data w0 E H0' such that the solution of problem
L2
x J0,e[)
IiOtw+IXw=0 inlx]0,e[ I.
x ]0,e[ inIZ
on w= w(.,O)=w0
(2.2)
satisfies
Corollary 2.2. We suppose there is no infinite order of contact between x ]0, T[ and the bicharacteristics of 8? — & If the function the boundary 9: (x, t) E(x) 0(t) controls geometrically, then for all e > 0, for all L2(8f1 x ]0,e[) such initial data w0 H0' (Il), there exists a control that the solution of problem
inflx]0,e[ (2.3) I.
w(.,0)=w0 intl
satisfies w 0 in x {t e}. Furthermore, we have on estimate of the control as follows
(1 +
(2.4)
Remark 2.3. Corollary 2.2 comes from Theorem 2.1 and the work of C. Bardos, G. Lebeau and J. Rauch [1] or of N. Burq and P. Gerard [4] on the exact controllability of the wave equation from a microlocal analysis. The is given by an observability estimate in the one dimensional constant case. Our result is not optimal in norm in the sense that it is sufficient to have an exact control result for to choose initial data ti,0 E the Schrodinger equation, with hypothesis of the multiplier method [12] [13] [5]. Also, G. Lebeau [9] has proved the exact controLlability for the Schrodinger equation with the geometrical control condition of the wave equation [1] and an analytic boundary. Furthermore, there exist open sets which do not satisfy the geometrical control condition and in which it is possible to control exactly with regular initial data [3]. Here, our goal is to use knowledge of the exact controllability for the wave equation to obtain an exact control result for the Schrodinger equation.
H'
K.-D. Phung
168
Proof of the unstable observability results for the Schrödinger equation
3
The parabolic problem
3.1
The proof of Theorem 1.1 comes from the work of Lebeau and Robbiano [11] or of Fursikov and Imanuviov [7] on the exact controllability for the heat equation from Carleman inequalities. We recall the result in (11] to be complete: be a Riemanian compact manifold with a boundary Let
0 (resp.
when
:
C0°°(wx]a,b[) with eventually
:
of class
For all 0 < a < b < T, there exists
and let be the laplacian on a continuous operator Sr L2(ffl
=
0)
such that for all
Vo E L2(Q), the solution of the heat equation
inIlx]0,T[
(resp.
v=Sr(vo) (resp. =0) v(,0)=vo satisfies v(.,T) 0. We have the following estimates: Lemma 3.1. Let w be the solution of the following evolution problem:
w=0 Then,
>0
f Iw(-?
0)12
dx S CT
(j
jT2) +f
1T
(3.2)
>0
f lw(,
0)12
dx CT
(j
1T 1w12 dtdx
+
j j 1fI2
dtdx). (3.3)
Furthermore if w(.,T) E H2flH01(1Z), then >0
f Iw(,0)12
dx
0, such that for all T> 0, if w(•, T)
exp (c (i +
f
1))
j
L2(fz), we have
fT wI2 dtdx
(3.6)
and, there is C> 0, such that for all T> 0, if w(.,T) E H2 n
we
have
j
exp
dx
(c (1 +
dtdx).
(1 JT
(3.7)
Thus the constant Ce of estimate (1.6) could be written explicitly in e.
Proof of Theorem 1.1 Let F(z) = fReedr; then IF(z)I = 3.2
Also, let
A > 0, and FA(z) = AF(Az) = We have
=
(3.8)
2ir
Let s,io
Rand W10,A(s,x)
=
f
+ is —
We remark that
where and thus
= =
f
+ is — £)
—ia,FA(eo + is — + is — £)
=
+ is —
+ 4(e)
de.
As u is the solution of (1.1), Wt0,,, satisfies
x) + I.
£)dt
x) = fR iFA(€O + is — W,0,A(s,x) = 0 Vx
W1(,..\(0, x) = (FA
*
4u(x,.)) (e0)
Vx
(3.9) ci.
K.-D.Phung
170
We define
E
ion
4'
'
and
[—
such
=
and dist(K;K0) =
Let L >0. We choose 'I' E Cr(]0,L[), 0 'i 1, < We take K = that mesK = =K -rJ• So, mesK0 =
We will choose to
J I(FA
*
(10)12 dx
f I
+
K0.
satisfies the following estimate:
By application of (3.2),
CT
T'
[ JrJo
1
z)12 dsdx 2
I I iFA(1o + is — t)4"(t)u(x,i)dtl dsdx
(3.10)
I
I I IOnWe0,A(s,x)l2dsdx irfo =
j
1T
dsdx
j FA(t0 + is —
Jo JrIJR 211 2
T
II
< — —4
2
dx 1L
< —
3ds)
Isup'112 L
4ir
f JrJo 2
+is Jo
all u
with compact sUpport,
E
+ where
(2.5)
C9 is a positive constant depending only on n and 0.
Proof of Theorem 2.1. Let u be as in Theorem 2.1. We may suppose that xO = 0, and using a weak unique continuation theorem of Wolff [11], it suffices to prove that Vu 0 in a neighborhood of 0. Let R0 > 0 be sufficiently large such that B(O, Ri') C For lxi > Ro define ü by ii(x) = . Then it suffices to prove that 0 for large But since the problem is rotation-invariant, it will suffice to prove that lxi. ii 0 in the cone: } (Ro large enough).
An easy computation shows that
J
(2.6)
0 as R —i 00.
(2.7)
Izi > R
The estimate (2.5) in Lemma 2.3 was stated for functions in with compact supports, but a standard limiting argument using (2.6) and (2.7) shows that it is also true for the function where E C00° such that = 0 if lxi 2R0 . Then, by choosing e > 0 such that 0 = n(n-1)'
+
Let M >
where
0
iiekZV(cbii)11L2(E)
(2.8)
sufficiently large to be chosen later, and let
is the unit vector (0,... , 1) Thus B is a convex body of where C is a positive constant depending only on n.
with 181 =
Sets of Positive Measure
By the Leibniz formula we have = by using (2.6), we get from (2.8), for all k E
183
Hence
+ + with pk E 8,
+ (MT1IEI) + 2(n —
V4'11fl1, +
+ 1Z)(2.9)
where
=
VÜIILP +
II
Thus if R0 is large enough, we have
1. Then it
follows from (2.9) that
C
+
+ 2(n —
i').
(2.10)
suppose that ü 0 in r, and we will prove that this leads to a contradiction. We claim that We
1?. IIekx(IWI + 2(n — Indeed
,we have IIekx(IWI + 2(n
2
—
f
but for x E r and pk
zer B we have pk x 6MR0. Hence
IIelcx(IWI + 2(n
—
JxEr(iW1 + 2(n — On the other hand, for pk E 8, we have R.
1 and M"IEk,I 1, we get (M"IEk, We
9=
+
recall that 8 is any real number such that 9> we have 8 + and e = = C
We
In particular, for obtain then
+
that is
C (iiw
Y) +
where C is a positive constant depending only on n.
Since are pairwise disjoint, by taking the sum over j and using (2.13), we get, with a new positive constant C depending only on n, fl+E
L'(IxI>Ro) +
X
-ltifl+t
> C —
which is a contradiction since + when R0 —. oo. This completes the proof of Theorem 2.1.
3
Proof of the main results
In this section we will prove Theorem 1.1 and Theorem 1.2. First we show that if u is as in Theorem 1.1 and x0 is an infinite order zero of u which lies in the Lebesgue set of IWI", then u has an exponential decay of the form (2.1) at ro. Thus we apply Theorem 2.1 to get Theorem 1.1.
Proposition 3.1. Let
p
=
n
3, and let u E
be a solution Suppose that for some xO E
and W E u satisfies (1.2), i.e., x0 is a zero of infinite order for u. Then
of (1.1) with V E
for all lol S 1.
f
Ix
= 0 as R
0.
(3.1)
186
R. Regbaoui
To prove Proposition 3.1 we need the following Carleman estimate which is a combination of two estimates: the first is the well-known Jerison—Kenig estimate (cf. [5]) and the second is proved in Wolff [101.
Lemma 3.2.
Let
p=
P'
k€N,
n
=
3. Then for all
llixI7VuIILa
+ where
=
=
(3.2)
C is a positive constant depending only on n.
Proof of Proposition 3.1. We may suppose x0 = 0. Let be as in Lemma 3.2 and set R = Let x E such that x(x) = 1 if lxi 2R . Thus x satisfies also
CR1"1.
(3.3)
The estimate (3.2) in Lemma 3.2 was stated for functions in C8°(R't \ {0}), but a standard limiting argument using (1.1) and (1.2), shows that it is also true for the function xu. Then LP'
Ii
+
Ii
lita
lxi
lxi
LP(IxkR) + CII
lxi
ii
IILP(IzI>R)
which by using (1.1) gives II
LP' (IzI 2. We introduce the convex conjugate f and we check that
f(s) —. p
exp
s
(is i")
,
as
Is
00.
(2.7)
Let us assume for the moment that there exists a function p E V(f?) such 0 in a neighborhood of w, p 0, pdx = 1 and
that p =
lie) E L'(cfl.
I
We shall return to condition (2.8) later on. Multiplying by p in (1.1) and integrating in
(2.8)
it follows, after integration
by parts, that
f pudx =
j
—
j pf(u)dx.
(2.9)
Note that in (2.9) the control v does not appear. This is due to the fact that 0 in w.
p
Applying Young and Jensen's inequalities and using the fact that f(s) = 1(1 s I) we obtain (2.10)
where
k= It is easy to see .that: if
(2.11)
j puodx is large enough, the solution of (2.10) blows up in finite time. Moreover, given any 0 < T < oo, by taking —
196
E. Zuazua
puodx large enough, one can guarantee that the solution of (2.6) blows — f0 up in time t 1 (in fact p> 2). It is then clear that the statement of Theorem 2.1 holds.
Note that, at least apparently, we have not used so far the fact that p> 2. But this condition is needed to ensure that (2.8) holds. Indeed, let us analyze (2.8) in the one-dimensional case. Of course, the only difficulty for (2.8) to be true is at the points where p vanishes. Assume for instance that p vanishes at x = 0. If p is flat enough, of the order of
p(x) = exp(—xm) we have
pf*
—
lip)
I)/i
exp(_x_m)
x
provided
which is bounded as x —.
m>(2m+2)/p. Of course, such a choice of m > o is always possible when p> 2, but not otherwise. This concludes the sketch of the proof of Theorem 2.1. We refer to [11] for more details.
Remark 2.3.
We did not check that (2.8) fails as soon as
logy IsI
for p 2. However, the existing results on the blow-up literature (see e.g.,
[14J and [151) show that when f is as above and 1
and f E C'(R). We fix the initial datum Uo and the control time T > We then introduce the function —
f f(s)/s, ifif s=0.0 S
0
0.
2 12
Linear and Semilinear Heat Equations
197
We rewrite system (1.1) as in
u=0
x (O,T)
on in
I,u(0)=uo
(2.13)
x LO,TD: z = 0 on 0S1x (0,T)} we introduce For any z E X = {z the linearized control problem:
S1x(0,T)
in on in
u=0
(u(O)=uo
0S1 x (0,T)
L°°(w x (0, T)) for system
As we shall see, there exists a control v (2.14) such that its solution u satisfies
u(T) = 0
(2.14)
S1.
(2.15)
in 51.
Moreover, the following bound on v holds: There exists C> 0 such that II V
Cexp (c (i+
II g(z)
ii u0
.
(2.16)
In this way we build a nonlinear map H: X —. X such that u = H(z) where u is the solution of (2.14) satisfying (2.15) with the control v verifying
the bound (2.16).
It is easy to see that the map H: X X is continuous and compact. On the other hand, we observe that u solves (2.13) when u is a fixed point of H. Thus, it is sufficient to prove that H has a fixed point. We apply Schauder's fixed point theorem. To do this we have to show
that
Vz X : II Z
N(z) ll,c, R,
R
(2.17)
for a suitable R. In view of (2.16), using classical energy estimates and the fact that, as a consequence of (2.4), lim sup IsI—.oc
I
I
=
0
(2.18)
s I)
deduce that (2.17) holds for R > 0 large enough. Therefore the problem is reduced to proving the existence of the control v for (2.14) satisfying (2.16). we
Step 2. Control of the linearized equation To analyze the controllability of the linearized equation (2.14) and in order to simplify the notation, we set a = g(z).
198
E. Zuazua
System (2.14) then takes the form in
u=
Ilx(0,T) x (0, T)
on in
0
(u(0)=uo
(2.20)
To analyze the controllability of (2.20) we consider the adjoint system
=
in on in
0
1lx(0,T) x (0,T)
(2.21)
The following observability inequality holds:
Lemma 2.4. There exists a constant C > 0 such that g exp
(c(T+
+
a
+a (2.22)
/
I
w
x (0, T)) and all T >
for every solution of (2.21) for all a E
0.
This observability inequality has been proved in [11J as a refinement
of those in [10] in which, on the right hand side of (2.22), we had
II
instead of The main ingredient of the proof of (2.22) is the Global Carleman Inequality in [13]. As a consequence of Lemma 2.4, by duality, the following holds: 'P
Lemma 2.5. Given any T >
L2()), 0, a E L°°(fZ x (0,T)) and there exists a control v E L°°(w x (0, T)) such that the solution u of (2.20) satisfies (2.15). Moreover, we have the following bound on v: V IIL"°(.,x(O.T))
+
where C >
0
1
a
II
exP(C (T +
+
T)
a (2.23)
uo 111,2(n)
is a constant that only depends on fi and w.
In a first approach, (2.23) does not imply (2.16). Indeed, in (2.23) the leading term in what concerns the growth rate of the observability constant 00 is of the order of exp (c as a + T) a 100). However, note that condition (2.15) is also satisfied if u verifies =
0
in
Linear and Semilinear Heat Equations
199
for some T < T and the control v is extended by zero to the interval [T, T]. Obviously, one can always choose T small enough so that 1100
x (0, T)) with C > 0 bounded above by C a for all a E independent of a. This is the key remark in the proof of (2.16) and Theorem 2.2. This is
strategy is in agreement with common sense: In order to avoid the blow-up phenomena, we control the system fast, before the blow-up mechanism is developed. This concludes the sketch of the proof of Theorem 2.2.
Remark 2.6. (a) Inequality (2.22) may be improved to obtain a global bound on provided we introduce a weight vanishing at t = T. Indeed, one can get inequalities of the form (2.22) with II hand side replaced by the weighted global quantity I
Jo
I1L2(n) on the left-
I
(2.24)
Jr1
We refer to Section 4 for a discussion on the best constant > 0 in (2.24). (b) Analyzing the proof of Theorem 2.2 one sees that the main obstacle to improving the growth condition (2.4) in Theorem 2.2 is the presence of the factor exp (c a in the observability inequality. Indeed, if we
had exp (C a II") instead of exp (c a with p> 3/2, then one would be able to extend the null-controllability result of Theorem 2.2 to nonlinearities satisfying the weakened growth condition
If(s)I
lim sup I
s I log'
I
s
=
0.
I
However, this seems to be out of reach with the L2-Global Carleman Inequalities in [13]. We shall return to this open problem in Section 4.
3
3.1
Lack of null-controllability for the heat equation on the half line Main result
In this section we discuss the following one-dimensional control problem:
I
0<x
