Carleman Estimates and Applications to Uniqueness and Control Theory (Progress in Nonlinear Differential Equations and Their Applications)

Progrecs in Nonlinear Differential Fqiiations and Their Applications Carleman Estimates and Applications to Uniqueness...

Author: Feruccio Colombini | Claude Zuily

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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

2001025975

CIP

AMS Subject Classifications: 35B60. 35315, 35Q30, 35310, 35R45, 34135. 35L05, 35L70

Printed on acid-flee paper.

02001 Birkhâuser Boston

Bzrkha user

All rights reserved. This work may not be translated or copied in whole orin part without the written permission of the publisher (Birkhäuscr Boston. do Springer-Verlag New York. Inc.. 175 Fifth Avenue, New York, NY 10010, USA). except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 0-8176-4230-7 ISBN 3-7643-4230.7

SPIN 10832661

Reformatted from editors' files by Inc., Cambridge, MA Printed and bound by Hamilton Printing Company. Rensselaer. NY Printed in the United States of America

987654321

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 <0}

we write

such that Uo(t)f

In what follows, we will write D+ and subspaces

with domain

=

t + R,t 0}

in IxI

instead

<—t

of D+,R

and D... of H0 have the following properties:

+ R,t

0).

and D...,R. The

Stabilization on Exterior Domains

3

a) D÷ and D_ are closed in H0.

=

c)

=

and D_ are orthogonal and

b) In odd dimension,

= {O}.

et fl

H0

teiR

H0.

tEIR

We now modify the system (L) by introducing an obstacle. Let 0 be a bounded open set of with a boundary 80 of class COC. And let = R"\0 be the exterior of 0. The system (L) becomes

withf=(fl,f2)EH=HDXL2

(E)

=

0

where H is the completion of

=

for the norm

j(t Vf (x)12 +

(2.3)

The equation (E) has a unique solution u E C(R, HD) fl C'(R. L2) and

the one parameter operator U(t), defined on H by U(t)f =

(8u%)

forms a unitary, strongly continuous group on H, generated by the operator

A=

fol\

with domain

D(A) =

{f E

E L2.f2 E HD}.

(2.4)

We can identify the space H with a subspace of H0 in the following way. For f E H, we write 1(x) = f(x) if x Q and 0 otherwise. It is then easy to see that f E Ho. And we identify I with f. Let R > 0 be such that 0 c BR = {x E < R}. To study the

perturbation caused by the obstacle 0, Lax and Phillips introduced the operator Z(t) = where P) is the projection on (resp Di). They also showed that the family forms a semigroup of contractions. This operator allows the study of the local energy behaviour for the perturbed solution. In the case of a nontrapping obstacle, Melrose [11) proves that Z(T) is compact for T = TR + 3R, where TR is the escape time. And he deduces exponential decay for local energy.

3

Statement of the result

We consider the damped wave equation

inR÷xfl (A)

u(0)

= fi,Otu(O) = 12 =0

with f = (11,12) E H

L. Aloui, M. Khenissi

4

where a(s) E and we define = {x E > 0}. We will use the notion of a generalized bicharacteristic ray. and we refer for example to [101 or [21, for a precise definition of these rays.

Definition 3.1. Exterior geometric control (E.G.C.) Let R and T be two real numbers> 0. We say that the couple (w. T) satisfies the E.G.C. over BR, if and only if every generalized bicharacteristic ray -y. x BR) is such that issued from *) y leaves lR1. x BR before the time T, or *)

x w between the times 0 and T.

intersects

We are now able to state our result.

Theorem 3.2. Let R >

0 and TR > 0 be such that satisfies the E.G.C. over BR. Then there exist c> 0 and ck > 0 such that

ER(u(t))

for any solution u of(A) with initial data

E

H supported in BR.

Remarks. i) If the obstacle is nontrapping, we obtain Melrose's result [11], by

taking a(x) = ii)

4

0.

If there is a trapped ray which does not intersect x w, the energy decay is not uniform, due to Ralston's theorem [14]. In this context. our result is thus optimal.

Local energy and the Lax—Phillips semigroup

In this section, we will prove that the solution of the damped wave equation is generated by a semigroup of contractions denoted by {U0(t): t 0}. We

will then introduce the semigroup of Lax—Phillips Z0(t) =

P and the orthogonal complements of the spaces of incoming and outgoing data. This semigroup role is to measure the effect of the obstacle 0 on the solution of the free wave equation. In our case, we will show that exponential decay of the local energy is equivalent to the one of (t) norm. Proposition 4.1. The operator Aa =

f0I\. —a)

.

maxzmal dzsszpatzve.

Proof. Note that D(Aa) = D(A0 = A) =

{f = (11,12)

liD x

L2,f2 E HD}.

Stabilization on Exterior Domains

Let f E D(A), and let us verify that Re(A0f,f) 0. Re(A0f, 1) = Re(Af, 1) + Re(Bf. f) where Bf =

= (Bf, f) =

(0. —af2)

j a(x)1f2(x)I2dx 0. —

Thus A0 is dissipative. It remains to show that Im(Id — A0) = H. Let

g E H. We look for f E D(A) satisfying

f—A0f=g. To do so. we write g = Since the set H/gj E L2} is dense in H. we can suppose that g, E H1(Q) (see [61); the equation (*) is then equivalent to

f

f1f2=9t

I —&fi

+12 =92

which gives (1) (2)

112=11—91 Let us define the bilinear form b(h, i,&) on H'(fz) by

b is

then continuous, and Ib(h. h)I = E L2, the linear form

i.e., b is coercive. Since (1 + a)gj +92

H' —.

C i&)

is continuous. According to the Lax—Milgraxn theorem, there exists a unique

H' such that = (g,lb) for all ui E H1. Then the equation (2) has a unique solution f' E H' in the sense of distributions. From the equation verified by f1 we deduce that E L2 and f2 E H0 fl L2, so h

E D(A) and satisfies the equation (*). 0 A0 is maximal dissipative; according to the Hille-Yoshida theorem, it generates a semigroup of contractions U0(t) such that if / E H, U0(t), then f is the unique solution of the following equation:

I=

(E)

(

OV_A

I.

with U0(t)f E C([0, +oc[,H). If we write U0(t)f ,

=

u,(t)E H

we obtain

L. Aloui, M. Khenissi

6

Then we deduce that the dissipative wave equation (A) has for H 12) a unique solution u E C([O, +oc[, HD) C'([O, +oc[, L2). where and P are respecFor t 0 we write = and The following proposition tively the orthogonal projectors on gives some properties of the operator Z0(t).

Proposition 4.2. We a)

= Za(t)D_ = {0}.

b) Za(t) operates

c)

have

is a

on K =

e

semi-group on K.

Proof. a) * Let f D_; from the definition of P, we have Z0(t)f = 0. * Let and D_ are orthogonal ([6]), hence Pf = I. Then, IE to deduce that Z0(t)f =0, it remains to verify that Ua(t)D+ C Let I E

and

be

=

u(t) the solution of

u(O) = fi, I.. U/nfl = 0.

c

Since f E thus u verifies

then

u(t,x) =

0

in lxi

the corresponding solution, with

Otu(O)

=

t + R,

12

t

0. But Supp(a) C BR,

u(0)=f1, Otu(0)=f2 I.

uion = 0.

Taking into account that the solution of the dissipative wave equation in exterior domains is unique (see [6]), we conclude that U0(t)f = U(t)f. Then we have U0(t)f E since U(t)f E ([61). fl and let us show that Z0(t)f E K. It is easy b) Let I E K = So, it remains to check that Z0(t)f D±. to see that Z0(t)f E

Z0(t)f = Let

=

g E D_; then we have

(Z0(t)f,g) = = (Ua(t)f, = (f,u:(t)9). To complete the proof of b) we give the following lemma.

Lemma 4.3. Let f E D...; then U(t)f = U(—t)f, for all t U(t) is the adjoint of the semigroup U0(t).

0, where

Stabilization on Exterior Domains Proof.

Since

7

is a semigroup generated by Aa, then U(t) is the

U0(t)

—A + B. We denote

semigroup generated by

f E D_: then we have V0(t)f

= U(t). Let

where

=

f OtVi=—V2 1 /

SO

\ where v(t) solves

v(t)

=

ViO 1

Similarly, U(—t)f

(v(O).Otv(O))

= (fl,—f2).

where w(t) is the solution of

=

VtO (w(O),Otw(O)) =

0 we write satisfy the equations For t

(Jion=O.

= w(—t); then i3(t) and tii(t)

= v(—t) and

= (fl,f2)

= (f1,f2)

= D_ we obtain U(—t)f = 0 for lxi t+R, t 0. and then —t + R, t 0. But Supp(a) C BR; then, due to the uniqueness for t 0, so w(t) = v(t) for = of the solution, we deduce that Since f E o in lxi

0

t 0. Consequently U(t)f = U(—t)f.

Finally, according to Lax—Phillips theory [6], we have U(—t)D.. C D...

D_ which proves that Za(t)f

for all t 0. This leads to U(t)g

c)Lets.tOandfEK. Wehave Za(t)Za(8)f = =

But on

=

(because

—

I) is the orthogonal projector

Then

Z0(t)Z0(s)f =

= Za(t+S)f which completes the proof of Proposition 4.2. We recall in the following proposition the relation between the semigroup Z0(t) and local energy (see [6]).

L. Aloui, M. Khenissi

8

Proposition 4.4. The following condition.s are equivalent:

a) VR>0, 3c,o >0, iIZa(t)fii

ce°tIIfIi, VIE H

b) Vp> 0, 3c',a' > 0,

with

= {g

Vg E

H, supported in B, D O} and + 192(x)12)dx.

= JxI
(4.2)

Proof.

H. So

in B,,, Vh

Z0(t)g =

= = U0(t)g in B,,, hence b)

= Let R> 0; since (Za(t))tO is a semigroup, it is sufficient to < 1. To do so, we will prove that there exists T > 0 such that a)

make use of the following lemma:

Lemma 4.5. We have U(t)D± C D±

a)

c

b)

c) If we set

D÷

for all t 0.

C

for all t 2R.

= U0(2R)

—

Uo(2R),

then M0f =

0

in lxi > 3R and

l1M01 Ii <2111 il5R

d)

=

—

Vt 4R.

Proof. a) Let! E and g D_; then we have (U0(t)f,g) = (f.U(t)g). Since U(t)g D_, (Ua(t)f,g) = 0 and therefore U0(t)f E D±. In the same way we prove the other inclusions. According to the theory b) It is sufficient to prove that Uo(2R)D± C correspond respectively of representation ([6]), the subspaces D... and Since the and L2([R. to the subspaces L2(] — oc, —R] x group Uo(t) operates as a right translation in L2, Uo(2R)D± is represented And that proves the point b). by c) Let f E H; due to the domain dependence principle, Uo(t)f = Ua(t)f in lxi > t + R, t 0; in particular for t = 2R, U0(2R)f = U0(2R)f in lxi > 3R. Using the same argument, we obtain that iiUo(2R)f113R S ill l15R and 11U0(2R)I113R UI 11SR, thus ii = iIMafii3R 2111 iisR.

Stabilization on Exterior Domains

9

d) We have —

=

4R)M0Pf +

4R)Uo(2R)Pf 2R)U0(2R)Pf — —

—

By b). Uo(2R)Pf

—

so the second and the third terms of the — 2R)Pf e Using then b). we

previous equality are zero. By a). obtain U0(2R)Ua(t — 2R)P—f also zero.

which shows that the fourth term is

o

We come back now to the proof of Proposition 4.4. We choose in assertion

b) p = 5R and T large enough such that for all g

:S

(4.3)

H3R.

Let f E H. Thanks to the previous lemma, we have: HZ0(T + 4R)f II =

If MaUa(T)M0P Ill S lIUc(T)M0PIIkR S

s Let T' = T + 4R: and t > 0. There exists k E N such that kT' t (k + 1)T'. And we deduce:

lI(Z(TI)f)IIk

11Z0(t)fII

<

5

0

Proof of the theorem

The main point of the proof is the following result.

Proposition 5.1. If the couple over BR and

satisfies the E.G.C assumption 1 and

is a sequence of K satisfying IIfnII =

still dc. for T = TR + 3R, then there exists a subsequence of and and data f of H such that: —i / ( t—T+R;t T}. where K(T) = {(t.x) E Proof. Let be a sequence of K satisfying the assumptions of the —+

1

noted by in

proposition. (fe) is bounded in H. so it has a subsequence weakly converging to some element f of H. We call the corresponding sequence

10

1,. Aloui, M. Khenissi

of solutions of system (A), and denote by Pa a microlocal defect measure associated to [3]. We will show that = 0 in TK(T). Let T(K(T)) and -y be a generalized bicharacteristic issued from q; we q have to consider two cases:

1st Case: followed in the time negative sense, does not meet or meets for the first time at to > 2R. In the two subcases -y(O) B(0, 2R). Since w C B(0, R), = near -y(O) and then = where is a microlocal defect measure P0 on a neighbourhood of associated to the sequence (Uo(t)ffl)flEN (here is identified with its first component). Recall now that EK, so, by Lemma 4.5, b), so po = 0 microlocally in TK(2R). Set E q

,_fq —

forthelstsubcase

y(ti)

for the 2nd snbcase

with t1 < to and -y(ti) TK(2R). Mo = 0 near q', so, by the measure propagation theorem [3], p0 = 0 near -y(O). And the same fact holds for Ma• Using then the measure propagation on the boundary of Lebeau [8], we deduce that = 0 near q.

2nd Case: meets the boundary 01? at t0 2R; in this case, y is a trapped ray, since E BR and T — to > TR. By the E.G.C. assumption, -v then intersects the stabilization region [t0, 7'] x w. On the other hand, —i 1 and = 1 imply that — 1. And since IIUa(T)fnII2 — 11h112 = we obtain that converges to 0 in L2([0, T] x w). Using the result of Gerard [3], we deduce that x 1?); r = 0}. But C Car(Ot) = E satisfies also a wave equation, so Supp C {r2 = which gives Pa = 0 on T ([0,1'] x w). Applying then the measure propagation theorem of Lebeau [8], we conclude that Pa = 0 near q. 0 Proposition 5.2. For T = TR + 9R, 112'a(T)II < 1.

Proof. We argue by contradiction and we suppose this inequality false. Then there exists a sequence of K such that = 1 and 1 when n —. +00. (fn)nEN is bounded in H. so it has a subsequence, still denoted by (f,). weakly converging to some element I in H. The operator Za(T) is continuous in H, when n —. +00. Let us prove that this convergence is strong.

= = Since f,,

= +

—

+

By Lemma 4.5, b), UO(2R)Ua(TR +

K, U0(TR + so

+

+

= 0, and we obtain

=

Stabilization on Exterior Domains

On the other hand.

IIZ0(TR+3R)fnl)

— 1. Using Proposition 5.1, we deduce that

+

1. so

11

—i (J0(t)f in ± 3R)). and this leads to U0(TR + 7R)f in H(111<SR). According to c) of Lemma 4.5. we have + M0U0(TR + 7R)f and thus + 7R)f. Since E K and in H, I E K

I

and •

Finally, we obtain a data I verifying (5.3)

11Z0(T)fll = 11111 = 1.

= 1 and

Indeed, we have

11111 = 11111: then 111112

f. 11111

hf 112 and 1) 1. Moreover

So 11111

1IZ0(T)fIl: so IhZa(T)fII = 1 11111

1 and I12'a(T)fnlI 11Z0(T)flI = 11111 = 1.

Now we will show the contradiction. Let

=

{f

1)1

—

1

E K/11Z4(T)f II =

Ill II}.

1st step: Cr is of finite dimension. Indeed, we will prove that its unity sphere is compact. Let E Sr; then llZa(T)fnhI = itfnII = 1. has a subsequence weakly converging to some element I of H. Since IIZa(T)fnIl = 1 and f, f. as previously, we can show that Z0(T)f, we Za(T)f and lIZa(T)f ii = Of II = 1. Since and IIf,11 = 1 = obtain the strong convergence of 2nd step: In this assertion we identify the initial operates on data with the corresponding solution and we give a characterization of the subspace CT.

I

Lemma 5.3. t

>0.

= {f E K/Ua(t)f E

and

= 0 in [0,t] x w}, for

*) Otu has finite energy. since flf II + hA! II and hf II are two equivalent norms on the finite dimensional space

a) If u is a solution of the dissipative wave equation. so is Otu. Since = 0 in [0.t] x w, = 0 in [0,t] x w. a)

For data I such that Ogu = 0 in [0, t] x u,, we have = U(t)f. Since 01(U(t)f) = AU(t)f = U(t)Af, OtZt is a solution of a free wave equation with initial data Af.

fl D(A); (U(t)Af,g) = (AU(t)f,g) = —(U(t)f,Ag). since Let g E A' = —A. It is then sufficient to prove that Ag E D÷: but g E D÷ so U(t)g = 0 in In R + t. t 0. Moreover U(t)Ag = Ot(U(t)g); then IL)4. U(t)Ag = 0 in In R + t. t 0 and consequently Ag

L. Alow, M. Khenissi

12

3rd step: End of the proof of Proposition 5.2. We have C7

{O}.

operates on GT which is a finite dimensional space. has an eigenvalue A. Let v be an associated eigenfunction which has the form v(t,x) = eAtf(x). This function verifies the free wave equation; then we Since

have is

=0 f/ôfl=O

that is, the data (f, Af) is an eigenfunction of A, which contradicts the fact that A does not have eigenvalues (see [6]). 0

End of the theorem's proof Combining the results of Propositions 4.4 and 5.2, we obtain the inequality (s) of the theorem.

Proof of Lemma 5.3. Let f E K.

IE =

11Z0(t)fll = Dill

=

0

in [0,t] x

llZ(t)fll = 11111

Now let us show the equivalence: IIZ(t)f II = III II then Z(t)f = U(t)f. Since U(t) is unitary, then DZ(t)fD = llU(t)ffl = 11111. On the other hand, if llZ(t)fD = 0 DZ(t)fll = = IIU(t)f II; therefore U(t)f D4. (since

U(t)f

E

(J0).

If U(t)f

References

[1] C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation, control and stabilisation of waves from the boundary, SIAM J. Control Optim. 305 (1992), 1024—1065. [2] N. Burq, Décroissance de lénergie locale de l'équation des ondes pour le problème extérieur et absence de resonance au voisinage du reel, Acta Math. 180 (1998) 16—29. [3] P. Gerard, Microlocal defect measures, Comm. Partial Duff. Equations 16 (1991), 1761—1794.

[4] P. Gerard, Oscillations and concentrations effects in semi-linear dispersive wave equation, J. Funct. Analysis 41(1) (1996), 60—98. [5] P.D. Lax, C.S. Morawetz, and R.S. Philips, Exponential decay of so-

lution of the wave equation in the exterior of a star shaped obstacle, Comm. Pure. AppI. Math. 16 (1963) 477-486.

Stabilization on Exterior Domains

13

[6] PD. Lax and R.S. Phillips. Scattering Theory Decay. Academic Press, New York. 1967.

[7) G. Lebeau. Control for hyperbolic equations. Actes du Colloque de Saint Jean de Monts (1991). (8] C. Lebeau. Equations des ondes amorties. A. Boutet de Monvel and V. Marchenko (eds), Algebraic and Geometric Methods in Math. Physics. 1996. Kluwer Academic Publisher. 73—109. [91

J.L. Lions, Contrôlabilité exacte. Perturbation et stabilisation des

systèmes distribués. R.M.A. Masson. 1988.

[101 R. Meirose and J. Sjostrand. Singularites of boundary value problems I. Comm. Pure. Appl. Math. 31 (1978). 593—617: IL C.P.A .M35 (1982). 129—168.

[11] R. Metrose. Singularities and energy decay in acoustical scat Duke Math. J. 46 (1979) 43—59.

[12] C.S. Morawetz. Decay for solutions of the exterior problem for the wave equation. Comm. Pure. Appi. Math. 28 (1975). 229—264.

[13) C.S. Morawetz. J. Ralston, and \V. Strauss. Decay of solutions of the wave equation outside non-trapping obstacles. Comm. Pure. Appi. Math. 30 (1977), 447—508.

[141 J. Ralston. Solution of the wave equation with localized energy. Comm. Pure. Appi. Math. 22 (1969). 807-823.

[15] J. Rauch and M. Taylor, Exponential decay of solutions for the hyperbolic equation in bouded domain. Indiana University Math. J. 24 (1972). 74—86.

[16] \V.A. Strauss. Dispersal of waves vanishing on the boundary of aextenor domain. Comm. Pure. AppL Math. 28 (1975) 265—278. Faculté des Sciences de Gabès Département de Mathématiques et Informatique Route de Médenine 6029 Gabès. Tunisia [email protected]

Carleman Estimate and Decay Rate of the Local Energy for the Neumann Problem of Elasticity Mourad Bellassoued ABSTRACT The asymptotic behavior of the local energy and the poles of the resolvent (scattering poles) associated to the elasticity operator in the exterior of an arbitrary obstacle with Neumann or Dirichlet boundary conditions are considered. We prove that there exists an exponentially small neighborhood of the real axis free of resonances. Consequently we prove that for regular data. the energy decays at least as fast as the inverse of the logarithm of the time. According to Stephanov—Vodev ([17J. [18]), our results are optima) in the case of

a Neumann boundary condition, even when the obstacle is a ball of The fundamental difference between our case and the case of the scalar laplaclan (see Burq 111) is that the phenomenon of Rayleigh waves is connected to the failure of the Lopatinskii condition. Key words. Carleman estimate, resonances, energy decay, elasticity system. AMS Subject Classification. 35 P20. 35 L20. 73 C02.

0. Introduction and main results In order to state our results more precisely, let Q = iR3\O be an exterior domain in 1R3 with and compact boundary ôfl, g = a positive definite symmetric matrix equal to the identity matrix outside a compact set. We consider the elastic wave equation with Neumann or Dirichlet boundary condition

onRxOQ I.u(O.x)=uo(r); Here. Ae(x.

and B(x.

are of the form

Ae(x. D2) =

+ (p + .X)V(div).

Throughout this paper. we suppose p > 0, 3) + 2i > 0 and div are the Laplacian, gradian and divergence operators defined by

V and =

M. Bellassoued

16

and divu = trace(Vu).

8,(g,,c93u); Vu =

N(x,D)u = o(u).n

(0.3)

where n(x) is the unit outer-normal vector to Q at x

811. In equation

(0.3),

cr(u) =

is the stress tensor, where c(u) = Vu) is the strain tensor. The purpose of this paper is to consider the asymptotic behavior of the local energy and the distribution of resonances. For R> 0, we define the local energy E(u. R, t) for the solution of (0.1) in

E(u.R,t) = j

(.xlthvu(t.x)12

+

say that problem (0.1) has the uniform local energy property when, for

any R> 0, there exists a continuous function f(t) satisfying Limf(t) = as t —, oc

0.1

0

such that E(u,R,t) f(t)E(uo.u1) holds for any t 0.

Relation to the literature

Shibata and Soga [16] formulate a scattering theory for the elastic wave equation which is analogous to the theory of Lax and Phillips [8]. In fact, the proof given by [16] implies the local energy decay property; that is, for any initial data uo. u1 the local energy for the solution of (0.1) in 1R decays as t tends to infinity. In the case of the scalar-valued wave equation with Dirichlet or Neumann

boundary condition or the elastic wave equation with Dirichiet boundary condition, if the obstacle satisfies a nontrapping condition in some sense, then it has the uniform local energy decay property. Furthermore, we can take the rate f(t) as eot (see Meirose [11], Melrose—Sjöstrand [13], Morawetz [14], Iwashita—Shibata [4) and Yamamoto [24]). In the case when the obstacle is trapping Ralston's example (see [15]) proves that we cannot generally expect local energy decays to occur uniformly.

For the elastic wave equation with the Neumann condition, however, there is an interesting phenomenon. It is the existence of the Rayleigh surface wave which seems to propagate along the boundary. Taylor [21) gives a rigorous treatment of the singularity and he proves that there are three types of rays that carry singularities. The first two types are classical rays reflecting at the boundary according to the laws of geometrical optics. and the singularities propagate along them with speeds =

Carleman Estimate and Decay Rate

17

+ The third type of rays lies on the boundary, and singularc2 = ities propagate along them with a slower propagation speed CR > 0 (the Rayleigh speed). Thus any obstacle is trapping for the problem (0.1) (even a ball of R3) from the point of view of propagation of singularities. Consequently Ikeheta—Nakamura [3] and Stephanov—Vodev [17] show that the problem with the Neumann condition does not have the uniform local energy decay property if the obstacle is a ball in R3 and the result is extended by Kawashita [5), [6] for any nontrapping obstacle. But for regular data, we can set P,n.R(t) =

E(u, R;t)

{

II(uo. 121)11 D(Am)

: (120. u1)

o}

(0.6)

where

_( 0

—ild 0

with m > 0. however, we can show that limpm.R(t) = as for the rate 0 by the method of Walker [23]. Indeed, his proof is based on the Rellich theorem and the local energy decay property, that is, R, t) = 0.

An important problem in this direction is to know how fast pm,R(t)

converges to zero as t oc. Ikeheta and Nakaniura [3) show that for any < > 0 we cannot get the estimate of the form even if is a unit sphere in R3, and Kawashita [5] shows that = holds for any 7 > 0. The second important problem is to know how fast Pm.R(t) converges to zero if the obstacle is trapping.

Main results In this present paper, we show a similar result for the solution of (0.1) 0.2

independent of the geometries of the obstacle; precisely, we prove pm.R(t) S Iog(2+t)2m.

Theorem 0.1. For any R1.R2 > 0 and m > 0 there exists C > 0 such that for any data (uo, u1) D(Am) supported in we have

E(u. R2, t) where

(log(2± t))2m

Il(uo,

u solves (0.1).

We have the previous result as a consequence of the existence of an exponentially small neighborhood of the real axis, free from the resonances (scattering poles). More precisely, the purpose of this paper is to give some information about the location of the poles of the outgoing resolvent R(z).

18

M. Bellassoued

We say that z

C is a resonance associated to the obstacle 0 if the

following problem has a nontrivial solution:

jflfl v(x. z)

on8'Z is z-outgoing.

The function v is said to be z-outgoing if for some a >> 1 there exist such that IE iz(x.z) = (Ro(z)f)(x)

in

B(0.a)c.

(0.10)

It is known that the resolvent R(z) acting on functions v E in H2 is holomorphic in Im z < 0 and can be extended as a meromorphic function from Im z <0 to the whole complex plane C with possible poles the in mi z > 0. Let X1.2 E be a cutoff function equal to 1 near are called resonances. In the case of the Laplacian with Dirichlet or Neumann boundary condition, it is well known that (see Melrose and Sjostrand [13]) for nontrapping obstacles the resonances lie above logarithmic curves of the type 1m2 = aLn(IRezI) — 8, a > 0. For trapping obstacles Burq [1] shows the existence of an exponentially small neighborhood of the real axis containing no resonances. Stephanov and Vodev [17] prove that, for the elasticity operator with Neumann boundary condition, there exists a sequence of resonances tending exponentially to the real axis for an arbitrary obstacle. In this paper we show that: poles of

Theorem 0.2. There exist C1. C2. C3 > 0 such that the outgoing resolvent is holomorphic in the region

U = {z

C:Imz

Moreover, there exist C >0 ondô >0,

{IzI

> C3).

j = 1.2 such that in the

region

U we have the estimate of R(z)

IIXIR(z)x211c(L2.H2) < To prove Theorem 0.2. we make use of an idea due to Lebeau and Robbiano [10] which has been adapted by Burq [1] for this kind of problem; it consists in using Carleman estimates to obtain information about the resolvent in a bounded domain. The cost is to use phase functions satisfying Hörmander's assumption and thus growing fast, far from the obstacle. The point is to show estimations for the solution of (Ae + r2)u = where r is a real parameter and f is compactly supported. Let be a bounded domain with smooth boundary aci0 and let A(x. D) — r2. Let = = Ae(x,D) — r2 with principal symbol and denote We define A(x,D.r) = p(x)

f

Carleman Estimate and Decay Rate

19

We assume that for satisfies — E + = Hörmander's assumption Hi,, 3C > 0 such that + iiip') =0 2p + > C whenever a1,(x,e + E + and we assume that 0.

Theorem 0.3. Assume that

< —C0 (where C0 > 0 i3 large enough) on E C Oft0. Then there exists C> 0 3uch that for any u E and Bu = 0 on E we have

J IA(x, D. r)u12 + lb

rJ

(r21u12 +

OCo\E

Crj (r21u12 + IVuI2)

(0.13)

for large enough r.

Remark 0.4. i) According to Ikeheta—Nakaxnura [3] and Stephanov— Vodev [171 Theorem 0.1 and Theorem 0.2, with Neumann boundary condition, are optimal even when the obstacle is a ball of ii) Theorem 0.1 follows from Theorem 0.2, which is proved in the general case by Burq (see [1]).

1

Spherical harmonics

Our purpose in this section is to give some information about the outgoing solutions of + z2)u = 0 outside a ball B(0, R1) (where = Ar lfl B(0, R1)c).

Preliminaries Let (r. 9, be the polar coordinates in R3. Denote by 1.1

m

n, m

n Ferrer's function defined by

( Set Ymfl (0,

=

Oz

(cosO) and define the vector spherical harmonics

Pmn, Bmn, Cmn by Pm,, = Ymn(0.

Cm,, = (n(n + 1))4 curl

s')),

By,,,, = W X Cmn

where the symbol (x) denotes the exterior product and

=

x E S2(0, R). It is known (see Morse—Feshbach 112), pp. 1898) that

for

M. Bellassoued

20

=

Bmn, Cmn} form an orthogonal basis in L2(S2). Denote by the spherical Hankel function of first order where

j

— i,r

(1.3)

= We have the following lemma for the properties of and 2.4 of Burq [1]).

(see Lemmas 2.3

i) For any a < 1, there exists C > 0, r0 > 0 such that for Lemma 1.1. any rf > r0 and < ITiar where r = Rek > r0, flmk( 1 we have

I H'(kr)\

C,

and

IH'(kr)\

C

(1.4)

ii) For any a < 1 and 0 < R1 < R2, there exists C > 0,c > 0.rO >0 such that for any Iki > TO and'y >

flmkl

1 we have

Cet'IH,(kRi)i

f

( 1.5

1.

Define now Lmn(r; k). Mmn(r; k). Nmn(r; k) as

I Lmn(r;k) = ?t!mn(r;k) = I.

1.2

(1.6)

Nmnfr;k) = k'curl(Mmn(r:k)).

Study of the outgoing solutions far from the obstacle

Since here we have two sound speeds, we have to consider the following five regions (i) Hyperbolic region: n(n + 1) (ii) Glancing region (I): r?r2.

Proposition 1.2. For any R2 > R1 > 0. there exists C1.C2,e and r0, such that for any z E C; JImzI 1, IRezi = fri > ro the outgoing solution for + z2)u = 0 outside a ball B(0,R1) satisfy

f

(IuI2 + 1r1Vu12)da (—ImNuii)dc InC1 f rR3 —

2 2.1

Jr=R1

(1.7)

+ IT1VuI2)da. (1.8)

Proof of Theorem 0.2 Construction and properties of the phase function

The purpose of this section is to construct two phases which satisfy Hörmanders conditions, except in a finite number of balls, such that on a

Carleman Estimate and Decay Rate

21

ball where one of them does not satisfy these conditions, the second does

and is strictly greater. Moreover, far from the obstacle, the functions = ,c. for > 0 arbitrarily small. coincide, are radial. and satisfy 2.1.1

Construction in a bounded region

Let R> 0 be such that B(O. R) contains space perturbation and operator perturbation i.e., the obstacle C) C B(0, R) and g = Id outside B(O, R).

Proposition 2.1. (see Burq [1]). There exist two functions ti'j.

<0

E

stands for the unit outer-normal vector field at 811), only having no degenerate critical points such that on 0 and . x > 0 and, when = 0, S(O. R) we have > are radiols and = W2 in a neighborhood of S(0. R). Finally = satisfying

Thenthereexistsatmnitenumberofpointsx1 E 11,i = 1,2,j = 1,... and e > 0, such that B(x,), 2e) C (1R, B(xj.3, 2e) fl B(x2.,, 2E) = 0 and = (where 1113 = in Denote Let us search for

in the form

=

=

for each large enough

+

—

Taking from

—

(2.2)

If a-1, (x. r) = 0, a simple computation gives = 0 and

=

+

(2.3)

Taking into account (2.3) we obtain {Rea.1,, Ima,,}(x. Hence ?b'

det 2.2

r) =

+ Q(j33)),

(2.4)

r) C.

(2.5)

0 and g CId: then we have r)

0

implies

{Rea,, Ima,} (x.

End of the proof of Theorem 0.2

Let u(x, z) solve the problem

bill onOll (,u-outgoing

(2.6)

M. Bellassoued

22

= fl B(0. R1). Let Xi. i = 1.2 be two cutoff functions equal to 1 in (uB(x,3.2e)c) and supported in (uB(x1,.eY).

where f is supported in

We apply Theorem 0.3 for the function efl"

= Re(z2)and

J

in a domain

+

J

Crj Using the properties of the phases large enough r we have

f

+

+

Combining

+ 2e).

for any

+ Vu12).

(2.7)

in U

>

.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
Denote K = {x

a(x, D, r) =

op(a).

(3.7)

Denote by (3.8)

the principal symbol of the operator, and set

=

+ op(a)):

=

(3.9)

—

its real and imaginary part. Then we have

fop(a) = where by

and

r).

+

310

are tangential symbols of order respectively 1 and 2 given f q2(x.

and

=

+

r) = r(x.

—

)2

—

r2r(x.

(3 11)

q') is the bilinear form attached to the quadratic form

3.1.2 Reduction of the elasticity system

In the system of normal geodesic coordinates the principal symbol of elasticity operator can be written as Ae(X,

=

+ (.\ + 1i)e(x.

(3.12)

M. Bellassoued

26

is defined by (3.5) and £(x, jection onto the space spanned by We set

is the orthogonal pro-

where E(x,

A(x, D, r) =

—

r2Id.

(3.13)

The principal symbol of A(x, D, 'r) is given by

A(z,E,r) = where A, (x, by

(3.14)

r) are tangential symbols in Si

and are defined

x

A0 = 1Jd + (i' + A1 =

A0 +

+

A2 =

p(ttitt)Id +

cd

+ —

r2

A0 +

+

—

r2Id. (3.15)

For fixed

E

T (0Q0), let

E C such that

+

+

+

=

—

icr);

(3.16)

then we have also by (3.6)

= = (3.17) + + + Here assume that 0, and let (w1,... ,"n—2) be the basis of {to, We can define the smooth matrix H = homogeneous of degree zero in (c', r), such that on a conic Lo, neighborhood of (x, r) we get A=

H'AH =

(3.18) 0

0

(2,L+)I)a—T2+(p+A)(iQ)2

0

jia —r2 —(z+A)(ia)2

0 We set now

j Then we have =0 1, £); we obtain

= 319 + (ick)to. + = and we can decouple the system by P = (Wi,.

A= P'AP—

(3.20)

Therefore, we obtain

det(A(x, where

r)) = ii"'(2p + r) =

—

r)

r)) and-y

+ A}.

(3.21)

Carleman Estimate and Decay Rate

27

3.1.3 Study of the eigenvalues

The proof of Carlemazf a estimate relies on a cutoff argument based on the nature of the roots with respect to of Let use now introduce the following microlocal regions:

K x S"':

(i)

=

E

(ii)

=

EKx

(iii)

=

EKx

{

+

—

+

—

> o}.

)2

= o}.

+

<0<

E

And for fixed

—

C such that

let T2

=

—

—

+

=

+

+

—

Taking into account (3.16) (3.17) we have

(ia,)2 =

+ —:

(3.22)

then we get by (3.12) and (3.17)

=

(r)2 +q2 —

For a fixed (x.e.r). decompose

+2irq1.

(3.23)

a polynomial in

then we

have the following lemma:

Lemma 3.1.

1)

denoted by

2) For any

For any (x,

r)

E

the roots of

4 satisfying ±1m4 > 0.

and

are

r) E Z.,, one of the root,s of a.,. is real.

3) For any (x. r) E. the roots of a, lie in the upper half-plane if <0 (resp. in the lower half-plane if > 0). 4) For any (x,

the roots of

r) E M. the roots Of 02p+). satisfying ±1m4 > 0 and satisfy 3).

28

M. Bellassoued

3.2

Carleman estimates in the region

Define the following weighted norms in H'

=

respectively in H' (&lo); (3.24)

I1LI?,. =

and define the norms 8tt 2 (3.25) + 8721L2 The purpose of this section it to get the Carleman estimates in the region Precisely, we take a cutoff function r) homogeneous of degree

=

zero in the region

purpose here is to prove the following proposition.

Proposition 3.2. There exist C> 0 such that for any large enough r we have IIA(x,

r)u[j2 +

whenever u E large enough) on

+

(3.26)

Furthermore if we assume that

> C0 (Co > 0

= 0} fl suppXo, then we have

IIA(x.D,r)u112 + I

op(

+

whenever u E

3.2.1

Scalar estimates

In this section we give a scalar Carleman estimate for the operator with principal symbol where This estimation is E + proved essentially in Lebeau—Robbiano [9] and [10]. Furthermore we assume

that H

satisfies the assumption ç For any x E K; 1 {Rea.,, Imo,} > 0

0

whenever

r) = 0

EKx

Lemma 3.3. There exist C> 0 such that for any large enough r we have IIop(a,A)v112 +

+

(3.27)

whenever t' If we assume that > 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<

i)

ii)

+

>

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:

=<

+

(4.20)

then we have by a simple calculation

=

(4.21)

0.

(b) Calculus of eigenvector associated to We get

= Let

be

+

+

÷

+

(4.22)

defined by

=

(4.23)

= 0. Using (4.20) we denote

then we have

I.

+ e1:

(4.24)

< e,r

=

and the principal symbol of B by

in and r Lemma 4.3. For any U r) is an i,9omOrphism under the assumption

0 the operator C0

for large

Proof. We will keep some of the notation from Section 3. Let B =

(A'B1,

Co

>

>0.

B0) be the principal symbol of the Neumann operator, where

(425)

jB Denote

= (bt

then we have by elementary calculations

j=1,... !n—2 + A3[2p(ia)2 + r2Jt1 = = A2[21t(ia)2 + r2leo + (4.26)

Then we get

=

r))

(4.27)

34

M. Bellassoued

where R is a function given by

=

(1— 2s2)2

—

452

— —

(4.28)

2p± A)

It is well known that there is only one simple root 8 = of R(s) = 0, s> 1 (see Taylor [21]) and we can prove that it has no roots in Rez > 0. Let E be the characteristic variety defined by

E=

E

T0c; 2

= {(x',e',r)

= except for (z', where CR

2irq1 —

is elliptic outside E

the Rayleigh speed. Therefore

r)

= o}

Z, U (+; then we have

+

+

(4.29)

= diag(0, —plm(71));

(4.30)

—

For the second part of Lemma 4.2, we get

then we obtain 1). Let now w = (w+,w_)

C"

C"; we have

8w = 8+w+ + Bw. Taking into account that 8+ is an isomorphism, then there exist C' >

0

such that

+

(4.31)

This shows that —(lZw,w) = p1w12

—

C"18w12

(4.32)

for large p. This concludes the proof of Lemma 4.2.

References [1] N. Burg, Décroissance de l'énergie locale de Féquation des ondes pour le problème extérieur et absence de resonance au voisinage du reel, Acta Math. 180 (1998), 1—29.

[2] L. Hörmander, The Analysis of Linear Partial Differential Operators, I, II. Springer-Verlag. 1985.

Carleinan Estimate and Decay Rate

35

[3] N. Ikehata and C. Nakarnura, Decaying and nondecaying properties of the local energy of an elastic wave outside an obstacle. Japan J. Appi. Math. 6 (1989), 83—95.

[4] H. Iwashita and Y. Shibata, On the analyticity of spectral functions for exterior boundary value problems, Cbs. Math. Ser. III 23(43) (1988). 29 1—3 13.

[5] M. Kawashita, On the local-energy decay property for the elastic wave equation with the Neumann boundary conditions, Duke Math. J. 87 (1992). 333—351.

[6] M. Kawashita, On a region free from the poles of resolvent and decay rate of the local energy for the elastic wave equation, Indiana Univ. Moth. J. 43 (1994). 1013—1043. [7] V. Kupradze. Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoebasticity, North Holland. Amsterdam. 1979.

[8) P.D. Lax and R.S. Phillips. Scattering Theory. New York, Academic Press, 1967.

[9] C. Lebeau and L. Robbiano, Contróle exact de léquation de Ia chaleur. Comm. Part. Duff. Eq. 20 (1995). 335-356.

[10] G. Lebeau and L. Robbiano. Stabilisation de l'équation des oxides par le bord. Duke Math. J. 86(3) (1997). 465—491. [11] R.B. Melrose. Singularities and energy decay in acoustical scattering. Duke Math. J. 46 (1979). 43—59.

[12] P. Morse and H. Feshbach. Methods of Theoretical Physics. McGrawHill. New York. 1953.

[13] R.B. Meirose and 3. Sjöstrand. Singularities of boundary value problems I, Comm Pure Appi. Math. 31 (1978), 593-617. [14] C.S. Morawetz, The decay of solutions of the exterior initial-boundary

value problem for the wave equation, comm. Pure Appl. Math. 28 (1975), 229—264.

[15) J.V. Ralston. Solutions of the wave equation with localized energy. Comm. Pure App!. Math. 22 (1969). 807—823.

[16] Y. Shibata and H. Soga, Scattering theory for the elastic wave equation, Pubi. RIMS Kyoto Univ. 25 (1989), 861—887.

36

M. Bellassoued

[17] P. Stefanov and C. Vodev, Distribution of the resonances for the Neumann problem in linear elasticity in the exterior of a ball, Ann. Inst. H. Poincaré, Phys. Th. 60 (1994). 303—321. [18J P. Stefanov and G. Vodev, Distribution of the resonances for the Neumann problem in linear elasticity outside a strictly convex body. Duke Math. J. 78(3) (1995), 677—714. [19] P. Stefanov and C. Vodev. Neumann resonances in linear elasticity for an arbitrary body, Comm. in Math. Phy. 176 (1996). 645—659.

[20] D.Tataru, Carleman estimates and unique continuation for solutions to boundary value problems. J. Math. Pures AppI. 9(4), 75 (1996), 367—408.

[21] M. Taylor. Rayleigh waves in linear elasticity as a propagation of singularities phenomenon, in Proceedings of the Conference on Partial Duff. Equa. Ceo.. Marcel Dekker, New York, 1979, pp. 273—291. [22] M. Taylor, Reflection of singularities of solution to systems of differential equations, Comm. Pure Appl. Math. 29 (1976), 1—38. [23] H. Walker. Some remarks on the local energy decay of solutions of the

initial boundary value problem for the wave equation in unbounded domains. J. Duff. Eqs. 23 (1977). 459—471.

[24] K. Yamamoto, Singularities of solutions to the boundary value problems for elastic and Maxwell's equations. Japan J. Math. 14(1), (1988). 119—163.

Université de Paris Sud Mathdmatiques. Bat. 425 91405 Orsay Cedex. France email: [email protected]

Microlocal Defect Measures for Systems Nicolas Burq ABSTRACT We define the microlocal defect measures for boundary value systems satisfying the strong Lopatinski condition and we apply these notions to the study of the asymptotic propagation of the energy for the solutions of the Lamé system.

ResuME. On définit des mesures de défaut de compacité pour les systèmes aux limites vérifiant Ia condition de Lopatinski uniforme au bord et on utilise ces notions pour étudier Ia propagation de Penergie pour les solutions du système de Lamé.

1

Introduction

The purpose of this paper is the study of solutions of boundary systems of second order partial differential equations with scalar principal symbol, satisfying strong Lopatinskii conditions at the boundary. The study of the propagation of singularities for such systems has been made by N. Denker [4) in a domain without boundary and by C. Gerard [51 near strictly gliding or strictly diffractive rays in a framework.

Our study will be limited to the propagation at the energy level (H'), but it allows us to generalize the above results (we shall suppress (almost) any hypothesis about the microlocal nature of the points where the analysis is performed) and furthermore, it will allow us to give a quantitatwe version of the propagation. More precisely, for any sequence of solutions to such a system, with bounded H'-norm, we will define a inicrolocal defect measure (or H-measure) giving a quantitative description of the asymptotic polarization of the sequence. This definition generalizes to boundary systems the notions introduced by P. Gerard [6] et L. Tartar [121. We shall prove a propagation result allowing the calculus of the measure (and hence of the polarization) along a certain flow (see Theorem 3.1). In the particular case of a unique wave equation, this program has been fulfilled by P. Gerard (6] et L. Tartar [12] without boundary, P. Gerard, E. Leichtnam [7], C. Lebeau [9] for Dirichlet boundary condition and H. Tataru (8] for absorbing boundary condition (in a different framework) (see also [11]).

We also show how our result can be applied to the study of the energy

38

N. Burq

decay for the thermoelastic system to prove a conjecture by G. Lebeau et E. Zuazua [10] about the uniform decay of the energy. The results presented here are to be published in [1].

2 2.1

Preliminaries Notation

Let Y = {y Rd, <1} be the unit ball in R" and X =]0, 1[xY. X is a manifold with boundary X = [0, 1[xY, OX = {x = 0) x Y. For N E N note L2 = L2(X;C"),

H' =

with norms (2.2)

=f Elttjl2dxdy j=1 =

IVu,I2dxdy

+J

(2.3)

with V

_(8f Of

Of

24 (.)

Let R = R(x, y,

be a second order scalar, seif-adjoint, classical, tangential, and smooth pseudodifferential operator, defined near 10,11 x Y, with real principal symbol r(x, y, ti); such that

for (x,y)€X and

(2.5)

Let be N x N matrices of smooth classical tangential pseudo-differential operators defined near 10,11 x Y, of orders 0 and

1, and principal symbols mo(x,y,tj), MoOx + M1. The principal symbol of P is = + r(x, y, Note (uk)kEN a sequence bounded in 0 and such that

Note P =

+ R)Id + (2.6)

1[xY), converging weakly to

=o(1) in L2(]0,1[xY), —B

0x

o(1) in L2(Y),

(2.7)

Microlocal Defect Measures for Systems

39

with B a pseudodifferential operator of order —1 satisfying the strong Lopatinskii condition. Note A as the space of N x N matrices of pseudodifferential operators with Q classical pseudodifferential operator with compact Q = Qt + support in X (i.e., Qt = for some w C000(X)) and Qa a classical tangential pseudodifferential with compact support in X. Note A8 the sorder elements of A.

Geometry TOX is the disjoint union £ U U N where (ro r 2.2

fl={ro>0}.

E={ro
(2.8)

that = {(y,?7) E Q; OrT Ix=o (y,'i) > 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<

E fi u g2.÷, there exists

>

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 <

IIUOIIHI(cz)d + IIU1IIL2(fl)a

jTj

div ul2dxdt.

(5.6)

In case d =

2, they deduced from this result a geometric necessary and sufficient condition for the uniform decay to hold. The following results

generalize this:

Theorem 5.2. (sufficient condition) Suppose that C R' has no infinite order contact with its tangents and (i) is fulfilled. Suppose also that

Microlocal Defect Measures for Systems

(iii,) There exists T > 0 such that any ray of P1' = 40?

\ {lInII = 0) fl

—

45

hits

{t EJO,T(}

(5.7)

at least two times at points where the critical directions are not connected by the flow of the propagation described above. Then the uniform decay holds.

Remark 5.3. If (iii) is fulfilled, then (i) is fulfilled, except for a finite dimensionnal space of functions.

Theorem 5.4. (necessary condition) Fix T> 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,
to [4) and [5).

Results

2

In polar coordinates IxI = r E (0, +oo) and operator assumes the form

=

the Laplace

where &, is the Laplace—Beltrami operator.

We note that &, is an unbounded, selfadjoint and negative operator in L2(S"'). We note also that the eigenvalues of the operator —&, are k(k + n — 2) with k E N, and the corresponding eigenspaces Ek are finite

dimensional. The spectrum o-(—&,) coincides with the set of all eigenval-

is the direct sum of the Ek, that is H =

ues. Also H =

k>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
and

:

Theorem 2.1. (see [4)) Let u E C

)uJ2 + C IrOruI2 + C

+

+

) with C, C,.., v = 0,... , h, positive constants. If (2.2)

Laplace and Bi-Laplace Operators

then u

0. (... in (2.1) indicate derivatives of u of order h —

1).

In cartesian coordinates the preceding theorem becomes

Theorem 2.2. (see [4]) If u E

lul + with

. .

—2

and satisfies the estimate IVuI2 +

.

+

positive constants and

ID°u12, (2.3) 2

<

,

then u

0.

The result that shows that the bounds for the constants in Theorem 2.1 are optimal is the following:

Theorem 2.3. (see [5]) Let h E N be a positive integer; then, for any E > 0 we can find two functions w E Cr(R2) with supp wR2 and a E C'°(R2 \ {0}) with hail00 <(2h — 1)!! + E such that =

+

in R2 \

o

{0}.

(2.4)

More precisely

2)

Formerly we have studied the tangential endpoint case for the Laplace operator, that is Theorem 2.4. (see [4)) Let u E Gb°° (Q) and satisfy (x)12

92;

ti (x)(2 +

(x)12

xE

(2.5)

infl.

Now we state new results for Laplace and bi-Laplace operators. We denote by C, C', C" positive constants that may be different in different equations or inequalities. At first we study the border radial case for the Laplace operator; in this case, differently from the tangential case above, we need no bound for the constant C.

Theorem 2.5. In R" with n

3, if u E

Cr(fl)

Ciu(x)h2+ lrOru(x)F2 for some positive constant C, then u

0 in

satisfies (2.6)

52

F. Colombini, C. Crammatico

We are now ready to state some border cases related to (2.2) for the hi-Laplace operator. The first one is the following.

Theorem 2.6. If u Cr(cZ)

satisfies

with C <

—

C' =

0,

ici,u12 + 9 I&u12

+ C"

C 1u12 + C'

C"

—

2)2,

then u

(2.7)

0 in

Remark 2.7. In R2 we have that if u 3

satisfies cannot be assumed. in Theorem 2.3 for

For the radial case, that is when Co =

9

and C1 =

=

C2

0

in (2.2), we

have the following.

Theorem 2.8. In R" with n

5, if u

satisfies +91(fOr)2U1

for some positive constants C, C', C", then u

(2.8)

0 in

Remark 2.9. As it will be seen below by using Carleman estimates, in this case, contrary to Theorem 2.6, the dimension n of the space does not influence the constants C, C', C", provided n 5. Finally, in the tangential-radial case we have the following.

Theorem 2.10. In R" with n C

5, if U E Cr(cl) satisfies

+ C' IrOrul2 + C" >•

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
• for k

m, m —

1,

from (3.8) we have

Cr,k

• for k =

9r4+rk(k+n—2)+r3;

m, r

C,.,,, = [3

2

r (n — 4) +

n—3

I

2

)]

2

• for k=m---l,

=

[3

5)] 2

.

+ r (n —4) +

(3.12)

Therefore from (3.7), taking into account that, if U = eTTV,

=

2

r4_21

Jf JJc2TT we

107'U12

= r2 ff

dTdw

ff +

Jf

dTdw,

have

ff

IPUI2 dTdi.,.'

r Then, if n 5, we can conclude as in Theorem 2.5 that the strong unique continuation holds for the estimate (2.8) without bounds for C, C', C". On the other hand, it is seen at once that if vi = 4 there does not exist r E [m, m + 1] such that

58

F. Colombini, C. Crammatico

• Cr.m

• Cr.m_i Thus, in this case we cannot give a Carleman estimate for the endpoint. Proof of Theorem 2.10. In this case we choose different values of 'r according

to dimension n of the space.

1. If n 6 we can take r as in (3.10), while if n = such that

5

r(r+n—4)=m(m+n—2)—m+1,

we choose r >

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

<(1 + e)

I(r8r)2 wI + C IOowI + C' IrOrwI

(3.16)

and S (1

+

e)

+ ClOowl + C' IrOrwl.

From (2.4) and (3.16)—(3.17) we have

+

+ 2 ITOrOOWI2 + (rOr)2 w12)

+ C Oowj2 + C' IrOrwI2.

Writing (3.18) in cartesian coordinates, the proof follows.

References [1] S. Alinhac and M.S. Baouendi, Uniqueness for the characteristic Cauchy problem and strong unique continuation for higher order partial differential inequalities, Amer. J. Math. 102 (1980), 179—217. [2] S. Alinhac and M.S.Baouendi, A counterexample to strong uniqueness for partial differential equations of Schrödinger's type, Comm. Partial Differential Equations 19 (1994), 1727—1733.

[3] B. Barcelo, C.E. Kenig, A. Ruiz, and C.D. Sogge, Weighted Sobolev inequalities for the Laplacian plus lower order terms, Illinois 3. Math. 32 (1988), 230—245.

[4] F. Colombini and C. Grammatico, Some remarks on strong unique continuation for the Laplace operator and its powers, Comm. Partial Differential Equations, 24 (1999), 1079—1094.

[5] F. Colombini and C. Grammatico, A counterexample to strong uniqueness for all powers of the Laplace operator, Comm. Partial Differential Equations 25 (2000), 585—600. [6] C. Grammatico, A result on strong unique continuation for the Laplace operator, Comm. Partial Differential Equations 22 (1997), 1475—1491. [7] L. Hörmander, Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations 8 (1983), 21—64.

[8) D. Jerison and C.E. Kenig, Unique continuation and absence of positive eigenvalues for Schrodinger operator, Ann. of Math. 121 (1985), 463—494.

F. Colombini, C. Crammatico

60

191

P.

Le Borgne, Unicité forte pour le produit de deux opérateurs

d'ordre 2, Indiana Univ. Math. J. 50 (2001); (http://inca.math.indiazia.edu/iumj/Papers/renderToc.php3).

elliptiques

[101 R. Regbaoui, Strong continuation for second order elliptic differential operators, J. Differential Equations 141 (1997), 201—217. 111] T. Wolff, A counterexample in a unique continuation problem, Comm. Anal. Geom. 2 (1994), 79—102. F. Colombini Dipartimento di Matematica-Università di Pisa via F. Buonarroti, 2 - 56127 Pisa, Italy email: colombini©dm.unipi.it and C. Grammatico Dipartimento di Matematica-Università di Bologna piazza di Porta S. Donato, 5 - 40127 Bologna, Italy email: [email protected]

Stabilization for the Semilinear Wave Equation in Bounded Domains B. Dehman 1

Introduction

The aim of this article is to prove a stabilization theorem for the semilinear wave equation on a bounded open domain of Ri', d 1 with boundary Dirichiet condition. More precisely. we study systems of the type

(Du +

+ f(u) =

on u(O,z) = u°(x) E

0

on JO. +oo[xfl

and Otu(O,x) = u'(x) E L2(Q)

where the nonlinearity f satisfies some conditions which will be specified later. In the linear framework, the study of stabilization, i.e., the exponential decay of energy, has been investigated by many authors. We will principally mention the work of Rauch and Taylor [7] and of Bardos, Lebeau and Rauch [1) who systematically used microlocal analysis techniques. in particular, the geometric control property. On the other hand, the literature is much less developed for nonlinear equations. Besides the article of Haraux [6] which describes a simple decay to 0 of the energy, we essentially quote the result of E. Zuazua [121 for the semilinear case. In the present work, we consider the article [12) and we relax some structure conditions imposed on the nonlinearity f. More precisely, instead of taking f globally Lipschitz or superlinear, we assign to it natural growth conditions, which are more "standard", and guarantee global existence for the solution and a good definition of the nonlinear energy. In return, we only produce a local stabilization theorem, i.e., if R > 0 is fixed, we establish the existence of C> 1 and > 0 such that

E(u)(t)

t

0

for any solution u of (1.1) whose initial energy satisfies E(u)(0) R. Contrary to the result obtained in [12], the constants appearing in this stabilization inequality (in particular the stabilization rate depend on

62

B. Dehman

R. i.e., on the ball of the energy space in which we choose the initial data. They are, of course, uniform on every ball. The other assumptions we have used are essentially the geometric control property (almost necessary and sufficient in the linear case) and a unique continuation condition. Besides the fact that the nonlinearities under consideration are more general, the principal interest of this work.in our opinion, lies in the method used in the proofs, which differs from the one used by Zuazua in [12]. It unifies the proofs of the global Lipschitz and superlinear cases. In particular, in the critical case d 3 and p = d/(d— 2) (see (2.3)), our approach is of microlocal nature and rests on properties of microlocal defect measures associated to sequences of solutions of the wave equation with bounded energy. We refer to the works [4J and [5) of P. Gerard for a complete presentation of these measures.

2

Notation and statement of the result

Throughout this paper, will denote a bounded open set of Rd. connected, with a boundary t9S1. We will call any set that is equal to the inin Rd a neighbourhood of the tersection of fi with a neighbourhood of boundary We denote by x = (x1, xd) the common point of Rd x and by 0 the wave operator on

We consider the semilinear system

(Du + a(x)0tu + f(u) = 0 on ]0.

u=O

on

u(O.x) = u°(x)

and Ottt(O.x) = u'(z)

E

E

Here. f is a function defined from R to R, of class C1, such that

sf(s) c(1

OVs E R

+

(2.2)

Vs E R, j = 0, 1

for some constant c > 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

<2E sup tE[0.TJ pT—c

+ JeI 2CE+j

J

pT—c

+Jt

I p

J pT—c e

p

J

nfl). Applying again the beginning 0 in H'(JO. T[ x —# 0 in H'(w, nfl) of the argument used in (4.13). we deduce that for almost every t ]0,T{. i.e.. 0 in L2P(wi nfl), and, by Lebesgue's theorem fT n (4.18) Ivn(t,x)12'dtdx —' 0, +oc. I I which implies

JO

8. Dehman

70

It remains now to treat the interior points of IL To do it. we consider a = satistruncature function The sequence fies:

=

—

an

in ]O. T[XRd

+ [0.

(4.19)

The right-hand member of this where [0. = — equation is clearly bounded in L' ([0. T]. L2(Rd)) and the initial datas are satisfies then the Strichartz inequality bounded in the energy space. (cf. [101):

with 1/q + dir = d/2 —

1

(4.20)

with q = > 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 <

1.

(4.21)

= The sequence (va) satisfies a Let = IIunIIL2(tO.7lxQ) and system analogous to (4.9). Moreover, we have: = 1.

(4.22)

Stabilization for the Semilinear Wave Equation

71

çT

/

Jo

/ a(x)I8tvnl2dtdx—#O,

(4.23)

Jci

(va) is bounded in H1 (JO,

(4.24)

So, there exists a subsequence of (tie), still denoted by (vu), and a function such that: v in v,

(4.25)

v in

v almost everywhere in 10. T( x

On the other hand, 0 a.e. in JO, it is important to point out that, due to assumption (2.6), the sequence is bounded in [0, i-oc[. So it has a converging subsequence. Passing then to the limit in system (4.9), and deriving in time the new equation, we obtain satisfies that the function w = We deduce that II vilLa = 1 and

f Ow + b(t. x)w =

0

in JO. T[ X

4 27

T[, Ld(1Z)). Using an estimation analogous to for some potential b E (4.16). one can see that w E H'(]O. T[xQ). And condition (UC) gives w 0 which leads to a of 11v11L2 = 1 and completes the on JO, proof of Theorems 2.1 and 2.2.

References [1] C. Bardos. C. Lebeau. and J. Rauch, Sharp sufficient conditions for the observation, control and stabilisation of waves from the boundary. SIAM J. Control Optim. 305 (1992), 1024—1065.

[2) N. Burq, Contrôlabilité exacte des ondes dana des ouverts peu réguliers, Asymptotic Analysis 14 (1997), 157—191. [31

N.

Burq and P. Gerard, C.N.S pour la contrôlabilité exacte de

I'equation des ondes. C.R.A.S. 325, série I (1997), 794—752. (4] P. Gerard, Microlocal defect measures. Comm. Partial Duff. Equations 16 (1991), 1761—1794.

[5] P. Gerard, Oscillations and concentrations effects in semi-linear dispersive wave equation. J. of Funct. Analysis 41(1) (1996), 60—98. [6] A. Haraux. Stabilization of trajectories for some weakly damped hyperbolic equations. J. Math. Pures et Appliquées 68 (1989), 145—154.

B. Dehman

72 [7]

J. Rauch and M. Taylor, Exponential decay of solutions for the hyperbolic equation in bounded domain. Indiana University Math. J. 24 (1972). 74—86.

adapté au contrôle des solutions des [8] L. Robbiano, Théorème problèmes hyperboliques. Comm. Partial Duff. Equations 16 (1991), 789—800.

[9] A. Ruiz. Unique continuation for weak solutions of the wave equation plus a potential. J.M.P.A. 71 (1992) 455—467.

[10] R. Strichartz. Restriction of Fourier transform to quadratic surfaces and decay of solutions of the wave equation. Duke Math. J. 44 (1977), 705—714.

spaces and unique continuation for solutions to the semilinear wave equation. Comm. Partial Duff. Equations 21(5. 6)

[11] D. Tataru, The

(1996). 841—887.

[12] E. Zuazua, Exponential decay for the semi linear wave equation with locally distributed damping. Comm. Partial Duff. Equations 15(2) (1990). 205—235.

Faculté des Sciences de Tunis Campus Univ. 1060 Tunis. Tunisia email:

Recent Results on Unique Continuation for Second Order Elliptic Equations Herbert Koch and Daniel Tataru 1

Introduction

The aim of this article is to describe some recent work [7, 8] on unique continuation for second order elliptic equations. Consider the second order elliptic operator in

P= the potential V and the vector fields W1 and W2. To these we

associate the differential equation

Pu= Vu-4-W1Vu+V(W2u). The (weak) unique continuation property (UCP) is defined as follows: Let u be a solution to (1.1) which vanishes in an open set.

(UCP)

Then u =0.

A stronger property is the strong unique continuation property (SIJCP) Let u be a solution to (1.1) which vanishes of infinite order at some point (SUCP) E R". Then u = 0. A smooth function vanishes of infinite order at x0 if all its derivatives vanish at For nonsmooth functions, we need an alternative definition. Given a function u E and So e R" we say that u vanishes of infinite order at

if there exists R so that for each integer N we have

I

luI2dx_
JB.-(xo)

If the potentials V, W1, W2 are zero and the coefficients of the principal part are constant, then the solution u to (1.1) must be analytic, therefore (SUCP) holds. The natural question is which are the class of coefficients gt) and potentials for which either (UCP) or (SUCP) hold. Ideally one

74

H. Koch, D. Tataru

would like to have positive results for (SUCP) complemented by counterexamples for (UCP). A natural first attempt is to use scaling arguments to try to "guess" the correct setup for this problem. More precisely, we look at the dilations which preserve the unique continuation property. If u solves (1.1), then u(Ax) solves a similar equation with coefficients x

giJ(Ax)

AW,(Ax),

)s2V(Ax).

Then we can look for coefficients in spaces that are invariant with respect to such a scaling, for instance

g'3EL°°, One might guess that unique continuation holds above this threshold and fails below. In dimension n = 2 this seems to be quite close to what actually happens.

Conjecture

1.1. Assume

g*JEL00,

that n =

2 and that

W,EL",p>1

Then (SUCP) holds. This was proved in Jerison—Kenig [5] for the case g2 = V E The gradient potentials E L" are discussed in Wolff [19]. Lipschitz coefficients g' can also be handled as in Koch—Tataru [7]. On the other hand, for bounded measurable coefficients g" with V = W = 0 the result was proved by Alessandrini [1] and later extended in Schulz [16] to the case

V,W EL°°. Sharp counterexamples for the corresponding weak unique continuation problem have been obtained only recently:

Theorem 1.2. a) (Kenig-Nadirashvili [6]) There are compactly supported functions u with E L' so that E L'. b) (Mandache [91) There are smooth compactly supported functions u so that for all p < 2. E

However, as it turns out, the case n =

2

is special in many ways. The

most striking difference is related to the counterexamples of Plis [14], Miller [12) and Mandache [10]. In dimension n 3 they produce coefficients which are of class for all s < 1 and for which (UCP) does not hold.

Thus the best one can do for n 3 is to try to obtain positive results in the case when g3 are Lipschitz continuous. The first positive result was obtained by Carleman [4] for ii = 2. He proved (SUCP) for g E C2 and V, W L°°. His idea for the proof, which remains until today one of the main tools in unique continuation, is to obtain a family of uniform weighted estimates, of the form

Second Order Elliptic Equations

75

The essential features of these estimates are that (i) the weight has a logarithmic blow-up at the point so where u vanishes of infinite order and (ii) they are uniform with respect to the (large) parameter r. The norms used are L2 norms, and the proof of the estimates relies on an integration by parts argument. Later on Aronszajn [2] and Aronszajn-Krzywicki--Szarski [3] observed that Carleman's argument can he carried over to higher dimensions and that only Lipschitz regularity of the coefficients is needed. However, one cannot hope to work with V E in the L2 estimates. This is because the V estimates are subelliptic with a loss of derivative, so they are ineffective close to scaling. This problem was solved by Jerison— Kenig [5] who proved instead scale invariant estimates,1 —

which

= —in lxi

=

is exactly what is needed in order to work with V

This argument was later extended by Sogge [17] to smooth coefficients gd.'. To deal with W1 in L" one would need to add a gradient bound in (1.3) of the form

+

lie

Unfortunately such an estimate fails regardless of the choice of the (nonconstant) function (It is, however, almost true in dimension n = 2). This

apparently intractable difficulty can be overcome using an ingenious idea due to Wolff [18). This starts with two simple observations: (i) estimates like (1.3) hold uniformly for a family of functions and (ii) the estimate (1.4) holds provided that is localized on a sufficiently small set. Then the idea is to osculate the function and show that for some choice of it we get the appropriate concentration for The osculation argument uses linear weights.

Lemma 1.3. (Wolff [18]) Let

be a positive compactly supported measure in Define lik by dILk(x) = Suppose B is a convex body in R". Then there is a sequence {k2} C B and, for each i, a convex body Ek. with

such that {

} are pairwise disjoint and

>iEkJ1 CIBI 'Here

i- has

to stay away from half integers, for reasons which are explained later.

H. Koch, D. Tataru

76

where C is a positive constant depending only on n, and where IBI, denote the Lebesgue measures of B and Ek..

One can see that we do not get directly the concentration on a small set. Instead we get a dichotomy: either the concentration occurs on a small set or it occurs on somewhat larger disjoint sets for many choices of the weight. hi the latter case this has to be combined with the LTh summability for the norm of W1 on the disjoint sets Eke. Using this localization argument, one obtains:

Theorem 1.4. (Wolff [18]) Assume that n 9u E Lip,

3 and that V

W1 E

Then (UCP) holds. One would hope to have a similar result for the strong unique continuation problem. The difficulty is that in that case one needs to deal with weight functions which are singular at the origin. Using radially symmetric weights and a one dimensional osculation argument, Wolff [19] proved and V E The that (SUCP) holds for = W1 E exponent for W1 was later improved by Regbaoui [15] to W1 for

n >6. Our recent result in Koch—Tataru [7] completes the expected results on

(SUCP). We prove that (SUCP) holds under sharp scale invariant assumptions on the metric g and on the potentials V, W1 and W2. For simplicity we assume that Zo = 0. To state our assumptions on g, V, Wi and with norms W2 we introduce the spaces 1

>

1

< q < 00. (1.7)

go

over Z. For the

JEZ

Here for the sake of uniformity in notation we let j

strong unique continuation property only sufficiently small j's are relevant. In a similar manner we define the spaces l°°(LP), co(LP) and the weak jq spaces, Then we consider metrics g uniformly bounded from above and below and satisfying <E,

e small.

This does not imply that g is close to the Eudidean metric. However, in our estimates later on we use a perturbation argument starting from estimates for the Euclidean metric. This requires a stronger form of (1.8), namely 119 —

+

<e,

e small.

The reduction of (1.8) to (1.9) is carried out using a suitable change of coordinates.

Second Order Elliptic Equations

77

For the potentiaLs V, W1 and W2 we consider the following assumptions:

YE

E small,

respectively II W1

+

e

small.

A simpler replacement of (1.8), (1.10) and (1.11) is li(L00),

V

W1,W2 E

If this holds, then the smallness condition in (1.8), (1.10) is satisfied in a small neighborhood of the origin. Now we can state our result.

Theorem 1.5. Assume that (1.8), (1.10) and (1.11)hold. Then (SIJCP) holds at 0 for H' solutions u to (1.1). There are two reasons we add a second gradient potential W2 to the usual

problem. On the one hand this is a natural term to add due to a certain symmetry in our estimates. On the other hand, it is exactly the kind of term which arises in the study of the unique continuation problem for the Dirac operator. Our results are essentially sharp. On the one hand the counterexamples of Miller [11], Plis [14] and Mandache 110] show that the metric g" has

to be at least Lipschitz. On the other hand, the functions provide a straightforward counterexample with V E L", p < or with W p < n. The smallness assumption on V in is necessary due to a counterexample of Wolff [20]. However, uniqueness holds for V = for large C, see Pan [13]. Wolff [20] constructs counterexamples to V = W2 = 0. The only gap which is left in (SUCP) with W, E

our results is therefore the gap between and W1 E E This gap can be filled, at least to a certain extent, but only at the expense of making the proofs considerably more technical. The other longstanding open problem is that of finding sharp counterexamples for (UCP). In two dimensions this was recently achieved by KenigNadirashvili [6] and Mandache [9]. Our recent result in Koch—Tataru [8] is the sharp one for n 3:

Theorem 1.8. a) Let n

Then there exists a nontrivial smooth

3, p <

compactly supported function tz so that E

b)Let n so

L9(R").

3. There exists a nontrivial compactly supported function u E that W and €

H.Koch,D.Tataru

78

The rest of the article contains sketches of the proofs of Theorems 1.5 and 1.6.

2

The unique continuation result

The unique continuation result is a consequence of certain Carleman estimates. To motivate our setup we first take a closer look at Jerison and Kenig's estimate (1.3). It is easier to do that in polar coordinates

x = e'O,

(s,9)

Rx

Then

=

dsd9.

On the other hand, one can compute the form of the Laplace operator in the new coordinates, =

— (n

—

2)0, +

=

or, after further conjugation by

(n_2)2 Using the transformation v = Ixj"1u, Jerison and Kenig's result (1.3), for instance, becomes V

LP(C)

<

e

LP'(C)

where L

=

+

—

(fl — 2)2

The assumption is that u vanishes of infinite order at 0 and oo translates

into a faster than exponential decay for v when s approaches ±00. is + N)2. Therefore (2.1) The spectrum of + hold if

This accounts for the restriction that r should stay away from ±( + N). One can replace the exponential weight i-s by any other weight h(s) which is convex in s. The above argument shows that W must not stay

Second Order Elliptic Equations

79

for a long time near half-integers. The function h must also be globally Lipschitz in order for both sides of the estimate to be finite. if we look instead at the problem with variable coefficients, then the convexity of h must be sufficient to allow for the difference g — I,,. Also we want to consider perturbations of h depending on the angular variable, which must also be controlled by the convexity of h. These considerations motivate the choice of the family of weights we use. Given a large parameter r, we consider weights of the form

= h(—ln lxi) +k(x) where h is a convex function satisfying [r, r2],

h'

ih"'i h",

h" + 12h'

>

—

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 <
relations

+

—

<
respectively TI?

"1 L'(B(O,2r)\B(O,r)) <<

We call such functions

In the case W1 =

LIII

1

r>

admissible weights. W2

=

0

the result follows from a Carleman type

estimate.

Theorem 2.1. Assume that (1.9) holds. Then for each r > 0 and each admissible weight

we have (LP)

(2.2)

for all u vanishing of infinite order at 0 and oc. To deal with W1 and W2 we use Wolff's lemma locally in dyadic annuli, using the allowed range for the k component of

H.Koch,D.Tataru

80

Theorem 2.2. Assume that (1.9) holds. Then for each r > 0, W1, W2 E and each function u vanishing of infinite order at 0 and oo there exists an admissible function so that Ie

uII,p'(x

)

+

(L')

+

< e

U

(Lu')

(2.3)

The estimates of Theorem 2.2 deviate from classical Carleman estimates: we have to use different functions for each r. It is an Instead of essential feature of our estimates that the choice of the weight function ço depends on u, W1 and W2. It is known that such an estimate cannot be true for all r with independent of u. The estimates (2.2) and (2.3) are just a simplified formulation of the estimates we actually prove. These also contain bounds for Vu and gradients of integrable functions on the right-hand side of the equation. More importantly, they contain sharp L2 norms. To keep things reasonably simple, consider a range where h' = 0(r), h" = 0(er), 0 E 1. The parameter E measures the amount of convexity available in the weight. Then we introwhich of functions defined on the cylinder R x duce the spaces are to be used for v:

Xr.e={VEI/flT_3(1+er)_*L2, For the right-hand side of the equation we use the dual space,

= L"

+ V(L2

+tr)+VL2.

In R" we introduce corresponding norms by reverting the transformation we described earlier. Thus, we set

= which also implies that

= Then the stronger replacement of (2.2) is (2.4)

Now we are ready to describe the main steps in the proof. STEP 1. We prove (2.4) in the special case when k=0 (i.e., is spherically symmetric) and P = & This we do in polar coordinates, using a spectral decomposition with respect to the eigenvalues of the spherical laplacian. Then one needs to solve the corresponding ode's in s and combine this with Sogge's estimates for the spectral projectors.

Second Order Elliptic Equations

81

STEP 2. We use a localization argument to transfer (2.4) to the case

of variable coefficients and nonzero k. Here only the L2 estimates are important, as they allow us to localize the estimates to sets which are small enough so that we can freeze the coefficients and the function k. STEP 3. We start with the estimates (2.4) on dyadic annuli in x (which corresponds to intervals of fixed length in s) and glue them together using a suitable partition of unit.

STEP 4. We use Wolff's lemma in n dimensions to localize the terms containing the gradient potentials to small sets, where they can be controlled using (2.4). This is achieved by osculating the function k within its allowed range. Note that for this part of the argument the norms involving e are not needed.

3

The counterexamples

The construction of the counterexamples improves the argument in Kenig— Nadirashvili [6]. We start with a bounded sequence of disjoint increasing annuli centered at the origin Ak so

= {rk —ak

that the thickness ak of Ak is equal to its the distance to Ak+1,

ro=aoj

rk÷1—rk=ak+ak+1, Here ak is

a decreasing slowly varying sequence so that

Eak =1. Corresponding to the sequence Ak we define a sequence of compactly supported cutoff functions Xk so that x(°) = 1 and VXk is supported in Ak. Then we define inductively a sequence as follows. Set u1 = Xi. For the inductive step we start with some fk of the form 1k

which is "close" to

in Ak. Here

=

where

is smooth,

nonnegative, supported in the unit ball and having integral 1. Let Vk be the solution to the elliptic problem

Vk

I

outside Ak

Ilk

iflAk

= 0 on OB(0,rk+1).

H. Koch, D. Tataru

82

Then set Uk+i = Xk+iVIc. The counterexample u is obtained as the limit of Uk and solves

= k

j

To guarantee the convergence it suffices to ensure that Iuk+i — ukI

<2_k outside Ak.

Similar bounds can also be obtained for the derivatives. This can be achieved by choosing sufficiently many The limit u is smooth if 4 are large enough. and On the other hand, we want to obtain bounds for

this we need to ensure that within each ball B(4, 4) the contribution to u coming from all other mollified 5's is negligible, in other words that we can essentially replace u locally by the solution to For

This works provided that (i) the size 4 of the support of the mollifiers is small compared to the distance between different and (ii) the stay away from 0. To ensure that the summation with respect to j causes no problems we need to use as few 's as possible. Thus we have two opposite requirements to balance, concerning the num-

ber of xL 's, respectively the radii 4. There is an optimal way to choose the xi's, namely on a Sak lattice. Here S is a small parameter. To "cover" Ak we need about such points. Then the need to be as large as possible modulo the summability condition. For instance one can take

k',

ak

< —1.

The choice of 4 is more delicate. To see why, compute

<(4y2,

in

that for the second bound to hold we need to make sure that the (x — is supported away from the local zero of Vu. This zero is close to the center of the ball so it suffices to choose Note

function

nonnegative, spherically symmetric and supported away from 0. Compute

then the above exponent is positive. Then in order If we choose p < for large to sum up the norms in all balls it suffices to choose 4 =

Second Order Elliptic Equations

83

N. On the other hand a polynomial dependence on k is overridden by the exponential decay in k in the convergence argument, so the limit function u is smooth. then all these norms are 0(1). Then all we could hope However, if p = overall norm. Even this can only be achieved if we ensure that for is an the are in a decaying geometric progression. This exponential decay is no longer controlled by the exponential decay in the convergence of the uk's, so in this case we cannot get the smoothness of u. The argument for the ratio is similar.

Acknowledgements. The first author's research was partially supported by the DFG and the DAAD. The second author's research was partially supported by NSF grants DMS-9970297 and INT-9815286.

References [11 Giovanni Alessandrini, A simple proof of the unique continuation property for two dimensional elliptic equations in divergence form, Quaderni Matematici If serie 276 (1992), Dipartimento di Scienze Matematiche,

Trieste.

[21 N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appi. 36(9) (1957), 235—249.

[3] N. Aronszajn, A. Krzywicki, and J. Szarski, A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. Mat. 4 (1962), 417—453.

[4] T. Carleman, Sur un problème pur les systémes d'équations aux dérivées partielles a deux variables indépendantes, Ark. Mat.. Astr. Fys. 26(17):9 (1939). [5] David Jerison and Carlos E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. 2 (1985), 121(3):463—494.

[61 Carlos Kenig and Nikolai Nadirashvili, A counterexample in unique continuation, preprint. [7] Herbert Koch and Daniel Tataru, Carleman estimates and unique continuation for second order elliptic equations with nonsmooth coefficients, CPAM, to appear.

[8] Herbert Koch and Daniel Tataru, Compactly supported solutions to + Vu = 0, http://wvw.math.nuu.edu/ tataru/sucp.htrnl.

84

lLKoch,D.Tataru

[9] Niculae Mandache, A counterexample to unique continuation in dimension two, preprint. 1101 Niculae Mandache, On a counterexample concerning unique continuation for elliptic equations in divergence form, Math. Phys. Anal. Geom. 1(3) (1998), 273—292.

(11] Keith Miller, Nonunique continuation for uniformly parabolic and elliptic equations in selfadjoint divergence form with Hoelder continuous coefficients, Bizil. Am. Math. Soc. 79 (1973), 350—354.

[12] Keith Miller, Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hoelder continuous coefficients, Arch. Rat. Mech. Analysis 54 (1974), 105—117. [13] Yifei Pan, Unique continuation for Schroedinger operators with singular potentials. [14] A. Plls, Unique continuation theorems for solutions of partial differential equations. In: Proc. Internat. Congr. Mathematicians (Stockholm, 1962), pp. 397—402, Inst. Mittag-Leffler, Djursholm, 1963. forte pour les opérateurs de Schrödinger avec potentiels de Kato, J. Funet. Anal. 134(2) (1995), 281—296.

[15] Rachid R.egbaoui,

[161 Friedmar Schulz, On the unique continuation property of elliptic divergence form equations in the plane, Math. Z. 228(2) (1998), 201—206.

117] Christopher D. Sogge, Oscillatory integrals and unique continuation for second order elliptic differential equations, J. Amer. Math. Soc. 2(3) (1989), 491—515.

and an application to unique [18] T.H. Wolff, A property of measures in continuation, Ceom. Funct. Anal. 2(2) (1992), 225—284. [19] Thomas H. Wolff, Unique continuation for V1VuI and related problems, Rev. Mat. Iberoamericana 6(3-4) (1990), 155—200. [20] Thomas H. Wolff, Note on counterexamples in strong unique continuation problems, Proc. Amer. Math. Soc. 114(2) (1992), 351—356.

Hebert Koch Institut für Angewandte Mathematik Universität Heidelberg email: Koch©iwr.uni-heidelberg.de and Daniel Tataru Department of Mathematics Northwestern University email:

Strong Uniqueness for Fourth Order Elliptic Differential Operators Philippe Le Borgne ABSTRACT This article describes a new strong uniqueness result for fourth order elliptic differential operators. The main theorem extends the classical results for second order operators to fourth order differential operators which are factorized into two second order operators. Uniqueness is associated with the differential inequality u

Vul

+ C4

fri 2

frI

In

E _3ID0ui

(e > 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 <

ID°u(x)I

86

P. Le Borgne

where P(x, D) is an elliptic differential operator of order n with complex valued functions as coefficients (in our case n = 2 or n = 4); we are interested in obtaining the following property: for all function u E HTh(Cl), if u verifies (1.1) and (1.2), then u 0 in a neighbourhood of 0. We say that P has the strong uniqueness property at zero associated to the differential inequality (1.2). Naturally, the more singular at 0 is the potential V0(x), the more difficult it is to obtain uniqueness. 1.2

Assumptions and classical results

The most important assumptions are the following:

(Al) the coefficients of the operator P(x, D) are Lipschitz continuous;

(A2) P(0, D) = (A3) P(x. D) is factorized in two second order elliptic operators. Assumption (Al)

This assumption appears for the first time in the result of N. Aronszajn, A. Krzywcki and J. Szarski (1962) see [5]; they improved older results from

N. Aronszajn (1957) [4], H.O. Cordes (1956) [9], and also T. Carleman (1938) [61 who solved the uniqueness problem for systems of PDE with two

independent variables. In the two last cases, strong uniqueness is proved for second order operators; the coefficients were C2. Most recently, L. Hörmander (1983) [16] obtained strong uniqueness for elliptic second order operators P(x, D) such that P(0, D) is real; the coefficients being Lipschitz continuous. According to the notation of (1.2), we note that in 116], V0(x) = In his thesis, B.. Regbaoui (1995) [23] solved strong uniqueness for the second order operator as we mentioned above, and this time the improvement concerned the potential: V0(x) = C0 We mention also C. Grammatico (1997) 111] who improved B.. Regbaoui's inequality in the case of the Laplacian. Assumption (A 2)

In his work, S. Alinhac (1980) [1] constructed a strong uniqueness counterexample for all differential operators P of any order, elliptic or not, in R2, under the condition that P has two simple, nonconjugate complex caracteristics. Consequently the assumption "P(O, D) is real" according to L. Hörmander's result 116] cannot be extended to fourth order differential

operators. As soon as the order of the operator is higher than two, the ellipticity of P and the assumption according to which the coefficients are

Fourth Order Differential Operators

87

real in zero are not enough to obtain strong uniqueness; one can mention the example of the operator

P= which does not have the strong unique continuation property: it is possible to construct smooth functions u and a such that Pu — au = 0 in a neighbourhood of 0 and supp u is a neighbourhood of 0. Of course, under the assumption P(0, D) = &, S. Alinhac's theorem cannot be applied.

To conclude this result we recall that elliptic operators with simple characteristics which verifies Cauchy's uniqueness from any hypersurfaces (Calderon's theorem) do not verify strong uniqueness from any submanifolds of codimension greater than one. This illustrates that strong uniqueness is an exceptional property in comparison with unique continuation. To obtain more precise results about strong uniqueness from manifolds, the reader should consult S. Alinhac and N. Lerner (1981) [3]. Assumption (AS)

This assumption arises perhaps for technical reasons; we can find it in the paper published in 1980 by S. Alinhac and M.S. Baouendi (21. This work collects many results about strong uniqueness for differential operators of any order. In particular, Alinhac and Baouendi proved the strong uniqueness property for any fourth order operator with smooth coefficients verifying P = QIQ2 + R where and Q2 are two second order elliptic differential operators such that Q1(O, D) = (i = 1, 2) and R is a third order operator. The differential inequality relating to strong uniqueness is

IP(x,D)uI SCPxr

I Iul

ID°ul\

IVul IoI=2

in

/

this inequality, u is flat at the origin. The proof is based on pseudo-

differential calculus ; in this way, the regularity of the coefficients is obvious.

Strong uniqueness for fourth order operator More recent are the results of F. Colombini and C. Grammatico (1998) [8]; they give advanced results concerning strong uniqueness for the Laplacian and its powers. The inequality relating to the case of the bi-Laplacian is 2

2

1u12

C1 —i + C2 lxi

IVul2 + C3 lxi 6

lDauI IaI=2

where

C3 <

9

-, 4

and where is is smooth.

Results associated with more precise differential inequalities, written in polar coordinates in are shown there. This work was the continuation

88

P. Le Borgne

of a study (see [7]) providing a counterexample of strong uniqueness for a flat function w satisfying a differential inequality of the type

with h N and V a singular function at 0. In 1960, T. Shirota [24] proved a result of strong uniqueness for the product of two second order elliptic operators. The coefficients are of class C2 and the differential inequality is weaker than that of the principal theorem, i.e., the inequality (1.3). This result uses the techniques of H.O. Cordes [9] and L. Hörmander [13]. Cauchy 's uniqueness and unique continuation property for a fourth order operator

It is said that the operator P has the unique continuation property associated with the differential inequality (1.2) if any solution u of (1.2) that vanishes in a neighbourhood of 0 vanishes identically. This property associated with the strong uniqueness property as defined above allows us to obtain that u vanishes identically in !l. Unique continuation for the product P of two second order elliptic operators is a consequence of the result of P.M. Goorjian [10] which proved in 1969 Cauchy uniqueness for the operators that we consider here. The associated differential inequality is the It was then an improvement of an following one: IPul S C(E10;<3 older result obtained by M.H. Protter [22] for operators with principal part equal to the bi-Laplacian. It is necessary to replace these results in works concerning differential operators with characteristics higher than 2: uniqueness depends on the mutiplicities of the complex characteristics. We can also mention R.N. Pederson [20] for a result similar to that of P.M. Goorjian, R.N. Pederson [19], and also L. Hörmander [15] for the uniqueness of the Cauchy problem relative to operators with regular coefficients.

Counterexam pie of unique continuation property concerning operators of orders higher than 4 In R3, A. [21] gives the example of an elliptic linear partial differential equation, with real coefficients of class having a nontrivial solution

of class C°° with compact support, vanishing outside the unit ball. This example does not contradict the theorem of S. Alinhac [1]. Two examples of nonuniqueness relative to the Cauchy problem are also shown for operators

of order 4 and 6 defined in R2. These results were extended to elliptic operators in R2: for these operators the multiplicities of the characteristic roots are high; we return to C. Zuily [25] and L. Hörmander [12] for other details.

Fourth Order Differential Operators

89

Main result

1.3

Two differential operators are considered, Q' and Q2. whose properties are identical to those of the operators studied by R. Regbaoui [23); precisely, they are defined as follows:

• Q1(x, D) and Q2(x, D) are two second order elliptic operators with complex coefficients in

• • the coefficients of Qi (x, D) are Lipschitz continuous;

• the coefficients of Q2(x, D) are C2 in this condition allows us to easily define the product Q,(x, D) Q2(x, D).

If we note P(x, D) =

Q1(x, D) Q2(x, D), we can define a new operator of order 4 by setting A(x, D) = (x)D° and P(x, D) = A(x, D) + R(x, D), where R(x, D) is a third order operator. The operator A(x, D)

has Lipschitz continuous coefficients and the operator R(x, D) has bounded coefficients.

The main result of our study is the following theorem:'

Theorem 1.1. Let P(x, D) be a fourth order elliptic differential operator in fl, a connected open subset of (n 2); P has Lipschitz continuous complex coefficients in We suppose moreover that P(x, D) is factorized into two elliptic second order operators, Q, (x, D) and Q2(x, D), as defined above. I/there exists C1 (1 154) ande such that C1? 0, C3 < e >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 <4

verifying the assumptions of Theorem 1.1,

= where C is a positive constant and

when R —'0, is the constant appearing in (1.3).

This theorem allows us to use a weight stronger than the weights exploited by L. Hörmander [16] and R. Regbaoui [23] to obtain an inequality of Carleman. We choose the weight introduced by N. Lerner [18]:

The use of this last is justified by the exponential decay in integral evaluated in Theorem 2.1.

of the

Fourth Order Differential Operators

91

We thus pose:

with O
Theorem 2.2. (Carleman inequality for P(x, D)) For any large enough and for any function v E C000(X \ {O}) where X is a neighbourhood of 0 sufficiently small, there is the following estimate: i-' X

4

(\D1

2

,

where

(2.2)

C is a positive constant independent of

Theorem 1.1 is easily proved starting from the two theorems above. Let us notice first that the inequality (2.2) is still true for a function u and satisfying the inequality with compact support belonging to

(2.1). It is enough in fact to show (2.2) for any function u E verifying the inequality above where X is a neighbourhood of 0. To obtain with compact support, finally the same result for any function u it will suffice to use a regularization. Therefore we suppose that u E supp(u) c B(0, R), and verifies (2.1). For > 0, let be a function defined by =1

if lxi 2ij

=

and

It is also supposed that for all a E

al

0

if

lxi
4, we have

where CQ is a positive constant. We can apply (2.2) to

Each

term appearing in the inequality (2.2) has the form dx,

(2.3)

N. It is sufficient to show that after a development using Leibniz's formula in the preceding expression, the commutators (i.e., the terms appear) tend to 0 when tends to 0. obtained when the derivatives of Since —. D"u when ,j —. 0, the theorem of Beppo—Levi will where k

then allow us to conclude the convergence of the terms (2.3). One can further show that

I

Jfl(121
is bounded for any This results easily from the estimate (2.1) of Theorem 2.1 which allows us to treat the terms resulting from the derivation of the function in (2.3). Thus the inequality (2.2) is true for u.

P. Le Borgne

92

Proof of Theorem 1.1 Let u verify the assumptions of Theorem 1.1. According to Remark 1.4. it is enough to prove that the function u vanishes and in a neighbourhood of 0. By Theorem 2.1, we know that u E 4. The inequality (2.2) applies to the function satisfies (2.1) for all Iai where E is such that = 0 for ri 21?o, and = 1 for ixl
cf

dx

dx +

+ y4 J

101=2

+

dx

JzI
dx

J

IzI
(2.4)

1aI3 JIxI
We also have:

f

dx

=

f

+

JIxI>Ro

dx

IzI
Combining (1.3) and (2.5), we obtain:

f c (f IxI(Ro +

IzI
J

IoI=2 Ixl
+

J

1QI3 kI
J

+ IrI>Ro

(2.6)

where C is a new positive constant. This time, the inequality (2.4) can be

Fourth Order Differential Operators

93

used to estimate the last term of (2.6),

cf

Izl>Ro

+

IxI
(74

+ (72

f

—

dx

—

IaI=2

f

+ (1 IoI=3

dx

JlxI
for C a new strictly positive constant. positive. Then, We choose first R0 possibly small to obtain 1— is a decreasing function of xi and thus we have

it is pointed out that if

and if jxj >

>

lxi

w,(x)

the existence of new constants C

These two last remarks allow us to

and C' < I such that

f

dx> (76

—

C)

IxI>Ro

f

dx

JxI
+

(74_C)f < R0

+ (72 +

—

(1 —

C)

dx

JIxI
C') >

f

_n+SiDCkuI2 dx.

While making 7 tend to +00, we obtain u

3

0 in B(O,Ro).

Proof of Theorem 2.1

Usual system oipoiar coordinates Let x by setting r = lxi > 0 and w = x/r 0 in 3.1

where we define a C°° diffeomorphism from \ {O} to (0; +oo) x S"'. In the polar coordinates (r,wi,w2,... thus introduced, we have

is the unit sphere in

0

0

(/Xj

(fT

for

P. Le Borgne

94

is a vector field on S"'. By setting and 1l,r = 0, which means that t = iogr, one can substitute for r a new coordinate t such that 0

0

=e

When r —i

0, t —p bourhood of —oc.

—oo,

for x

+

0.

(3.2)

we will be interested in values of t in a neigh-

The system of coordinates thus defined is used by L. Hörmander (see 1)

[16]). The operators Q3 operate in L2

with the following properties:

(i)

(ii) >,

= 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
ff

dt

+ dt dw,

C" Jf

(3.7)

and C" are strictly positive constants. where Ck, Thereafter, we look for majorations of I(r, in) and J(r, in) by some terms with the largest power of r possible: intuitively the term which has the lowest power of r gives the quality of minoration of f dx after return to the cartesian coordinates. Thus estimations concerning I(r, in) are more precise than those concerning J(r, in). To obtain these estimates we use the harmonic spherical functions (see for example R. Regbaoui 123] or C. Grammatico [11]). It is known that the spectrum of as operator of is k(k + n — 2), k E N and each eigenspace can be identified with the space, denoted Ek, of the spherical harmonic functions of degree k. Therefore for all p, q = 1, 2, we have

=

+n—

where ink is the projection of in on Ek.

Let us give some arguments that involve the majoration of I(r, in). By using projections of in on Ek, there are

I(r,w) = kEN

Fourth Order Differential Operators

97

with

cQr,n(k)=(k—r)2(2+k—r)2(r+k+n—4)2(r+k+n--2)2. We denote

the function defined by

= where

is

(3.8)

—

+n

—

2))'

a positive constant and it is shown that, if 0

Jf +

<

9,

then

for any r chosen sufficiently large and such that

r E {n + n N}. This result gives that for any strictly positive real large, there exists a constant C > 0 such that

I(r, w)

(3.9)

— m)

for r sufficiently

dtdw +

JJ i&,w12 dtdw + Cr2

ff

dt dw. We

proceed in the same manner to obtain a minoration of J(r, w). The

passage to cartesian coordinates is now possible with the two precise details which follow. The first elementary computation using the properties of the vector fields allows us to obtain

f

=

r2fflwl2dtdw + ff!Otwl2dtdw

For the term concerning the second order partial derivative of v, a more precise calculation is necessary; it is shown that the most important terms of dr are written in polar coordinates as follows: >1oI—2

f

r4fJfwI2dtdw; 2r2EfJJIZiwI2dtdw; These two last computations are essential; we use them with (3.10): since minorization of J(r, w) is better than the niinorization of I(r, w), we obtain the Carleman inequality (3.5) after transformation in Cartesian coordinates.

98

P. Le Borgne

3.3

Proof of Theorem 2.1

We now show the inequality (2.1) of Theorem 2.1. This inequality is independent of the factorization of the operator P(x, D). We show the proof for an operator A(x, D) having only terms of order 4. Taking into account the inequality (1.3), the conclusion obtained remains valid for the operator P(x, D). The first arguments of the proof repeats those already used by L. Hör-

mander in [14J. By using the ellipticity of P(x, D) and the lemma of Friedrich, we show that if u E is locally integrable and verifies almost everywhere a rather general inequality of the type (1.2), then and, for lol 4, uE

f

ID°ul2dx = O(RN)

V N >0,

R —. 0.

It is shown then that Theorem 3.1 applies to any function v E that satisfies the estimate (3.11). We apply in fact the estimate obtained in Theorem 3.1 to yR = XRU, where u is a solution of (1.3) (and then verifies (3.11)) and where confirms that E

XRl

and

for

XR=0,

for

IxIR.

The value of the parameter R will be specified later on according to the

constant appearing in the differential inequality (1.3). The function XR is built such as

4.

(3.12)

By assumption on A(x, D), we have that IDavR(r)12.

IAt.x, D)(VR)(x)

(3.13)

IaI=4

The majorization (3.13) clearly uses the fact that the operator A(x, D) has

only fourth order terms. The following inequalities prove that one could have ignored this assumption under the condition that the coefficients of the lower terms are Lipschitz continuous. By combining the estimate obtained

Fourth Order Differential Operators

99

in Theorem 3.1 and the inequality (3.13), we obtain:

f —C0R2)

f

+ +

f

dx +

lxi

f

—

dx

dx

IoI=3

is a strictly positive constant. We thus deduce for some r sufficiently large:

where

J

D)(XRU)121x1" dx —

C0R2)

dx

JIxI
+ C,7r4 f

+ Cqr2 J IxkR

xI
+

+

—

dx

!zkR

f

(3.14)

Also we have

J

dx

+

=

f

1x18_2nIA(x,

dx

Jxt>R xl82niA(x,

dx,

(3.15)

and as u is a solution of (1.3), +

+

+

whereÔ, >1 for 1
(3.16)

100

P. Le Borgne

Combining (3.15) and (3.16) with (3.14) we get

f

}xIS_2nJA(x,

dx

IxI>R

(C,1r4 — C0R2) IaI=4

+ (C,,r4

J

—

dx

J

f dx

IoI=2

+ (C,?r2 —

Since

C3 <

we

dx. (3.17)

JIzI
have for C3 sufficiently close to 1, and ij sufficiently close

toO, —77—

> 0;

then we choose I? to have —

C0R2

> 0 and

—

> 0.

(3.18)

the inequalities (3.18) are well verified for a real Choosing R = Thus we have, with C, a 6 > 0 rather small (it is supposed that < new strictly positive constant:

cJt:I>R

D)xRuI2IxI" dx (3.19)

In addition, according to (3.12), we get

J

IxI>R

dx

(3.20)

Fourth Order Differential Operators

And also:

f

lDaul2 dx

IxI
—2r+2(aI—n

>

(3.21)

J

We combine then the estimate (3.21) with (3.19) and (3.20) to obtain 1

for all al

Ia?4

4; that is, still

f

=

,

when

1? —s 0.

(3.22)

lxI
We recall the constraint r E {k + k N} coming from Theorem 3.1. Since R, = 6r+, Rk —p 0 when k —' +00, the estimate (3.22) is still valid for R sufficiently small while replacing for example C by This finishes the proof of Theorem 2.1.

4 4.1

Proof of Theorem 2.2 Proof of Carleman's inequality for a second order operator

Since P(x, D) is factorized in two second order operators, it is natural to take advantage of an inequality of Carleman in poiar coordinates verified by each member of the product. As in the proof of Theorem 3.1, we use the system of poiar coordinates (t,w) definite for all x R" \ {0}, by x = where (t, w) E R x S" '.The following notation is adopted:

• for x = =

we define the function

in Rx S"' by setting

• for all differential operators P, we denote P., the operator when P is written with the vector fields when P is written with aE

a

(or

=

102

P. Le Borgne

Theorem 4.1. (Carleman inequality for Q1(x,D) and Q2(x,D)) Let Q(x, D) be an elliptic second order differential operator with Lipschitz confor y > 0 sufficiently large, tinuous coefficients verifying Q(0, D) =

\ 0) x

for all functions w E

C is a positive constant independent of Proof of Theorem 4.1 The proof again takes on certain arguments used by we get R. Regbaoui in [23J. While using + = 0? + (n —

+

=

+ &,

+

+ (n —

and then

e2tP.., =

+

+ j+IaI2

where the functions

verify

with d€

C,,0(t,w) =O(et) and d(C,,0) =O(et)

It1 ...

be the operator obtained from Q..1 when and C,,0 are reLet placed respectively by —Os, We want to obtain minorizations

and.

of

=

IIe2tQ,w112 — IIe2tQwIl2

S(y, w) =

+

We have D(-y, w) = 4Re((O? + + (n —

—

a27e°t

+

+n—

+ R(-y, w)

where R('y, w) is a sum of terms of the form _y4_I0I_Ii3IRe((TAOw,

with

IckI

< 2,

t—.-oo.

( 2 and T satisfies for IaI

—

A°v))

1, IA°TI = O(e_4atet) when

For t <0 and Iti sufficiently large, by using a result of L. Hörmander [14] (Lemma 17.2.4. p. 12), one can show, IR('y,w)I < 01<2

Fourth Order Differential Operators

103

verifies MQ(t,w) = Integrating by parts, we obtain for any function v with compact support

where

D(y,w)

+ — L(y, w) +

—

w),

where

= 1

and is a polynomial of degree lower than 1 — Concerning S(-y, w), a direct calculation yields

S('y,w)

+

+ +

+2

> IIe_4atIliwII2

—

+

—

where the polynomial qc, is a polynomial of degree lower than or equal to 3—

and

=

when t —i

—co.

One obtains then

+

'yD('y,w) +

+ 2>

+

—

lie

atIl1Wli2

+ 13a414lie_btwli2

—

where the polynomial is a polynomial of degree lower than 1 = satisfies For any e > 0, we have that e

=

= —

+ —

—

lai, and

104

P. Le Borgne

We deduce that w) + S(7, w)

+ (1

+

—

+ (13

+

2

—

(4.2)

— —

I°F1

Choosing for example 13 —

<

thus

>0

and

In addition, using the ellipticity of

> 0.

1—

(4.3)

there is a strictly positive constant

K such that

K

(4.4) IaI=2

and since

=

>

+

(4.5)

Taking into account (4.3), (4.4) and (4.5), we have

C

< (1 where

—

w112 + (13 —

C is a new strictly positive constant. Since for t <

—To

where T0 is

sufficiently large,

(a
>

+>

(4.6)

Fourth Order Differential Operators

105

According to (4.2), we thus deduce for some new strictly positive constant C, w) + S(i', w)) The awaited estimate (4.1) then reduces the preceding inequality since

7D(7,w) +

S e•v

+ l)IIe2tQ,wtI2.

Remarks

1. The Carleman inequality obtained remains unchanged with a modifica any operator tion to the constant C if we add to the operator of the form for

< 1.

(4.7)

This means that, in the expression of Q, the partial derivative of order

lower than two play a minor role to obtain (4.1). 2. As already noticed by L. Hörmander [16] and R. R.egbaoui [23], the assumptions on the coefficients of Q can be appreciably improved. Write Q D, D3; the proof above reveals that Theorem as follows: Q(x, D) = 4.1 is still valid when the functions are lipschitz in \ {O} and verify lVa,jI 0, if we choose a sufficiently small, we have e_10t >> e_2ate46t. It is easy to see that these conditions would not modify the result of strong uniqueness for the operator P.

4.2

Proof of Carleman inequality for P(x, D)

Let us notice first that to obtain (2.2), it is enough to prove the following inequality expressed in polar coordinates for any function w E C0°° (J — oo,To [ x (4.8)

112

IoI3 To see it. we use the expression in polar coordinates of the vector fields

(1 5 i S n) as in the end of the proof of Theorem 3.1; it is not difficult in this case.

Theorem 2.2 comes from the application of the inequality (4.1) to the operators Qi and Q2 in order to obtain a similar estimate for the operator P(x, D) = Q1(x, D) Q2(x, D). We show the following inequality, stronger — oo,To[xS"'): than (4.8), valid for any function w E CIIe4tyPl,1,__,w112

106

P. Le Borgne

The proof is easy. We have

= We apply the inequality (4.1) to the operator

This gives us

k2 The commutator

eZt} is

written in the form

e4t

where the functions

are the coefficients of the operator except for certain constants, these constants coming from the derivation of the exponential. It is a negligible operator as is those of the remark 1 above. According to (4.10), we thus have

CIle4tQ,w112

(e2tQ2_,w) 112.

IaI2

In the same way, one has tiIOIAa (e2tQ2

_IaI)atjtapAa,

=

+ and

= 1P1S2

iaI
where the functions

(i

QIe—1 _IoI)otAaw)

IaI2

the operator is still negligible: is a new bounded function. Thus in the inequality (4.11), according to remark 1 above, via a possible modification of the constant C one can replace the operator and

3

(e2tQ2.y)

by

_IoI)otAa.

Fourth Order Differential Operators

By applying again Theorem 4.1 to Q2, we obtain (4.9), where C is a new positive constant.

References [1]

S.

Alinhac, Non-unicitd pour des opérateurs différentiels

Ca-

ractéristiques complexes simples, Ann. Sci. Ec. Norm. Sup. 13 (1980), 385—393. [2]

S. Alinhac and M.S. Baouendi, Uniqueness for the characteristic Cauchy problem and strong unique continuation for higher order partial differential inequalities. Amer. J. Math. 102 (1980), 179—217.

[3)

S. Alinhac and N. Lerner, Unicitd forte

d'une variété de di-

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
Degenerate Cauchy—Riemann Equations

111

developments on higher order microlocalization are given in the paper f4J. We could formulate a more geometric result than dealing only with operators of type (1.1). Anyhow, it is important to stress that the straightening canonical transformations should be performed prior to the use of a pseudodifferential calculus: it is a standard result to reduce the study of a principal type operator with a complex valued symbol to studying

+iq(t,x,D2) where q is real-valued of order one. Now, if we knew that our operator satisfies the Nirenberg—Treves condition (P), it was proved by Beals and Fefferman in (1J that a nonhomogeneous microlocalization procedure led to

= a(t,x,e)61(x,e) with a nonnegative a of order 0, and a nondegenerate b1 of order 1 (db1 0 at b1 = 0). We need to perform a non homogeneous canonical transforma-

tion in the

variables to get a reduction as in (1.1). Then, only after this straightening of b1, we are able to use our two-microlocal calculus.4 In particular, it does not seem possible to work directly with ab1 as above and to find a pseudo-differential calculus tailored to the geometry of b1 in these coordinates. In order to prove that (1.5) describes a well-posed problem and to get for u the inequalities analogous to (1.3—4), we need first to prove some Hilbertian lemmas. This is the purpose of the next section.

2

Sharp estimates for abstract evolution equations

Lemma 2.1. Let 1111 be a Hubert space, t i—i Q(t) a tueakly continuous mapping from I = [T_, T÷] C R in £(E) such that there exists t E I, ReQ(t) =

+ Q(t))
41n fact, the (lrst microlocalization gives, after a homogeneous canonical transforma-

tion, the operator D, + iq(t, x, Di), then a nonhomogeneous microlocalization, due to Beals and Fefferman, gives the models + io(t, x, D1)b1 (x, Di). After this, we perform a nonhomogeneous canonical transformation to get 1), + iao(t, x, So the second microlocalization that we are using by quantifying the Heaviside function of appears in fact as a third step of microlocalization performed after two canonical transformations. 5This lemma is essentially trivial, but we want to emphasize that the constants occuring in (2.2) are the same as for a scalar ODE.

N. Lerner

112

u(t) be a C' mapping from I to El and define I = Let t it = du/dt. Then, for t E I, we have

+

J

(2.2)

stands for the norm in El. If p =

where

— Qu with

0,

one can take T_ =

—00.

Proof We assume first that p =0 and we calculate d

2

= 2Re(u(t),u(t))

= 2 Re (Q(t)u(t), u(t)) + 2 Re (f(t), u(t))

2Re(f(t),u(t)),

(2.3)

2Re(f(s),u(s))ds, and

so that fort El, Iu(t)12 — Iu(T_)12 Iu(t)12 < lu(T_)12

+

Iu(s)I ds = a(t).

J

Thus we obtain6 & = 211 lu < 2IfIi"2, i.e., lioh/2 Iu(T_ )I ,—.-—.,.——-..'

Iu(t)I

which is (2.2) when p =

yielding

ft

+j 0.

(2.4)

If(s)I ds,

To get (2.2) when p > 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 <

+ fT. / 21f(s)flu(s)Ids,

yielding (2.5) by the same method as before. We shall use also the analogous7

Lemma 2.2. Let El be a Hubert space, t Q(t) a weakly continuous mapping from I = [T_, T+] c R in £(EI) such that there exists js 0 satisfying for oil t E I,

+

ReQ(t)

Let t '—. u(t) be a C' mapping from I to H and define f =

(2.6) — Qu with

ü=du/dt. Then, fortE I, we have

+j If

=0, we can take

(2.7)

= +oo.

Remark 2.3. It may be also interesting to notice that the proof of (2.2) and (2.7) for a specific function u(t) requires only checking (2.1) and (2.6)

on u(t), e.g., for (2.1), Re(Q(t)u(t),u(t)) Remark 2.4. In the same vein, it is also easy to prove that the condition t

Re(Q(t)u(t), u(t))

7Lemmas 2.2—2.2' are obtained by changing t in .-t in Lemmas 2.1—2.1'.

N. Lerner

114 does

not change sign from + to — is

sufficient8 to get the estimate

2fIDeu+iQ(t)uldt

for u E

(2.8)

In fact, we calculate with the nondecreasing9 a(t) = sign(Re(Q(t)u(t), u(t)))

f 2 Re(Deu(t) + iQ(t)u(t), i(a(t) + sign(t — T))u(t))dt + + We

2sign(t

f

- T)Re(Q(t)u(t),u(t))]dt

get also the twin estimate of (2.5).

Lemma 2.2'. Under the o.sstimptions of Lemma

have, for t E

2.2, we

(T_,T÷1, —t)

1/2

+j

2

< Iu(T+)I

_3)ds]

Re( (Q(s) + 1u)u(s),

+

f

(2.9)

— Q(s)u(s)Ids.

Remark 2.5. It will be important in the sequel to notice that once the assumptions (2.1) or (2.6) are satisfied for some nonnegative they still hold for larger This implies that the estimates (2.5) and (2.9) provide — at the cost of changing the weight — an unlimited quantity of L2 norm.

3

Second microlocalization metric

Let values in

a continuous function on R x RTh x

x

[1,+oo)

with

such that sup

=

< +00.

8Although this condition looks quite useless, since it depends heavily on the distinguished function u, it contains essentially the results of the previous lemmas, e.g., in Lemma 2.2, this function is always nonnegative when p = 0. 91n fact, using the previous condition, one can find 0 such that sign(t — We take then o(t) = sign(t — 0). 0) R.c(Q(t)u(t), u(t)) = Re(Q(t)u(t),

Degenerate Cauchy—Riemann Equations

We assume that a0 =

115

A. We omit below the A-dependence of

0 for

C°°(R, [0, 1)) such that

00. Let

=

1/2,

1 for

=

so that

1

be in

=

1

We set also with

+Y(e1) = for

—1/2, ? = 0 A

C=

=0

—1/2,

[0,1)) such

for

—1/2,

0 for

=

and with

Let

=

(3.2)

1,

for

1/2.

that

—1, so that Yi' = Y.

for

(3.3)

=

1,

+ g

= jdxI2

+

+

(34)

with 1A 2

Lemma 3.1.

The

(3.5)

.

otherwise.

metrics C,g sati.sfy the inequalities C g gC

and

=

Moreover the metric g is slowly varying and temperate and is a gweight. The function Y belongs to S(1 , g) and for a S( 1, C) vanühing for A, the symbol

Proof The

belongs to

inequalities

are obviously satisfied since 1

over if A2,

with

On

=

= Y, we have

the other hand , if 1 +

A2 1 +

we get

since the triangle inequality implies —rn)2).

More-

N. Lerner

116

Moreover, if 1 +

we have

<1+77? <1. A2

—

1+77?), we obtain andg is thus proven = temperate. To prove that g is slowly varying, we assume that (X — Y) — 77112 r2, which implies Assuming 1 + A2, we get

Finally, if A2

< and

1/2, the inequality 1 +

then if 2r2

\1+77?

—

A2

2 +477? implies

j

1+77?

—

On the other hand, if 1 +e?

our assumption gives r2A2, implying A2 1 1 + 271? +2r2A2, which gives A2 < 2 + 471?, provided that 2r2 < 1/2. We obtain A2 —

\i +

J

—

'¼ 1

I

+ 'i?

—

— Y) 1/4 implies yr <4gx, which implies that g is slowly varying. To prove that is a g-weight, we assume first that max(1 + e?1 1 + n?) A2. We get then

Finally, we proved that

<

—

1+77?

<2 — + 2 9x( X—V ).

Ontheotherhand,ifl+e? A2 1

+ A2

<1 —

=

1+71?) A2 istrivialsincethen 1.

The function Y belongs to S(1,g) since (VP 1 and y(k)

is com-

pactly supported. If a satisfies the assumptions of the lemma, we have C g implies = where x

S(1,G) c S(1,g) and

E

the lemma follows from the

0

Leibniz formula.

Lemma 3.2.

Let a be

in S(1,C), vanishing for

A. Then, we have

+p

Degenerate Cauchy—Riemann Equations

117

with p and

the same result holds with?,? or? replacing Y. Moreover we have and for all m €R,

= Id,

+

C

Proof. The Weyl composition formula and Lemma 3.1 give for all N 1, )a(x,

y(k)

+

a)(z,

1
When k 1, the symbol beg) since y(c) is compactly supported. Since Y +? = 1, Y, Y S(1,g) and YY is compactly supported, we obtain the last statements of the lemma. 0 withpN longs to

E

Lemma 3.3. Let a be a nonnegative symbol in S(1, G), vanishing for A. There exists C, depending only on a finite number of semi-norms of a, such that

+ C 0 on

+ C 0, The same result holds with?, —? or

replacing Y.

belongs to The symbol is nonnegative, with the same semi-norms as a in S(1, C). CArding's inequality with gain of one derivative and Lemma 3.1 give the first result. The second result follows from the first one and from Lemmas 3.2 and 3.1, which implies the L2-boundedness of operators with symbols in S(1, g). 0 Proof.

4

Energy estimates

Let us now consider L=

where a0 is given in (3.1) and r0 E S(1,G) uniformly in t. Let I = [T_,T+] be a compact interval of R. We have for u using Lemma 3.2, YWLU =

+ = DtYu7u + +i =

+

+ + YWfWYWu

+ + Ytu

Y"u +

N. Lerner

118

Moreover, using Lemma 3.3, we get that there with P1 E depending only on a finite number of semi-norms of exists a constant and r0 such that —

jti

Lemma 2.2 then gives for

jyWu(t)

zywrw) >

-a) (yWLU)(s) Ids i

IYwu(T÷ ) I +f

+j

(4.2)

In the same way, using Lemma 2.1 and Lemma 3.3, we get with p2 E and a constant depending only on a finite number of semi-norms of a0 and r0, such that —

for

+J +

j

(4.3)

(s)Ids.

Thus, we have, assuming p max(p1,p2), +

+ pt

+

+j

J

pt

(4.4)

+ I

+J Jt Moreover, using (2.5) and (2.9) instead of Lemmas 2.2 and 2.3, we obtain also the estimates r

—t)2

)W —

YWU(j))dt

+

LI r

)W

+ LI If

+

+

+ +

j

+j

I(ywLu)(s) Ids —a)

+j

Ids + j

Ids

Ids.

Degenerate Cauchy—Riemann Equations

119

Choosing large enough with respect to a finite number of semi-norms of a0 and r0, we ensure

+

0

+

and

0

so that we obtain —t)2

f

)W

i 1/2

Y1vu(t))dtj

+

I

r , + 1]

+

LI

+

f +f +

+f +

f

Moreover, using Lemma 3.3. and choosing J2 large enough with respect to a finite number of semi-norms of 00 and r0, we get and

and thus r

if LI

—t)

)W

+ ,1)ywu(t) ywu(t))dt

r I

+ + +

f

+f

j

+j

(4.5)

According to Lemma 3.3, we can choose j.z large enough with respect to a finite number of semi-norms of 00, r0 to secure

+

0

and

+

0.

(4.6)

120

N. Lerner

We obtain, adding (4.4) and (4.5)

+ 1/2

(j

+

1/2

+2h,.iV2 (j r

+ 11J +

t)

)W

)yWu(t)

+

1/2

[fe_2

+

2

+

f

—.)j (pru)(s)

j

Ids

Ids + j

Ids.

(4.7)

From Lemma 3.3, (3.2), there exists C 0 depending only on a finite number of semi-norms of a0, such that Q=

=

C)WYW =A>O

+

—

+

(4.8)

=B>0

with E

can note also that from the results of symbolic calculus of Section 3, are L2 bounded operators. This and justifies the notation for the nonnegative operator We

= We write (4.7) as

+ C)WYW +

+

(4.10)

Degenerate Cauchy—Riemann Equations —e)

+

Y'1'u(t)

+ 2_'p"2

121

IIe_hL(t_T_ )

L2(LL2(It"))

+ +

I i.ii p

r

)W

If LI

+

2

j

+

e_SA(T+_8)I(YWLu)(s)lds

+f

/L

+

2j

+j

(4.11)

We need to get rid of the last two terms in (4.11), using the same argument as in [Hi) (Lemma 26.7.2).

Lemma 4.1. Let A and d be positive numbers with A 1. Let u(t,x) be a function in S(R x with suppu C {kzl d/2}. Then for N> 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},

<0}),

then E L2(R"). Moreover, we define Lrm,m —

n Hm.fh

—

Let us consider the oriented hypersurface

= 0}

71

in

and the nonzero conormal bundle of 'K, T*(fl). This is a 2nparametrized by

dimensional submanifold of

x R"'\{O} x R\{0}

—'

x Rx x

x R\{0} x

R'1 (5.6)

The conormal bundle N(71) is identified with 71 x {—1, +1). The Hm,m' wave- front set is defined naturally as a subset of N (71), WFm,m'(U) =

It is a closed conic subset of

P=

Z) E Nl(7I) u (71).

(5.7)

Let us consider the operator

=

+iQ(t)

(5.8)

Degenerate Cauchy—Riemann Equations

0. Assume that

uniformly in t and ao(t, x,

where a0, r0

Pu u(T4,.) u(t,.)

u(T_,.) for alit E

E

Then we want to prove that

for all t€l.

u(t.) E

We return to the energy method. We calculate, with T E [T_,T+], Xo E 5° and setting v(t) = supported in a conic neighborhood of (x0, 2

=

2

=

j

Re(Dtv + iQ(t)v, ie_2T+_t)1(T,T+)(t)YWYwv)

dt

+

T JYWv(T)12

+

j

—

IY't'v(T+ )12

—

lYuJv(T+)12.

2

T

rT+

+ jT (which is purely imaginary) mod-

The symbol of [yw,Q) is ulo This implies that

=

with wo,_i E

On the other hand, using the notation (3.3) and Lemma 3.3, we get that Re

+

+

= Re —

C)YW

if iz 2C (C depends only on a finite number of semi-norms of ao and r0). As a consequence, we get 2

+ 2 Re([P,

)

+

j

T

+ JTI

+ &2,4(T+ —T)

126

N. Lerner

Once again, we examine the commutator LP, xow] = i[Q, xou] with a realvalued symbol in 50.0 modulo terms in S°''. Since the operator we get with r0 —i E has a real-valued symbol in S°'° up to terms in —t)

2

+

)

JT

+

)2

—T) I

12 I

+ jT and thus

2j

sup

t)IYWX0WUI + —

Cje_21A(T+_t)

dt.

Consequently, we obtain

/

2 (J

2

+

cj

dt,

as well as

2 (f e_p(t_T_

sup

+

dt,

yielding the following theorem.

Theorem 5.3. Let I = defined on I x

[T_, T÷] be a compact interval of the real line. Let be classical symbols of order 0, i.e., smooth functions x such that

sup

+

x R"

Let us assume that for all (t, x, the operutor P as

P=

+ E I x R'1 x

ao(t, x,

=

0. We define

Degenerate Cauchy-Riemann Equations Let s, s' be real numbers, and (so,

Vt E I,u(t,.) E

u(T+,.) e

127

E W' x (R"\{O}) and assume that tz(T...,.)

Pu E L'(I,

Then we obtain that

u(t,.) E

for all t E I.

References [1] R. Beals and C. Fefferman, On local solvability of linear partial differential equations, Ann. of Math., 97 (1973), 482—498. 121 J.M. Bony, Second microlocalization and propagation of singularities for semi-linear hyperbolic equations. In Hyperbolic Equations and Related Topics (Mizohata, ed.), Kinokuniya, 1986, pp. 11—49.

[3] J.M. Bony and J.Y. Chemin, Espaces fonctionnels associés au calcul de Weyl—Hörmander, Bull. S.M.F., 122 (1994), 77—118.

[4) J.M. Bony and N. Lerner, Quantification asymtotique et rnicrolocalisations d'ordre supérieur, Ann. Sc. ENS, 22 (1989), 377—483. [5] J.M. Delort, FBI Transformation, Second Microlocalizotion and SemiLinear Caustics, Lect. Notes in Math., 1522, Springer, 1992.

[6] L. Hörmander, The Analysis of Linear Partial Differential Operators, Springer-Verlag, 1985.

171 L. Hörmander, On the solvability of pseudodifferential equations. In Structure of Solutions of Differential Equations (M. Morimoto and T. Kawai (eds.), World Scientific, Singapore, New Jersey, London, Hong Kong, 1996, pp. 183—213.

[8] B. Lascar and R. Lascar, Propagation of singularities for a class of non-real pseudo-differential operators, C. R. Acad. Sci. Paris Sér. I Math., 321(9) (1995) 1183—1187.

[9] B. Lascar, R. Lascar, and N. Lerner, Propagation of singularities for non-real pseudo-differential operators, .J. Anal. Math., 64 (1994), 263— 289.

[101 B. Lascar and N. Lerner, Resolution de l'equation de Cauchy-Riemann dans des espaces de Cevrey, Soumis a publication. [11] G. Lebeau, Deuxième microlocalisation sur les sous-varietés isotropes, Ann.Inst. Fourier, 35(2) (1985), 145—216.

128

N. Lerner

[121 N. Lerner, When is a pseudo-differential equation solvable? Ann. Inst. Fourier, (2000) 50, 2(Cinquantenaire), 443—460. [13] L. Nirenberg and F. On local solvability of linear partial differential equations, Comm. Pure Appl. Moth., 23, 1-38, (1970), 459—509; 24 (1971), 279—288.

[14] J. Sjöstrand Singularités analytiques microlocales, Astérisque, 95, (1982).

University of Rennes Un. Rennes 1, Irmar, Campus de Beaulieu 35042 Rennes cedex, France email: lerner©univ-rennesl .fr

A Gârding Inequality on a Manifold with Boundary Nicolas Lerner and Xavier Saint Raymond 1

The main result

x and m E Let a E one has estimates of order in if

R.

The function a is said to be a symbol +

such a symbol we x for all multi-indices a, fi associate the pselAdo-differential operator (of order m) a(x, D) : defined by uniformly on

a(x, D)u(x) =

f

a(x,

We thus get a whole pseudo-differential calculus with the following classical properties:

• The differential operators a(x, D) =

with smooth aQ(x) (bounded) coefficients are pseudo-differentTal operators with symbols = >ioi<m aa(x)r (this follows from the inverse Fourier formula).

• This class of operators is stable under adj unction and composition, and this means that we have a(x, D)' = a'(r, D) and a(x, D)b(x, D) = D) for some symbols a' and altb that can be written explicitly in terms of a and b.

• The operator a(x, D) associated with a symbol a of order in is continuous from the Sobolev space into for all s which means that there are constants C8 such that Ila(x, C3

R,

:= C,(f(1 +

Such a pseudo-differential calculus allows us to replace computations on operators with more algebraic type computations on symbols, for which usual techniques of analysis are available: localization by multiplication by cutoff functions, Taylor expansions, etc. From this point of view, Gdrding

130

N. Lerner, X. Saint Raymond

inequalities provide a link between inequalities on symbols and inequalities on operators. independent of x, for which The very simple example of a symbol

is the Fourier transform of a(x, D)u and that can be treated through the Plancherel formula, shows that one can expect an estimate on f a(x,D)u(x)u(x)dx when the L2 scalar product lZe(a(x,D)u, u) = the symbol 1?.e a(x, is assumed to be nonnegative. In the case of a symbol actually depending on both x and we first observe that for any symbol a of order 2m,

I(a(x,D)u, u)I Ija(x,D)uI}_,,, IltSiIm since

the operator a(x, D) is continuous in Sobolev spaces, whence

Re(a(x,D)u, u) The sharp Gárding inequality states that this inequality still holds true for

symbols a of order 2m + 1 provided that iZe a(x, 0 on x W'. In the early work of GArding [2], this estimate was proved under the elliptic assumption for large by freezing the coefficients at some points, by using the Plancherel formula for the operator with constant coefficients thus obtained, and by estimating the difference from the actual operator with variable coefficients. The ellipticity assumption was removed by Hörmander [5) who proved this inequality by means of integration by parts, but in a microlocal way,

that is after cutting the symbol into pieces in the phase space. However, it is known that the differential operators are the only pseudo-differential operators satisfying the local property supp(a(x, D)u) c supp u for all u E S(R") and therefore, Hörmander's proof did not allow him to control the support of the function u under study. This is why it is necessary to work globally on However, in the applications, inequalities on the symbol a can be estabFx lished (usually) only locally, say for (x, where F is a closed

subset of If there is a security distance between the support of the function u in the inequality and the complement of F in we get the result by using cutoff functions which allow us to work with everywhere nonnegative symbols without modifying the action on u of the operator a(x, D) more than by well-controlled terms. But does this inequality still hold true for functions u supported in the whole of F? Our main result, proved in [7], is an answer to this question in a very particular situation. and leta be a symbol of order2m+1 satisfying Theorem 1.1. Letm E iZea = b + cx,, with (everywhere) nonnegative symbols b and c of order

2m + 1. Then there exists a constant C such that

lZe(a(x,D)u, u)

A Carding Inequality on a Manifold with Boundary

for

131

O}.

with

It is possible to give a more geometric statement of our result by considering a pseudo-differential operator on a manifold with boundary acting on half-densities, hut it is useless to give such a statement here. Let us point out that the situation considered in Theorem 1.1 is particular for three main reasons: first, because the closed subset F is the half-space with smooth boundary and not any closed subset of RTh, second, because the symbol is assumed to satisfy iZe a = with nonnegative symbols b and c, which is more than just assuming that iZe a(x, 0 when 0,

and third, because the order of the operator has to be an odd positive integer.

But actually, it turns out that this particular situation is exactly what we needed to apply this result to the uniqueness problem that motivated our work.

2

Motivation: Uniqueness in the Cauchy problem

Let u be a solution of a partial differential equation a(x, D)u = 1. We want to know if, localLy, u is completely determined by its values in a half-space { x E RYL; < 0 }. Since the equation is linear, this means that considering the solutions of the equation a(x, D)u = 0 supported in 0), we ask whether these solutions u are automatically { x E Wt; zero in a neighborhood of the point x0 E { x E Rn; = 0 }. We can already prove the following qualitative theorem without any com-

putation, just by waving hands: the uniqueness property is linked with convexity of the half-space { x 0). Indeed, if the uniqueness property at x0 is valid for the half-space { x E W'; 0 }, that is if all the solutions supported in this half-space vanish near xo, then it is also valid for any more convex half-space { x 0 }, that is {x satisfying Zo 0} C {x R"; 0}, since then supp u C { x E R'; 0 } implies supp u C { x R"; 0). There exists a quantitative form of this theorem that is due to Hörmander [4], and

that can be stated as follows: the uniqueness property at x0 holds for the half-space { x E R'1; 0) as soon as a condition of strnng pseudoconvexity is satisfied. This condition of strong pseudo-convexity more precisely means that

+

>0

for all tangent, complex characteristics ( E + i iR \ (0). In this formula, the +••• means additional terms that make it invariant under a change of variables, but that do not deserve being explicitly written here. Let us sketch the proof of this result. Roughly speaking, the idea is to integrate by parts in the integral f Ja(x, D)u(x) 12 dx to get an estimate of

132

N.

Lerner, X. Saint Raymond

the form

dx,

J Ia(x, D)u(x)12 dx

from which we can conclude that a(x, D)u = 0 implies u = 0. But it is a local problem and we have only local assumptions, so that the integral must be limited at a neighborhood X of ZO, and then we get from integration by parts a boundary term that prevents us from concluding. To get rid of this boundary term, we introduce in the integrals a weight function in order to get an estimate of the form: Vy>> 1,

f e

IBu(x)l Iu(x)I dx,

Iu(x)i2 dx

called a Carleman estimate. Assuming that the family of open sets

is a basis of neighborhoods of x0, we can write our Carleman estimate with the solution of a(x, D)u = 0 and the open set X26 to get IBu(x)I Iu(x)Idx

It follows that Iu(x)12 dx < E1 0 in X5 by letting 1, and then we get u

Iu(x)I2dx

go

IBu(x)I Iu(x)I dx for all y>> to infinity.

Actually, instead of integrating on X25 and getting boundary terms through an integration by parts, one can use a cutoff function x satisfying x 1 in X6, supp x C X26 and supp x' close to OX26, write our Carleman estimate for x u and replace integration by parts with a Gârding inequal.. ity. We thus have to prove an inequality a(x, u for all C8°(X26), that is the inequality V using the notation v = u, and A,, = a(x, D) (the operator a(x, D) conjugated by the weight function To get this last inequality, we write

e

= ((AA,,—A,,A;)v,v) = ([A,A1]v,v), and when computing the symbol of the commutator [A, A,,), we find that it looks much like Hörmander's expression () + •: we () thus get the link with the condition of strong pseudo-convexity. Thus, the conclusion of these computations is that

A CArding Inequality on a Manifold with Boundary

133

• first, it is possible to prove a Carleman inequality with the weight function provided that the half-spaces { x E 5 } are sufficiently convex; and

• second, it is possible to use them in the proof of the uniqueness property at xO for the half-space { x 0 } provided that this half-space is itself even more convex (so that the family of X6's is a basis of neighborhoods of Xo). On the other hand, there are results due to Alinhac [1), Robbiano 18] and Saint Raymond [9] showing that the uniqueness property does not hold any more when Hörmander's expression + takes negative values at x = xO, and this leads us to discuss the case of

weak pseudo-convexity where the positivity assumption is weakened into a nonnegativity assumption on the surface { x RTh; = 0 }. To prove a

Carleman estimate in this case, we would have to establish the nonnegativity (everywhere) of the symbol of the commutator [A, A variant of this method is to remark that, by convexifying the level surfaces of the function tb, we can force this symbol to become nonnegative on the 0 side, but it is at the cost of making it negative on the sa(x) <0 side. And this is where we need a GAMing inequality in a half-space such as that of Theorem 1.1. Let us quote one of the results we get in this way.

Theorem 2.1. Let a(x, D) be a differential operator with real principal symboip of order three. We assume that

=

=

{p, 0

\

E

and

{{p,

{0}, 0

0,

and that the surface S = { x E RTh; = 0 } is noncharactenstic at Xo (these are transversality assumptions). We also assume that the surface S is weakly pseudo-convex, that is 0

0

for all x close to x0 on the surface S. Then the compact uniqueness property holds at xo for the Cauchy problem, that is: any solution u E of a(x, D)u = 0 supported in {x E > 0) U {x0} vanishes in a whole neighborhood of xo. Sketch of the proof. Since we want to prove a compact uniqueness property, it is sufficient to use the function itself as a weight function. However, as explained above, to force the principal symbol of the commutator [A,

to be nonnegative on the 0 side, we have to convexify the level surfaces of the weight function there. To get this, we will use a weight function of the form = F(A(x) ço(x)) for suitable functions F and A.

134

N. Lerner, X. Saint Raymond

with )t(X) = l—B1x12, for More precisely, we set = positive parameters A and B to be chosen. Then, the conjugate operator a(x, D) e1, lies in a class of pseudo-differential operators A1, = e7 with large parameter and the principal symbol of A.1 in this class is we develop the = — i'-yil/(x)). Then writing r = then —i as polynomials in r. Finally, discussing the different terms thus obtained with arguments adapted to the different points (x0, r), we can construct three symbols q, r and s such that

polynomials

e)) = r(x,

+ 2 lZe(q(x,

—i

-y)

+ w(x) s(z,

'y)

for (x, -y) E x x[i, oo[, where is a neighborhood of x0, and r and a are everywhere nonnegative symbols. Let us explain how this can be done in the most significant situation, that is near the points (xo, 0) where

= 0. Because of our transversality assumption, {p, = {p, {p, the functions ço, p and {p, can be taken nearby as local coordinates in the phase space, and this gives {p, in a + ri + C0 neighborhood, which leads to a second estimate

p=

+ ri this estimate and from the development of third estimate

—

r 3/

we get a 2

from which we can construct the symbols q, r and a by using a Taylor formula.

Once these symbols q, r and s are constructed, we can use the Girding inequality of Theorem 1.1 for the symbol r + a (or rather a variant of this inequality adapted to the pseudo-differential calculus with large parameter -y), and this finally gives the Carleman estimate that implies the compact uniqueness property. 0 The same method also allows us to prove a uniqueness result for principally normal operators of biprincipal type of any order, but under a technical and rather intricate pseudo-convexity assumption. The interested reader is referred to our paper 171.

3

The proof of the Gârding inequality

The proof of Theorem 1.1 we now describe mixes arguments of two different types. On the one hand we use very standard pseudo-differential calculus,

but we also need some results from abstract operator theory. Indeed, the

A Carding Inequality on a Manifold with Boundary

135

main problem is to estimate the commutator of a pseudo-differential opwhich is not even erator with the operator of multiplication by Lipschitz continuous, but only in the Holder space C1'3. Actually, it is possible to get an estimate in this situation only because there is another factor that controls the commutator. But this business leads us definitely out of the pseudo-differential calculus, and our commutator then needs to be treated thanks to an abstract estimate due to Nirenberg and Treves. However, these operations can be performed only microlocally, and there-

fore we need to cut our symbol into pieces, and this produces remainder terms that also have to be estimated. At this point, we use the classical Cotlar's lemma on almost orthogonal series of operators, and this tool requires an additional estimate that must be understood at a second microlocal level x,, =0 }. Actually, around the conormal bundle of the boundary { x E this last step is not surprising since all the problem is located at points where the symbol becomes negative when x gets across this boundary. Let us now discuss these different steps more precisely. We begin with several reductions. The first one shows that it is sufficient to treat the case in = 0, that is the case of a symbol x8 a of the first order,

where a 0 is also a symbol of the first order. Usually this reduction comes from the composition with an elliptic pseudo-differential operator, but here, we must be more careful since we want to keep a precise control on the supports of functions, and this is why we need to assume an odd positive integer order in Theorem 1.1. The next reduction shows that we may assume that supp u C { x R"; 0 I }. This seems clear since the problem is located around { x E R"; x,, = 0 }, and this reduction as a bounded operator on L2. allows us to consider multiplication by Finally, instead of the usual quantization formula, we will use rather the Weyl quantized operator a's' defined by

a u(x)=(2ir) —n

j

e

i(x

\

V'

2

and satisfying a's' — a(x, D) bounded on L2(R") for all symbols a of the

first order. In this context, the inequality to be proved can be written S(R") supported in the strip u {x ER"; 0 : 1}. By using the properties of Weyl quantization, then by noting that x,, U = stands for the operator of multiplication by u where I + Zn), we have

u) =

u) +

Moreover,

=

u),tt) =

a" + a'° x+)u ,u).

136

N. Lerner, X. Saint Raymond

Thanks to the usual (sharp) Girding inequality,

if NT = this gives

((xna)"u, u)

—

IINTuIIL2

11u11L2

is bounded on the subspace of L2(Rtl) functions supported The operator 05 S 1 }, and an inequality of in the strip { x E in the abstract theory of operators bounded on a Hubert space states that we would have IiNTuIIL2

II [xn,aW]u

2

II [Zn,

if &" were an operator that is bounded on

However, this is not the case for our operator a" since a is a symbol of the first order. Therefore, we have to proceed microlocally, that is by writing our operbounded on L2 (Rn) with a ator a"' as a (divergent) series of operators where For this, we set control on the norms in = = are compactly supfor ii 1 and =1— and we ported functions. Now, the operators are bounded on inequality, but to guarantee the summamay apply the bility with respect to ii of the estimates, we still have to "tune" frequences where in u, that is to let the operator au)" act on the function taking the value 1 on the = for a function E support of Then we can write f

.,W_ —

fWJ

\W fW

i'>O

for an operator R bounded on L2(R"), but now, it is not possible any longer since here we have the with the operator to replace multiplication by that makes a screen between and u. Multiplication by Zn operator —x_, and this gives two terms to be estimated: one is therefore equal to that can be estimated thanks to the Nirenberg—Treves term containing inequality as explained before, and another term of the form

u) =

u)

H, an expression where we when setting A,, = H (x_ a,,W + a,,W x...) have added the operator H of multiplication by the Heaviside function to keep in mind that the support of u is contained in the half-space {x Zn O}. To estimate this last term, we use Cotlar's lemma that reduces the proof A,, to an estimate of of the continuity in of the operator 1/2 1'

C(L2)

+ A Ap

1/2

A Carding Inequality on a Manifold with Boundary

137

Since x_ is the operator of multiplication by the Lipschitz continuous function = H( — xc), we have Hx_ = 0 and + [x_, H, which is a useful expression because it is easy to get esH timates for commutators of a pseudo-differential operator with the operator of multiplication by a Lipschitz continuous function. Moreover, this shows

and AA,,I =

that we can write AVA =

for operators B,, and C,, that are uniformly bounded on L2(R"). Therefore, we are thus led to estimate and prove its summabifity with respect to ii. This is the purpose of our second microlocalization estimate that can be stated as follows. Lemma 3.1. Let H be the operator of multiplication by the Heaviside ftsnction let be a smooth function with support contained in the domain R'2; 4}, and set = Then there ezists E a constant C independent of 0 and ii 0 such that

I/A—v( 6

Since we may assume that v +6, the supports of and of are disjoint, whence H = (H — and since the Fourier transform of H — is equal, up to a constant, to Plancherel ® formula gives Proof.

= Again since z'

cJ on the domain

,u -4- 6, we have (r —

of integration, so that we get an estimate

H

v) I

C' 2" f

r)

r)I dr)

And then, using the Cauchy—Schwarz inequality first inside the parentheses

and second for the integral with respect to

C'

sup

f <

that proves the lemma.

(

we

finally get the estimate

r)2dT)"2

r)2 dr)"2)

r)12 II1LHL2 II VilLa

138

N. Lerner, X. Saint Raymond

N.B. In his Ph.D. thesis [3], F. Hérau announces an improvement of our Carding inequality stated in Theorem 1.1, where he considers symbols with a limited smoothness with respect to the variables x.

References [1] S. Alinhac, Non-unicité du problème de Cauchy, Ann. of Math., 117 (1983), 77—108.

[2] L. Carding, Dirichlet's problem for linear elliptic partial differential equations, Math. Scand., 1 (1953), 55—72.

[3] F. Hérau, Opérateurs pseudo-différentiels serni-bornés, Ph.D. thesis, Rennes, September 1999.

[4] L. Hörmander, Linear Partial Differential Operators, Springer, Berlin, 1963.

[5] L. Hörmander, Pseudo-differential operators and non-elliptic boundary problems, Ann. of Math., 83 (1966), 129—209.

[6] L. Hörmander, The Analysis of Linear Partial Differential Operators III & IV, Springer, Berlin, 1985.

[7] N. Lerner and X. Saint Raymond, Une inégalité de Carding sur une variété a bord, J. Math. Pures AppI., 77 (1998), 949—963. [8] L. Robbiano, Sur les conditions de pseudo-convexité et l'unicité de Cauchy, Indiana Univ. Math. J., 36 (1987), 333—347.

[9] X. Saint Raymond, Non-unicité de Cauchy pour des opérateurs principalement normaux, Indiana Univ. Math. J., 33 (1984), 847—858. Nicolas Lerner Université de Rennes 1 Département de Mathématiques Irmar Campus de Beaulieu, 35042 Rennes Cedex, France .fr

Xavier Saint Raymond Université de Nantes Département de Mathématiques 2 chemin de la Houssinière BP 92208, 44322 Nantes Cedex 3, France and UMR 6629 du CNRS

Some Necessary Conditions for Hyperbolic Systems Tatsuo Nishitani 1

Introduction

In this article some necessary conditions in order for the Cauchy problem for hyperbolic systems to be well posed will be studied. In the scalar case, from Ivrii—Petkov [1J, for the well-posedness of the Cauchy problem, a set

of vanishing conditions on the lower order terms must be verified at a multiple characteristic. Our purpose is to find some necessary conditions which correspond to the Ivrii—Petkov conditions for systems. In 12], we obtained a necessary condition in this direction. Here we continue this study.

Let m, N be positive integers and let fi be an open neighborhood of the with coordinatesz = (x0,xi,... = origin of We denote

by P(x, D) a differential operator of order rn with N x N smooth matrix valued coefficients. We write

where Pm..1(X, D) is the homogeneous part of degree rn—j. We assume that = 0 has only real roots with respect to for all x E fI and det Pm(X,

E W'. We also assume that the hyperplane {z0 = const.) = (b,... is noncharacteristic for P. Let (0, be a characteristic of order r, that is

d1Pm(0,en)=O, j=0

r—1

where 0 stands for the zero matrix. Before stating our necessary conditions we recall a sufficient condition for the Cauchy problem to be well posed (microlocally) ([3]) when Pm(x, D) = h(x, D)IN is a scalar operator, which may give an idea for understanding our necessary conditions.

Let us assume that the characteristic set of order r of h is a manifold, which we denote by E. Assume that Vz =

E E,

is strictly hyperbolic on

(1.1)

where the localization h at z is the first nontrivial term in the Taylor expansion of h around z, which is a polynomial on We assume also that Vz

E, the propagation cone of h; is transversal to E at z

140

T. Nishitani

where the propagation cone is the dual cone of the hyperbolic cone of with respect to the canonical symplectic structure on (Tscz). We finally assume that Pm...j (x,

vanishes

of order at least r — 2j on E.

Then we have

Theorem 1.1. Assume (1.1), for P is (microlocally) well posed.

and (1.3). Then the Cauchy problem

While the condition (1.3) is necessary for the well-posedness in the scalar case (the Ivrii—Petkov condition), it is not necessary for the system case, which is easily seen by taking h(x, D) for which (1.1) and (1.2) are verified and upper triangular Pm3 with zero diagonals, not verifying (1.3). We now turn to necessary conditions. Let = ('co, , E with

0<'c,
ic=max,c2

and study

= where

=

+ etc. Recall that

,

Proposition 1.2. Assume that 0 E

and the Cauchy problem for P is well posed near the origin. Then for every compact W C and every positive T> 0 there are C > 0, A > 0 and p E N such that IUICO(WL)

foranyuECo°°(W),AA,ItI
+ A"D)

= —Atm

m—j($)'. F(j,caj3;c)>—M

+ O(Xm_M)

x

where

F(j,,13;c) =

—('c,f3)

+

—j

—

Some Necessary Conditions for Hyperbolic Systems

141

with a ci which will be determined later and M is a sufficiently large integer, and by 0(Am_M) we denote a differential operator of order m whose coefficients are bounded by Am_M on any preassigned open set. We now assume that —('c,/3)+(ic,a)

—'co)

and there are some j 1, a,

+

—

—

> 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)
= Then we have

Theorem 1.3. Assume (1.7) and (1.8). Assume also that det = 0 has nonreal roots at least one of which has multiplicity at most 3 with respect to at some (x, a'). Then the Cauchy problem for P is not well posed.

In a particular case 'c = (1/2,... , 1/2), (1.7) is of course verified and (1.8) implies that there are i 1, a, j3 such that

Ia+/31
(x,

is

given by

Let us consider another case in which the localization the form = Q(xk+1,...

(x,

has

T. Nishitani

142

with a polynomial Q of homogeneous degree r. Take 1

0 is small. On the other hand, (1.8) implies that

then (1.7) holds if

there are j r—

such

1

1, a, /3 with

2j

—

a

+ /31

+ 2e(r +

+

—

—

113(2)1) > 0

0 where a = = (ao,... ak+1, = (J$l),j3(2)). Note that this occurs if there exist j 1, a, /3

that an),

such that

r — 2j with

Ia +

2

0.

Inductive lemma

To prove Theorem 1.3 we follow exactly the same arguments as in [2). Let us set

._.L

A) =

(x,

(0,

Then it is clear that PA(x,D) = Am+fG(O)(x,A_(7oD;A) + 0(Am_M). We rewrite

(x,

A) as

f 0(°) (x, 10 = where o,(C(°)) see that

E

A)

=

and

= F(ja$ao)=f

A6'

(x,

<... are polynomials in

From (2.1) we

=

We say that a differential operator P(x, D; A) with a parameter A belong to if there are z' Q+ and a finite number of differential opera-

tors P,(x, D) with coefficients in C°°(U) so that P(x, D; A) is a sum of D). Recall a lemma in [2).

Lemma 2.1. Let G(z, D) be a differential operator with coefficients in C°°(U) and Let a, 9 E be such that a 9 > 0 and let C C°°(U). Then

Some Necessary Conditions for Hyperbolic Systems

)c°D;

= C(x,

(i) A) with

If

E

=

fl.(U). as

0

uniformly with respect to x

U, then

+ )c°D)) +

=

(ii)

143

r(x, )c°D; A). We

now recall a lemma for a rank reduction procedure ([2]).

Lemma 2.2. Consider the differential operator + where

0
i)

—

+

0,,,

E

7?.(U), which verifies

=

5

p

(2.2)

sum being finite and (x, D) denoting a differential operator with real analytic coefficients and (x) is a real analytic function in U such that 4(x) is a root of = 0 with uniform multiplicity qp; that is the

in U

=

(ii),,

= (

—

,(x,

in U x

r,,(x,

in U

with some positive integer (iv),,

a,, 0,,,

j 1

belong to N/k(p).

Then we can find a r,,+t x r,,+j matrix valued differential operator (2.3)

A)

such

that

(I) the

5 F(P) (x,

A)

=

(x,

sum being finite and

(II),, det

= e(x,

det

+ .V°e) +

144

T. Nishitani

for any 0< 0

where e(x,O)

(III),,

=

0 and

0.

there is k'(p) E N such that

j 1

(IV),,

belong to N/k'(p)

and, the most important thing, the construction of an asymptotic null solution to (2.2) is reduced to the construction of an asymptotic null solution to (2.3).

We turn to the next step. We are now concerned with the construction of a null asymptotic solution to (2.3). One can express =

where 4

+

(2.4)

I from (III),,. Define mm

,,

I

\ so

(A—8)]

that, in particular,

—

0,,. Set

=

— Op+1.

0 our present purpose we shall assume that > 0. If make a different argument in the following. We make the assumption For

q,, =

we (2.5)

+ =

+O(A_0)] (2.6)

which results from (ii),, and det

(x,

=

[det

(x, 4) +

by (2.4), it follows from (If),, that

= c(x)

(2.7)

Some Necessary Conditions for Hyperbolic Systems

with some c(x)

145

0. Applying Lemma 2.1 we compute (2.8)

which yields

D))

+ 320

A_öJ(F

+

A_ep+1 D; A)

j0 where

Define

E I

(x,

by

and

A) =

(x,

A)

= 220

and

A) =

>

A).

120

Here

Then (2.8) can be written as

note that 53(F(P)) + +

D; A)].

D) + A_cp+l

(2.10)

It is easy to see that

= Then we have

Lemma 2.3. Assume (2.5). Then one can find a E Q+ with S 0,,, a real analytic function and r,,+i x r,,+i matrix valued differential operator

(x, D; A) with

1> (x, D; A),

1) (x,

A) E

such that the construction of an asymptotic null solution to (2.3) is reduced to the construction of an asymptotic null solution to

+ where

verified with

=

—

+

A) (2.12)

Furthermore the conditions (i),,+1 — (iv),,+1 are

q,,.

From (2.11) it follows that = + ..., where the dot denotes a polynomial of degree less than = q,,. Then it is easy to find verifying (ii),,+1, (iii),,+1 in some open set. (iv),,+j is obvious.

146

3

T. Nishitani

Proof of Theorem 1.3

We prove Theorem 1.3 assuming that = 0 has a nonreal double root so that we may start off assuming qo = 2. We first confirm that the hypothesis (2.5) holds. general we get either r,4.

1 and

Since 2 = qo qj = 2 or = 1. If

< q,, iii

= 1, then from the proof of Lemma 2.2 we conclude that (2.5) holds necessarily. On the other hand if

= 2, then necessarily 4 = 1 and hence (2.5). Thus one can apply

Lemma 2.2 and Lemma 2.3 repeatedly. Here we remark the following:

Lemma 3.1. The iteration procedure of Lemma 2.2 and Lemma 2.3 occurs only finitely many times before reaching a point where p

=

<0

—

for some integer j3 E N.

We now may assume that for a certain positive integer t

Ot>O,

Ot+1=Ot—Ot+IO.

Our task is to construct an asymptotic solution to

G(t)(z,)(x) +

+

A).

We apply Lemma 2.2 again to get an operator of size re+1 x rt+I

F(t)(x,

A) =

jo Writing written as:

=

F(t)(z, A°' D; A) =

Remarking that Ot+j

+ O(A'))

[t)(z jo implies

o,(F(t))

(7t4

we can write

F(t)(r =

the above (3.1) can be

+

D) + O(A°' )].

Some Necessary Conditions for Hyperbolic Systems

147

with some 0 < <62 <••, E Q+ and K,(x,D), which are differential operators. One can seek an asymptotic solution to this operator in the form

jo and this is a well-known procedure. From the assumption we may assume

that Im ro(x,e') < —c with some c> 0. We solve

under the condition

= where (±,

UxV

ilx' — ±112

+

(z',.)

is a suitable point in U x V. Then it is easy to see that

(x)

verifies

Im

c{±o — x0 + p2:' — I'L2}

xo

±o

near ± with some c > 0. Thus the constructed asymptotic solution contradicts the inequality in Proposition 1.2.

References 1] V.Ja. Ivrii and V.M. Petkov, Necessary conditions for the Cauchy problem for nonstrictly hyperbolic equations to be well posed, Uspehi Mat. Nauka, 29 (1974), 3—70.

[2J A. Bove and T. Nishitani, Necessary conditions for the well posedness of the Cauchy problem for hyperbolic systems, preprint, 1999. T. Nishitani, Strongly hyperbolic systems of maximal rank, Pub!. RIMS Kyoto Univ., 33 (1997), 765—773.

Department of Mathematics Osaka University Machikaneyama 1-16 Toyonaka Osaka, Japan tatsuo©math.wani.osaka-u .ac.jp

Strong Unique Continuation Property for First Order Elliptic Systems Takashi Okaji 1

Introduction

There is a long history about the strong unique continuation property, going back to the works of P. Carleman, C. Muller, E. Heinz, N. Aronszajrt and H. 0. Cordes. After their works many advances were made, among them, differential inequalities with critical singularities as well as subcritical ones

were intensively investigated in connection with the absence of positive eigenvalues in the continuous spectrum ([2], [15], [16), [17], [8], [4]). One of the standard approaches to the study of systems is to reduce them

to (quasi-) diagonal ones. However, this approach requires both smoothness and less multiplicity of their eigenvalues ([3], [5], 121 and [11]). Thus, only few attempts have so far been made at studying systems without any assumption on both the regularity and the multiplicity ([9]). In this paper we are concerned with the strong unique continuation prop-

erty for three types of systems that are beyond the scope of the standard approach. Let L be (sub-) elliptic systems of first order differential operators defined on a nonempty, open, connected, subset of R". Without loss of generality, we may assume that SI contains the origin. The differential inequality with critical singularity is expressed as

[Lul

x E Sl\{O}.

First of all, we deal with a certain general elliptic system with two indeIn particular, we can improve a result due to G.N. pendent variables Hue and M.H. Protter ([7]). We emphasis that no regularity assumptions on its eigenvalues are imposed. Next we limit the discussion to two physically important systems. For the three dimensional case, we deal with the time harmonic Maxwell equations Strictly speaking, it is a in an inhomogeneous anisotropic medium nonelliptic system, but it has a nice structure compensating for its fault. We also treat the Dirac operator in the general dimension and summarize The multiplicity a part of the result obtained jointly with L. De Carl

T. Okaji

150

of characteristics of the (n-dimensional) Dirac operator becomes higher as the dimension increases. We approach these systems by the same principle based on Carleman inequalities. Operator calculus instead of symbolic calculus is an essential tool to derive the fundamental inequalities as in the theory of the single equations ([15), [81 and [17)). We describe the idea briefly in

2

Elliptic systems in two dimensions

In [7J, G.N. Hue and M.H. Protter obtained an interesting result on the unique continuation property for a class of elliptic systems in two independent variables. Let Cl be a nonempty open connected subset of R2. Without loss of generality, we may assume that it contains the origin. In what follows, Cl denotes CZ\{O} and r = + y2 for (x, y) Cl. They considered a system of the form )u1 +

N is an m x rn matrix with complex entries of class C' (Cl) and M is a constant. They proved, roughly speaking, that if N is a normal elliptic matrix, any solution of (2.1), satisfying =

0,

V$> 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
and it is continuously differentiable in Cl\{O} such that

+

urn sup

p—.O

We

consider the system of differential operators A(z,

+ B(x,

= 0.

Cl

Strong Unique Continuation Property

151

where A and B are in x in normal matrices defined in ci. Further, we assume

that they commute with each other, and either A or B is invertible at any point of ci. Thus, locally, it is equivalent to the system

Lu =

+ N(x,

y) is also an m x m normal matrix and u is a function on ci

with range in Ctm. We shall assume the following properties (2.3), (2.4) and (2.5):

N(x,y)

NN = NN,

L3Llc(ci).

(2.3)

V(r,y)

(2.4)

11,

where N is the conjugate transposed matrix of N. Let A,, j = 1,2,... , m be the eigenvalues of N. Then, there exists a positive number t5 such that

JImAj(x,y)I5,

j=1,2,...

,m

(2.5)

for all (x,y) E ci. We write them as A, = 11j + ia,,

v3

E R.

If all the eigenvalues of N(O, 0) are simple, we can smoothly diagonalize

N(x, y) near the origin. In this case, the equation Lu = 0 is equivalent to a family of first order single equations. On the other hand, if N(0, 0) has multiple eigenvalues, there is no smooth diagonaiization of N(z, y) in general. In particular, we shall treat the case when there exists a nonreal complex number such that for each j = 1,... , m, or

(2.6)

In what follows the positive number M0 is equal to

= max (x2 +

+ y)2),

(r,y)€S

(2.7)

r denotes the distance between (x, y) and the origin and let Il stand for ci\{0}. We say that u E L2(ci) vanishes of infinite order at the origin if

I

urn Ri" Iuj2dxdy =0, R-.0 Jir
'IN >0.

Theorem 2.1. Suppose (2.3,)-(2.6). Let u lLul

KoIuI/r,

V(x,y)

satisfy ci.

(2.8)

If K0 < (2M0)' and u vanishes of infinite order at the origin, then u is identically zero in ci.

Remark 2.2. When in = 1, Y.F. Pan showed that the conclusion holds for any large K0 (Lemma 7 in [14]).

T. Okaji

152

3

Maxwell's equation

In this section we proceed to the study of optical systems in three dimensions. We consider Maxwell's equations in continuous medium C R3:

f

0gB

+ curiE = 0, divB = 0 + curiH = J, divD = p.

B is the magnetic induction, E is the electric field, D is called the electric

induction, H is the magnetic field, J is the current density and p is the charge density. The constitutive relations are given by

H=

D = cE,

p'B, J = aE.

(3.2)

In isotropic and homogeneous media, we assume that e and are positive constants characteristic of the medium considered, called respectively the permittivity (or the dielectric constant) and magnetic permeability. Furthermore, in this case a is a nonnegative constant called the constant of conductivity. The third relation of (3.2) is called Ohm's law. In inhomogeneous anisotropic media such as crystals, e and are symmetric matrices depending only on the position z. Moreover, a is also a nonnegative symmetric matrix, called a conductivity tensor. The problem in which we are interested is the stationary one. Let us consider the functions of the following forms:

E(x,t) =

Eo(x)etAt,

B(x,t) =

(33)

where A is a nonzero real constant. Substituting these functions into the equations (3.1) and dropping the sufiix "C)", we obtain the time harmonic Maxwell equations in an anisotropic inhomogeneous medium Q: f —curiE = iAM(r)H, curlH = iAE(x)E + o-(x)E.

(3 4)

Here, t(x), e(x) and a(x) denote 3 x 3 nonnegative definite symmetric

matrix-valued functions in fl, and A R\{0}. In what follows, we call them constitutive matrices. This is a 6 x 6 system with a weakly coupling lower order term. Its principal part is not elliptic. In fact,

= 'r2(r —

det where

fO =

0 0

+

Strong Unique Continuation Property

In what follows, we denote the smallest eigenvalue of

153

and e(O) by

/Amin and Emin, respectively. For a domain fl containing the origin, stands denotes the subset of C' (fl) consisting of all functions for and C'

f satisfying Urn

=

sup (lxi

0.

P—•° O
stands for the norm of matrices as a multiplication on C3. We say that a function u E vanishes of infinite order at the origin

The symbol ii

lim

r—O

We

[

J111
ul2dx = 0,

VN E N.

denote by C6 (el) the space of Holder continuous functions in ft Namely,

if I E

then there exists a positive constant C such that

lf(x) — f(y)l

y

—

ft

(3.5)

Then, we have

Theorem 3.1. ([12]) Let o(x) 0. Suppose that two matrices p(x) and i(x) are nonsingular in Il such that > 0.

p(O) = Aoe(0),

(3.6)

In addition, we assume that every component of these two matrices belongs to C6(1l) n with 0 <6 < 1. Then the time harmonic Maxwell equations (3.4) have the strong unique continuation property in a neighborhood of the origin.

These results are a consequence of the result for the next differential inequalities of Maxwell type:

J

Icurlul

Ailtzl/lxl xEQ,

1

(37)

a(x) is a real positive symmetric matrix defined on and A1, A2 are some positive constants. We assume that for a positive number 5 1, where

fl

(3.8)

In what follows, 0mm denotes the smallest eigenvalue of o(0), which is denotes the largest elgenvalue positive by our assumption. Moreover, of a(O).

Theorem 3.2. ([12]) Let o satisfy (3.8). Suppose u E (3.7). If AiOmax + A2 <0mm!2

satisfies (3.9)

and u vanishes of infinite order at the origin, then u is identically zero.

154

T. Okaji

Since the proofs of these results for Maxwell equations require a lot of space, we shall give only a sketch of a proof of Theorem 3.2 for Maxwell's

equations in Section 7. We have to mention the work 1191 due to V. Voge[sang who has obtained a similar result for the Maxwell system with subcritical singularities like (c > 0).

The Dirac equation

4

We consider the Dirac equation in any dimensional space. Let = a0 be anticommuting matrices satisfying the relations

j

= 1,2,3,

a2=I, The Dirac equation that describes free relativistic electrons is represented by

iho1tp(t,x) = where H0 is given explicitly by the 4 x 4 matrix-valued differential expression

H0 =

+/3mc2.

Here, c is the speed of light, rn is a mass of a particle and h is Planck's constant. More generally, let j = 0,... ,n be N x N symmetric matrices forming a basis of the Clifford algebra. Namely, they satisfy (4.1). It is known Throughout this paper, we that N is in the set DZ, where D = only consider the irreducible cases. Namely, we assume that

N = 212+h1

(4.2)

The (n-dimensional) Dirac operator with which we are concerned is

L=

(4.3) 5=1

It is easy to see that

L2 = -M. Furthermore, a simple calculation shows that d et(

ifn=2,N=2 ifn=3,N=4.

The multiplicity of the characteristics becomes higher and higher as the space dimension becomes large. This causes some difficulty in the study of the Dirac operator by using the matrix diagonalization method ([3] and [5]).

Strong Unique Continuation Property

155

Theorem 4.1. (De Carli and Okaji [6]) Let n 2. Let u E be a solution of the differential inequality

xEfl,

(4.4)

C is a positive constant strictly less than 1/2. Then, u is identically zero if it vanishes of infinite order at some point of ft where

Remark 4.2. There are related works by Vogelsang or Jerison. The former considered the same problem in the three dimensional case when the righthand side of (4.4) was replaced by C/IxIl_E with e> 0 ([18]). The latter studied the problem in the scale of spaces ([9]).

Remark 4.3. As for the same problem for the differential equations instead of (4.4), the situation is changed. We refer to [6] for the detail.

5

Basic principle

Although the three systems mentioned in the previous section have different outlooks, we can deal with them in a unified way. We are going to describe quickly our approach to these problems. For analyzing the strong unique continuation property, it is the most convenient to use polar coordinates (r, x The first thing we have to do is to reduce a given E system to a certain system of first order differential operators Lu in which the radial derivatives and the angular derivatives are completely separated. More precisely, we expect that it should be written as

where C is a d x d matrix whose components consist of vector fields on the 1, smoothly depending on the radial variable r. What we unit sphere want to do is to derive a Carleman inequality with a certain weight where y is a large parameter and 0(r) is a smooth real valued function that

is both monotone decreasing and blows up at the origin. Unfortunately, we are often confronted by a problem that prevents us from deriving the Carleman inequality at one stretch. It means that we often have to make two steps to derive the final Carleman estimate. At the first step, we consider the operator L0 obtained by restricting the coefficients of L at the origin r = 0: L0 = Or +

We have to show that L0 gives a good approximation of the original op-

erator L. The next important step is to estimate the gaps in the spectrum of the angular part C0. Suppose that C0 is a selfadjoint operator in

T. Okaji

156

with D(G0) a complete orthonormal basis

and there exists in consisting of eigenvectors of Go:

jEZ, where A2, j

Z, are numbered in nondecreasing order such that

Define E = {A,;j = 1,2,. . }, which is a subset of positive real numbers. .

Proposition 5.1. ((12]) Besides the above conditions, suppose

x,yE Efl[R,oo)}} =

Aim inf{Ix—yI: Then, we can find a sequence as j —, oo such that

f

i

of positive numbers that tends to oo

—rOru+Coul2dx

for all u E

< lxi

>0.

—

and all sufficiently large j E N.

If E is equally distributed in the positive real line, we can improve the preceding result. Let us denote the elements of E by ... in < increasing order.

Corollary 5.2. Suppose that for all sufficiently large j, — lAj

=

(5.2)

— ,ij+i.

Denote the last positive number (5.2) by 8 and set i', = Then, it holds that

f

+

(5)2f I—

rOru + GouI2dx

luI2dx

(5.3)

for all u E (Cr(0 < xJ < i)}d and all sufficiently large j E N. Proof of Corollary 5.2. From the hypothesis, it follows that

u(x) =

in

(5.4)

jEz

y = log r and let I denote an interval (—oo, R). The next lemma gives a crucial step for estimating each coefficient u2 (r) of the eigenfunction Set

expansion (5.4).

Strong Unique Continuation Property

157

Lemma 5.3. Let ii E R. Then, it holds that

j

{(—-v + v)2

+

}

dy

for all! ECr(I).

This identity can be easily verified by use of an integration by parts. we obtain Applying this lemma with P = and 1(y) =

Jr

+ C0u, I2r2dr

)2

+

for all positive 7. In view of (5.3) since it follows that

I7jILkIö/2,

f

=

I2r2dr

(5.5)

(5.5) implies Vk

if j is suflciently large. This completes the proof of Corollary 5.2.

As for the proof of Proposition 5.1, we refer to [12). From Proposition 5.1 or Corollary 5.2, it follows that if C < uo/2, then ILoul

xE

(5.6)

has the strong unique continuation property. Namely, u E satisfying (5.6) is identically zero provided that u vanishes of infinite order at the origin. Once we obtain the Carleman inequality for L0, we can show that the solution of exponential order has to vanish for the sake of the ellipticity of the system, as Regbaoui showed in his article [17] for the single elliptic equation of second order. This technique is also well known in another context concerning the problem of spectral analysis, for instance, the limiting absorption principle or the absence of positive eigenvalues. However, in the spectral problem, we are interested in a neighborhood of infinity instead of the origin. In any case, improving the vanishing order of the solution enables us to use a powerful weight for which the good Carleman inequality holds.

In the subsequent sections, we shall apply this principle for three kind of systems as mentioned before.

6

Outline of proof of Theorem 2.1

To prove Theorem 2.1, we shall use two types of Carleman inequalities. Let

R(x,y) = {x2 +(ImC)_2(_ReCx+y)2}hI'2.

T. Okaji

158

First of all, we shall derive a Carleman inequality with some remainder terms. Then, we have

Theorem 6.1. For an arbitrary small positive number E, there exists a positive constant C such that

f

:S (1 + E) f

+ C(1 + E') f for any u(x,y)

(6.1)

N.

and any -y

As a direct consequence of Theorem 6.1 and the ellipticity of L, we have

Theorem 6.2. Suppose that K0 < (2M0)'. If u satisfying (2.8) vanishes of infinite order at the origin, then there exist positive constants B and C such that

+

J0< R(x,y)p

dxdy

+

for any small positive p.

For the sake of Theorem 6.2, we can use another Carleman inequality with a stronger weight function.

Theorem 6.3. For a sufficiently small -y f R21 log RI exp{y(log R)2}Iul2dxdy

for any u(x, y) E

and any large positive 'y.

Theorem 2.1 follows from Theorem 6.3 by the standard procedure. We refer to 1131 for more details.

7

Sketch of proof of Theorem 3.1

Without loss of generality, by use of an orthogonal transformation and a dilation transformation, we may assume that a(O) is equal to the identity matrix. It is not difficult to verify this, but it takes plenty of time, so we omit it. Then, we see that

Jcurlu =

OrU

+ div(u)k,

+

where

/

0 W3

W3

W2

0

0

Strong Unique Continuation Property

/

G= Here,

\L2

—L2\

L3 0

0 —L3

and

L1

01

—L1

w2

\W3

iL = (iLi, iL2, iL3) are the angular momentum operators: L3=x102—x201.

L1 =x203—x302,

Lemma 7.1. C is equal to G as the operator in

Moreover, G

has only discrete spectra which form a subset of Z.

We use the spherical harmonics to prove this lemma. We refer to [12} for more details. For the sake of Lemma 7.1, we can deduce the following Carleman inequality with remainder terms.

= I. Let B be any nonempty open ball with center at the origin of R3. Then, there e:rist a small positive number r and a positive constant C such that

Theorem 7.2. Suppose that

1/2

1

{f

1/2

<{J 1/2

+

{f

+c

1/2

{f

+ iui2) dx} 1/2

+ A0 {f for all u

lxi

(7.2)

Z.

Lemma 7.3. There exists a positive constant K such that for all

>1

and all u 7_2

f

<

K

+

J

(7.3)

J

Theorem 7.4. Suppose that (8.9) holds. Then, there exist a small positive number r and a positive constant C such that

f

1xi2 log lxii

111)2

lul2dx

< Cfeboshmfl2lcurlul2dx +

for all u E

< lxi
(7.4)

N.

160

T.Okaji

To prove this theorem, we use the idea of [19]. Define the scalar operat

Qu =

a01u

and decompose curlu into its radial and angular parts:

Lu=VAu—wAQu. For each

E R3, we define the matrix

as

Jvu=VAU.

and

The next identity is an analogue to (7.1) in the isotropic case.

Proposition 7.5. = —Q(au) + — +(aw,Qu)(a — 1)aw +

+ +

This proposition follows from the next lemma (cf. [19]).

Lemma 7.6. Forg E {cl(U)}3, we have L((aw) Ag) = (divg)aw — (Q(w

—

—

r'g + o(r')g.

Now, we are in a position to prove Theorem 7.4. Define

=

—

and

RQU = (a

—

1)Qu + (aw, Qu)(a — 1)aw

+

J (Ou,v)dr = f (u,Ov)dr + Jofr_1)IuflvIdr. For v =

and p' = (d/dr)cp(r), we see that = (Q — = RQv

—

—

= Lv

1)v + (aw,v)(a — 1)aw +

(7.6) Defining

Sv = (a

—

1)v + (aw, v)(a — 1)aw +

Strong Unique Continuation Property

we obtain

= (Q

+ Gv +

—

+ O(r')v.

+ RQV +

We write the right-hand side of the last identity as the sum of the following two terms:

=(Q+r' =

+ RQV + O(r')v.

It follows that

f

f

f

=

f

—

f

(7.7)

+ r' )vl2dx + J Icy —

+ 2Re f ((Q +

)v, Ov — p'v

+ + cdSv) dx.

(7.8)

Choose cD(r) = r)2/2. Using an integration by parts carefully, we arrive at Theorem 7.4. Finally, we can show Theorem 3.1 by adding the two Carleman estimates for E and H. We refer to 112] for more details.

8

Outline of proof of Theorem 4.1

Let n 2. We shall use the polar coordinates:

r=lxI, Set Pj = Rx

and

y=logr.

P1 is a skew symmetric matrix. In these coordinates (y, the Dirac operator can be written as follows.

L= where

= >JPhPk

—

E

162

Note

T. Okaji

that A is a unitary matrix and A =

—A.

A calculation easily shows

that C(G+n—2)= —A. Thus, we know that the eigenvalues of —A are k(k + n —2), k =

0,

1,2

Therefore, it turns out that the eigenvalues of C are a subset of Z. In particular, when n = 2, the spectrum of C is equal to Z. When n 3, it coincides with the set Z\{—1, —2,... ,2 — n}. This gives

Theorem 8.1. Let n 2 and let r = lxi. There exists a positive number i'o such that

andy >'Yo inN orN+ /2 if n is odd

holds for everyuE or even, respectively.

We refer to [61 for more details.

Appendix Finally we would like to give an interesting remark on the well known similarity between the Dirac equation and the Maxwell equation in a vacuum:

fcurlE+

divH =0, =

curlH —

divE = 4irp,

where p is the charge density and 3 the electric current. This system can be written in the form analogous to the Dirac equation 3

4ir

1

= ——'I'. C

Indeed,

we define the components of i/.'

as

chcp, and

x0 = ct, x1 = x, x2 = y, x3 =

0 o\ oiool ooiol' 000ij 1

0

z.

k=1,2,3

Furthermore, we have 0

—1

0

0

—1000 0

00—i'

0

0

i

0

Strong Unique Continuation Property

00—10 02=

0 —1

0 0

0 0

163

000—1

i

00—i 0

0

0

i

0

0

—1000

0—i 00

Then, it is easily seen that jfk, £ = 1,2,3, 0k01 + 0102 =

0203 =

= 25k11

0301 = io2.

(A.2)

Acknowledgement. The author would like to thank the organizers of this nice conference for their kind invitation and hospitality.

References [1]

S. Alinhac and M.S. Baouendi, Uniqueness for the characteristic Cauchy problem and strong unique continuation for higher order partial differential inequalities, Amer. J. Math., 102 (1980), 179—217.

[2] S. Alinhac and M.S. Baouendi, A counterexample to strong uniqueness for partial differential equations of Schrödinger type, Comm. in Partial Duff. Eq., 19 (1994), 1727—1733.

[3] T. Carleman, Sur un problem d'unicité pour les systemes d'équations aux dérivées partielles a deux variables indépendantes, Arkiv for Matematik, Astr. Fys., 26B (1939), 1—9.

(4] 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.

(5] A, Douglis, Uniqueness in Cauchy problems for elliptic systems of equations, Comm. Ptire Appi. Math., 6 (1953), 291—298.

[6] L. De Carli and T. Okaji, Strong unique continuation property for the Dirac equation, Pubi. RIMS, Kyoto Univ., 35(6) (1999), 825—846.

[7] G.N. Hile and M.H. Protter, Unique continuation and the Cauchy problem for first order systems of partial differential equations, Comm. P.D.E., 1 (1976), 437—465.

[8] C. Grammatico, Unicitá forte per operatori ellittici, Tesi di Dottorato, Univ. degli Studi di Pisa, 1997.

(9] D. Jerison, Carleman inequalities for the Dirac and Laplace operator and unique continuation, Adv. Math., 63 (1986), 118—134.

164

T.Okaji

[10] H. Kaif and 0. Yamada, Note on the paper by De Carli and Okaji on the strong unique continuation property for the Dirac equation, Publ. RIMS, Kyoto Univ., 35(6) (1999), 847—852. [11] T. Okaji, Uniqueness of the Cauchy problem for elliptic operators with fourfold characteristics of constant multiplicity, Comm. P.D.E., 22(1— 2) (1997), 269—305.

[12] T. Okaji, Strong unique continuation property for time harmonic Maxwell equations, to appear in J. Math. Soc. Japan.

[13] T. Okaji, Strong unique continuation property for elliptic systems of normal type in two independent variables, preprint. [14] Y. F. Pan, Unique continuation for Schrodinger operators with singular potentials, Comm. Partial Differential Equations 17 (1992), 953—965.

[15] R. Regbaoui, Strong unique continuation results for differential inequalities, J. Funct. Anal., 148 (1997), 508—523.

[16] R. Regbaow, Strong unique continuation for second order elliptic differential operators, J. Duff. Eqs., 141 (1997), 201—217. [17] R. Regbaoui, Strong unique continuation for second order elliptic differential operators, Séminaire sur les equations aux dérivées partielles, 1996—1997, Exp. No. Ill, Ecole Polytech., Palaiseau, 1997. [18] V. Vogelsang, Absence of embedded eigenvalues of the Dirac equation for long range potentials, Analysis, 7 (1987), 259—274.

[19] V. Vogelsang, On the strong continuation principle for inequalities of Maxwell type, Math. Ann., 289 (1991) 285—295. Division of Mathematics Graduate School of Science Kyoto University Kyoto, 606-8502, Japan email: [email protected]

Observability of the Schrödinger Equation K.-D. Phung ABSTRACT The goal here is to present two approaches concerning results on observability and control of the Schrodinger equation in a bounded domain. Our results are obtained from different works on the control of the heat equation or of the wave equation. From the theory of exact and approximate controllability, introduced by J.L. Lions [101, we know that observation is equivalent to approximate controllability and stable observation is equivalent to exact controllability. Our first result is based on a gaussian transform which traduces any estimate of stable observability of the heat equation to an estimate of unstable observability for the Schrodinger equation (see Section 1 below). This work is similar to those

done by L. Robbiano [14J for hyperbolic problems on the domain where the geometrical control condition of C. Bardos, C. Lebeau and J. R.auch on the exact controllability of the wave equation [1) is not satisfied.

Our second result is about exact control for the Schrödinger equation (see Section 2) and is inspired by a transform introduced by L. Boutet de Monvel [21 for the study of the propagation of singularities of an analogous solution of the Schrodinger equation. Our strategy is to construct an exact control for the Schrodinger equation from an exact controllability result for the wave equation.

1

Observability results for the Schrodinger equation

be an open bounded domain of R'2 with a boundary ôft We Let consider the Schrödinger equation with the Dirichiet boundary condition:

inQxRt

u=0 u(.,0)=u0 md,

where the solution u E C(R;HJ(d)) if u0 E We say that we have stable boundary observability of the heat equation if for all open I', non-empty set of Od, such that F C Od, for all T> 0, there exists CT > 0 such that the solution w of the evolution problem

w=0 w(.,T)

indxJO,T[ L2(cZ)

K.-D. Phung

166

satisfies

(11r18

j

lw(, 0)12 dx

CT

w12 dt&c

+

j

1T1112

dtdx).

We propose to establish the following observability estimates:

Theorem 1.1. If a stable boundary obseruability of the heat equation is satisfied, then we have a boundary observability estimate in logarithmic type for the Schrödinger equation.

Corollary 1.2. For all

0, there is > 0 such that for all initial data of problem (1.1), we have

uo E H2(1l) fl

p / t I Vuol2dxexpiCcJO

/ f

dx ) JrJo

\

Jo

'

UO

Remark 1.3. The estimate (1.4) is equivalent to

I

in (2 +

2dx

I

Jo

fr Corollary 1.2 comes from Theorem 1.1 and the work of G. Lebeau and L. Robbiano 1111 or of A.V. Fursikov and O.Yu. Imanuvilov [7J on the exact controllability of the heat equation obtained from Carleman inequalities. Note that D. Tataru 1151 gives a directly unique continuation estimate for the Schrodinger equation with Dirichlet boundary condition from Carleman's type inequalities. Corollary 1.2 is also valid for internal observability of the Schrodinger equation: Let w be a non-empty open set included in fi. For all e > 0, there is > 0 such that for all initial data H2(1?) fl of problem (1.1), we have ito E

/

P

\ 2

f0IuoI2 dx J

I I Iu(x,t)I2dxdt.

Exact control result for the Schrödinger equation

be a bounded open set in R", n > 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

(,

j

loT1112

dtdx). (3.4)

Observability of the Schrodinger Equation

169

Another approach, based on the work of Fursikov and lmanuvilov gives the following uniform time estimates [6]:

with boundary be a connected bounded open set included in C C°°. Let w be the solution of the following evolution problem:

Let

35

w = 0 on

Then, there is C > 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

<

A2

—

4ir

dsdx

I4"(l)I Iu(x,t)Idtldxds

JcIIJR 211 2

I

I

(1 e_41t0_h12 I'V(€)12 Iu(x,t)12d1) mes(K)dx

4dist(K,Ko)2

< <

sup I'I'(t)l2mes(K) f

(K,Ko)2

J0 1A21

—

(L)2)]

42L2

—

(L)2)] jIuoI2dx.

exp

[-i-

F) 2dtdx

1 IttoI2dx

4ir A2T

—

I

JQJK

—

Observability of the Schrödinger Equation

171

Thus, inequality (3.10) becomes:

Jn(FA *

dx

.))

2

A2TL exp 4ir

[1 Jrjo

I

A2T

+ CT—eXP

1A2 /

—

L

(L)2)]

j

IuoI2dx

(3.11)

By the Parseval relation, we have:

j

(x,

— (FA * 4!u (x,•)) (€o)12 deo

_—.

I

=

2

2

2

I

j

<

f 1c1'(e0)u (x,

+ $(€o)810u (x,

—— I < A2[(L) JK

I

—

By integrating on

in JRI

I

JO

(x,.)) &o)12 d€odx

— (FA *

[(4)7 L

1L

Join

KJfl 2

[(i) So,

L

we obtain

(x, 4

deo

L [12 dx + L I in

dx].

(3.12)

from (3.11) and (3.12) =

mes(Ko) I

in

I I

JK0 in

mes(Ko)Cr

A2TL 4ir

/ A2

\2

A2T

+

mes(Ko)CT—exp

+

A2

4 2L \LJ —

I

)

JrJo —

jIuoI2dx

(T2

f

K.-D. Phung

172

Finally

L

f

4ir

[8fj] —2

+Cr—exp 1 42

By choosing L = 8AT, with

j

11

—

(1 —

A2)

——

(8),ij Jn Iuol2dx

<0 and T <

I JrI jo

Itiol2dx

this becomes

1,

8AT

+

12

(A2_

+

+

j

JJSAT 1 2

1

1

lj Iuol2dx

+

Iuol2dx

j

We write 8AT

I Iuol2dx < Cexp(A2T2) I

1

t)I2dedx +

j

(3.13)

Now, we take 1

f

< Cexp

T

T31

dx

Ji

rJo

conclude by choosing T = 1, with e > 0. The estimate (1.4) of Theorem 1.1 is obtained by interpolation. We

Observability of the Schrödinger Equation

173

Proof of the exact control result for the Schrödinger equation

4 4.1

The Schrödinger equation in one space dimension

We give two results on the Schrödinger equation in one dimension.

Proposition 4.1. If n = set there is we have

1, then for all w C = ]A, B[ non-empty open in IR and a neighbourhood of the point x = B, for all E > 0, > 0 such that for all u0 E initial data of problem (1.1),

II re

2

I'

2

j IuI dxdt.

j

IlUolILa(n)

JO Jw

Proposition 4.2. There exists a triplet (f, u, F) such that +

=

f•1113T,2,2T1

—

® 6(s + 2T)

+83u®5(s—2T) in ]0,efxR3

F(e,)=0

(4.2)

in 1—T,T1.

Proof of the Proposition 4.1. Comes from the multiplier method [12J, [13], [5). The constant 13e could be written explicitly in c, from (3.6).

Proof of the Proposition 4.2. From the HUM method and Proposition 4.1, for all data u0 L2 (]—2T, 2T[), there exists a control I L2a0,e[ x ]3T/2, 2T[) such that the solution U: (t,s) '—p u(t,s) C([0,e]; L2(J — 2T, 2T[)) satisfies + = f•1113T/2,2T1 in )0,e[ x ]—2T, 2T[ u (., —2T) = 0, u (., 2T) = 0 on 10, Cf u(0,.)=0 in ]—2T,2T[ u(e,.)=u0 in ]—2T,2T[

3

and (4.4)

If IIL2(JO,c(x)3T/2,2T1) In particular, we take u0(e,s) = (]—2T,2T[), 0 X 1, XU-T.rl = x

where s E ]—2T,2T[, 1.

Thus,

e

2e

(4.5)

Let

H ( ,s)

—

u(t,s)

—

0

in [0,e] x [—2T,2T] in [0,e] x (]—oo,—2T[u]2T,+oo[)

46

174

K.-D. Phung

where u is the solution of (4.3). Thus,

= f.11j3r12,2r1 +89u®5(s—2T)

+

—

O,u ® 5 (s + 2T)

(4.7)

H(0,)=0 mR3

H(e,s) = and

(4.8)

IIHIIL2(lo,E(xJ_2r,2T1) < e

Let E (I, s) be the fundamental solution of the Schrodinger equation in one dimension: 2

E(t,s) =

(4.9)

The solution E E C°° ({t > 0) x R,) n C ([0, +oo[;

j

(R3)) satisfies

in {t>0}xR8 E (o,)

= S (.)

H—'1'2' (R3).

(4.10)

We finally choose F (t, s) = E (t, s) + H (t, s), which is the solution of (4.2).

4.2

The hyperbolic problem

We give a result for the exact control of the wave equation.

Proposition 4.3. If the function 9 : (x,t) ]0, T[

x

; R) controls exactly il for the wave equation with partially null ini-

tial data then, for all initial data

(il), there exists a control C(R;

E H'(]—T,T[;L2(OIfl) such that the solutiony L2(1l)) satisfies

inIlxR on Oil X

= =

=

0

IR

in

4 11

Futherinore IIQIIL2(rxj_T,TL)

Proof of Proposition metry: y

'

+ IIOtUIILa(rx)_T,T() < CT

We extend the solution

(4.12)

(x, t) of (2.1) by sym-

'I(x,t) inilx[O,T) x [—T, O[.

413 ) .

And from the HUM method, we have I19l1L2(aflx)_T,T() + IIOtOIIL2(9flxI_T,TI) CT

(4.14)

Observability of the Schrodinger Equation

175

Proof of Theorem 2.1 Let 0< t Wedefinew(x,t) such that

4.3

w(x,t) =

(4.15)

where y (x,e) '—i y(x,€) and F : (t,€) u—' F(t,€) are solutions of the problems y (x, t)IanxI—T,Tt

(x)

=

£) •1Iaczxl—r,r(

infl

(4 16

(]—oo,—T]U[T,-i-oo[)

and

in ]0,e[x(—T,T]

F(0,.)=o(.)

F(e,)=0

(4.17)

in [—T,T].

The existence of y is given by the Proposition 4.3 with the hypothesis of exact controllability for the wave equation with partially initial data. The existence of F is given by the Proposition 4.2 where the support of the second member of (4.2) does not meet 1O,E[ x 1—T,TE. w (x, t): We calculate (jôt +

iôtw(x,t)

= =

f

(x, £) de

(t, £) y

=0. (4.18)

Conclusion

w(x,t) =

on

x ]0,e[

419

w(.,0)=w0

w(.,e)=0

with an estimate of the control = = =

(x, t) on

x 10, e[, given by

fTr_(E+H)(t,i)0()de

(4.20)

+

where IIt9t,2IIL2(anxjo.cL)

=

IfTT H (t, £) f0 IIH (t, )11L2(I

(x,

dt

del dxdt 2

Ik)(x, )11L2(I_T.r() dx

< (4.21)

K.-D. Phung

176

and IIt'e,111L2(ôflxIoeE) =

10

(t, €) p(x,t)

fsfl

=

(4.22)

dxdt

JR

and del

Ilk

dx

+

le(x,t)(e)I +

flelM

ScM

()12

+

+ cM'

JR

5

cM

!1QIIL2(anxR)

deds

< (4.23)

Remark 4.4. The proofs of Theorems 1.1 and Theorem 2.1 are still true if we change the laplacien operator by an elliptic, autoadjomt, regular in espace operator. We complete the result of control in [8].

References [1] C. Bardos, C. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30(5) (1992), 1024—1065.

[2] L. Boutet de Monvel, Propagation des singularités des solutions d 'equations analogues a i 'equation de Schrödinger, Lecture Notes in Mathematics, 459, 1975.

[3] N. Burq, Contrôle de l'dquation des plaques en presence d'obstacles strictement convexes, Mémoires S.M.F., 55, nouvelle série, 1993.

[4) N. Burq and P. Gerard, condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Acad. Sci. Paris, t. 325, série 1, 1997, 749—752.

Observability of the Schrödinger Equation

177

[5] C. Fabre, Resultats de contrôlabilite exacte interne pour l'équation de Schrodinger et leurs limites asymptotiques: application a certaines equations de plaques vibrantes, Asymptotic Analysis, 5 (1992), 343— 379.

[6] E. Fernandez-Cara and E. Zuazua, Conference a Cortona, 1999. A.V. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, no. 34.

(8] M.A. Horn and W. Littman, Boundary Control of a Schrödinger Equation with Nonconstant Principal Part, Lecture Notes in Pure and Applied Mathematics, no. 174.

[9] C. Lebeau, Contrôle de l'équation de Schrodinger, J. Math. Pures et Applic., 71 (1992), 267—291.

1101 J.-L. Lions, Contrôlabilité exacte, stabilisation et perturbation des systèrnes distribués, 1, CoIl. RMA, Masson, Paris, 1998.

[11] G. Lebeau and L. Robbiano, Contrôle exacte de l'equation de Ia chaleur, Comm. Part. Duff. Eq., 20 (1995), 335—356. [12]

I. Lasiecka and R.Triggiani, Optimal regularity, exact controllability and uniform stabilisation of Schrodinger equations with Dirichlet control, Differential and Integral Equations, 5 (1992), 521—535.

[131 E. Machtyngier, Exact controllability for the Schrodinger equation, SIAM J. Control Optm., 32(1) (1994), 24—34.

[14] L. Robbiano, Fonction de coüt et contrôle des solutions des equations hyperboliques, Asymptotic Analysis, 10 (1995), 95—115.

[15] D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems, J. Math. Pures et Applic., 75 (1996), 367—408.

Kim dang Phung 17 rue Leonard Mafrand 92320 Chatillon, France Kim-Dang.Phung©cmla.ens-cachan.fr

Unique Continuation from Sets of Positive Measure Rachid Regbaoui 1

Introduction

Let fi be a connected open subset of We say that the differential inequality

and let V, W be functions on

IvuI + IWVuI the weak unique continuation property (w.u.c.p) if any solution u of (1.1) which vanishes on an open subset of is identically zero. And we say that (1.1) has the strong unique continuation property (s.u.c.p) if any has

solution u is identically zero whenever it vanishes of infinite order at a point is said to vanish of infinite order of We recall that a function u E at a point x0 (or that u is flat at x0) if for all N > 0, pj

f

=

as R —+ 0.

(1.2)

lx—xol
There is an extensive literature on the unique continuation for differential inequalities like (1.1). We refer the reader to ([1, 4, 5, 8, 10, 11]) for more details. The study of the above properties becomes interesting when V and W are and singular functions. For example, one could ask for which exponents P2 the condition V E and W E implies the unique continuation properties for (1.1). By applying Sobolev inequalities to solutions of (1.1) with sufficiently small supports, a natural question arises: the condition n (for n 3) is sufficient to get unique continuation P1 n/2 and p2

properties for (1.1)? In (11], Wolff proved that this is the case for the (w.u.c.p). This result seems to be optimal, but it is not known if there exist counterexamples. As can be seen by elementary examples, the condition n is necessary for the (s.u.c.p) to hold. When W 0, pj n/2 and Jerison and Kenig [5] showed that this is also sufficient. For nontrivial W, the problem is solved in dimension 3 and 4 by Wolff 1101. In higher (see [4], [9]. dimensions, there are several results giving conditions on [10]). However, as is shown by Wolff (12], the (s.u.c.p) fails in general if P2 = n and ;z 5.

R. Regbaoui

180

One of our purposes in this paper is to prove that the condition Pi n/2 i.e., there and p2 n suffices to get the (s.u.c.p) almost everywhere in is a set N c with zero Lebesgue measure such that if u vanishes of infinite order at some point xO E \ N, then it vanishes identically in As a consequence, we prove that the above condition gives also the unique continuation from sets of positive measure. Before stating our main theorem, let us introduce some notation: If f then it is well known that for almost every x0 we have

urn RTh J

1(z)

—

=

0.

Ix—xol
is called the Lebesgue set of f and we will denote it by L(f). it is clear that from (1.3) we have The set of such

f

IX—XoI
L(f). Theorem 1.1. Let p = for all

E

(1.1) with V at some point

3, and t€t u E

be a solution of and W E If u vanishes of infinite order then u is identically i.e., satisfies (1.2) at

zero in CL

As we shall see in Sections 2 and 3, Theorem 1.1 is a combination of two results : first, we will prove that if u is as in Theorem 1.1, then u cannot have zeros of "exponential" order, i.e., there is no point such that

J

Iu(x)P2dx =

0

for all N >

0

(as R

0)

(1.5)

Ix—xokR

unless u is identically zero. For this, we follow strongly a modified version of the Carleman method due to Wolff [111. Next, we show that if so E L(IWI") is zero of "exponential" order, i.e., is a zero of infinite order for u, then

satisfies (1.5). Actually, as it can be seen in the proof of Proposition 3.1 (see Section 3), to get (1.5), one needs only to assume that

I

lim R0 which is weaker than So

IW(x)I"dx =

0

JIz—xol
L(IWI'1).

As a consequence of Theorem 1.1, we prove also the following:

Theorem 1.2.

Let p = be a solution of 3, and let u E (1.1) with V E and W If u vanishes on a set of positive measure, then u vanishes identically in Cl.

Sets of Positive Measure

181

When W 0, Theorem 1.2 is proved in De Figueiredo—Gossez [2]. (See also a similar result by Gossez—Loulit [3] in dimension 2.) The arguments of [2] consist in showing that if (1.1) enjoys the (s.u.c.p) then it is also true for the unique continuation from sets of positive measure. The difficulty here is that when W E the (s.u.c.p) fails in general. But as we shall see, Theorem 1.1 suffices to use the ideas of (2] and [31 to get Theorem 1.2. For some interesting applications of unique continuation from sets of positive measure, see [2].

Exponentially decaying solutions

2

In this section we prove a unique continuation property for exponentially decaying solutions of (1.1). This is a crucial point in the proof of Theorem 1.1.

Theorem 2.1. Let p = of (1.1) with V E

3, and let u

n

be a solution Suppose that u satisfies, for

and W

some xo

f

for aUN>0 asR—i0.

(2.1)

Ix—xol < R

Then u vanishes identically in Cl.

The proof of Theorem 2.1 is based on Carleman's method. This is a modification of the proof of Theorem 1.3 in the author's work [9]. We follow closely the ideas of Wolff [11], where the main tool is the following lemma:

Lemma 2.2. (cf. [11]) Let

be a positive measure in

with faster than

exponential decay, i.e.,

irnT'logz((x: lxi

T}) =

—00.

(2.2)

by djzk(x) = ekxd,z(x). Suppose 13 is a convex body in R?Z. Then there is a sequence {k, } C B and, for each j, a convex body Ek, with

Define

\ Ek3) such

that

ii

(2.3)

} are pairwise disjoint and satisfying

C181 where C is a positive constant depending only denote the Lebesgize measures of B and Ek,.

(2.4)

on n, and where (BI, lEk,

182

R. Regbaoui

We need also the following Carleman-type estimate which is proved in [11] (see also [6J).

Lemma2.3. Letp= then for

E C R" with iEl and for a119>

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 <

\ B(O, Ro)) and satisfies

E

+ (iW(x)l + 2(n

—

2)lxl_1) iVü(x)l

where V E \ B(0, Ro)) and W Since u satisfies (2.1), then

B(0, Ro)).

= 0 (e_NR) for all N >

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. <

+

where IIiiIIwl,p is the of ii in the set {x is the L,"-norm of W3i in the same set.

<

<2Ro}, and

184

R. Regbaoui

It follows that + 2(n

—

+ IIWiIILP)

I"

+ 2(n —

note here that since we have supposed ü 0 on F, then Vu by (2.7)). R0 being fixed, if M is large enough we obtain (we

1?.

+ 2(n

0 on 1'

<1,

—

—

that is

+ 2(n —

1?. <

as claimed. Then (2.10) becomes

+ 2(n

—

Define a positive measure ji by dp(x) = ((IWI +2(n—2) -1 "dx. By using (2.7) and standard interpolation inequalities, one can easily check

that p has faster than exponential decay. Hence by Lemma 2.2 there is a sequence {k, } with {pk, } c 8, for each j a convex body Ek, such that are pairwise disjoint and satisfying: + 2(n —

<

(2.12)

+ 2(n —

and

C,

(2.13)

where C is a positive constant depending only on n. We may suppose IEk, I

(else drop all of them but one and enlarge it to have volume

=

If we take E = Ek, in (2.11) and combine with (2.12) we get I)

<

+ 2(n —

.

(2.14)

Sets of Positive Measure

185

But by Holder's inequality we have + 2(n

<

(II

—

IIL"(Y,) + 2(n

—

x

Then by comparing with (2.14)

where Y, = Ek,

(M"IEk, 1Y°

+ Ek3

Since M>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

=
for all N > 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
+

Cfl lxi

II

lxi

LP(IxI
+

L2(IxI
LP(IxI
+

(3.4)

We have by Holder's inequality

< Cli VIL..,a(IzI
if R is small enough (that is -y big enough) we get, since V E

(3.5)

C111x11'VUIILP(111
In the same manner, we have Cli WIIL..(lxI
C1R

Sets of Positive Measure

where

C1

187

is a new positive constant. Hence

C1Rfl xl'VullLa(IxI
(3.6)

Thus, by combining (3.4), (3.5) and (3.6), we obtain

lXl'UIILP'(,rI
<

It remains to estimate the right-hand side of (3.7). We have

but by using the Leibniz formula and (3.3),

where lull w2.p is the

of u in the bail B(O,2R). We have then

C3R211u11w2.p,

CII

which together with (3.7) give + (R2/P' — C1R)

C3R2llullw2.p. On

(3.8)

the other hand + (R2/P' — C1R) + (R2/P'

—

C1R)

Hence, by (3.8) we get

+

—

CIR)IlVullL2(IXI
(3.9)

which is better than the desired result since -y = R2. (We note here that k N), then R2 must be in A, but one can check since 'y E A = without difficulty that (3.9) remains valid for all small R> 0.) Proof of Theorem 1.1. It is clear that this is a direct consequence of Theorem 2.1 and Proposition 3.1.

For the proof of Theorem 1.2 we need the following form of Poincaré's inequality.

R. Regbaoui

188

Lemma 3.3. (cf. [7]) Let U E W1' (B(xo,r)), r > 0, and suppose that there exists E C B(xo,r) of positive measure such that u 0 on E. Then for all measurable sets A C B(xo, r),

f Aluldx

JB(xo,r)

IEI

where IEI and IAI denote the Lebesgue measure of E and A.

Proof of Theorem 1.2. Let u be as in Theorem 1.2. By assumption there is a set E of positive measure such that u 0 on E. By using Theorem 1.1, it is clear that it suffices to show that almost every point of E is a zero of infinite order for u. We follow closely the arguments of [2] and [3J. First let us prove the following:

Claim: For all

and all r > 0 with B(xo, 2r) C Q, we have

e

f

B(xo,r)

lVu(x)l2dx

JB(xo,2r) Iu(x)l2dx

r

where C is a positive constant depending on V and W. Indeed, Let E such that = 1 if lxi ( r . Thus qS satisfies also By using (1.1) and Sobolev inequalities in R'1, it is not hard to check that (3.10) follows from the elementary inequality

J IV&u)I2dx <

dx.

J

We know that almost every point of E is a point of density of E. Let x0 be such a point, that is 1EflB(xo,r)/1B(xo,r)I —+ 1 as r —'0. Thus, given E > 0, there exists r ro, then

1EflB(xo,r)l/IB(xo,r)l 1—c

and

IECflB(xo,r)l/IB(xo,r)I e,

where EC is the complement set of E in IL We have by Lemma 3.3 (applied

to u2 instead of u)

I

JB(xo,r)

lui2dx =

I

ul2dx

I

JB(zo,r)

—

lVui2dx

which by using (3.10) and (3.11) gives for all r TO,

I

JB(xo,r)

C,

j' —

I E)'2 JB(x,j,2r)

(3.12)

Sets of Positive Measure

189

Set g(r) = fB(xo,r) ul2dx and let us fix N E N. We choose e > 0 such that = 2-h' (note here that r0 depends on N). We can then rewrite (3.12) as

g(r)

2_Ng(2r)

for r

r0 .

(3.13)

Now, by iterating (3.13) we get for

g(p)

Thus, if we fix r < we get from (3.14)

To

and

r/rO,

we

(3.14)

choose k E N such that 2_kro < r g(r)

and since 2—k

r0.

2*kNg(2ro)

obtain

g(r)

(r/ro)Ng(2ro),

that is g(r) = O(r"). This shows that so is a zero of infinite order for u. The proof of Theorem 1.2 is then complete.

References [1] B. Bercelo, C. Kenig, A. Ruiz and C.D. Sogge, Weighted Sobolev inequalities and unique continuation for the Laplacian plus lower order terms, JU. J. Math., 32 (1988), 230—245. [2] D. De Figueiredo and i-P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations, 17 (1992), 339—346.

[3] i-P. Gossez and A. Loulit, A note on two notions of unique continuation, Bull. Soc. Math. Beig. Ser. B, 45(3) (1993), 257—268. [4]

L. Hörmander, Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations, 8 (1983), 21—64.

[5] D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math., 121 (1985), 463494.

[6] C. Kenig, A. Ruiz and C. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J., bf 55 (1987), 329—347.

[7] 0. Ladyzenskaya and N. Uraltzeva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.

190

R. Regbaoui

[8] R. Regbaoui, Strong unique continuation results for differential inequalities, J. Funct. Anal., 148 (1997), 508—523. 191

Unique continuation for differential equations of R. Schrodinger's type, Comm. Anal. Geom., 7 (1999), 303—323.

[10] T. Wolff, Unique continuation for ( VIVuI and related problems, Revista Math. Iberoamericana, 6 (1990), 155-200.

[111 T. Wolff, A property of measures in R7' and an application to unique continuation, Geom. Funct. Anal., 2 (1992), 225—284. 112] T. Wolff, A counterexaxnple in a unique continuation problem, Comm. Anal. Geom., 2 (1994), 79—102.

Département de Mathématiques Université de Brest 6, Avenue Le Gorgeu 29285 Brest, France email:

Some Results and Open Problems on the Controllability of Linear and Semilinear Heat Equations Enrique Zuazua ABSTRACT In this paper we address the problem of null-controllability of heat equations in two different cases: (a) The semilinear heat equation in bounded domains and (b) The linear heat equation in the half line. Concerning the first problem (a) we show that a number of systems in which blow-up arises may be controlled by means of external forces which are localized in an arbitrarily small open set. In the frame of problem (b) we prove that compactly supported initial data may not be driven to zero if the control is supported in a bounded set. This shows that although the velocity of propagation in the heat equation is infinite, this is not sufficient to guarantee null-controllability properties. We also include a list of open problems.

1

Introduction

In this paper we present some recent results on the controllability of Linear

and semilinear heat equations in bounded and unbounded domains. We describe mainly the results that have been developed in full detail in joint works with E. Fernández-Cara [10, 111 and S. Micu 122, 23J. To unify the presentation, let us consider a smooth domain of R'1, n 1. Note that we do not assume to be bounded. In fact, in part of this paper, will be assumed to be the half-line. Let f : R —.. R be a locally Lipschitz function. Let w be an open nonempty subset w C Given T > 0, consider the semilinear heat equation: in

u=

0

(u(x,0)=uo(x)

flx(0,T)

x (0,T) on in fI.

(1.1)

In (1.1), u = u(x, t) is the state, which is assumed to be a scalar function, denotes the characteristic function of w so that 1 in w and 0 in and v = v(x, t) is the control. System (1.1) is a model example of a

192

E. Zuazua

semilinear parabolic equation or system with an exterior source, v, acting on the system as a control through the subset w. We assume w to be bounded, even when fl is unbounded. This assumption is natural since we would like to see what is the minimum amount of control that is needed to control the system. To simplify the presentation we also assume that

1(0) = 0.

Under the assumption (1.2), u

0 is an equilibrium solution of (1.1).

Indeed, when (1.2) holds, (1.1) is satisfied with u 0 and v 0. The problem of null-controllability we address is as follows: Given uo E to find a control v E L2(w x (0,T)) such that the solution n of(1.1) is globally defined in x (0, T) and satisfies

u(x,T)

0 in fl.

Some comments are in order:

Remark 1.1.

(a) Some minimal assumptions are needed on I to guarantee the local well-posedness of system (1.1). All through this paper we shall assume that I

f'(s)

a.e.

SEIR

with C >0 and

p 1 + 4/n, so that for every u0 E L2(Q) and v E L2(w x (0, T)) there exists a maximal

time of existence 0
(i) Either T = T, or

(ii) T
00

as t

/ Tt.

(b) According to comment (a) above we see that the first requirement in the null-controllability problem is guaranteeing that the control v is such that the solution of (1.1) is globally defined in the time interval [0, TI.

But, in addition to this, we require the control to be such that (1.3) holds, i.e., the solution reaches the equilibrium configuration at time T. (c) Note that when null-controllability holds, extending the solution and

control for t T as (1.7)

Linear and Semilinear Heat Equations

193

u is a solution of the semilinear heat equation (1.1) for all t 0. (d) If the nonlinearity f satisfies a suitable growth condition at infinity, the first requirement in the null-controllability property, i.e., the fact that the solution is defined in the whole interval [0, T] is immediately satisfied. More precisely, as shown in [6], if (1.4), (1.5) are satisfied and, moreover, for some C > 0,

If(sflCIsI(1+loglsl) holds for all s 0, then, for all T> 0, uo E L2(cl) and v there exists a unique globally defined solution ii

(1.8)

L2(tu x (0,T)),

C ([0, T]; L2(1)) of (1.1).

In this paper we focus on two particular problems:

Problem 1. Null-controllability for the semilinear equation when ci is bounded.

Problem 2. Null-controllability for the linear equation (with I one space dimension when ci =

0) in

(0, oo).

These two choices are justified by the fact that the problem is by now rather well understood when f 0, i.e., for the linear heat equation, when ci is bounded. More precisely. the following is known (see, for instance, [13] and [20]): Let ci be a bounded smooth domain of n 1 and let w be any open nonempty subset of ci. Let T > 0. Then, for any uo L2(cl) there exists V E x (0, T)) such that the solution u of —

u=0

= vL

(u(x,0)=uo(x)

in on in

ci x (0,T)

OIlx(0,T) ci,

satisfies (1.3).

In view of this result it is natural to study the two problems above. In Problem 1 we intend to analyze to what extent the null-controllability is kept in the presence of nonlinear terms. In Problem 2, in view of the infinite speed of propagation involved in the heat equation, we intend to see whether the null-controllability is kept when ci is an unbounded domain, w being bounded.

In Section 2 we analyze Problem 1 and we present the main results in [10, 11, 12J. Section 3 is devoted to Problem 2 and the results in [22, 23] are presented. In Section 4 we present some open problems, closely related to the topics addressed in these notes. There is a large literature on controllability problems for heat equations. Here we focus on two particular aspects of the theory. The interested reader may learn more from the bibliography at the end of the paper. Acknowledgements. The author acknowledges the invitation of F. Colombini and C. Zuily to deliver a series of lectures at the meeting in Cortona.

E, Zuazua

194

The hospitality and support are also warmly acknowledged. This work was supported by grant PB96-0663 of the DGES (Spain).

Null-controllability for the semilinear heat equation

2

All through this section we assume nonempty open subset such that w 2.1

to be a bounded domain and w a

ft

Main results

The first main result in [11] is a negative one:

Theorem 2.1. There exist smooth nonlinearities such that 1(0) =

0

and

(1.4)—(1.5) hold with

(2.1)

and

p > 2,

(2.2)

for which system (1.1) fails to be null-controllable for all T > 0. According to this result, if one wants to get null-controllability properties for all non-linearities in a suitable class, roughly, one has to impose a growth condition of the form (2.3)

The second main result in [11] guarantees the null-controllability under a stronger restriction on the growth rate.

Theorem 2.2. Let f be a locally Lipschitz function such that 1(0) =

0

and (1.4)—(1.5) hold.

Assume that

f(s)

urn sup IaI—oo

I

s

I

=

(2.4)

0.

log312 I s

Then, for every T > 0, system (1.1) is null-controllable. Whether the system is null-controllable or not when I behaves at infinity in the range C1

I

s I log3'2

s Il f(s)

is an interesting open problem.

l

C2 I s

log2 I s I, as

s i—' 00

(2.5)

Linear and Semilinear Heat Equations

195

As far as we know, in the blow-up literature there is no evidence of the failure of null-controllability for f as in (2.3) (see Subsection 2.2 below) but the technique of proof we employ for Theorem 2.2 fails when (2.5) holds with C1 > 0 large enough (see Subsection 2.3).

2.2

Sketch of the proof of Theorem 2.1

We consider the nonlinearity

f's' / log"(l + c)do,

f(s)

Vs

Jo

(2.6)

ER

with p> 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

Sketch of the proof of Theorem 2.2

Here we briefly describe the main steps of the proof of Theorem 2.2. We refer to [11) for a complete proof.

Step 1. Description of the fixed point method To simplify the presentation we assume that uo C(Ifl for some o > 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).

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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
(u(x,0)uo(x), 0<x
200

E. Zuazua

Here u = u(x, t) is the state and v = v(t) is the control that acts on the system through the extreme x = 0. The main difference of system (3.1) with respect to those that we discussed above is that the equation holds in the infinite domain = (0, oo) =

We consider the boundary control problem for simplicity. But the same negative results hold when the control acts on a bounded subinterval of (0,oo). The null-controllability problem may be formulated as follows: Given E L2 (0,oo), tofindv E L2(0,T) such that the solutionu of(3.1) satisfies

u(x,T) =

0,

Vx > 0.

(3.2)

In order to make the formulation of the problem precise we have to clearly identify the solution of (3.1) we are working with. We shall analyze the solution defined by transposition that turns out to be inC ([0,T]; H1 (R+)) (see (21] and [221). There are however other smooth solutions of (3.1). They grow very fast as x —+ co and therefore they are not achievable by transposition (see [18] and [22]).

Let us now analyze the observability inequality corresponding to this null-control problem. Consider the adjoint system

0<x
0
(3.3)

0<x 0 such that II

cj

T

dt

(3.4)

holds for every solution of (3.3). If (3.4) holds it is easy to see that the sytem is null-controllable. However, it is easy to see that (3.4) is not true. Indeed, given L2(k+) we consider the sequence of initial data = — k). Let be the solution of

(3.3) associated with this datum. It is easy to see that T

1IL2(R+) is 2

k

bounded below by a posLtlve constant while f0 t) dt tends to zero as k oo. However, one could think that a variant of (3.4) in which the )0) is replaced by a suitable weighted norm could be true. But norm of the situation is much worse than that. Indeed, assume that there exists a positive smooth function p such that the weighted observability inequality

IIw(o)

I

12 dt

(3.5)

Linear and Semilinear Heat Equations

201

holds for every solution of (3.3). Here we are using the notation

j

II I

f2p(x)dx.

If (3.5) were true, we would deduce by duality that for any Uo

L2(R+;

there exists a control v E L2(0, T) such that the solution of (3.1) satisfies (3.2).

But, as the following result from 1221 shows, there is no positive weight Consequently the obp for which null-controllability holds in servability inequality (3.5) does not hold:

Theorem 3.1. There is no v

D(R÷), uo 0 for which there exists L2(0, T) such that the solution of (3.1) satisfies (3.2).

This result is in strong contrast with those of the case where the domain is bounded. Even if the information propagates at infinite speed on the

heat equation, the null-control condition is not achievable for any nice smooth and compactly supported initial data. In a first reading this result might seem to be in contradiction with those in [18) that, in particular, guarantee that: For any bounded and continuous u0 there exists a continuous control v C([0, T]) and a smooth solution of (3.1) such that (3.2) holds. However, both results are compatible. The solution in 118) is not the "physical" solution in the sense of transposition but another one that grows extremely fast as x H In the following sections we give a sketch of the proof of Theorem 3.1. We refer the interested reader to 1221 for the details. We first reformulate the control problem as a moment problem. When doing so we use in an essential way the similarity variables and the weighted Sobolev spaces introduced in 171 when analyzing asymptotic properties of solutions of semilinear heat equations. We show that the moment problem is critical and, in fact, that one may not expect an L2-solution for "nice" data. More precisely, the control problem turns out to be equivalent to the following moment problem: Given S > 0 and a sequence {am }m to fifld

v e L2(0,S) such that Cs

= am, Vm 1.

I

Jo

(3.6)

As we shall see, it turns out that (3.6) may only have a nontrivial solution v

L2(0,S) when the coefficients {am}mi grow exponentially fast at

infinity. In fact, the following holds:

Theorem 3.2. Let S > 0 be given. Assume that lim

m—.oo emö

=0

(3.7)

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E. Zuazua

for all 5 > 0. Then, if (3.6) admits a nontrivial solution v E L2(0, S), necessarily am 0, for all m 1. As a consequence of Theorem 3.2, Theorem 3.1 holds immediately since,

roughly speaking, (am) are the Fourier coefficients of the datum to be controlled and, when the initial datum is C°° and of compact support, the sequence {am}mi is bounded and (3.7) holds for all 5 > 0. In [221, using Paley—Wiener's Theorem we prove that there are exponentially growing coefficients {am} for which (3.6) admits a solution v E L2(0, S). As an immediate consequence we deduce the existence of ini-

tial data with exponentially growing Fourier coefficients that are nullcontrollable. In [23] we have extended these results to the case of the boundary control in the half space of The similarity variables apply in any conical domain

but our analysis does not apply in such a general setting. We shall return to this question in Section 4. The rest of this section is organized as follows. In Subsection 3.2 we describe the similarity variables and reduce the control problem to the moment problem (3.6). In Subsection 3.3 we prove the negative result of Theorem 3.2 on the moment problem. 3.2

Reduction of the control problem

We introduce the similarity variables

y = z/y'flT,

log(t + 1)

(3.8)

and the new unknown:

w(y,s) = e"2u (e3/2,ea

—

i)

.

(3.9)

Then, u solves (3.1) if and only if w solves I

w(O,s) = ii(s),

Lw(y,0)=uo(y),

w=0, y>O, 0 < s <S y>O

0<s<S (3.10)

with

iJ(s) =

S = log(T+

—

1).

1)

(3.11)

(3.12)

L2(0, T) if and only if E L2(0, S). Consequently, the Obviously v null-controllability of system (3.1) and (3.10) are equivalent properties.

Linear and Semilinear Heat Equations

203

Therefore, in the sequel we shall analyze the null-controllability of system (3.10).

Let us now analyze the elliptic operator involved in (3.10). We set 1

Lw =

(3.13)

—

Observe that the operator L may also be written in the symmetric form Lw =

(3.14)

K(y) = exp(y2/4).

(3.15)

where

Therefore, in order to analyze (3.10), it is natural to introduce the following weighted spaces:

•

L2(R+;K)={f:R+ _iIR:jf2K(y)dy
• These spaces, endowed with their canonical norms, are Hubert spaces. We also introduce the dual of H0' K) that we shall denote by H' K). The following properties are well known (see [7] and [22]):

•

j

S

16j

2 I

• the embedding H0' (1R4.; K)

• L : H0' (IR÷;K)

K(y)dy, 'If E L2

(]R÷;K);

K) is compact;

H' (IR+;K) is an isomorphism;

• the domain of L as unbounded operator in L2

K) is D(L) =

K) fl H0'(IR÷; K);

• the eigenvalues of L are

= j, j 1 and the corresponding eigen-

functions where

=

K'.

= d22'[4J/dy22', j

1

• The eigenfunctions may be normalized to constitute an orthonormal basis of L2 (R+; K). Let us now consider system (3.10). We define the solution of it by transposition. For this, we introduce the adjoint system I

y>0, =

0,

0 <8 < S

y>0.

0<s<S (3.16)

E. Zuazua

204

Multiplying in (3.10) by we deduce that 1.S

/ Jo

and integrating by parts (formally by now),

ç

/ whKdyds = / w(y, S)eo(y)K(y)dy I, —

(3.17)

/

0

We adopt (3.17) as the definition of solution of (3.10) in the sense of transposition. More precisely, given w0 E L2 K) and v L2(0, S), w E

C([0, S]; H' (R÷; K)) is said to be a solution of (3.10) in the sense of transH0' K) position if (3.17) holds for every solution e of (3.16) with K)). We refer to f22] for the proof of the existence and h L' (0, S; H0' and uniqueness of this solution. In the sequel, when discussing the null-controllability of system (3.10) we will be studying the null-controllability of the solution of (3.10) defined, as above, by transposition.

With the precise definition of solution in mind we can transform the control problem into a moment problem. Indeed, according to (3.17), if w(S)

0 we have

jf whKdyds =

j

(3.18) —

By taking h is then

0 and

=

L' (0, S; H0' K)). taking into account that the solution of

H0' (R÷; K) and h

for all solutions of (3.16) with and

= we

deduce that, if

=

j1 then (3.18) holds if and only if I

Jo

=

Vj 1.

(3.19)

On the other hand, it can be seen that

asj—'oo. Consequently, concerning the statement of Theorem 3.2, the moment problems (3.6) and (3.19) are equivalent. The functions v in (3.19) and (3.6) are related by the multiplicative factor e8/2. We

conclude this section by proving the main result on the moment

problem.

Linear and Semilinear Heat Equations

3.3

205

Proof of the main result on the moment problem

This subsection is devoted to proving Theorem 3.2. We follow very closely a method by Rill—Nardzewski described in [25, pp. 166—167]. First of all we observe that for any g E L2(0, S) the following identity holds:

=

j9(u)du.

(3.20)

To prove (3.20) it is sufficient to observe that

(—1'

f

= .L8

{i — exp

}

g(u)du

oo by means of the dominated convergence theorem. Using this identity it is easy to deduce that if v E L2(O, S) is such that

and to pass to the limit on the last integral as x —'

6

/

Jo

(3.21)

then, necessarily, the support of v is contained in [0,6]. Indeed, according to (3.20), we have J0Seki(3_u)v(S

—

u)du =

fv(S

—

u)du

(3.22)

but, in view of (3.21),

JS

f

—

S)ekJ6

=

[exp

—

ii

Taking into account that the last expression tends to zero as j s S — 6 we deduce that

oo

for

S—6 and consequently

v(S—s) =0,VsS—6 v(s) =

0,

Vs 8.

(3.23)

According to (3.23), we deduce that if (3.21) holds for all 5 > 0, then v(s) = 0 for all .s 0.

206

E. Zuazua

Open problems

4

In this section we present some open problems related to the issues we have addressed above.

4.1

Sharp constants on the observabillty inequality for the constant coefficient heat equation

Consider the linear, constant coefficient heat equation in Ilx(0,T) on Oflx(0,T) in

where is a bounded smooth domain, and T > 0. Let nonempty subset of Q.

be an open

As mentioned above (see Section 2), there exist positive constant y > 0 and C > 0 such that

fT

jT

f

f

ço2dxdt

(4.2)

for every solution of (4.1). It would be interesting to characterize the best

constant -y in (4.2). This problem might be easier to solve when T = 00. Note that (4.2) still holds for T = oo and that both integrals converge since the L2-norm of solutions of (4.2) tend exponentially to zero as time goes to infinity. The inequality then reads as

cj

ff

f

(4.3)

and characterizing the best constant -y > 0 in (4.3) is an interesting open problem.

Note that solutions of (4.1) may be written in Fourier series as

= k> I

with orthonormal basis of L2(fl)

where {ak } are the Fourier coefficients, {Ak } k 1 the eigenvalues of

Dirichiet boundary conditions and constituted by eigenfunctions. Then 2

L2(fl)

}k

_V'

an

2 —2Akt

k> 1

ff

=

4f

Linear and Semilinear Heat Equations

207

On the other hand =

J

k,j1

J

J

0

I kj Thus, getting the best constant -y > 0 in (4.3) is equivalent to finding it so that the inequality aka3

f

J

(4.4)

It is not difficult to get a lower bound on the constant (4.3). Indeed, as shown in [101, if we set

> 0 in (4.2) and

k>1

k

0

w

holds for all {ak} E £2

exp (p2/4t) G(x, t),

w = cos

(4.5)

where C is the fundamental solution of the heat equation

G(x, t) =

exp (— I x 12 /4t),

then a very singular solution of the heat equation in x (0, oo). Let B(xo,p) be a ball contained in and consider = with as in (4.5) for this value of p. It can be checked that t,b is bounded on Oft x (0, oo) and in w x (0, oo), but 11L2(n) increases exponentially Ot This solution may be modified on the boundary so that it as t satisfies the Dirichlet homogeneous boundary condition, without changing

the other properties. In this way we see that the constant y > 0 in (4.2) and/or (4.3) necessarily satisfies y> p2/4. Therefore, we have the following lower bound on y:

-y

sup

p2/4.

This lower bound may be sharp in radially symmetric geometries. But this remains to be proved.

4.2

Sharp observability constant for the heat equation plus a potential

Let us consider now the linear heat equation with a bounded potential I

+ ap

—

=

0

=0

in on in

ft x (0, T) Oft x (0, T) 1,

(4.6)

208

E. Zuazua

x (0,T)). where a = a(x,t) there As we have shown above, given a nonempty open subset C > 0 such that the following observexists a positive constant C = ability inequality holds: exp

[c(T +

+Ta

a

+

j j ço2dxdt (4.7)

for every solution of (4.6), for all T > 0 and all potential a E L°°(l1 x (0, T)). in the We have also seen that the exponential factor exp (c a

observability inequality (4.7) is the main obstacle for proving the nullcontrollability of the semilinear heat equation for nonlinearities that grow at infinfty faster than isi log312 Isi. However, there is no evidence of the lack of null-controllability in the class of nonlinearities that grow at infinity slower than 181 log2(s).

This suggests that the factor exp (c

a

could possibly be replaced

However, as we have shown above, this does not seem by exp(C a possible to be done with the global Carleman inequalities we have used. Note however that these are Carleman inequalities in L2(Q x (0, T)). One (0, T; could think of Carleman inequalities in spaces This could and consequently lead to an improvement of the factor exp (c a to a relaxation of the growth condition on the nonlinearity ensuring nullcontrollability.

4.3

Relaxing the growth condition on the nonlinearity ensuring null-controllability

As we have mentioned in Problem 4.2 above, there is no evidence that null-controllability fails for nonlinearities that grow at infinity less than sJ log2 Isi. In fact there is no evidence of faillure of null-controllability for

nonlinearities f such that VI F(s) is not integrable at infinity, F being the primitive of f, i.e., F(z) = f(s)ds. I

However, by now, we only have a proof of null-controllability under the condition that (4.8)

or, under the slightly weaker condition that lim sup lsl—.oo

with e small enough.

I

Isi

=

(4.9)

Linear and Semilinear Heat Equations

209

Relaxing the growth rates (4.8)—(4.9) is an interesting open problem that is closely related to the improvement of the observability inequality (4.8), as we mentioned in Problem 4.2 above.

4.4

On the lack of null-controllability on general unbounded domains

In Section 3 we proved that "nice" initial data (compactly supported and smooth) are not null-controllable for the heat equation on the half-line R+ with a Dirichlet boundary control at x = 0. (see [23]). The same results can be proved on the half-space On the other hand, the similarity variables apply in any conical domain Q. However, the methods in [231 do not apply in such a general geometric setting because of the lack of orthogonality of the traces of the normal derivatives of the eigenfunctions on It would be interesting to know if the negative result of Section 3, extended to in [23], holds in any conical domain. 4.5

Lack of null-controllability for general 1— d equations on the half tine

Let us consider the 1 —

d,

variable coefficient, linear heat equation on the

half-line:

(p(x)tLt — u(0,t) = v(t), u(x,0) = uo(x), 1

0,

x>

0,

t >0

t>

0

(4.10)

x > 0.

It would be interesting to see if the negative results of Section 3 may be extended to this more general equation (with, say, a and p positive, bounded and smooth functions). Of course, the same problem arises for more general equations: coefficients depending both on x and t, equations with lower order potentials, equations involving semilinear terms

References [1] S. Anita and V. Barbu, Internal null controllability of nonlinear heat equations, ESAIM: COCV, 5 (2000), 157—173.

[2] S.A. Avdonin and S.A. Ivanov, Families of ExponentiaL9. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge Univ. Press, 1995.

[31 V. Barbu, Exact controllability of the superlinear heat equation, Appi. Math. Optim., 42(1) (2000), 73—89.

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Bezerra, Controlabilidade aproximada para a equaçao de calor em dominios nao limitados. Controlabilidade nula, Ph.D. dissertation,

[41 S.

Universidade Federal do Rio de Janeiro, July 1999.

[5] T. Cazenave and A. Haraux, Equations d'évolution avec nonlinéarité logarithmique, Ann. Fac. Sci. Toulouse, 2 (1980), 21—51. [6) T. Cazenave and A. Haraux, Introduction aux problèmes d 'evolution semilinéaires, Mathematiques & Applications, Ellipses, Paris 1989.

(7] M. Escobedo and 0. Kavian, Variational problems related to selfsimilar solutions of the heat equation, Nonlinear Analysis TMA, 11(10) (1987), 1103—1133.

[81 C. Fabre, J.P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc. Royal Soc. Edinburgh, 125A (1995), 31—61.

[9] E. Fernández-Cara, Null controllability of the semilinear heat equation, ESAIM: COCV, 2 (1997), 87—107.

110] E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Advances Duff. Eqs., 5 (4—6) (2000), 51—85.

[11] E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing-up semilinear heat equations, Annales de I 'IHP, Analyse non-hnéaire, to appear. [121 E. Fernández-Cara and E. Zuazua, Control of weakly blowing up semilinear heat equations. In Nonlinear PDE's and Physical Modelling (H.

Berestycki and Y. Pomeau, eds.), Klüwer, NATO-AS! series, to appear.

113] A.V. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equation8, Lecture Notes Series #34, Research Institute of Mathematics, Global Analysis Research Center, Seoul National University, 1996.

[14] V. Galaktionov, On blow-up and degeneracy for the semilinear heat equation with source, Proc. Royal Soc. Edinburgh, liSA, (1990), 19— 24.

1151 V. Galaktionov and J.L. Vázquez, Regional blow-up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation, SIAM J. Math. Anal., 24(5) (1993), 1254—1276. (16]

J. Henry, Contrôle d'un réacteur enzymatique a l'aide de modèles a paramètres distribués: Quelques problèmes de contrôlabilité de systèmes paraboliques, Ph.D. Thesis, Université Paris VI, 1978.

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[17] O.Yu. Imanuvilov and M. Yamamoto, On Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Preprint #98-46, University of Tokyo, Graduate School of Mathematics, Komobo, Tokyo, Japan, November 1998. [181 B.F. Jones Jr., A fundamental solution of the heat equation which is supported in a strip, J. Math. Anal. AppI., 60 (1977), 314—324. [19] A. Khapalov, Some aspects of the asymptotic behavior of the solutions

of the semiinear heat equation and approximate controllability, J. Math. Anal. AppI., 194 (1995), 858—882.

[20] 0. Lebeau and L. Robbiano, Contróle exact de l'équation de Ia chaleur, Comm. P.D.E., 20 (1995), 335—356.

[21] J.-L. Lions, Contrôlabilité exacte, stabili.sation et perturbations de systèmes distribués. Tomes 1

2. Masson, RMA 8 & 9, Paris, 1988.

[22] S. Micu and E. Zua.zua, On the lack of null-controllability of the heat equation in the half-line, Trans. Amer. Math. Soc., 353(4)(2000), 1635—1659.

[23] S. Micu and E. Zuazua, On the lack of null-controllability of the heat equation in the half-space, Portugaliae Mathematica, 58(1) (2001), 1— 24.

[24J L. de Teresa and E. Zuazua, Approximate controllability of the heat equation in unbounded domains, Nonlinear Anal., T. M. A., 37(8) (1999), 1059—1090.

[25] K. Yosida, Functional Analysis, Springer Verlag, 1980.

[26] E. Zuazua, Some problems and results on the controllability of Partial Differential Equations, Proceedings of the Second European Conference of Mathematics, Budapest, July 1996, Progress in Mathematics, 169, 1998, Birkhäuser Verlag Basel/Switzerland, pp. 276—311.

[27] E. Zuazua, Controllability of Partial Differential Equations and its Semi-Discrete Approximations. In EDP-Chile (C. Conca et al. eds.), to appear. Departamento de Matemática Aplicada Universidad Complutense 28040 Madrid. Spain [email protected]

Progress in Nonlinear Differential Equations and Their Applications Editor Haim Brezis de Mathématiques P. et M. Curie 4. Place Jussieu 75252 Paris Cedex 05 France

and Department of Mathematics Rutgers University New Brunswick, NJ 08903 U.S.A. Progress in Nonlinear Differential Equations and Their Applications is a book series that

lies at the interface of pure and applied mathematics. Many differential equations are motivated by problems arising in such diversified fields as Mechanics, Physics, Differential Geometry, Engineering, Control Theory, Biology, and Economics. This series is open to both the theoretical and applied aspects, hopefully stimulating a fruitful interaction between the two sides. It will publish monographs. polished notes arising from lectures and seminars, graduate level texts, and proceedings of focused and refereed conferences. We encourage preparation of manuscripts in some form of TeX for delivery in camera-ready copy, which leads to rapid publication, or in electronic form for interfacing with laser printers or typesetters. Proposals should be sent directly to the editor or to: Birkhauser Boston. 675 Massachusetts Avenue, Cambridge, MA 02139

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Parametrized Measures and Variational Principles Pablo Ped regal

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Topics in Mathematical Modelling of Composite Materials Andrej Cherkaev and Robert Kohn, editors

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Geometrical Optics and Related Topics Perruccio Colombini and Nicolas Lerner, editors

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Entire Solutions of Semilinear Elliptic Equations I. Kuzin and S. Pohozaev

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Stability of Functional Equations in Several Variables Donald H. Hyers'. George Isac and Themiszocles M. Rassias

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Topics in Nonlinear Analysis: The Herbert Amann Anniversary Volume Joachim Escher and Gieri Simoneft, editors

PNLDE 36 Variational Problems with Concentration Marlin Flucher PNLDE 37

Implicit Partial Differential Equations Bernard Dacorogna and Paolo Marcellini

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Systems of Conservation Laws: Two-Dimensional Riemann Problems Yuxi Zheng

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Linear and Nonlinear Aspects of Vortices: The Ginzburg—Landau Model Frank Pacard and Tristan Rivière

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RecenL Trends in Nonlinear Analysis: Festschrift Dedicated to Alfonso \rignoli on the Occasion of his Sixtieth Birthday Jurgen Appell. editor

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Quasi-hydrodynamic Semiconductor Equations Ansgar Jüngel

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Semigroups of Operators: Theory and Applications A. V Balakrishnan, editor

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Nonlinear Analysis and its Applications to Differential Equations M.R. Grosshino, M. Ramos, C. Rebelo, and L Sanchez, editors

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The Monge-Ampère Equation Cristian £ Gutiérrez

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Spatial Patterns: Higher Order Models in Physics and Mechanics LA. Peletier and WC. Troy

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Carleman Estimates and Applications to Uniqueness and Control Theory Ferrucio Colombini and Claude Zuily, editors

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Recent Advances in the Theory of Shock Waves Hem rich Freistühler and Anders Szepessy. editors

('arleman Estimates and Applications to tniqueness and Control Ferruccio ('olombini and Claude Zuils. Editors

This solurne consists of 14 research articles that are an outgrossth of a scientific meeting held in Cortona on the subject of( arleman Estimates and Control \eis results are l)rcseflted on Carkman estimates and their applications to uniqueness and controllability of partial differential equations and stems.

The main topics are unique continuation for elliptic Pl)Fs and control theory and ins erse problems. \eo results on strong tilliqueness for second or higher order operators arc explored in detail in seseral papers. In the area of control theory. the reader ssill lind applications ofCarlenian estimates to stabilitation. and esact control for the ssase and Schrüdinger equations. final paper presents a challenging list ol open problems on the topic of of linear and semilinear heat equations. The articles contain eshaustise and accessible to mathematicians. elenientars background in I'DIs.

sell-contained proofs and graduate students nith an

Contributors: '5. t•erner 1. Sishitani

•

Bellassouvd

5. Burq

T. Okaji

F. (olombini

Phung K. Regbaoui X. Saint Raymond D. Tataru

B. Dehman

C. Grammatico SI. Khenissi II. Koch

F. /uaiua

P. Fe Bisrgne

Birkl,äuser ISBN 0-8176-4230-7 w,sss.birkhauser.cum

III

I

.

Carleman Estimates and Applications to Uniqueness and Control Theory (Progress in Nonlinear Differential Equations and Their Applications)

Carleman estimates and applications to uniqueness and control theory

Nonlinear Equations: Methods, Models and Applications (Progress in Nonlinear Differential Equations and Their Applications)

Minimax Theorems (Progress in Nonlinear Differential Equations and Their Applications)

Nonlinear Analysis and Differential Equations (Progress in Nonlinear Differential Equations and Their Applications)

Nonlinear partial differential equations and their applications

introduction to differential equations and their applications

Trends in Partial Differential Equations of Mathematical Physics (Progress in Nonlinear Differential Equations and Their Applications)

Differential Equations: Theory and Applications

Differential equations: Theory and applications

Differential equations and applications

Differential equations and applications

Nonlinear Partial Differential Equations and Applications

Control of partial differential equations and applications

Nonlinear Partial Differential Equations and Their Applications (Studies in Mathematics and its Applications, Vol 31)

Variational Methods in Shape Optimization Problems (Progress in Nonlinear Differential Equations and Their Applications)

Topics in Mathematic Modelling of Composite Materials (Progress in Nonlinear Differential Equations and Their Applications)

Quantization Methods in the Theory of Differential Equations (Differential and Integral Equations and Their Applications)

Stochastic differential equations theory and applications

Stochastic Differential Equations: Theory and Applications

Theory and applications of fractional differential equations

Engineering differential equations: Theory and applications

Integral equations and their applications

Stochastic differential equations and applications

Stochastic differential equations and applications

Delay differential equations and applications

Integral Equations and their Applications

Partial differential equations and applications

Delay Differential Equations and Applications

Stochastic differential equations and applications,

Delay differential equations and applications

Carleman Estimates and Applications to Uniqueness and Control Theory (Progress in Nonlinear Differential Equations and Their Applications)

Carleman estimates and applications to uniqueness and control theory

Nonlinear Equations: Methods, Models and Applications (Progress in Nonlinear Differential Equations and Their Applications)

Minimax Theorems (Progress in Nonlinear Differential Equations and Their Applications)

Nonlinear Analysis and Differential Equations (Progress in Nonlinear Differential Equations and Their Applications)

Nonlinear partial differential equations and their applications

introduction to differential equations and their applications

Trends in Partial Differential Equations of Mathematical Physics (Progress in Nonlinear Differential Equations and Their Applications)

Differential Equations: Theory and Applications

Differential equations: Theory and applications

Differential equations and applications

Differential equations and applications

Nonlinear Partial Differential Equations and Applications

Control of partial differential equations and applications

Nonlinear Partial Differential Equations and Their Applications (Studies in Mathematics and its Applications, Vol 31)

Variational Methods in Shape Optimization Problems (Progress in Nonlinear Differential Equations and Their Applications)

Topics in Mathematic Modelling of Composite Materials (Progress in Nonlinear Differential Equations and Their Applications)

Quantization Methods in the Theory of Differential Equations (Differential and Integral Equations and Their Applications)

Stochastic differential equations theory and applications

Stochastic Differential Equations: Theory and Applications

Theory and applications of fractional differential equations

Engineering differential equations: Theory and applications

Integral equations and their applications

Stochastic differential equations and applications

Stochastic differential equations and applications

Delay differential equations and applications

Integral Equations and their Applications

Partial differential equations and applications

Delay Differential Equations and Applications

Stochastic differential equations and applications,

Delay differential equations and applications

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