proceedings

Nicole Berline

Claude Sabbah

Groupes finis. Préface des éditeurs

Nikolay Tzvetkov

Bilinear estimates related to the KP equations

Ari Laptev

Timo Weidl

Recent results on Lieb-Thirring inequalities

Johannes Sjöstrand

JOURNÉES ÉQUATIONS AUX DÉRIVÉES PARTIELLES
JOHANNES SJÖSTRAND
Asymptotic distribution of eigenfrequencies for damped wave equations
Journées Équations aux dérivées partielles (2000), p. 1-8
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Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
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Journees Equations aux derivees partielles
Nantes, 5-9 juin 2000
GDR 1151 (CNRS)
Asymptotic distribution of eigenfrequencies
for damped wave equations
Johannes SJOSTRAND
Resume
II est bien connu que les frequences propres associees a un d^Alembertien
amorti sont confinees dans une bande parallele a Paxe reel. Nous rappelons
Pasymptotique de Weyl pour la distribution des parties reelles des frequences
propres, nous montrons que "presque toutes" les frequences propres appar￾tiennent a une bande determinee par la limite de Birkhoff du coefBcient
d^amortissement. Nous montrons aussi que certaines moyennes des parties
imaginaires convergent vers la moyenne du coefficient d'amortissement.
Abstract
The eigenfrequencies associated to a damped wave equation, are known to
belong to a band parallel to the real axis. We review Weyl asymptotics for the
distribution of the real parts of the eigenfrequencies, we show that up to a set
of density 0, the eigenfrequencies are confined to a band determined by the
Birkhoff limits of the damping coefficient. We also show that certain averages
of the imaginary parts converge to the average of the damping coefficient.
1. Introduction and results
Let M be a smooth compact Riemannian manifold of dimension n and let A be
the corresponding Laplace Beltrami operator. In control theory one is interested in
the long time behaviour (t —> +00) of solutions to
(<9(2
 - A + 2a{x)9t)v(t, x) = 0 on R x M. (1.1)
Here we let a € COO{M^Ti) be the "damping" coefficient (where true damping
corresponds to taking a > 0). Much more general problems can and have been
considered : M could have a boundary, a could have discontinuities, we could
replace the scalar equation (1.1) by a system and so on. The reason for us to
Acknowledgements: We are greatful to G. Lebeau, P. Freitas and A. Laptev for stimulating discussions, and to M.
Solomjak for drawing our attention to the work of Markus-Matsaev
MSC 2000 : 35P15, 35P20
Keywords : non selfadjoint, eigenvalue
XVI-l
look at this eqution is that it leads to simplified model problems for resonances for
strictly convex obstacles.
In this talk we will only discuss the stationary problem obtained by putting
v(t,x) = e^u^x} :
(-A - r2
 + 2ia(x)T)u(x) = 0 (1.2)
If there exists a non-trivial solution to (1.2), we call r € C an eigenfrequency.
Equivalently r is an eigenvalue of
= f ° 1 ^•.^xH^^xH 0
,
\ —A 2za[x) ) P=
with domain H 2
x H l
, where H8
 = H^M) is the standard Sobolev space on M. The
spectrum (i.e. the set of eigenfrequencies) is discrete and symmetric under reflection
in the imaginary axis. There are several equivalent definitions of the multiplicity
m(r) € {l,2,..}ofan eigenfrequency r (see [7] for more details) and we shall always
count the eigenfrequencies with their multiplicity. If r is an eigenfrequency, then it
is easy to see that
r inf a < Im r < sup a, Re r ^ 0,
t2min(infa,0) < Imr < 2max(supa,0), Rer =0. ^ )
In the case a > 0, the energy of solutions to (1.1) is non-increasing when t —^ +00
and Lebeau [4] has obtained a lower bound on the rate of decay in terms of inf Im r
and A_ introduced below, and an earlier result in the same direction was obtained
by J. Rauch and M. Taylor [6]. Lebeau also obtained a sharpening of (1.3) for large
eigenfrequencies: Put
1 (T
{a}T = ^f S-T a ° ^PW)^ T > °5
where p{x, ^) = <^2
 denotes the principal symbol of —A, and Hp is the corresponding
Hamilton field. Let
A+ := inf sup (a)r = lim sup (0)7^
T>0p-i^) T^OO^-I(I)
A_ :== sup inf (0)7- == lim inf (a)^
TX^-KI) r-^p-^i^
where the second equalities follow from subadditivity arguments. The following
result is essentially due to [4]:
Theorem 1.1 For every e > 0 all except possibly finitely many eigenfrequencies
belong to R + %]A_ - e, A+ + e[
We refer to M. Asch, G. Lebeau [1] for two interesting refinements of this result
and to P. Freitas [2] for various estimates. Both papers contain interesting numerical
results.
The results presented here concern the asymptotic distribution of eigenfrequen￾cies inside the bands appearing in Theorem 1.1. Two of them are analogous to
XVI-2
results concerning the distribution of resonances for strictly convex obstacles ob￾tained by the author [8] and by M. Zworski and the author [10]. The third result
has not yet any corresponding analogue, and Zworski and the author intend to look
into that question, as well as the question of getting remainder estimates in the
main result of [10]. The first result gives the standard Weyl asymptotics, and can
probably be deduced from [5].
Theorem 1.2 The number of eig en frequencies r, with 0 < Rer < A is equal to
{^-T I I drr^+^A71
-
1
), A->oo.
27T J Jp(x^)<l
During this conference M. Solomjak indicted to us some work of A.S. Markus
and V.I. Matseev, and the most relevant paper seems to be [5]. Theorem 2.1 in
that paper looks very much like a generalization of the preceding result. The proof
in that paper seems to be very close to the one we give below (in the case when
/ = 1) and uses finite rank perturbations to open a gap in the spectrum as well as
considerations of relative determinants. We have not found any result like Theorem
1.4 below and it would be interesting to see what our proof gives for more general
elliptic operators.
Notice that Theorem 1.2 is the standard (and in general optimal) result in the
selfadjoint case, a = 0. Also notice that it implies that the number of eigenfrequen￾cies with A<ReT<A4-li s (^(A71"
1
), when A -^ +00.
In order to state the second result, we introduce the almost everywhere limit on
p"^!), given by the Birkhoff ergodic theorem:
(^)oo == lim (^T.
T—>oo
Then
A_ < essinf(a)oo
 < esssup(a)oo
 < A^. (1-4)
At each place the inequality may be strict. If the geodesic flow is ergodic, we have
equality in the middle.
Theorem 1.3 For every e > 0, the number of eig en frequencies in [A, A + 1] + z(R\
[essinf {a)oo — e, ess sup (a)oo + ^]) is o{Xn
~
l
), A —> oo.
Somewhat vaguely, one can say that the relative density of the eigenfrequencies
outside R + z[essinf {a)oo — e, ess sup (a)oo + ^] is equal to 0.
The last result concerns the meanvalue distribution of the imaginary parts.
Theorem 1.4 Fix some CQ > 1 and let Ai, As € R satisfy
1 < Ai < As, -
2
 < Co, \2 - Ai > logAi. (1.5)
^i
Let N{\\, \^) be the number of eigenfrequencies r with Ai < Rer < A2. Then
———r. E ^--^f a^
dx+
OW-^gxY- (L6) A(Ai,A2 ) ,^p) vol(A/)7.v A2 - Ai
AI <^Rer<\^
XVI-3
M. Hitrik (personal communication) has obtained further results in the case
when the geodesic flow is periodic. A more detailed study of the distribution of the
imaginary parts may be possible in this case.
In the following, we outline the proofs of the theorems 1.2,1.4 which use some
recent trace formula techniques, here in the semiclassical setting of [9]. The proof
of Theorem 1.3 is more technical. For more details, see [7].
2. Ideas of the proofs of Theorem 1 and 3.
Write T = \/z//i, 0 < h < 1, where z belongs to the fixed domain Q :=
e^^a, /3[, with 0<a<l<(3<oo,0<0o<^ . From (1.2), we get
(P - z)v = 0, where P = P + ihQ(z), P = -^A, Q{z) == 2a(x)^/z. Everything
works in a more general /^-pseudodifferential framework and the essential features
are that P is elliptic selfadjoint, dp ^ 0 on p-^o:,/?]) and that Q is holomorphic
in z and formally selfadjoint for z > 0. Let a < Ei < E^ < /? with E^ - Ei >, 4h,
£1 - a, /? - £2 > Const. > 0. Put E» = (£\ + E^/2, ro = (£'2 - £i)/2.
Lemma 2.1 For every C > 0 there exists a selfadjoint operator P = P+ h8P with
the same domain as P such that {Ej + [-Ch, Ch}) no-(P) = 0, j = 1, 2, \\6P\\ < C
ll^lltr ^ C(C')^1
-". - '
Here || • ||tr denotes the trace norm. This lemma follows easily form the fact that
P has ^^(h1
'"-) eigenvalues in intervals of length h.
Put P = P + ihQ(z) = P + h6P. We have
IK^-.IK￾h +
W
 \lmz\
for
z C D(Eo, ro + 2h) \ {z € D(EQ, ro - 2h) ; |Im z\ < Ch},
where D(E, r) denotes the open disc in C with center E and radius r, and C > 0 is
sufficiently large. Write
z - P == (z - P)(l + h(z - P)-^?), D(z) = det(l + h{z - P)-^?).
Using a convexity estimate of H. Weyl (see [3]), we get
\D{z)\ < exp \\h{z - Pr^P^ < exp(0(l)——'2——)
h + \lmz
In the subset where \lmz\ > Ch (for a sufficiently large constant C), we also
have
li^ _w-i|l ^ _°(1
)
K^
-
pr•"<
^|I^„•
and writing
(1 + h{z - P)-^?)-1
 = {z - -P)-\z - -P} = 1 - h(z - P)-^P,
XVI-4
we get a similar upper bound for the inverse of D, i.e. a lower bound for D\
mz)} ^ ^'-^'rfe)-
Using Jensen's formula in a standard way, we see that the number of eigenvalues
Zi, ..,ZN ofP in
D(E^ro+h)\{zeD{E^ro-h) ; \lmz\<Ch} (2.1)
is O^1
-
71). Let b^(z) be the Blaschke factor associated to Zj and D(Eo,ro + j/i)
(having Zj as its only zero inside the disc and being of modulus one on the boundary).
Write
D{z) = G(z)D,(z)^ D,{z)=f[b^(z)^
.7=1
so that G(z) is holomorphic and non-vanishing on the domain (2.1). Standard
arguments (used in a similar context in [9]), involving Harnack's inequality, show
that
llog?)ll
<o(l)/^Im^ (2
-
2
)
^loaG^)|<______________0
(
l
)
fe2•"______________ ^
'dz ° v " - (h+\lmz\)mm(h+\lmz\,ro+^-\z\y [ '
in
P(Eo,ro + J/i) \ {z C D(E^ro - ^) ; \lmz\ < Ch}. (2.4)
Let 7 be a hexagonal positively oriented contour in the domain (2.4) with vertices
at the points EQ±"^, Eo±r(h)±i'2Ch, j = l,2withr(/i) = ro+a(h)h where a(h)
is small and chosen so that 7 avoids the z,. One can verify that if/is holomorphic
in D(Eo,ro+2h'), then
E /(A) = tr (^ y f{z)(l - W{z - P)-1
^), tr (2.5)
Xea'CP^mt (7) 'r
where cr(P) denotes the set of eigenvalues of P, i.e. the complex numbers z such
that Ker ("P - z) / 0, and int 7 denotes the interior of 7. We have the analogous
relation for P. Moreover,
tr ^i/(z)(l
 ~
 9ZP)(Z
 ~
 prldz
 ~ f /(^)(l
 ~
 9ZP){Z
 ~
 p)
-ldz)) (2
-
6
)
=iJ^i^D
^
dz
'..S/^^^^^^10^^
Assume that |/| < 1. Then
^ /(^)=0(/i1
-"),
2j£int(-r)
XVI-5
^ ^ f(^ logG(.) = 0(1)/>- /; ^A = 0(A-").
Choosing / = 1, we get
#(a(P) n hit (7)) = #(a(P) n int (7)) + 0(/i1
-
71).
By an easy deformation argument and a well known result on spectral asymptotics:
#(a(P) n int (7)) = #(a(P) n int (7)) = ——,. ( // ^ + 0(/i)),
(JTTn)71 JJEi<p<E2
which implies Theorem 1.2.
To obtain Theorem 1.4, we take / holomorphic in D(EQ, rQ+2h) with /' == 0(1)
and with f(z) real when z is real. Since Imf(zj) = 0(/^), we get
Im2^//(^)^
logDb(^
=
^
2-")'
Im^^/(z)^(logG(^)^=
Im ^[f(z) logG(z)]^^^ - Im ^ ;, f'(z) \og{G(z))dz,
where we choose a branch of the logarithm of G{z) with a cut along £'0 + [0, +oo[.
Here we notice that the first term in the last expression vanishes, since /{EQ + r(/i))
is real and the two limiting branches of logG differ by a multiple of 27ri at that
point.
On the other hand, we have on 7:
|Re \ogG(z)\ = |log|G(z)|| < 0(1)-^———. (2.7)
a i jj.m z\
By (2.2), we have on 7:
, d , ... . 0(l)/i2
-"
'^^^^(^llm^-
We integrate this from £'0 ± iro/2 to z E 7, when ±Im z > 0, and get
|Im(logG(z)) - C±(/i)| ^ OW—^——— ±Im2 > 0, h + |Imz|
where C±(/z) 6 R. Hence, using also (2.7) for Re logG^-z), we get:
h
2
-
\og{G(z))=iC±(h)+0(l)
h + \lmz\
Here
——f zC^h}f'(z)dz
ZTTZ J-fn{±].m z>0]
XVI-6
is real, so
Im 2^2 / //(^
 log G{z)dz = o(
l
)
/l2-7l
 ^ ^
Taking imaginary parts of (2.5), (2.6), we get:
^ Im/(A)=Imtr 1
 f f(z)(l - 9,P)(z - P^dz + ^(l^log1
Aea(P)nint (7)
 zm •/7 /l
Thanks to the fact that we have a sufficiently good control over (z - P)~1
 along
all 0/7 (which is not the case for (z - P)~1
), we can analyze the integral in the
preceding equation and show that it is equal to
(2^ //., <_^w flw1'
m<l(x
-
(
-
p(x
-
wxdi; + °^l0^
with q(x,^z) = 2^za(x). Taking f(z) = 2^/z, leads to Theorem 1.4.
References
[1] M. Asch, G. Lebeau, The spectrum of the damped wave operator for a bounded
domain in R2
. Preprint.
[2] P. Freitas, Spectral sequences for quadratic pencils and the inverse problem for
the damped wave equation, J. Math. Pures et Appl., 78(1999), 965-980.
[3] I.C. Gohberg, M.G. Krein, Introduction to the theory of non-selfadjoint opera￾tors, Amer. Math. Soc., Providence, RI 1969.
[4] G. Lebeau, Equation des ondes amorties, Algebraic and geometric methods in
mathematical physics (Kaciveli, 1993), 73-109, Math. Phys. Stud., 19, Kluwer
Acad. PubL, Dordrecht, 1996.
[5] A.S. Markus, V.I. Matsaev, Comparison theorems for spectra of linear oper￾ators, and spectral asymptotics, Trans. Moscow Math. Soc. 1984(1), 139-187.
Russian original in Trudy Moscov. Obshch. 45(1982), 133-181.
[6] J. Rauch, M. Taylor, Decay of solutions to nondissipative hyperbolic systems on
compact manifolds, Comm. Pure. Appl. Math. 28(1975), 501-523.
[7] J. Sjostrand, Asymptotic distribution of eig en frequencies for damped wave equa￾tions, Publ. R.I.M.S., to appear.
[8] J. Sjostrand, Density of resonances for strictly convex analytic obstacles, Can.
J. Math., 48(2)(1996), 397-447.
[9] J. Sjostrand, A trace formula and review of some estimates for resonances,
p.377-437 in Microlocal Analysis and spectral theory, NATO ASI Series C, vol.
490, Kluwer 1997. See also Resoances for bottles and trace formulae, Math.
Nachr., to appear.
XVI-7
[10] J. Sjostrand, M. Zworski, Asymptotic distribution of resonances for convex ob￾stacles, Acta. Math., 183(2)(2000), 191
CENTRE DE MATHEMATIQUES, ECOLE POLYTECHNIQUE
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