![The conference in honor of Jean Vaillant
BERNARDGAVEAU
Laboratoire Analyse et Physique Mathematique,
Universite Pierre et Marie Curie, Paris, France
Since his thesis in 1964 prepared under the direction of J. Leray and A. Lichnerowicz
the main theme of the mathematical work of Jean Vaillant has been the study of systems of
hyperbolic or holomorphic partial differential equations. The basic example of an hyperbolic
equation is the wave equation which is the mathematical description of wave propagation at
finite velocity, as, for example, the propagation of small disturbances in fluids (the sound) or
of electromagnetic waves in vacuum. Examples of hyperbolic systems include the Maxwell
system for the propagation of electromagnetic waves, the Dirac system for the propagation of
spinors and Einstein equations in general relativity. The wave equation is the fundamental
example of a strictly hyperbolic equation, for which the propagation velocities are different
and do not vanish. An approximation of the solutions of a strictly hyperbolic equation is the
high frequency approximation or geometrical optics approximation : sound or light propagates
essentially along the trajectories of the Hamilton Jacobi equation associated to the partial
differential equation. One can say, in a rather unprecise manner, that « singularities are
propagated along ^characteristics », which, a posteriori, justifies the use of geometrical
optics, the laws of reflexion and refraction. On the other hand, interference and diffraction
phenomena show that sound or light can be described as fields which can be added, rather
than particles, but again a good approximation of these phenomena is the propagation along
bicharacteristics at least in the simplest situations. Dirac system is an example of a system
with multiple characteristics with constant multiplicities. Maxwell system in a non isotropic
medium, like a crystal, is an hyperbolic system with multiple characteristics, but their
multiplicity is non constant: The velocities of propagation depends of the direction of
propagation, but for special directions, some velocities may coincide. In this situation, the
approximation of geometrical optics is no more valid : The propagation along
bicharacteristics (or rays) is not a good description or approximation of the phenomenon, and
indeed this can be checked experimentally. A light ray falling on certain crystals, is, in
general refracted along a certain direction. Nevertheless for special incidence angles,
corresponding to the geometry of the crystal, the ray is refracted, not along another ray, but on
a whole conical surface. Until the end of 19 century, this experiment was the only proof of
the electromagnetic nature of light, because all the other light propagation phenomena could
be described by a wave equation, without the use of the complete Maxwell system (see [1],
[2]).](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-21-320.jpg)
![2 Gaveau
In the beginning of the 1960's, strictly hyperbolic equations with simple
characteristics (the velocities of propagation are distinct and non zero), are well understood
(see [1], [3]). Around that time, Jean Vaillant begins a systematic study of equations or
systems of equations which are non strict. In his thesis ([4], [5]), he introduces the notion of
localization with respect to a factor of the characteristic determinant of a system with constant
coefficients and he relates this notion to the equation of propagation along the
^characteristics. This seems to be the first attempt to apply the method of localization, in
particular using invariant factors. Following the article of Garding, Kotake, Leray (Probleme
de Cauchy VI, [6]), J. Vaillant defines a new invariant associated to systems with double
characteristics and gives an application to the Goursat problem [7], and to the localization for
systems with variables coefficients and double characteristics : This is the first example of a
Levi condition in this setting. J. Vaillant relates also the difference of the subcharacteristic
polynomial, and the second coefficient of exp (-icocp) P (x, DX) exp (icocp) to the Lie
derivative of the volume form along the ^characteristics [8]. This is an important invariant,
because it is well known that the existence and the regularity of the solution depend, in
degenerate cases not only of the principal symbol but also on the lower order terms of the
operator. This result was rederived by Duistermaat and Hormander.
In 1973-74, J. Vaillant constructs the asymptotic expansion of the solution of an
hyperbolic systems with characteristics of variable multiplicities [9]. He defines the
localization of an hyperbolic system at a multiple points with application to conical refraction
[9]. This work will be extended in 1978, when he constructs the parametrix for the Cauchy
problem with multiple characteristics [10], in relation with the invariants of the system.
In [11], J. Vaillant studies the symmetrisation of localized hyperbolic systems and
defines the notion of « reduced dimension » : The property of symmetry is proved in the case
of a maximal reduced dimension. These last few years, he has continued to study the
symmetric of strongly hyperbolic systems, in particular with T. Nishitani. If the reduced
dimension of a system of rank m is not less than 2, a constant coefficient
2
systems is symmetrizable ([7], [8]). For a system with non constant coefficients, if at any
point the reduced dimension is not less than — 2, the system is also symmetrizable
with a regular symmetrizer [18]. Recently he has determined the multiple points according to
the reduced dimension [19].
In 1982, in collaboration with D. Schiltz and C. Wagschal [12], J. Vaillant has
studied the ramification of the Cauchy problem for a system in involution with triple
characteristics. This problem reduces to the question of the singularities of integrals of](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-22-320.jpg)
![Conference in honor of Jean Vaillant 3
holomorphic forms depending of parameters, on chains depending also of parameters. The
problem is to determine the singularities of these integrals with respect to the parameters. The
first systematic work in this direction was the article of J. Leray [13] in the algebraic case. J.
Vaillant studies the ramification in the general holomorphic case, using a grassmann boundle
[14]. Since 1987, J. Vaillant has started the problem of the classification of systems with
constant multiplicities : definition of invariant Levi conditions, relations to the Cauchy
problem in the C°° and Gevrey classes. For any system, he obtains systematically the Levi
conditions [15], [16].
J. Vaillant has founded a research group and a seminar, which he has maintained, for
more than thirty years, independent of fashions « mots d'ordre » and which survives in
difficult conditions. Freedom of thought, which is a necessary condition for any creative
work, is paid a very high price. Creation, scientific or artistic, cannot be judged according to
economic or social criteria, measured in monetary values. Research is not a collective activity.
The highly mysterious activity of thought can only be a personal activity. During all his
career, Jean Vaillant, following the example of Leray, has tried to defend by his attitudes and
his work, the values of scientific creation and intellectual independence.
For more than forty years, J. Vaillant has developed many collaborations with his
Japanese and Italian friends, in particular Y. Hamada, Y. Ohya, K. Kajitani, T. Nishitani and
S. Spagnolo, F. Colombini, A. Bove and E. Bernardi. He has also developed many european
collaborations and he has created a european network of belgian, french, Italian, and
Portugese universities. All his friends know that they can rely on his help and his advises.
acknowledgment: We are very grateful to Anne Durrande, Evelyne Guilloux and Maryse
Loiseau for their help during the preparation of this conference. We also thank the Maison
Europeenne des Technologies, in particular Madame Muller for her help.
B. Gaveau
Laboratoire Analyse et Physique Mathematique
14 avenue Felix Faure
75015 PARIS](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-23-320.jpg)
![Gaveau
[1] Courant-Hilbert: Methods of Mathematical Physics Vol
Interscience 1962
[2] M. Born & E. Wolf : Principles of optics
Penjamon Press 1980
[3] J. Leray : Hyperbolic differential equations
Lectures notes, Princeton 1950
[4] J. Vaillant: Sur les discontinuites du tenseur de courbure en theorie
d'Einstein-Schrodinger
CR Acad Sci Paris - 10 juillet 1961, 30 octobre 1961, 15 Janvier 1962
[5] J. Vaillant: Caracteristiques multiples et bicaracteristiques des systemes d'equations
aux derivees partielles lineaires et a coefficients constants
Annales Institut Fourier 15 (1965) et 16 (1966)
[6] L. Garding, T. Kotake, J. Leray : Uniformation et developpement asymptotique de la
solution du probleme de Cauchy lineaire a donnees holomorphes ; analogic avec la
theorie des ondes asymptotiques et approchees (Probleme de Cauchy I bis et VI)
Bull. Sci. Math. France 92 1964, 263-361.
[7] J. Vaillant: Donnees de Cauchy portees par une caracteristique double : role des
bicaracteristiques
J. Maths Pures et Appliquees 47 (1968), 1-40
[8) J. Vaillant: Derivee de Lie de la forme element de volume le long des
bicaracteristiques et polynome sous-caracteristique de Garding-Kotake-Leray
CR Acad Sci Paris -10 mars 1969
[9] J. Vaillant: Solutions asymptotiques d'un systeme a caracteristiques de multiplicite
variable
J. Maths Pures et Appliquees 53 (1974), 71-98
[10] R. Berzin, J. Vaillant- Parametrix a caracteristiques multiples
Bull. Sci. Math 102 (1978), 287-294
[11] J. Vaillant: Symetrisation de matrices localisees
Annali della Scuola Normale Superiore di Pisa. Ser. IV, 5 (1978), 405-427
[12] D. Schiltz, J. Vaillant, C. Wagschal: Probleme de Cauchy ramifie
J. Math. Pures et appliquees (1982)
[13] J. Leray : Un complement au theoreme de N.Nilsson sur les integrales de formes
differentielles a support singulier algebrique](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-24-320.jpg)
![Conference in honor of Jean Vaillant 5
Bull. Soc. Math. Fr 95 (1967), 313-374
[14] J. Vaillant: Ramifications d'integrales holomorphes
J. Math. Pures et Appliquees 65 (1986), 343-402
[15] J. Vaillant: Conditions d'hyperbolicite pour les systemes
Bull. Sci Math 114 (1990), 243-328
Conditions de Levi in Travaux en cours 48. Herman (1994)
Analytic hyperbolic systems
Pitman, Research Notes in Mathematics 349 (1996), 209-229
Conditions invariantes sur les systemes d'equations aux derivees partielles et
probleme de Cauchy
in complex Analysis and microlocal analysis, RIMS Kokyuroku 1090,
p. 131-142
Kyoto University (1999)
Invariants des systemes d'operateurs differentiels et sommes formelles
asymptotiques
Japanese J. Math. 15 (1999), 1-153
[16] J. Vaillant,G. Taglialatela : Conditions invariantes d'hyperbolicite et reductiondes
systemes
Bull. Sci. Math. 120 (1996), 19-97
[17] J. Vaillant: Symetrie des operateurs hyperboliques 4 x 4 et dimensionreduite
Annali della Scuola Normale Superiore di Pisa 29 (2000), p. 839-890.
[18] J. Vaillant,T. Nishitani: Smoothy symmetrizable systems and the reduced
dimensions
Tsukuba Journal Vol. 25, n° 1,juin 2001, p. 165-177.
[19] J. Vaillant, H. Delquie : Dimension reduite et valeurs propres multiples d'une matrice
diagonal!sable 4 x 4
Bull. Sci. Math. 124 - 4 (2000).](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-25-320.jpg)
![Hyperbolic systems with nondegenerate characteristics
Tatsuo Nishitani
Department of Mathematics,Osaka University, Machikaneyama 1-16,Toyonaka Osaka, 560-0043,
Japan
1 Introduction
In this note we discuss the stability of non degenerate characteristics of hyperbolicsys-
tems of general order under hyperbolic perturbations. We also study the well posedness
of the Cauchy problem for hyperbolic systems of general order with non degenerate
characteristics.
For first order systems we have introduced non degenerate characteristics in[7],
[8]. According to this definition, simple characteristics are non degenerate and non
degenerate double characteristics coincide with those studied in [2], [3], [4]. We adapt
this definition for higher order systems with an obvious modification. For a system
of general order, in a standard manner, we can associate a first order system and we
prove in section 2 that non degenerate characteristics of the original system are also non
degenerate characteristics of the same order for the corresponding first order system
and vice versa (Proposition 2.1).
In section 3 we prove that one can not remove non degenerate characteristics by hy-
perbolic perturbations: any hyperbolic system which is sufficiently close to a hyperbolic
system with a non degenerate characteristic must have a non degenerate characteris-
tic of the same order. In particular, near a hyperbolic system with a non degenerate
multiple characteristic, there is no strictly hyperbolic system. Moreover we show that,
near a non degenerate characteristic of order r, the characteristics of order r form a
smooth manifold of codimension r(r + l)/2 (Theorem 3.1). The same result wasproved
for analytic first order systems in [7]and for systems with non degenerate double char-
acteristics in [2].
In section 4. the well posedness of the Cauchy problem for hyperbolic systems
with non degenerate characteristics is discussed. We prove that a hyperbolic system
of which every characteristic is non degenerate is smoothly symmetrized and hence
the Cauchy problem is C°° well posed for arbitrary lower order terms (Theorem 4.1.
Theorem 4.2). This generalizes a result in [2] (see also [4]) where the same result was
proved for hyperbolic systems with constant coefficients with non degenerate double
characteristics.
In the last section we restrict our considerations to 2 x 2 first order hyperbolic
systems with constsnt coefficients with n independent variables. We show that if n < 3](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-27-320.jpg)
![8 Nishitani
then such a system is a limit of strictly hyperbolic systems (Proposition 5.1). Contrary
if n > 3, by [5]there is no strictly hyperbolic 2 x 2 system.
2 Non degenerate characteristics
Let P(x) be a m x m matrix valued smooth function defined near x € Rn
. We assume
that P(x] is a polynomial in x so that
(2.1) P(x) = J>,(x>r
j=0
where x1
= (x2, .... xn). We say that P(x) is hyperbolic with respect to 9 = (1, 0, ..., 0) <E
Rn
if detA)(z') ^ 0 near x' = x' and
(2.2) detP(x + A0)= 0 =» Ais real.
We call x is a characteristic of order r if
(2.3) ^(detP)(f) = 0, V|a < r, <9£(detP)(x) ^ 0, 3|a| = r.
Following [7], [8], we introduce non degenerate characteristics. To do so we first define
the localization of P(x) at a multiple characteristic x verifying
(2.4) KerP(x) n ImP(x) = {0}.
Let dimKerP(x) = r and let v,...,vr be a basis for KerP(x). Taking (2.4) into account
we can choose linear forms £]_,...,£r so that
^(ImP(x)) = 0, ^(^) = <5y
where 5jj is the Kronecker's delta. Then we define a r x r matrix PX(X) by
(2.5) (li(P(x + HxWh&jzr = Ai[P4(i) + 0(/z)].
It is easy to see that the definition of PX(X) is independent of the choice of v,...,vr,
that is r
Pt(x)v = ^[UjCQ^-^XPCx + ^)^)]^
is a well defined map from KerP(x) to KerP(x). We denote
(2.6) P-x = (Px(x) x € Rn
} C M(r- R)
which is a subspace of M(r; R), the space of all real r x r matrices. We first show
Lemma 2.1 Assume (2,2) and let x bea characteristic verifying (2.4) with dimKerP(x)
r. Then we have
(2.7) detP(x + //a;) = ff[c detPg(z) + O(fj,)] withc^O.
Assume further that detPx(x) 7^ 0 then
(2.8) detPj(^) ^ 0,
(2.9) detPi(x + X9] = 0 =*• A 25 rea/ /or a// x e Rn
.](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-28-320.jpg)
![Hyperbolic systems with nondegenerate characteristics 9
Proof: In view of (2.4) we can choose a non singular constant matrix T so that
where G is a non singular (m —r}x(m —r] matrix. With P(x) = T~l
P(x)T we write
P(X + HX} = P(X)
Denoting
it is clear P±(x} = Pi(x] which follows from the definition. Since Px(z) = P$(x) we
have
(2.10) P*(x) = PU(X).
Note that
(2.11) detP(z + AW:) = detP(z + fjuc) = //[detG detPn(s) + O(/z)]
which shows the first assertion. To prove the second assertion suppose that detPz(0) =
0 so that detP(x + //#) = o(/O by (2.7). Since detP(o:) is hyperbolic in the sense(2.2)
it follows that
[S£detP](z) = 0, V|a| < r.
This implies detP5(x) = 0 which is a contradiction. We turn to the third assertion.
Since
detP(x + A*(X 4- A0)) = pfc detPs(x + A0) + O(A*)]
if detPx(x + A0) = 0 has a non real root A, then taking p, ^ 0 sufficiently small the
equation
c detPfx + 0 + O * = 0
admits a non real root. This contradicts (2.2).
We now generalize the notion of non degenerate characteristics for P(x) in (2.1)
which is defined in [7], [8] for the case q = 1, making an obviousmodification.
DEFINITION 2.1: We say that x is a non degenerate characteristic (of order r} of P(x)
if the following conditions are verified.
(2.12) KerP(x) n ImP(x) = {0},
(2.13) dimPi = r(r + l)/2, r = dimKerP(x),
(2.14) detPx(^) ^ 0, Px(0)~1
Ps(x) is diagonalizable for every x € Rn
.
Simple characteristics verify (2.12)-(2.14) with r = 1 and hence non degenerate. Non
degenerate double characteristics have a special feature:](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-29-320.jpg)
![10 Nishitani
Lemma 2.2 Assume that dimKerP(z) = 2 and KerP(x) n ImP(z) = {0}. Then a
double characteristic x is non degenerate if and only if the rank of the Hessian of P(x)
at x is maximal, that is 3.
Proof: Assume that rankHesssP = 3. From Lemma 2.1 and the assumption it is clear
that detP5(0) ^ 0. Hence, by Lemma 2.1again, Pi(^)-1
Pi(x) has only real eigenvalues
for every x. From Lemma 4.1 in [6]there exists a constant 2 x 2 matrix T such that
T~l
(P£(0}~1
PX(X}}T is symmetric for every x so that one can write
(2.15) T-i(Ps(0rl
Pt(x}}T
and obviously Px(0]~l
Px(x] is diagonalizable for every x. Since the rank of the
quadratic form det(Pi(^)~1
Pi(x)) is 3 and hence 0;(x), i — 1,2,3 are linearly in-
dependent. Thus it is clear that dimP^ = 3. Conversely assume dimP^ = 3 and
detPx(9) 7^ 0. Since P2(9}~1
PX(X] has only real eigenvalues for every x, (2.15) still
holds. Then it follows that <& are linearly independent. Hence rankHess^P = 3 by
(2.7) again. This proves the assertion.
REMARK 2.1: Assume that q = 1 and AI(X'} is symmetric in (2.1). Then (2.12) and
(2.14) are always verified.
REMARK 2.2: By definition, the order of non degenerate characteristics never exceed
m, the size of the matrix whatever q is.
To study P(x] we consider the following mq x mq matrix valued function
P(x) = xl
0 -/
0 0 -/
A(x']
where / is the identity matrix of order m. It is clear that
(2.16) detP(x) = detP(x).
Then the condition (2.2) implies that all eigenvalues of A(x'} are real, equivalently
(2.17) all eigenvalues of P(x] are real.
Our aim in this section is to prove
Proposition 2.1 Let x be a non degenerate characteristic of order r of P(x). Then x
is also a non degenerate characteristic of order r for P(x) and vice versa.](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-30-320.jpg)
![Hyperbolic systems with nondegenerate characteristics 11
Proof: Assume that x is a non degenerate characteristic of order r of P(x). We first
check
(2.18) 7;— (x) (KerP(z)) = KerP(z).
C/«Z/ J
Let vi,...,tv be a basis for KerP(x) and take ^ so that ^(ImP(z)) = 0 and ti(vj) = 6ij.
Then by definition we have
Hence
P.(*)
Then detP5(^) ^ 0 implies that
KerP(x) c
CXi
and hence the result.
We note that
KerP(z) = {t
(u,x1u,...,xri
u) u 6 KerP(z)}
and dunKeiP(x} = r of course. We next describe lmP(x). Write
0fc(x) = £^(x>r"fc
j=0
then it is easy to see that
9-1
lmP(x) = {t
(w(l
...,w(<}
-l
P(x)v-^M^w(k}
) ^...V'-^t; € Rm
}.
fc=i
We first show that
(2.19) KerP(x] n lmP(x] = {0}.
Let ^ be a linear form on Rm<?
. Writing v - f
(i;(1)
, ...,v(g)
) € Rm<?
one can write
where ^^') are linear forms on Rm
. Assume l(lmP(x}) = 0. This implies that
(2.20) t(j
')=£q
<i>j(x}-), l<j<q-l,
Assume moreover £(Ker"P(x)) = 0 so that
rVx^w +x?-1
") = o.](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-31-320.jpg)
![12 Nishitani
Prom this, noting the identity
•7-1 QD
E
7 _ 1 , , , n—1 *J 1 , v
zi ^j(x] +x l
= jrr(x]
3 —1 *
one gets
8P
(2.22) ^(<?}
(—-(x» = 0, VueKerP(x).
Rrom (2.18) it follows that £((?)
(KerP(x)) = 0. Since KerP(x) + ImP(x) = Rm
then
(2.20) shows that t^ = 0 and hence i = 0. This proves that
(2.23) KerP(x) + ImP(x) = Rm?
and hence (2.19).
We next examine (2.13). (2.14) for P(x). Let U = t
(u,xiu,...,x~l
u] € KerP(x}
where u € KerP(x). Consider P(x}U:
P(x]U = *((xi —Xi)w, (xi —Xijiiw, ..., (x — Xi)xq
l~"u, *)
where the last component * is
P(xl,xl
}u+(x].xrl
-x}u
= P(o:)n + [P(xi, x')- P(XI, x'}}u + xq
l~l
(xl - x^u.
Now it is easy to see that this is equal to
(2.24) P(x}u - ]T <t>k(x)wW +0((xi _ £l)2}_
k=l
Let i be a linear form on Rmq
with £(ImP(z)) = 0. Prom (2.24) it Mows that
9-1 g-l
t(P(x)U) =
; = 1 fc=l
(2.25) = W(P(x}u) +0((Xl -
by (2.20). Let us take Uj = i
(uj ,XiUj,...,xq
^l
Uj} € KerP(x) where {u-j} is a basis for
KerP(x) and write
dP r
-£—(x)uj =
®Xl
k=i
Take ^2 so that
£(ImP(x))=
Let us take 9)](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-32-320.jpg)
![Hyperbolic systems with nondegenerate characteristics 13
SO that r r
^(|_(*X-) = E ^E *iP*kM =<V
ax
l k=l p=l
We nowdefine linear forms ^(-) = '(^(^(x)-), ...^(^-i^)-)^-)) on Rm
« then
we have
(2.26) 4(ImP(x)) = 0,
as observed above. Prom (2.25) it follows that
Since B = (f
A)~1
= Px(0}~1
we conclude that
(2.28) Pt(x) = P*(6rl
P*(x).
Since Ps(9) = I then (2.13) and (2.14) for Ps(x) follow immediately.
Conversely assume (2.23). Let £M
be a linear form on Rm
such that fiq}
(lmP(x)} =
0, ^(KerP(z)) = 0anddefine t^ l<j< q-l by(2.20). Then we have t(lmP(x)) =
0 and moreover (2.20) shows ^(KerP(x)) = 0 and hence t - 0 by (2.23). Thus wehave
W = 0 which proves KerP(x) + ImP(x) = Rm
and hence (2.12). To check (2.13),
(2.14) for P(x) we note that Ker"P(x) n Im'P(x) = {0} implies that
dP
u € KerP(x), ^— (x]u € lmP(x) =^> u = 0.
OX i
Hence we have (2.18) again and thus (2.28). Then the rest of the proof is clear.
3 Stability of non degenerate characteristics
In this section we discuss the stability of non degenerate characteristics under hyper-
bolic perturbations.
Theorem 3.1 Assume that P(x] is a m x m real matrix valued smooth function of
the form (2. 1) verifying (2.2) in a neighborhood U of x and let x be a non degenerate
characteristic of order r of P. Let P(x] be another m x m real matrix valued smooth
function of the form (2.1) verifying (2.2) which is sufficiently close to P(x] in Cq+2
,
then P(x} has a non degenerate characteristic of the same order close to x. Moreover,
near x, the characteristics of order r are non degenerate and they form a smooth
manifold of codimension r(r + l)/2. In particular, near x the set of characteristics of
order r of P(x] itself consists of non degenerate ones which form a smooth manifold
of codimension r(r + l)/2.](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-33-320.jpg)
![14 Nishitani
To prove Theorem 3.1,taking Proposition 2.1 into account, we study P(x) of the
form
(3.1) P(z) = x + P^(x'}
where we assume that
(3.2) detP(x) = 0 => X! is real near x' = x'.
This is equivalent to say that all eigenvalues of P#
(x') are real. We extend the stability
result for P of the form (3.1) which is proved for real analytic P(x),P(x) in the case
r = m in [7]and in [2]when x is a non degenerate double characteristic.
Proposition 3.1 Assume that P(x) is a m x m real matrix valued smooth function
verifying (3.2) and x is a non degenerate characteristic of order r of P(x]. Let P(x)
be another mx m real matrix valued smooth function of the form (3.1) verifying (3.2)
which is sufficiently close to P(x) in C2
. Then P(x) has a non degenerate characteristic
of the same order close to x. Moreover, nearx, the characteristics of order r of P(x] are
non degenerate and form a smooth manifold of codimension r(r + l)/2. In particular,
the characteristics of order r of P(x) itself consists of non degenerate ones which form
a smooth manifold of codimension r(r + l)/2.
Proof: We first show that the proof is reduced to the case that P and P are r x r
matrix valued function. Without restrictions we may assume that x = 0. As in the
previous section, we take T so that one has
T-i
1
* w
" ~ ' 0 G
where G is non singular. Denote T~l
P(x]T and T~1
PT by P(x) and P(x) again.
Writing
Vz) PK(X]
we have
(3.3) PH(Z) = X! + £ AjXj + 0(|x|2
) = P0(x) + 0( x2
).
From the assumption PO(X) is diagonalizable for every x and {/, A^,..., An} span a
r(r -h l)/2 dimensional subspace in M(r;R). By Lemma 2.1 all eigenvalues of PO(X)
are real then one can apply Theorem 3.4 in [9]and conclude that: there is a constant
matrix S such that
where Aj are symmetric and {I,Ai,..., An} span Ms
(r; R), the space of all r x r real
symmetric matrices. We still denote
o-i n / c A / c-i A - / c
0
I P(x I I I I P(x) I
0 / ^(x)
0 / ' 0 I }n j
0 /](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-34-320.jpg)
![16 Nishitani
This gives A^G^i = A'n, AiG-2 = X22- Plugging these relations into the remaining
two equations, we have
A12 + AnGi2 = Gl2^22 + Gi2^42lGi2
.421 + Ay^Gii —GiA + G^iAi^G^l-
Let us set
then the equations become
{
j'-*
Fi(Gi2, G2i, A12, A-2i) = 0
F (G Go A A ) = 0
It is well known that
*/r
^(o,0,0,0)
is non singular iA and ^422 have no common eigenvalue. Then by the implicitfunction
theorem there exist smooth Gi2(^4i2, ^21) an
d Gi{A-&, AH) defined in li^ull + H^bi i <
6 with Gi2(0, 0) = 0, G2i(0,0) = 0 verifying (3.8). This proves the assertion.
We look for T(x) in the form
T(x) =To + T!(X), T0(x) = ( / °f } , T!(0) =0.
^21 1 )
The equation which is verified by T(x] is:
(3.9) (P0 H- Pi(x))(T0 + T^x)) = (T0 + T
where P0 = P(0), P0T0 = r0P0 and
O p f ~
i"£lX
Recall that
' Pn(x)
_
° " ' ~ P21(x)
Look for TI (x) in the form
0 T12(x)
>i(x) 0
Equating the off diagonal entries of both sides of (3.9) we get
(v in - V^iVi • 72)P22(x)
V'-)
-lu
^ A T_ . i D. .(~ _L r> _(~r<_ _L ^ _ _ . ~ , L
/ S~V t HH T-> / , r-r- / A ,
— (Gii •+• -ti2)P22(x) + ^i2(A22 +
A22T21 + P21(x) + P22(x)G21 + P22(x)T2:
= (G2i+T21)P11(x)+r21(A11 +](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-36-320.jpg)
![Hyperbolic systems with nondegenerate characteristics 17
On the other hand, equating the diagonal entries of both sides we have
22(z) =A21T12 + P22(x) + P21(x)(G12 + T12).
Plugging (3.11) into (3.10) we obtain
Fi(Tl2, x) = AuTiz - T12(A22 +A2lGi2) +Pu(x)Gi2 +Pl2(x) +Pn(x)T12
-(Gi2 +T12}(A21T12 + P21(x)[G12 + T12] + P22(x)) = 0
and
F2(T2i, z) = A22T21 - T2l(An + A12G2i) + P22(x)G21 + P21(x) + P22(x)T12
~(G2l +T21)(A12r21 + Pi2(x)[G21 + T21] + Pn(x)) = 0.
Since
*i(Ti2, 0) = AnTu - T12A22, F2(T21, 0) = A22T21 - T21AU
when A2i = 0, Ai2 = 0, x = 0, it is clear that
are non singular if ||^4i2|| + ||A21|| is small. Then by the implicit function theorem there
exist smooth Ti2(x) and T21(x) with T12(0) = 0, T2i(0) = 0 such that
P!(T12(x),x) = 0, F2(T21(x),x) = 0.
This proves the assertion.
We return to the proof of Proposition 3.1. Since P(x,y) is sufficiently close to
P(x, y] and
P ( 0
'°) =
(o G detG
^°
one can apply Lemma 3.3 to P(x, y] and find G(x, y) such that
(3.12) G(Xtvr*P(
Denote G(x,y)-l
P(x,y)G(x,y) and G(x,y)-1
P(x,t/)G(x,7/) by P(x,y) and P(x,y)
again. We summarize our arguments in
Proposition 3.2 Assume that P^g and P0ng verify the assumption in Proposition
3.1. Then we may assume that P^ig and Parig have the form
~
0 P22(x,y) ' > - P21(x,y)](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-37-320.jpg)
![18
with
Nishitani
- R ( x , y ) , R(xty)=0((x + M)2
)
3=1 3=1
where the following properties are verified: for any neighborhood U of the origin there
is a neighborhood W c U of the origin such that for any e > 0 one can find e > 0 so
that if Parig — Pong c2
(u) < ? then we have
(3.13)
(3.14) |
Moreover one has
Pn(x,y) -
I k
y}.
j=l
det(A -f Pn(x, j/)) = 0 => A is real.
Proof: Since P(x, y] and P(x, y) are obtained from Porig and P^ig by a smooth change
of basis and a linear change of coordinates then (3.13) is clear. Let us recall
which verifies (3.12) where ||G12(0,0)|| + |G2i(0,0)|| becomes as small as we please if
1 is small. Hence G(x,y} is enough close to the identity and then (3.14) follows from
(3.6). Note that
det(A + Pang] = det(A + Pn(x,y))det(A + Pv(x,y}}.
Then the last assertion follows immediately.
We proceed to the next step. Write
(3.15) Pu(x,y) = Pn(0,7/) + (^(x^y^Kij^r-
Let us denote
where
Lemma 3.4 Assume that Pu(x,y) —Pu(x.y)ci(w) < e ana
{(x,y) x ,y < e} C
W. Then for x , y < e we have](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-38-320.jpg)
![Hyperbolic systems with nondegenerate characteristics 19
Proof: Write
Pn(x,y} = Pu(Q,y} + f^Aj(y)xj + R(x,y)J R(x,y] = O(x2
)
j=i
so that
(3.16) T= (*j(z,y)) = £A,-(yto - E^ +*(*>*)•
j=i j=i
Noting dx,Pn(0,y) = Aj(y), dx,Pu(0,y) = ^ + dx,fl(0,y) and
dXiR(Q,y)<Cy<Ce if |y| < e
with C independent of P, one gets
(3.17) Aj(y}-Aj<C€ if|y|<e.
Now it is clear that
(3-18) l^.^-^I^C'e i f | y | < €
because of (3.14) and (3.17). On the other hand from
3-1
and (3.14) it follows that
|Pii(0,y)| < Ce|y| +Cy2
< Cey if |y| < e.
Moreover |Pn(0,y) - P.(Q,y)c*(w) < eshows
(3.19) |Pn(0,y)|<c + Ce|y|<C'e if |y| < e.
We now estimate T(x,y) = (t}(a:,y)) and dX ] T(x,y}. Note that |5XjJR(x,y)| < C|x|
since dXiR(Q,y) = 0 and |^(x,y)| < C for |a| = 2 with C independent of P. Then
by (3.16) and (3.18) one sees
(3.20) |r(x, y)| < Ce|x| + C|x|2
< C'ex if |x| < e,
|0,,r(z,y)| < Ce 4- Cx < C'c if |x|, |y| < e
which proves the assertion.
Let us study the map
where Ba = {x € Rfc
| |x| < a}. Let
A : R* 3 x .-» 0](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-39-320.jpg)
-I < 1/2 if x , y < a.
Let us write Pu(Q,y} = (^(y)) and we apply the implicit function theorem to
where 9 6 R*; y G R*. Then there exist a: > 0. a2 > 0 and a smooth g(y,0) defined
in 0 6 Bai , y e -Ba2 such that
Note that
(3-22) pfo,0)
if t/l, |0| < e because of (3.19). Set
(3-23) (q(y,9)) =
then from (3.15) and (3.21) it follows that
(3-24) ^(?/, £) = #}, z > j .
Let us write
tf5(y,0) = <5(j/) + x}M)
where cj(y) = ^-(y, 0) and Xj-(y, 9} = O(9}. We show that c£(y) = 0 for p < 5. This,
together with (3.24), implies
(3.25) ' Ai(0(2/,0),i/) = 0.
Let us put /i(A) = det(X+Pu(g(y, 9},y}}. From Proposition 3.2 it follows that h(] = 0
implies Ais real. Take #* = 0 for i > j unless (i,j] = (q,p). Then one has
If c^(y) 7^ 0 then h(X] =0 has a non real root for sufficiently small 9 because Xq(Ui 0) =
0. This is a contradiction and hence the assertion.
Here we extend Proposition 3.1 in [7]. Let
where FI = I and {Fi, ..., Fk] be a basis for M5
(m; R) and hence k = m(m + l)/2.
Proposition 3.3 Assume that P(x] is a m x m real matrix valued smooth function
defined in a neighborhood of the origin o/Rn
. Assume that all eigenvalues of P(x] are
real and
k ftp
(3.26) £sr(°to
j=l UX
3
is sufficiently close to F(x}. Then there is a 5 > 0 such that P(x] is diagonalizable for
every x with x < 6.](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-40-320.jpg)
![Hyperbolic systems with nondegenerate characteristics 21
Proof: It is enough to repeat the proof of Proposition 3.1 in [7]with a slight modifica-
tion. Let us write
P(u +x} = P(u} + Q(x,u}
so that Q(0,o;) = 0. For T € O(m), an orthogonal matrix of order m we consider
+ x)T = 'TP(u;)T
Denoting QT
(x,u) = (0j-(x,a;;T))i<iij<m we show that there exist a 6 > 0 and a
neighborhood W of the origin of Rfe
such that with x = (xa, x6), xa = (xi, ...,xfc),
is a diffeomorphism from W into {?/ € Rfc
| |y| < 6} for every T € O(m) and every xj,,
w with |x&|, |c«;| < 6. To see this we write
= E F
i*i + E
F)P
E
j=k+l
then it is clear that for any e > 0 one can find 6' > 0 such that
(3.27) ||^(x,w)||<e|x|
if |x|, |u;| < 5' and (3.26) is sufficiently close to F(x). Let us study
where lj(xa; T} are linear in xa. Since O(m) C Rm
is compact it is clear that we have
'(*!,..,**),_.
> c>0
with some c > 0 for every T e O(m). In view of (3.27), taking e > 0 so small we
conclude that
'^M
%^(0,0,0;T)
with some d > 0 for every T G O(m). By the implicit function theorem and the
compactness of O(m) there exists a smooth xa(ya,X6,u;;T) defined in |j/a|, |xb|, |w| < <5
and T e O(m) such that
for](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-41-320.jpg)
![22 Nishitani
where we have set ya — (yl
j]i>j € RA This proves the assertion.
We now show that P(^J] is diagonalizable for every u £ Rn
with u; < 6. Take
T € 0(m) so that
(3-28)
1=1
where {Aj are different from each other and Aij are Ti x TJ matrices such that Aij = 0
if i > j and A^ are upper triangular with zero diagonals. Let us set
s-l
- U((*>
P=I
< rp}
where TO = 0. As observed above one can take ((y,-)i>j, £&) as a new system of local
coordinates around the origin of Rn
. Denote
yn = y (
and, putting y/; = 0, 0:5 = 0. consider
det(A = det(A + PT
(w + x)) = [ det(A , u; T))
where
with sl — J, s0 = 0. Note that we have
if p>q,
Hence applying the same arguments as in the proof of Lemma 3.4 in [7]we get AH = 0.
Then from Lemma 3.5 in [7]it follows that PT
(u] is diagonalizable. Since a;, u < 5
is arbitrary we get the assertion.
We now prove that near (0,0) the characteristics of order r of P(x, y] form a smooth
manifold given by x ~ g(y, 0). Let (x, y} be a characteristic of order r of P(x, y) close
to (0,0). Then it is clear that (x, y) is a characteristic of the same order for Pu(x,y]
because detP22(x, y} ^ 0 near (0,0). Recalling that Pu(x,y] has the form
, y ) , x* = (x2,...,xfc)
we see that detPn(xi,x#
,y) = (xi —Xi)r
and hence
Thus the zero is an eigenvalue of order r of Pi(x,y}. On the other hand Proposition
3.2 gives
(3.29) •(0) - F3 , -(0) < Ce.](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-42-320.jpg)
![24 Nishitani
Let P«j(x,£) be the principal symbol of P(x, D):
a=q
and we assume that
(4.4) detP?(x,£) = 0 =Ki is real Vx 6 n,V£' - (&,...,£„) € R^1
.
We prove the following result which extends those in [2], [4].
Theorem 4.1 Assume that every characteristic over (0,f), )£' = 1 of Pq(x,£) is
at most double and non degenerate. Then the Cauchy problem for P(x, D] is C°°
well posed near the origin for arbitrary lower order terms. Moreover if P(x, D} is
another system of the form (4-1) verifying (4-4) with ^ne
principal symbol Pq(x,£) =
Ho=q AQ(x)£a
of which AQ(x) are sufficiently close to Aa(x] in C2
(Q) for a = q then
the Cauchy problem for P(x, D) is C°° well posed near the origin for arbitrary lower
order terms.
Assuming the analyticity of the coefficients we have
Theorem 4.2 Assume that Aa(x), a —q are real analytic in fi and every character-
istic of Pq(x,£) over (£,£'), 1 is non degenerate. Then the Cauchy problem for
P(x, D) is C°° well posed near the origin for arbitrary lower order terms.
The proof is very simple. We reduce the Cauchy problem for P(x, D} to that for a
first order system "P(x, D). Taking the invariance of non degeneracy of characteristics
proved in Proposition 3.1, it suffices to apply the previous results in [6] and [7]which
assert the existence of a smooth symmetrizer <S(x, D'} for P(x, D) defined near the
origin.
Let us write
(4.5) P(x, D}u = Dq
lU + E A
J(X
' D'}Dr°u = f.
J=2
Put
where (D'}2
= 1+ £"=2 D]. Then (4.5) is reduced to
^U +
where U =
0 -/
-I
Af(x,D')
(D'}U = P
, P = f
(0, ..., 0,/) and](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-44-320.jpg)
![Hyperbolic systems with nondegenerate characteristics 25
Let us denote by A°(z,£') the principal symbol of Af(x,E>'} and set
0 -/
0 0 -/
-/
(4.6)
Fix (0,1"'), If I = 1- Let (0, A,,!"'), z = 1, ...,p, be characteristics of & + -4(z,f ) where
(0, AJ,|>
) are at most double and non degenerate and A, are different from each other.
Then there exists a smooth T(x, £') defined near (0,£'), homogeneous of degree 0, such
that
T(z, rrU(z,r)r(x,O = -4i(x,O e • • • e A>(^ f)
where (0,Aj,f') is a non degenerate characteristic of "Pw
(rr,£) = fi + -^(x, £')._From
Proposition 3.1 it follows that the characteristic set of defP^ (z, £) near (0,At,<f') is a
smooth manifold of codimension 3 through (0, Ai,£'). Then one can apply Lemma 3.1
in [6]to get a smooth <Sw
(z, £') defined near (0,£'), homogeneous of degree 0 such that
is symmetric. This proves that A(x,£') is smoothly symmetrizable near (0,1"') by
<S(1)
(z, f) 0 • •• © «S(p)
(z, £'). By the usual argument of partition of unity one can
prove that there is a smooth <S(z, £') which symmetrizes ,A(x, £')• Thus the Cauchy
problem for P(x, D) is C°° well posed for arbitrary lower order terms and hence so is
forP(x,p).
Let Aa(x) be sufficiently close to Aa(x] for |a| = q in C2
(f2). Let ^4(x, ^') be
defined by (4.6) from P9(x,£'). Then it is clear that A(x,g) is sufficiently close to
-4(x,f') in C2
(fi x {1/2 < |^'| < 2}). To prove the last assertion it suffices to show
that every characteristic^ P(x, £) =£1+ A(x, £') over (0,f') is non degenerate and at
most double. Let (0, A, f ) be a characteristic of 7>(x, £)_ = Ci + ^(^, CO- If
(0,A, f') is
simple then the characteristic of P(x, £} close to (0,A,f ) is also simple. If (0, A, f) is
a non degenerate double characteristic, then from Proposition 3.1 it follows that the
characteristic of P(x,^} near (0, A,^') is simple except in a manifold of codimension 3
consisting of non degenerate double characteristics. This proves that the characteristic
of P(x, £) close to (0, A, £') is non degenerate and at most double. This completes the
proof of Theorem 4.1.
To prove Theorem 4.2, it suffices to apply Theorem 1.1 in [7]:
Proposition 4.1 ([7]) Assume that A(x,^'} is a m x m matrix valued real analytic
function defined near (0,f ) and (0,A, f ) is a non degenerate characteristic for £j +
.A(x,£') of order m. Then there exists a real analytic <S(x,£') defined near (0,<f') such
that
is symmetric.](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-45-320.jpg)
![26 Nishitani
5 A remark
As proved in Theorem 3.1, if
3=0
has a non degenerate multiple characteristic (x, £) then near P there is no strictly
hyperbolic system of the form (5.1). When q = 1 more detailed facts are known (see
[5], [1]). Let P be a first order (q = I) system with constant coefficients:
(5.2) P($=ti-EA&
j=2
where Aj are m x m constant matrices. We always assume that
(5.3) all eigenvalues of Z^=2 AJ£J are real for every £' = (£2, -•-, £n)-
Then from [5] P(£) can not be strictly hyperbohc if n > 3 and m = 2 modulo 4.
Contrary to this if q —2 it is clear that for any n and m there exist strictly hyperbolic
systems.
In this section we study 2 x 2 first order systems with constant coefficients: from
the result refered above if n > 3 then P(£) can not be strictly hyperbolic. On the other
hand we have
Proposition 5.1 Let m = 2 and P(£) verify (5.3). If n < 3 then P(£) can beap-
proximated by strictly hyperbolic systems: there are strictly hyperbolic P€(£) of the form
(5.2) converging to P(£) as e —* 0.
Proof: Since the assertion is clear for n — 2 we may assume that n = 3. We first
consider the case that P(£) is diagonalizable for every £. From Lemma 4.1 in [6]there
exists a constant matrix T such that
is symmetric for every £. Take a basis
for Ms
(2; R) and write with f ' = (^2,
If 02 and 03 are linearly independent then P(£) itself is strictly hyperbolic because](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-46-320.jpg)
![28 Nishitani
by (5.3). Note that 0 ^ 0 because l2 ^ 0. Then one can write i-& — k<£> with some k
because of (5.5) and hence
where 1- a2
k2
> 0 by (5.5).If
1 k
(5
'6)
- -a't -1
Ct A/ J_
is diagonalizable then so is P(£) for every £ contradicting the assumption. If (5.6) is
not diagonalizable and necessarily 1 —a2
/c2
= 0 one can take U so that
r,_i / i K TT
t/ I 2- , 1 U
I k / O 1
a2jfe _i J^ - ^ o o;•
Take 7/> so that ^ and 0 are linearly independent and define P£(£) by
(5.7)
It is clear that P€(^) is strictly hyperbolic and converges to P(£) as e —> 0 because
If 7 = 0 then we have £n = i^i from (5.4). Hence one can write
Since ^12(^)^21(^') > 0 by (5.3), assuming £12 7^ 0 we can write
with some constant A; > 0 so that
If k > 0 then P(^) is diagonalizable for every £ and contradicts the assumption. If
k = 0 it suffices to define P£(£) by the right-hand side of (5.7) with t = £n and 0 = ^12-
The proof for the case £2i 7^ 0 is just a repetition. Thus we complete the proof.
References
[1] S.FRIEDLAND, J.ROBBIN AND J.SYLVESTER, On the crossing rule, Comm. Pure
Appl. Math. 37 (1984), 19-37.](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-48-320.jpg)
![Hyperbolic systems with nondegenerate characteristics 29
[2] L.HORMANDER, Hyperbolic systems with double characteristics, Comm. Pure
Appl. Math. 46 (1993), 261-301.
[3] F.JOHN, Algebraic conditions for hyperbolicity of systems of partial differential
equations, Comm. Pure Appl. Math. 31 (1978), 89-106.
[4] F.JOHN, Addendum to : Algebraic conditions for hyperbolicity of systems of partial
differential equations, Comm. Pure Appl. Math. 31 (1978), 787-793.
[5] P.D.LAX, The multiplicity of eigenvalues, Bull. Amer. Math. Soc. 6 (1982), 213-
214.
[6] T.NlSHITANl, On strong hyperbolicity of systems, in Research Notes in Mathe-
matics, 158, pp. 102-114, 1987.
[7] T.NlSHITANl, Symmetrization of hyperbolic systems with non degenerate charac-
teristics, J. Func. Anal. 132 (1995), 92-120.
[8] T.NlSHITANl, Stability of symmetric systems under hyperbolic perturbations,
Hokkaido Math. J. 26 (1997), 509-527.
[9] T.NlSHITANl, Symmetrization of hyperbolic systems with real constant coefficients,
Scuola Norm. Sup. Pisa 21 (1994), 97-130.](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-49-320.jpg)
![The Cauchy problemfor hyperbolicoperators
dominated by the timefunction
Kunihiko Kajitani
Institute of Mathematics,University of Tsukuba,
Tsukuba, Ibaraki 305,Japan
In Honor of Jean Vaillant
0 Introduction
This note is devoted to the C*30
well posedness of the Cauchy problem for a hyperbolic
operator which are a generalization of the effectively hyperbolic operator. The result
in this note is a joint work with S. Wakabayashi and K. Yagdjian and the detail of the
proof will be appeared in [8]. Let
j+a<m
be a partial differential operator with smooth coefficients a^a(t,x) € Z?°°(Rn+1
). We
consider the Cauchy problem for P
P(t,x,Dt,Ds)u(t,x) =/(*,*), i€[0,T], z 6 R n
, . }
D}u(Q,x) =Uj(x], j = 0,...,m-l, x 6 Rra
. ( }
For the principal symbol Pm(t,x, A, £) of the operator P defined by
j+a=m
we assume that for all t e R, x € Rn
, A€ R, ^ € Rn
, the following representation
with the real-valued functions j(t, x,£), j ~ 1, . ..,m, and with an integer d > 2 and
a nonnegative Lipschitz continuous function y>(t,x) and ipo(t,x) € B1
(Rn+1
) satisfying
dt<Po 7^ 0 if (/?o = 0,
k, (0.2)
31](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-51-320.jpg)
![32 Kajitani
holds and v(t,x} does not vanish outside a compact set in FP+l
. Thus the operator
P(t, x,Dt, Dx] is a hyperbolic operator with the characteristics Xj(t, x,f), j = 1,..., m,
of variable multiplicity in a compact set and strictly hyperbolic outside a compactset.
It is well-known that the lower order terms ofthe operators with multiple characteristics
play crucial role in the well-posedness of the Cauchyproblem (see, for example, [2], [9]).
Therefore we make an assumption
dtd%ajt0t(t,x} < Ck£<f>(t,x)d
(3
--'XTn
-J
')-fc
-l/3
!. j + a=s, 1 < s < m, (0.3)
for (t,x) € Rn+1
, where y>(t,x) is given in (0.2). These kind of conditions for the
coefficients with 5 < m —1 are called Levi conditions. To describe a propagation
phenomena in the Cauchy problem we denote
A_ '^= sup A'(t x £)
and define a hyperbolic cone of symbol Pm by
T := { (A,£) € R"+1
; A> A^KI } (0.4)
while F* is dual cone of F that is
r :={(t,x] 6 Rn+1
; U 4- 1 • £ > 0 foraU (A,^) € T } (0.5)
and is called a propagation cone of symbol Pm.
For the Cauchy problem (0.1)we can prove the following theorem.
Theorem 0. 1 Assume (0.2) and (0.3). TTien /or any Uj € #°°(Rn
) and for any f
€ C°°([0,r];Jff00
(Rn
)) there exists a unique solution u € C°°([0,r];^00
(Rn
)) o/ t/ie
Cauchy problem (0.1). For t/ie support of the solution the following formula holds
U *+
(o,y)lj U f f o y ) , (0-6)
m-l
Do = U supp Ui , fio = supp /, K+(r, y] = (r, y) + T* . (0.7)
t=0
Thus according to this theorem the solution propagates along the propagation cone.
Example 0. 2 The second-order operator
P = JD? - (fc(t, x)24
satisfies the conditions (0.2), (0.3), if we take fa = x(^~£f^} and 02 € B(
R2
), where
X € C°°(R) satisfing x(t) = t, t < 1 and x(^) = ±2, ±£ > 2 ,and (^ = ($ + <pi)5
does not vanish outside a compact set and PQ = fa. See [3] for more general hyperbplic
operators with double characteristics.](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-52-320.jpg)
![Hyperbolic operators dominated by the time function 33
Example 0. 3 For the operator P = P3 + PI + PI, Ps = Ttj+a=saj,a(t)X:
'^a
, s =
1, 2, 3, with the principal symbol
the condition (0.3) will be satisfied if
oo.aMHEa^f-^"1
, N = 2,
.7=1
ai,a(t, x) = £M^'1
, <*>,«(*, *) = £ c^-J
$-2
, |a| = 1 ,
3=1 3=2
where $,&i are the same as in Example 0.2 and a;-, bjCj are in BCC
(R2
}.
The microlocal version of Theorem 0.1 will be given in [7]. For the operators with
the multiple characteristics a microlocal energy method developed in [4], [5]; allows to
prove the well-posedness and gives a complete pictureofthe propagation ofsingularities.
When wecan take tpo = t, the case that d in (0.2) depends on j is treated in [8]. When
d = 1, Theorem is already proved in [6]. Therefore we consider here the case of d > 2.
We shall describe the outline of the proof of Theorem 0.1 in the next section.
1 The outline of the proof
In this section we shall give the outline of proof of an a priori estimate for the Cauchy
problem (0.1) under the assumptions (0.2) and (0.3). The proof will be done in the
following seven steps.
Step 1: Reduction to the problem in usual Sobolev spaces
The equation P(t, x, Dt, Dx}u(t,x) = f(t, x} obtained from (0.1) one can replace by the
equation
P7(t, x, Dt, Dx)u^(t, x) = /7(i, x) , /7(t, x] := e-^f(t, x) , (1.1)
for new unknown function u^ (t,x) = e~^t
u(t,x], where /7 € H°°(Rn+i
}, while
P7(t, x,A, Dx)v = P(t, Dt - fy, Dx)v = e~^P(t, Dt, Dx}(e^v] . (1.2)
If we define v by
,x) := eA
(t,x,Dx}v(t,x) =
with weight function A(t,x,f) € C°°(R2n+1
) denned by (1.8) below then the function
v(t,x) satisfies
P^A(t,x,Dt,Dx)v(t,x) = /7>A(U), A,A(*,*) :=eA
(i,x,JDI)-1
/7(i,x), (1.3)](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-53-320.jpg)
![34 Kajitani
where
P7iA(i, z, Dt, Dx) := eA
(£, x, Dx}~l
P(t, x, Dt - ry, D,)eA
(t, or, I?) (1.4)
and the inverse eA
(t, x, Dx}~1
is assured by the ellipticity of the symbol eA(tjX
^. Our
aim is to obtain a priori estimate for the solutions of the differential equation (1.3):
ibl!^(R^i)<-^i|P7,A(^^A,^xX-)||^(R^i) for all v e H°°(Rn+l
} . (1.5)
v /
This estimate holds for all 7 > 70, where 70 will be chosen and fixed, while C is inde-
pendent of 7. Since P(t,x,Dt,Dx) = e^eA
(t,x,Dx}P^(t,x,DtlDx}e-^lt
eA
(t1x,Dx)-1
1
from the last inequality we obtain
^^} (1.6)
+l
Let d > 2 be an integer given in (0.2) and ipi functions given in (0.9) and define
1 . 2
8 = -±-f <0, = V + l^i2
(1-7)
(1-8)
(1.9)
x AQ(S, y,
(1.12)
where /i, 7, Af 6 R+
are parameters. The metric gt^,s(s, y,77, C) is a Riemann metric
in T*(R7H
"1
) and slowly varying in the sense of Definition 18.4.1 [1] that follows from
the fact that u(t,x,£) and r(t,x,£) are weights with respect to g. The constant
M counts the loss of regularity, while 7 controls the support of solution and reflect the
finite propagation speed.
Step 2: Leray- Carding's method
In this step we prove estimate (1.5). Denote HT£ = rd + £j=iCjC^ and (f, C) =
- Set Q(i
'*'A
'c) := (H
*Pm}(t
' XjA
~''(7+Af)
'e
"'Ai)for thesymbo1of
the separating operator. To prove estimate (1.5) it is enough to show that there exists
a constant CQ > 0 such that an estimate
Im (P7^v, Qv)ff.(R«-n) > C0||v||^(R^i} (1-13)
holds for all v € ^(R""1
). Wenote
Im (P^v, Qv)H,(nn+i} = — {(Q*P7,Av, u)H.(R-n) - (P7^Qv, v)H»(R-n)} • (1-14)](https://image.slidesharecdn.com/1007365-250518155939-e92e607c/85/Hyperbolic-Differential-Operators-And-Related-Problems-1st-Edition-Vincenzo-Ancona-Editor-54-320.jpg)
Hyperbolic Differential Operators And Related Problems 1st Edition Vincenzo Ancona Editor
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- 6. Hyperbolic Differential Operators and Related Problems edited by Vincenzo Ancona Universita degli Studi di Firenze Florence, Italy Jean Vaillant Universite Pierre et Marie Curie, Paris VI Paris, France MARCEL DEKKER, INC. NEW YORK • BASEL
- 7. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress. ISBN: 0-8247-0963-2 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue,New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-260-6300; fax: 41-61-260-6333 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquartersaddress above. Copyright © 2003 by Marcel Dekker, Inc. AHRights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, elec- tronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 1 0 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA
- 8. PURE AND APPLIEDMATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EXECUTIVE EDITORS Earl J. Taft Zuhair Nashed Rutgers University University of Delaware New Brunswick, New Jersey Newark, Delaware EDITORIAL BOARD M. S. Baouendi Anil Nerode University of California, Cornell University San Diego Donald Passman JaneCronin University of Wisconsin, Rutgers University Madison Jack K. Hale Fred S. Roberts Georgia Institute of Technology Rutgers University S. Kobayashi David L. Russell University of California, Virginia Polytechnic Institute Berkeley and State University Marvin Marcus Walter Schempp University of California, Universitdt Siegen Santa Barbara Mark Teply W. S. Massey University of Wisconsin, Yale University Milwaukee
- 9. LECTURE NOTESIN PURE AND APPLIED MATHEMATICS 1. N. Jacobson, Exceptional Lie Algebras 2. L-A Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis 3. /. Satake, Classification Theory of Semi-Simple Algebraic Groups 4. F. Hirzebnich et a/., Differentiable Manifolds and Quadratic Forms 5. I. Chavel, Riemannian Symmetric Spaces of Rank One 6. R. B. Burckel, Characterization of C(X) Among Its Subalgebras 7. B. R. McDonald et a/., Ring Theory 8. Y.-T. Siu, Techniques of Extension on Analytic Objects 9. S. R. Caradus et a/., Calkin Algebras and Algebras of Operatorson Banach Spaces 10. E. O. Roxin et a/., Differential Games and Control Theory 11. M. Orzech and C. Small, The Brauer Group of Commutative Rings 12. S. Thornier, Topology and Its Applications 13. J. M. Lopez and K. A. Ross, Sidon Sets 14. W. W. Comfort and S. Negrepontis, Continuous Pseudometrics 15. K. McKennon andJ. M. Robertson, Locally Convex Spaces 16. M. Carmeli and S. Malin, Representations of the Rotation and Lorentz Groups 17. G. B. Seligman, Rational Methods in Lie Algebras 18. D. G. de Figueiredo, Functional Analysis 19. L Cesari et a/., Nonlinear Functional Analysis and Differential Equations 20. J. J. Schaffer, Geometry of Spheres in Nomned Spaces 21. K. Yano and M. Kon, Anti-Invariant Submanifolds 22. W. V. Vasconcelos, The Rings of Dimension Two 23. R. E. Chandler, Hausdorff Compactifications 24. S. P. Franklin and B. V. S. Thomas, Topology 25. S. K. Jain, Ring Theory 26. B. R. McDonald and R. A. Morris, Ring Theory II 27. R. B. Mura and A. Rhemtulla, Orderable Groups 28. J. R. Graef, Stability of Dynamical Systems 29. H.-C.Wang, Homogeneous Branch Algebras 30. £ O. Roxin et a/., Differential Games and Control Theory II 31. R. D. Porter, Introduction to Fibre Bundles 32. M. Altman, Contractors and Contractor Directions Theory and Applications 33. J. S. Golan, Decomposition and Dimension in Module Categories 34. G. Fairweather, Finite Element Galerkin Methods for Differential Equations 35. J. D. Sally, Numbers of Generators of Ideals in Local Rings 36. S. S. Miller, Complex Analysis 37. R. Gordon, Representation Theory of Algebras 38. M. Goto and F. D. Grosshans, Semisimple Lie Algebras 39. A. I. Arruda et a/., Mathematical Logic 40. F. Van Oystaeyen, Ring Theory 41. F. Van Oystaeyen and A. Verschoren, Reflectors and Localization 42. M. Satyanarayana, Positively Ordered Semigroups 43. D. L Russell, Mathematics of Finite-Dimensional Control Systems 44. P.-T.Liu and E. Roxin, Differential Games and Control Theory III 45. A. Geramita and J. Seberry, Orthogonal Designs 46. J. Cigler, V. Losert, and P. Michor, Banach Modules and Functors on Categories of Banach Spaces 47. P.-T. Liu and J. G. Sutinen, Control Theory in Mathematical Economics 48. C. Byrnes, Partial Differential Equations and Geometry 49. G. Klambauer, Problems and Propositions in Analysis 50. J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields 51. F. Van Oystaeyen, RingTheory 52. B. Kadem, Binary Time Series 53. J. Barros-Neto and R. A. Artino, Hypoelliptic Boundary-ValueProblems 54. R L. Stemberg et a/., Nonlinear Partial Differential Equations in Engineering and Applied Science 55. 6. R McDonald, Ring Theory andAlgebra III 56. J. S. Golan, Structure Sheaves Over a Noncommutative Ring 57. T. V. Narayana et a/., Combinatorics, Representation Theory and Statistical Methods in Groups 58. T.A. Burton, Modeling and Differential Equations in Biology 59. K. H. Kim and F. W. Roush, Introduction to Mathematical Consensus Theory
- 10. 60. J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces 61. O. A. Me/son, Direct Integral Theory 62. J. E. Smith et a/., Ordered Groups 63. J. Cronin, Mathematicsof Cell Electrophysiology 64. J. W. Brewer, Power Series Over Commutative Rings 65. P. K. Kamthan and M. Gupta, Sequence Spaces and Series 66. 7. G. McLaughlin, RegressiveSets and the Theory of Isols 67. T. L. Herdman et a/., Integral and Functional Differential Equations 68. R. Draper, Commutative Algebra 69. W. G. McKay and J. Patera, Tables of Dimensions, Indices, and Branching Rules for Repre- sentations of Simple Lie Algebras 70. R. L. Devaney and Z. H. Nitecki, Classical Mechanics and Dynamical Systems 71. J. Van Gee/, Places and Valuations in Noncommutative Ring Theory 72. C. Faith, Injective Modules and Injective Quotient Rings 73. A. Fiacco, Mathematical Programming with Data Perturbations I 74. P. Schultz et a/., Algebraic Structures and Applications 75. L B/can et a/., Rings, Modules, and Preradicals 76. D. C. Kay and M. Breen, Convexity and Related Combinatorial Geometry 77. P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces 78. C.-C. Yang, Factorization Theory of Meromorphic Functions 79. O. Taussky, Ternary Quadratic Forms and Norms 80. S. P. Singh and J. H. Burry, Nonlinear Analysis and Applications 81. K. B. Hannsgen etal., Volterra and Functional Differential Equations 82. N. L. Johnson et a/., Finite Geometries 83. G. /. Zapata, Functional Analysis, Holomorphy, and Approximation Theory 84. S. Greco and G. Valla, Commutative Algebra 85. A. V. Fiacco, Mathematical Programming with Data Perturbations II 86. J.-B. Hiriart-Urruty et a/., Optimization 87. A. Figa Talamanca and M, A. Picardello, Harmonic Analysis on Free Groups 88. M. Harada, Factor Categories with Applications to Direct Decomposition of Modules 89. V. I. Istratescu, Strict Convexity and Complex Strict Convexity 90. V. Lakshmikantham, Trends in Theory and Practice of Nonlinear Differential Equations 91. H. L Manocha and J. B. Srivastava, Algebra and Its Applications 92. D. V. Chudnovsky and G. V. Chudnovsky, Classical and Quantum Models and Arithmetic Problems 93. J. W. Longley, Least Squares Computations Using Orthogonalization Methods 94. L P. de Alcantara, Mathematical Logic and Formal Systems 95. C. E. Aull, Rings of Continuous Functions 96. R. Chuaqui, Analysis, Geometry, and Probability 97. L. Fuchs and L. Salce, Modules Over Valuation Domains 98. P. Fischer and W. R. Smith, Chaos, Fractals, and Dynamics 99. W. B. Powell and C. Tsinakis, Ordered Algebraic Structures 100. G. M. Rassias and T. M. Rassias, Differential Geometry, Calculus of Variations, and Their Applications 101. R.-E.Hoffmann and K. H. Hofmann, Continuous Lattices and Their Applications 102. J. H. Lightboume III and S. M. Rankin III, Physical Mathematics and Nonlinear Partial Differential Equations 103. C. A. Baker and L. M. Batten, Finite Geometries 104. J. W. Brewer et a/., Linear Systems Over Commutative Rings 105. C. McCrory and T. Shifrin, Geometry and Topology 106. D. W. Kueke et a/., Mathematical Logic and Theoretical Computer Science 107. B.-L Lin and S. Simons, Nonlinear and ConvexAnalysis 108. S. J. Lee, Operator Methods for Optimal Control Problems 109. V. Lakshmikantham, Nonlinear Analysis and Applications 110. S. F. McCormick, Multigrid Methods 111. M. C. Tangora, Computers in Algebra 112. D. V. Chudnovsky and G. V. Chudnovsky, Search Theory 113. D. V. Chudnovsky and R. D. Jenks, ComputerAlgebra 114. M. C. Tangora, Computers in Geometry and Topology 115. P. Nelson et a/., Transport Theory, Invariant Imbedding, and Integral Equations 116. P. Clement et a/., Semigroup Theory and Applications 117. J. Vinuesa, Orthogonal Polynomials and Their Applications 118. C. M. Dafermos et a/., Differential Equations 119. E. O. Roxin, Modem Optimal Control 120. J. C. Diaz, Mathematicsfor Large Scale Computing
- 11. 121. P. S. Milojevft Nonlinear Functional Analysis 122. C. Sadosky, Analysis and Partial Differential Equations 123. R. M. Shortt, GeneralTopology and Applications 124. R. Wong, Asymptotic and Computational Analysis 125. D. V. Chudnovsky and R. D. Jenks, Computers in Mathematics 126. W. D. Wallis et at, Combinatorial Designs and Applications 127. S. Elaydi, DifferentialEquations 128. G. Chen et at, Distributed Parameter Control Systems 129. W. N. Everitt, Inequalities 130. H. G. Kaper and M. Garbey, Asymptotic Analysis and the Numerical Solution of Partial Differ- ential Equations 131. O. Anno et a/., Mathematical Population Dynamics 132. S. Coen, Geometry and Complex Variables 133. J. A. Goldstein et a/., Differential Equations with Applications in Biology, Physics, and Engineering 134. S. J. Andima et a/., General Topology and Applications 135. P Clement et a/., Semigroup Theory and Evolution Equations 136. K. Jarosz, Function Spaces 137. J. M. Bayod et a/., p-adic Functional Analysis 138. G. A. Anastassiou, Approximation Theory 139. R. S. Rees, Graphs, Matrices, and Designs 140. G.Abrams et a/., Methods in Module Theory 141. G. L Mullen and P. J.-S. Shiue, Finite Fields, Coding Theory, and Advances in Communications and Computing 142. M. C. JoshiandA. V. Balakrishnan, Mathematical Theory of Control 143. G. Komatsu and Y. Sakane, Complex Geometry 144. /. J. Bakelman, Geometric Analysis and Nonlinear Partial Differential Equations 145. T. Mabuchi and S. Mukai, Einstein Metrics and Yang-Mills Connections 146. L. Fuchs and R. Gdbel, Abelian Groups 147. A. D. Pollington and W. Moran, Number Theory with an Emphasis on the Markoff Spectrum 148. G. Dore et a/., Differential Equations in Banach Spaces 149. T. West, Continuum Theory and Dynamical Systems 150. K. D. Bierstedtetal., Functional Analysis 151. K. G. Fischer et at. Computational Algebra 152. K. D. Elworthy et a/., Differential Equations, Dynamical Systems, and Control Science 153. P.-J. Cahen, et a/., Commutative RingTheory 154. S. C. Cooper and W. J. Thron, Continued Fractions and Orthogonal Functions 155. P. Clement and G. Lumer, Evolution Equations, Control Theory, and Biomathematics 156. M. Gyllenberg and L. Persson, Analysis, Algebra, and Computers in Mathematical Research 157. W. O. Bray et at, Fourier Analysis 158. J. Bergen and S. Montgomery, Advances in Hopf Algebras 159. A. R. Magid, Rings, Extensions, and Cohomology 160. N. H. Pavel, Optimal Control of Differential Equations 161. M. Ikawa, Spectral and Scattering Theory 162. X. Liu and D. Siegel, Comparison Methods and Stability Theory 163. J.-P. Zolesio, Boundary Control and Variation 164. M. KHzeketat, Finite Element Methods 165. G. Da Prato and L. Tubaro, Control of Partial Differential Equations 166. E. Ballico, Projective Geometry with Applications 167. M. Costabeletal., Boundary Value Problems and Integral Equations in Nonsmooth Domains 168. G. Ferreyra, G. R. Goldstein, andF. Neubrander, Evolution Equations 169. S. Huggett, Twister Theory 170. H. Cooketal., Continue 171. D. F. Anderson and D. E. Dobbs, Zero-Dimensional Commutative Rings 172. K. Jarosz, Function Spaces 173. V.Ancona et at. Complex Analysis and Geometry 174. £ Casas, Control of Partial Differential Equations and Applications 175. N, Kalton et at, Interaction Between Functional Analysis, Harmonic Analysis, and Probability 176. Z. Deng ef at. Differential Equations and Control Theory 177. P. Marcellini et at Partial Differential Equations and Applications 178. A. Kartsatos, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type 179. M. Maruyama, Moduli of Vector Bundles 180. A, Ursini and P. Agliand, Logic and Algebra 181. X, H. Cao et at, Rings, Groups, andAlgebras 182. D. Arnold and R. M. Rangaswamy, Abelian Groups and Modules 183. S. R Chakravarthy and A. S. Alfa, Matrix-Analytic Methods in Stochastic Models
- 12. 184. J. E Andersen et al., Geometry and Physics 185. P.-J. Cahen et al., Commutative Ring Theory 186. J. A. Goldstein et al., Stochastic Processes and Functional Analysis 187. A. Sorbi, Complexity, Logic, and Recursion Theory 188. G. Da Prafo and J.-P. Zolesio, Partial Differential Equation Methods in Control and Shape Analysis 189. D. D. Anderson, Factorization in Integral Domains 190. N. L Johnson, Mostly Finite Geometries 191. D. Hinton and P. W. Schaefer, Spectral Theory and Computational Methods of Sturm-Liouville Problems 192. W. H. Schikhofet al., p-adic Functional Analysis 193. S. Sertoz, Algebraic Geometry 194. G. Caristi and E. Mitidieri, Reaction Diffusion Systems 195. A. V. Fiacco, Mathematical Programming with Data Perturbations 196. M. Kfizek et al., Finite Element Methods: Superconvergence, Post-Processing, and A Posteriori Estimates 197. S. Caenepeeland A. Verschoren, Rings, Hopf Algebras, and Brauer Groups 198. V. Drensky et al., Methods in Ring Theory 199. W. B. Jones and A. Sri Ranga, Orthogonal Functions, Moment Theory, and Continued Fractions 200. P. E. Newstead, Algebraic Geometry 201. D. Dikranjan and L. Salce, Abelian Groups, Module Theory, and Topology 202. Z. Chen et al., Advances in Computational Mathematics 203. X. Caicedo and C. H. Montenegro,Models, Algebras, and Proofs 204. C. Y. Yildinm and S. A. Stepanov, Number Theory and Its Applications 205. D. E. Dobbs et al., Advances in Commutative Ring Theory 206. F. Van Oystaeyen, Commutative Algebra and Algebraic Geometry 207. J. Kakol et al., p-adic Functional Analysis 208. M. Boulagouaz and J.-P. Tignol, Algebra and Number Theory 209. S. Caenepeeland F. Van Oystaeyen, Hopf Algebras and Quantum Groups 210. F. Van Oystaeyen and M. Saon'n, Interactions Between Ring Theory and Representations of Algebras 211. R. Costa et al., Nonassociative Algebra and Its Applications 212. T.-X. He, Wavelet Analysis and Multiresolution Methods 213. H. Hudzik and L. Skrzypczak, Function Spaces: The Fifth Conference 214. J. Kajiwara et al.. Finite or Infinite Dimensional Complex Analysis 215. G. Lumer and L. Weis, Evolution Equations and Their Applications in Physical and Life Sciences 216. J. Cagnoletal., Shape Optimization and Optimal Design 217. J. Herzog and G. Restuccia, Geometric and Combinatorial Aspects of Commutative Algebra 218. G. Chen et al., Control of Nonlinear Distributed Parameter Systems 219. F. AHMehmeti et al., Partial Differential Equations on Multistructures 220. D. D. Anderson and I. J. Papick, Ideal Theoretic Methods in Commutative Algebra 221. A. Granja et al., Ring Theory and Algebraic Geometry 222. A. K. Katsaras et al., p-adic Functional Analysis 223. R. Salvi, The Navier-Stokes Equations 224. F. U. Coelho and H. A. Merklen, Representations of Algebras 225. S. Aizicovici and N. H. Pavel, Differential Equations and Control Theory 226. G. Lyubeznik, Local Cohomology and Its Applications 227. G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications 228. W. A. Camiellietal., Paraconsistency 229. A. Benkirane and A. Touzani, Partial Differential Equations 230. A. Illanes et al., Continuum Theory 231. M. Fontana et al., Commutative Ring Theory and Applications 232. D. Mond and M. J. Saia, Real and Complex Singularities 233. V. Ancona and J. Vaillant, Hyperbolic Differential Operators Additional Volumes in Preparation
- 13. Preface The papers collected in this volume are concerned with hyperbolic problems, or problems the methods of which are related to hyperbolic techniques. T. NISHITANI introduces a notion of nondegenerate characteristic for systems of linear partial differential equations of general order. He shows that nondegenerate char- acteristics are stable under hyperbolic perturbations, and he proves that if the coefficients of the system are real analytic and all characteristics are nondegenerate then the Cauchy problem for the system is well posed in the class of smooth functions. K. KAJITANI studies a class of operators that generalize the linear hyperbolic op- erators, introducing the notion of time function, and proving the well-posedness of the Cauchy problem in the class of C°° functions. The Cauchyproblem is also the subject ofthe paper by A. BOVE and C. BERNARDI; they state some results for a class of linear hyperbolicoperators with double characteristics, not effectively hyperbolic. In particular they prove well-posedness in the C°° class under a geometric condition and a Levi condition, and well-posedness in the Gevrey class under more general assumptions. For a linear system whose principal part is hyperbolic and whose coefficients depend only on time, H. YAMAHARA establishes necessary and sufficient conditions for well- posedness in the Gevrey class, whatever the lower order terms are. L. MENCHERINI and S. SPAGNOLOconsider a first order hyperbolic system in two variables whose coefficients depend only on time; they define the notion of pseudosymmetry for matrix symbolsoforder zero, and determine the Gevrey class wherethe Cauchy problem is well-posed, according to the type of pseudosymmetry of the principal matrix symbol. The 2-phase Goursat problem has been solved by means of Bessel functions; here J. CARVALHO E SILVA considers the 3-phase Goursat problem, using instead some hyper- geometric functions in four variables. He also discusses the general problem, pointing out that the main difficulties are due to the lack of results on special functions. The Stricharz inequality for the classical linear wave equation has been generalized by M. REISSIG and Y.G. WANG to the case of time-dependent coefficients: the coefficient is the product of an increasing factor and an oscillatory factor. The interaction was studied by the authors; in the present paper they extend the result to a one-dimensional system describing thermoelasticity. The noncharacteristic, nonlinear Cauchy problem is the subject of the paper by M. TSUJI. The classical solution has some singularities, so that the problem arises of studying the extension of the solution beyond the singularities. The author constructs a Lagrangian solution in the cotangent bundle, getting a multivalued classical solution; then he explores how to construct a reasonable univalued solution with singularities. Y. CHOQUET considers the Einstein equations. By a suitable choice of the gauge, (for instance, an a priori hypothesis on coordinates choice) she obtains a hyperbolic system in the sense of Leray-Ohya, well-posed in the Gevrey class of index 2. She studies old and new cases where the system is strictly hyperbolic and well-posed in the C°° class. Necessary and sufficient conditions for the Cauchy-Kowalevskaya theorem on systems of linear PDEs were given by Matsumoto and Yamahara; on the other hand, Nagumo constructed a local solution, unique, for a higher order scalar Kowalevskian operator, whose coefficients are analyitic in x and continuous in t. Here W. MATSUMOTO, M. MURAI and T. NAGASE show that for a system whose coefficients are analytic in x and iii
- 14. 1V Preface C°° in t the above condition of Matsumoto and Yamahara is necessary and sufficient also when the order on d/dt is one, the order on d/dx is two, and the rank of the second order part is at most one. B.W. SCHULZE and N. TARKHANOV construct a general calculus of pseudodif- ferential operators on a smoothly stratified space, with local cone bundle neighborhood, including ellipticity and the Fredholm property. M. DREHER and I. WITT propose edge Sobolev spaces for the investigation ofweak hyperbolicity for linear and semilinear operators; propagation of singularities is stated. For the surface waves of water K. O. Friedrichs gave a systematic procedure to obtain the Airy equations from the Euler equations; a rigorous mathematical approach was more recently given by T. Kano in the analytic case. Here T. KANO and S. MIKI develop the theory in the Lagrange coordinate system. The wave equation in shallow water and the Boussineq equation, via Korteweg-de Vries, are obtained as approximate equations in Lagrangian coordinates. For certain equations of Schrodinger type, J. TAKEUCHI states necessary and suffi- cient conditions for the Cauchy problem to be well-posed in H00 ; he uses a time indepen- dent Z/2 symmetrization, with loss of regularity. D. GOURDIN studies a 2-evolution operator in the sense of Petrosky; subject to the condition that the real roots of the principal polynomial with constant coefficients have constant multiplicity. He finds sufficient conditions for the well-posedness of the Cauchy problem in Sobolev spaces. Some generalizations are also given. K. KAJITANI investigates the Gevrey smoothing effects of the solution to the Cauchy problem for Schrodinger- type equations: the decay of initial data is related to the Gevrey class with respect to the space variables of the solutions. The metaplectic representation allows the construction of the solutions of the Schrodinger equation for the quadratic hamiltonians. M de GOSSON is able to obtain the result more generally for any physical hamiltonian. F. COLOMBINI and C. GRAMMATICO consider the problem of the strong unique- ness of the solution (in a neighborhood of the origin) in Rn , for particular fourth order elliptic operators flat at the origin. In the second order case, the known result in R2 is obtained in Rn . In the case of a product of some model second order operators in R2 with Gevrey coefficients strong uniqueness is obtained under a condition on the Gevrey index, related to the form of the operator. A sharp condition on the modulus of continuity was obtained by Tarama for an elliptic operator of second order to have the local uniquenessproperty. D. DEL SANTOshows that this condition is necessary by constructing a nonuniqueness example. For certain holomorphic operators with polynomial coefficients Y. HAMADA con- structs some solutions whose domain of holomorphy has an exterior point. The results are based on the earier work by Hamada, Leray, Takeuchi, as well as Darboux, Halphen and Chazy. The remaining papers contain more theoretical results.
- 15. Preface v P. LAUBIN presents some known results and new advances on the topology of spaces of holomorphicfunctions in an open subset of a Prechet space; he gives a projective charac- terization of holomorphic germs using seminorms whose form is similar to the one appearing in the Whitney extension theorem for C°° functions. Let Y be a smooth submanifold of a C°° manifold X; a distribution u in the com- plement of Y has the property that the closure of its wave front in the cotangent bundle of X is orthogonal to the tangent bundle of Y. M.K.V. MURTHY describes its analytic behavior in a neighborhood of Y by the notion of microlocal scaling degree along y, and obtains a result similar to the Hormander theorem for homogeneous distributions. A. DEBIARD and B. GAVEAU in their paper apply stochastic methods to determine the ground state of an atomic molecular system. The system is represented by a Hamilto- nian operator acting on a Hilbert space. A lower bound for the ground state is obtained using the Feynman-Kac formula and the special homogeneity property of the Coulomb potential. L.S. SCHULMAN raises the difficulty of conceiving that a single dynamical system could contain subsystems, in significant mutual contact, possessing opposite thermodynam- ics arrows of time. By examining possible cosmological justification for the usual arrow of time he finds that a consistent way to establish such justification is by giving symmetric boundary conditions at two (cosmologically remote) times and seeking irreversible behav- ior in between. Other boundary conditions, modeling shorter periods in the evolution of the cosmos, can be found that allow the simultaneous existence of two thermodynamic arrows, notwithstanding moderate interaction between the two systems. Vincenzo Ancona Jean Vaillant
- 17. Contents Preface Hi Contributors ix 1. The conference in honor of Jean Vaillant 1 Bernard Gaveau 2. Hyperbolic systems with nondegenerate characteristics 7 Tatsuo Nishitani 3. The Cauchy problem for hyperbolic operators dominated by the time function 31 Kunihiko Kajitani 4. A remark on the Cauchy problem for a model hyperbolic operator 41 Enrico Bernardi and Antonio Bove 5. Gevrey well-posedness of the Cauchy problem for systems 53 Hideo Yamahara 6. Gevrey well-posedness for pseudosymmetric systems with lower order terms 67 Lorenzo Mencherini and Sergio Spagnolo 7. Le role des fonctions speciales dans les problemes de Goursat pour des equations aux derives partielles a coefficients constants 83 Jaime Carvalho e Silva 8. Influence of the hyperbolic part on decay rates in 1-d thermoelasticity 89 Ya-Guang Wang and Michael Reissig 9. Integration and singularities of solutions for nonlinear second order hyperbolic equation 109 Mikio Tsuji 10. Causal evolution for Einsteinian gravitation 129 Yvonne Choquet-Bruhat 11. On the Cauchy-Kowalevskaya theorem of Nagumo type for systems 145 Waichiro Matsumoto, Minoru Murai, and Takaaki Nagase 12. Differential analysis on stratified spaces 157 B. W. Schulze and N. Tarkhanov 13. Edge Sobolev spaces, weakly hyperbolic equations, and branching of singularities 179 Michael Dreher and Ingo Witt
- 18. viii Contents 14. Sur les ondes superficieles de 1'eau et le developement de Friedrichs dans le systeme de coordonnees de Lagrange 199 Tadayoshi Kano and Sae Miki 15. Probleme de Cauchy pour certains systemes de Leray-Volevich du type de Schrodinger 233 Jiro Takenchi 16. Systemes du type de Schrodinger a raciness caracteristiques multiples 255 Daniel Gourd in 17. Smoothing effect in Gevrey classes for Schrodinger equations 269 Kunihiko Kajitani 18. Semiclassical wavefunctions and Schrodinger equation 287 Maurice de Gossan 19. Strong uniqueness in Gevrey spaces for some elliptic operators 301 F. Colombini and G. Grammatico 20. A remark on nonuniqueness in the Cauchy problem for elliptic operator having non-Lipschitz coefficients 317 Daniele Del Santo 21. Sur le prolongement analytique de la solutiondu probleme de Cauchy 321 Yusaku Harnada 22. On the projective descriptions of the space of holomorphic germs 331 P. Laubin 23. Microlocal scaling and extension of distributions 339 M. K. Venkatesha Murthy 24. A lower bound for atomic Hamiltonians and Brownian motion 349 A. Debiard and B. Gaveaii 25. A compromised arrow of time 355 L. S. Schulman
- 19. Contributors Enrico Bernard! University of Bologna, Bologna, Italy Antonio Bove Universityof Bologna, Bologna, Italy Jaime Carvalho e Silva Universidade de Coimbra, Coimbra, Portugal Yvonne Choquet-Bruhat Universite de Paris 6, Paris, France F. Colombini Universita di Pisa, Pisa, Italy Maurice de Gosson Blekinge Instituteof Technology, Karlskrona, Sweden, and University of Colorado at Boulder, Boulder, Colorado, U.S.A. A. Debiard Laboratoire Analyse et Physique Mathematique, Universite Pierre et Marie Curie, Paris, France Daniele Del Santo Universita di Trieste, Trieste, Italy Michael Dreher University of Tsukuba, Tsukuba, Japan Bernard Gaveau Laboratoire Analyse et PhysiqueMathematique, Universite Pierre et Marie Curie, Paris, France Daniel Gourdin Universite de Paris 6, Paris, France C. Grammatico Universita di Bologna, Bologna, Italy Yusaku Hamada Kyoto, Japan Kunihiko Kajitani Universityof Tsukuba, Tsukuba, Japan Tadayoshi Kano University of Osaka, Toyonaka, Japan P. Laubint Universityof Liege, Liege, Belgium Waichiro Matsumoto RyukokuUniversity,Otsu, Japan Lorenzo Mencherini Universitadi Firenze, Florence, Italy Sae Mild University of Osaka, Toyonaka, Japan Minoru Murai Ryukoku University, Otsu, Japan M. K. Venkatesha Murthy Universita di Pisa, Pisa, Italy Deceased.
- 20. Contributors Takaaki Nagase Ryukoku University,Otsu, Japan Tatsuo Nishitani Osaka University, Osaka, Japan Michael Reissig TU Bergakademie Freiberg, Freiberg, Germany L. S. Schulman Clarkson University,Potsdam, New York, U.S.A. B. W. Schulze Universitat Potsdam, Potsdam, Germany Sergio Spagnolo Universita di Pisa, Pisa, Italy Jiro Takeuchi Science University of Tokyo, Hokkaido, Japan N. Tarkhanov University of Potsdam, Potsdam, Germany Mikio Tsuji Kyoto Sangyo University,Kyoto, Japan Ya-Guang Wang Shanghai Jiao Tong University, Shanghai, P.R. China Ingo Witt University of Potsdam, Potsdam, Germany Hideo Yamahara Osaka Electro-Communication University,Osaka, Japan
- 21. The conference in honor of Jean Vaillant BERNARDGAVEAU Laboratoire Analyse et Physique Mathematique, Universite Pierre et Marie Curie, Paris, France Since his thesis in 1964 prepared under the direction of J. Leray and A. Lichnerowicz the main theme of the mathematical work of Jean Vaillant has been the study of systems of hyperbolic or holomorphic partial differential equations. The basic example of an hyperbolic equation is the wave equation which is the mathematical description of wave propagation at finite velocity, as, for example, the propagation of small disturbances in fluids (the sound) or of electromagnetic waves in vacuum. Examples of hyperbolic systems include the Maxwell system for the propagation of electromagnetic waves, the Dirac system for the propagation of spinors and Einstein equations in general relativity. The wave equation is the fundamental example of a strictly hyperbolic equation, for which the propagation velocities are different and do not vanish. An approximation of the solutions of a strictly hyperbolic equation is the high frequency approximation or geometrical optics approximation : sound or light propagates essentially along the trajectories of the Hamilton Jacobi equation associated to the partial differential equation. One can say, in a rather unprecise manner, that « singularities are propagated along ^characteristics », which, a posteriori, justifies the use of geometrical optics, the laws of reflexion and refraction. On the other hand, interference and diffraction phenomena show that sound or light can be described as fields which can be added, rather than particles, but again a good approximation of these phenomena is the propagation along bicharacteristics at least in the simplest situations. Dirac system is an example of a system with multiple characteristics with constant multiplicities. Maxwell system in a non isotropic medium, like a crystal, is an hyperbolic system with multiple characteristics, but their multiplicity is non constant: The velocities of propagation depends of the direction of propagation, but for special directions, some velocities may coincide. In this situation, the approximation of geometrical optics is no more valid : The propagation along bicharacteristics (or rays) is not a good description or approximation of the phenomenon, and indeed this can be checked experimentally. A light ray falling on certain crystals, is, in general refracted along a certain direction. Nevertheless for special incidence angles, corresponding to the geometry of the crystal, the ray is refracted, not along another ray, but on a whole conical surface. Until the end of 19 century, this experiment was the only proof of the electromagnetic nature of light, because all the other light propagation phenomena could be described by a wave equation, without the use of the complete Maxwell system (see [1], [2]).
- 22. 2 Gaveau In the beginning of the 1960's, strictly hyperbolic equations with simple characteristics (the velocities of propagation are distinct and non zero), are well understood (see [1], [3]). Around that time, Jean Vaillant begins a systematic study of equations or systems of equations which are non strict. In his thesis ([4], [5]), he introduces the notion of localization with respect to a factor of the characteristic determinant of a system with constant coefficients and he relates this notion to the equation of propagation along the ^characteristics. This seems to be the first attempt to apply the method of localization, in particular using invariant factors. Following the article of Garding, Kotake, Leray (Probleme de Cauchy VI, [6]), J. Vaillant defines a new invariant associated to systems with double characteristics and gives an application to the Goursat problem [7], and to the localization for systems with variables coefficients and double characteristics : This is the first example of a Levi condition in this setting. J. Vaillant relates also the difference of the subcharacteristic polynomial, and the second coefficient of exp (-icocp) P (x, DX) exp (icocp) to the Lie derivative of the volume form along the ^characteristics [8]. This is an important invariant, because it is well known that the existence and the regularity of the solution depend, in degenerate cases not only of the principal symbol but also on the lower order terms of the operator. This result was rederived by Duistermaat and Hormander. In 1973-74, J. Vaillant constructs the asymptotic expansion of the solution of an hyperbolic systems with characteristics of variable multiplicities [9]. He defines the localization of an hyperbolic system at a multiple points with application to conical refraction [9]. This work will be extended in 1978, when he constructs the parametrix for the Cauchy problem with multiple characteristics [10], in relation with the invariants of the system. In [11], J. Vaillant studies the symmetrisation of localized hyperbolic systems and defines the notion of « reduced dimension » : The property of symmetry is proved in the case of a maximal reduced dimension. These last few years, he has continued to study the symmetric of strongly hyperbolic systems, in particular with T. Nishitani. If the reduced dimension of a system of rank m is not less than 2, a constant coefficient 2 systems is symmetrizable ([7], [8]). For a system with non constant coefficients, if at any point the reduced dimension is not less than — 2, the system is also symmetrizable with a regular symmetrizer [18]. Recently he has determined the multiple points according to the reduced dimension [19]. In 1982, in collaboration with D. Schiltz and C. Wagschal [12], J. Vaillant has studied the ramification of the Cauchy problem for a system in involution with triple characteristics. This problem reduces to the question of the singularities of integrals of
- 23. Conference in honor of Jean Vaillant 3 holomorphic forms depending of parameters, on chains depending also of parameters. The problem is to determine the singularities of these integrals with respect to the parameters. The first systematic work in this direction was the article of J. Leray [13] in the algebraic case. J. Vaillant studies the ramification in the general holomorphic case, using a grassmann boundle [14]. Since 1987, J. Vaillant has started the problem of the classification of systems with constant multiplicities : definition of invariant Levi conditions, relations to the Cauchy problem in the C°° and Gevrey classes. For any system, he obtains systematically the Levi conditions [15], [16]. J. Vaillant has founded a research group and a seminar, which he has maintained, for more than thirty years, independent of fashions « mots d'ordre » and which survives in difficult conditions. Freedom of thought, which is a necessary condition for any creative work, is paid a very high price. Creation, scientific or artistic, cannot be judged according to economic or social criteria, measured in monetary values. Research is not a collective activity. The highly mysterious activity of thought can only be a personal activity. During all his career, Jean Vaillant, following the example of Leray, has tried to defend by his attitudes and his work, the values of scientific creation and intellectual independence. For more than forty years, J. Vaillant has developed many collaborations with his Japanese and Italian friends, in particular Y. Hamada, Y. Ohya, K. Kajitani, T. Nishitani and S. Spagnolo, F. Colombini, A. Bove and E. Bernardi. He has also developed many european collaborations and he has created a european network of belgian, french, Italian, and Portugese universities. All his friends know that they can rely on his help and his advises. acknowledgment: We are very grateful to Anne Durrande, Evelyne Guilloux and Maryse Loiseau for their help during the preparation of this conference. We also thank the Maison Europeenne des Technologies, in particular Madame Muller for her help. B. Gaveau Laboratoire Analyse et Physique Mathematique 14 avenue Felix Faure 75015 PARIS
- 24. Gaveau [1] Courant-Hilbert: Methods of Mathematical Physics Vol Interscience 1962 [2] M. Born & E. Wolf : Principles of optics Penjamon Press 1980 [3] J. Leray : Hyperbolic differential equations Lectures notes, Princeton 1950 [4] J. Vaillant: Sur les discontinuites du tenseur de courbure en theorie d'Einstein-Schrodinger CR Acad Sci Paris - 10 juillet 1961, 30 octobre 1961, 15 Janvier 1962 [5] J. Vaillant: Caracteristiques multiples et bicaracteristiques des systemes d'equations aux derivees partielles lineaires et a coefficients constants Annales Institut Fourier 15 (1965) et 16 (1966) [6] L. Garding, T. Kotake, J. Leray : Uniformation et developpement asymptotique de la solution du probleme de Cauchy lineaire a donnees holomorphes ; analogic avec la theorie des ondes asymptotiques et approchees (Probleme de Cauchy I bis et VI) Bull. Sci. Math. France 92 1964, 263-361. [7] J. Vaillant: Donnees de Cauchy portees par une caracteristique double : role des bicaracteristiques J. Maths Pures et Appliquees 47 (1968), 1-40 [8) J. Vaillant: Derivee de Lie de la forme element de volume le long des bicaracteristiques et polynome sous-caracteristique de Garding-Kotake-Leray CR Acad Sci Paris -10 mars 1969 [9] J. Vaillant: Solutions asymptotiques d'un systeme a caracteristiques de multiplicite variable J. Maths Pures et Appliquees 53 (1974), 71-98 [10] R. Berzin, J. Vaillant- Parametrix a caracteristiques multiples Bull. Sci. Math 102 (1978), 287-294 [11] J. Vaillant: Symetrisation de matrices localisees Annali della Scuola Normale Superiore di Pisa. Ser. IV, 5 (1978), 405-427 [12] D. Schiltz, J. Vaillant, C. Wagschal: Probleme de Cauchy ramifie J. Math. Pures et appliquees (1982) [13] J. Leray : Un complement au theoreme de N.Nilsson sur les integrales de formes differentielles a support singulier algebrique
- 25. Conference in honor of Jean Vaillant 5 Bull. Soc. Math. Fr 95 (1967), 313-374 [14] J. Vaillant: Ramifications d'integrales holomorphes J. Math. Pures et Appliquees 65 (1986), 343-402 [15] J. Vaillant: Conditions d'hyperbolicite pour les systemes Bull. Sci Math 114 (1990), 243-328 Conditions de Levi in Travaux en cours 48. Herman (1994) Analytic hyperbolic systems Pitman, Research Notes in Mathematics 349 (1996), 209-229 Conditions invariantes sur les systemes d'equations aux derivees partielles et probleme de Cauchy in complex Analysis and microlocal analysis, RIMS Kokyuroku 1090, p. 131-142 Kyoto University (1999) Invariants des systemes d'operateurs differentiels et sommes formelles asymptotiques Japanese J. Math. 15 (1999), 1-153 [16] J. Vaillant,G. Taglialatela : Conditions invariantes d'hyperbolicite et reductiondes systemes Bull. Sci. Math. 120 (1996), 19-97 [17] J. Vaillant: Symetrie des operateurs hyperboliques 4 x 4 et dimensionreduite Annali della Scuola Normale Superiore di Pisa 29 (2000), p. 839-890. [18] J. Vaillant,T. Nishitani: Smoothy symmetrizable systems and the reduced dimensions Tsukuba Journal Vol. 25, n° 1,juin 2001, p. 165-177. [19] J. Vaillant, H. Delquie : Dimension reduite et valeurs propres multiples d'une matrice diagonal!sable 4 x 4 Bull. Sci. Math. 124 - 4 (2000).
- 27. Hyperbolic systems with nondegenerate characteristics Tatsuo Nishitani Department of Mathematics,Osaka University, Machikaneyama 1-16,Toyonaka Osaka, 560-0043, Japan 1 Introduction In this note we discuss the stability of non degenerate characteristics of hyperbolicsys- tems of general order under hyperbolic perturbations. We also study the well posedness of the Cauchy problem for hyperbolic systems of general order with non degenerate characteristics. For first order systems we have introduced non degenerate characteristics in[7], [8]. According to this definition, simple characteristics are non degenerate and non degenerate double characteristics coincide with those studied in [2], [3], [4]. We adapt this definition for higher order systems with an obvious modification. For a system of general order, in a standard manner, we can associate a first order system and we prove in section 2 that non degenerate characteristics of the original system are also non degenerate characteristics of the same order for the corresponding first order system and vice versa (Proposition 2.1). In section 3 we prove that one can not remove non degenerate characteristics by hy- perbolic perturbations: any hyperbolic system which is sufficiently close to a hyperbolic system with a non degenerate characteristic must have a non degenerate characteris- tic of the same order. In particular, near a hyperbolic system with a non degenerate multiple characteristic, there is no strictly hyperbolic system. Moreover we show that, near a non degenerate characteristic of order r, the characteristics of order r form a smooth manifold of codimension r(r + l)/2 (Theorem 3.1). The same result wasproved for analytic first order systems in [7]and for systems with non degenerate double char- acteristics in [2]. In section 4. the well posedness of the Cauchy problem for hyperbolic systems with non degenerate characteristics is discussed. We prove that a hyperbolic system of which every characteristic is non degenerate is smoothly symmetrized and hence the Cauchy problem is C°° well posed for arbitrary lower order terms (Theorem 4.1. Theorem 4.2). This generalizes a result in [2] (see also [4]) where the same result was proved for hyperbolic systems with constant coefficients with non degenerate double characteristics. In the last section we restrict our considerations to 2 x 2 first order hyperbolic systems with constsnt coefficients with n independent variables. We show that if n < 3
- 28. 8 Nishitani then such a system is a limit of strictly hyperbolic systems (Proposition 5.1). Contrary if n > 3, by [5]there is no strictly hyperbolic 2 x 2 system. 2 Non degenerate characteristics Let P(x) be a m x m matrix valued smooth function defined near x € Rn . We assume that P(x] is a polynomial in x so that (2.1) P(x) = J>,(x>r j=0 where x1 = (x2, .... xn). We say that P(x) is hyperbolic with respect to 9 = (1, 0, ..., 0) <E Rn if detA)(z') ^ 0 near x' = x' and (2.2) detP(x + A0)= 0 =» Ais real. We call x is a characteristic of order r if (2.3) ^(detP)(f) = 0, V|a < r, <9£(detP)(x) ^ 0, 3|a| = r. Following [7], [8], we introduce non degenerate characteristics. To do so we first define the localization of P(x) at a multiple characteristic x verifying (2.4) KerP(x) n ImP(x) = {0}. Let dimKerP(x) = r and let v,...,vr be a basis for KerP(x). Taking (2.4) into account we can choose linear forms £]_,...,£r so that ^(ImP(x)) = 0, ^(^) = <5y where 5jj is the Kronecker's delta. Then we define a r x r matrix PX(X) by (2.5) (li(P(x + HxWh&jzr = Ai[P4(i) + 0(/z)]. It is easy to see that the definition of PX(X) is independent of the choice of v,...,vr, that is r Pt(x)v = ^[UjCQ^-^XPCx + ^)^)]^ is a well defined map from KerP(x) to KerP(x). We denote (2.6) P-x = (Px(x) x € Rn } C M(r- R) which is a subspace of M(r; R), the space of all real r x r matrices. We first show Lemma 2.1 Assume (2,2) and let x bea characteristic verifying (2.4) with dimKerP(x) r. Then we have (2.7) detP(x + //a;) = ff[c detPg(z) + O(fj,)] withc^O. Assume further that detPx(x) 7^ 0 then (2.8) detPj(^) ^ 0, (2.9) detPi(x + X9] = 0 =*• A 25 rea/ /or a// x e Rn .
- 29. Hyperbolic systems with nondegenerate characteristics 9 Proof: In view of (2.4) we can choose a non singular constant matrix T so that where G is a non singular (m —r}x(m —r] matrix. With P(x) = T~l P(x)T we write P(X + HX} = P(X) Denoting it is clear P±(x} = Pi(x] which follows from the definition. Since Px(z) = P$(x) we have (2.10) P*(x) = PU(X). Note that (2.11) detP(z + AW:) = detP(z + fjuc) = //[detG detPn(s) + O(/z)] which shows the first assertion. To prove the second assertion suppose that detPz(0) = 0 so that detP(x + //#) = o(/O by (2.7). Since detP(o:) is hyperbolic in the sense(2.2) it follows that [S£detP](z) = 0, V|a| < r. This implies detP5(x) = 0 which is a contradiction. We turn to the third assertion. Since detP(x + A*(X 4- A0)) = pfc detPs(x + A0) + O(A*)] if detPx(x + A0) = 0 has a non real root A, then taking p, ^ 0 sufficiently small the equation c detPfx + 0 + O * = 0 admits a non real root. This contradicts (2.2). We now generalize the notion of non degenerate characteristics for P(x) in (2.1) which is defined in [7], [8] for the case q = 1, making an obviousmodification. DEFINITION 2.1: We say that x is a non degenerate characteristic (of order r} of P(x) if the following conditions are verified. (2.12) KerP(x) n ImP(x) = {0}, (2.13) dimPi = r(r + l)/2, r = dimKerP(x), (2.14) detPx(^) ^ 0, Px(0)~1 Ps(x) is diagonalizable for every x € Rn . Simple characteristics verify (2.12)-(2.14) with r = 1 and hence non degenerate. Non degenerate double characteristics have a special feature:
- 30. 10 Nishitani Lemma 2.2 Assume that dimKerP(z) = 2 and KerP(x) n ImP(z) = {0}. Then a double characteristic x is non degenerate if and only if the rank of the Hessian of P(x) at x is maximal, that is 3. Proof: Assume that rankHesssP = 3. From Lemma 2.1 and the assumption it is clear that detP5(0) ^ 0. Hence, by Lemma 2.1again, Pi(^)-1 Pi(x) has only real eigenvalues for every x. From Lemma 4.1 in [6]there exists a constant 2 x 2 matrix T such that T~l (P£(0}~1 PX(X}}T is symmetric for every x so that one can write (2.15) T-i(Ps(0rl Pt(x}}T and obviously Px(0]~l Px(x] is diagonalizable for every x. Since the rank of the quadratic form det(Pi(^)~1 Pi(x)) is 3 and hence 0;(x), i — 1,2,3 are linearly in- dependent. Thus it is clear that dimP^ = 3. Conversely assume dimP^ = 3 and detPx(9) 7^ 0. Since P2(9}~1 PX(X] has only real eigenvalues for every x, (2.15) still holds. Then it follows that <& are linearly independent. Hence rankHess^P = 3 by (2.7) again. This proves the assertion. REMARK 2.1: Assume that q = 1 and AI(X'} is symmetric in (2.1). Then (2.12) and (2.14) are always verified. REMARK 2.2: By definition, the order of non degenerate characteristics never exceed m, the size of the matrix whatever q is. To study P(x] we consider the following mq x mq matrix valued function P(x) = xl 0 -/ 0 0 -/ A(x'] where / is the identity matrix of order m. It is clear that (2.16) detP(x) = detP(x). Then the condition (2.2) implies that all eigenvalues of A(x'} are real, equivalently (2.17) all eigenvalues of P(x] are real. Our aim in this section is to prove Proposition 2.1 Let x be a non degenerate characteristic of order r of P(x). Then x is also a non degenerate characteristic of order r for P(x) and vice versa.
- 31. Hyperbolic systems with nondegenerate characteristics 11 Proof: Assume that x is a non degenerate characteristic of order r of P(x). We first check (2.18) 7;— (x) (KerP(z)) = KerP(z). C/«Z/ J Let vi,...,tv be a basis for KerP(x) and take ^ so that ^(ImP(z)) = 0 and ti(vj) = 6ij. Then by definition we have Hence P.(*) Then detP5(^) ^ 0 implies that KerP(x) c CXi and hence the result. We note that KerP(z) = {t (u,x1u,...,xri u) u 6 KerP(z)} and dunKeiP(x} = r of course. We next describe lmP(x). Write 0fc(x) = £^(x>r"fc j=0 then it is easy to see that 9-1 lmP(x) = {t (w(l ...,w(<} -l P(x)v-^M^w(k} ) ^...V'-^t; € Rm }. fc=i We first show that (2.19) KerP(x] n lmP(x] = {0}. Let ^ be a linear form on Rm<? . Writing v - f (i;(1) , ...,v(g) ) € Rm<? one can write where ^^') are linear forms on Rm . Assume l(lmP(x}) = 0. This implies that (2.20) t(j ')=£q <i>j(x}-), l<j<q-l, Assume moreover £(Ker"P(x)) = 0 so that rVx^w +x?-1 ") = o.
- 32. 12 Nishitani Prom this, noting the identity •7-1 QD E 7 _ 1 , , , n—1 *J 1 , v zi ^j(x] +x l = jrr(x] 3 —1 * one gets 8P (2.22) ^(<?} (—-(x» = 0, VueKerP(x). Rrom (2.18) it follows that £((?) (KerP(x)) = 0. Since KerP(x) + ImP(x) = Rm then (2.20) shows that t^ = 0 and hence i = 0. This proves that (2.23) KerP(x) + ImP(x) = Rm? and hence (2.19). We next examine (2.13). (2.14) for P(x). Let U = t (u,xiu,...,x~l u] € KerP(x} where u € KerP(x). Consider P(x}U: P(x]U = *((xi —Xi)w, (xi —Xijiiw, ..., (x — Xi)xq l~"u, *) where the last component * is P(xl,xl }u+(x].xrl -x}u = P(o:)n + [P(xi, x')- P(XI, x'}}u + xq l~l (xl - x^u. Now it is easy to see that this is equal to (2.24) P(x}u - ]T <t>k(x)wW +0((xi _ £l)2}_ k=l Let i be a linear form on Rmq with £(ImP(z)) = 0. Prom (2.24) it Mows that 9-1 g-l t(P(x)U) = ; = 1 fc=l (2.25) = W(P(x}u) +0((Xl - by (2.20). Let us take Uj = i (uj ,XiUj,...,xq ^l Uj} € KerP(x) where {u-j} is a basis for KerP(x) and write dP r -£—(x)uj = ®Xl k=i Take ^2 so that £(ImP(x))= Let us take 9)
- 33. Hyperbolic systems with nondegenerate characteristics 13 SO that r r ^(|_(*X-) = E ^E *iP*kM =<V ax l k=l p=l We nowdefine linear forms ^(-) = '(^(^(x)-), ...^(^-i^)-)^-)) on Rm « then we have (2.26) 4(ImP(x)) = 0, as observed above. Prom (2.25) it follows that Since B = (f A)~1 = Px(0}~1 we conclude that (2.28) Pt(x) = P*(6rl P*(x). Since Ps(9) = I then (2.13) and (2.14) for Ps(x) follow immediately. Conversely assume (2.23). Let £M be a linear form on Rm such that fiq} (lmP(x)} = 0, ^(KerP(z)) = 0anddefine t^ l<j< q-l by(2.20). Then we have t(lmP(x)) = 0 and moreover (2.20) shows ^(KerP(x)) = 0 and hence t - 0 by (2.23). Thus wehave W = 0 which proves KerP(x) + ImP(x) = Rm and hence (2.12). To check (2.13), (2.14) for P(x) we note that Ker"P(x) n Im'P(x) = {0} implies that dP u € KerP(x), ^— (x]u € lmP(x) =^> u = 0. OX i Hence we have (2.18) again and thus (2.28). Then the rest of the proof is clear. 3 Stability of non degenerate characteristics In this section we discuss the stability of non degenerate characteristics under hyper- bolic perturbations. Theorem 3.1 Assume that P(x] is a m x m real matrix valued smooth function of the form (2. 1) verifying (2.2) in a neighborhood U of x and let x be a non degenerate characteristic of order r of P. Let P(x] be another m x m real matrix valued smooth function of the form (2.1) verifying (2.2) which is sufficiently close to P(x] in Cq+2 , then P(x} has a non degenerate characteristic of the same order close to x. Moreover, near x, the characteristics of order r are non degenerate and they form a smooth manifold of codimension r(r + l)/2. In particular, near x the set of characteristics of order r of P(x] itself consists of non degenerate ones which form a smooth manifold of codimension r(r + l)/2.
- 34. 14 Nishitani To prove Theorem 3.1,taking Proposition 2.1 into account, we study P(x) of the form (3.1) P(z) = x + P^(x'} where we assume that (3.2) detP(x) = 0 => X! is real near x' = x'. This is equivalent to say that all eigenvalues of P# (x') are real. We extend the stability result for P of the form (3.1) which is proved for real analytic P(x),P(x) in the case r = m in [7]and in [2]when x is a non degenerate double characteristic. Proposition 3.1 Assume that P(x) is a m x m real matrix valued smooth function verifying (3.2) and x is a non degenerate characteristic of order r of P(x]. Let P(x) be another mx m real matrix valued smooth function of the form (3.1) verifying (3.2) which is sufficiently close to P(x) in C2 . Then P(x) has a non degenerate characteristic of the same order close to x. Moreover, nearx, the characteristics of order r of P(x] are non degenerate and form a smooth manifold of codimension r(r + l)/2. In particular, the characteristics of order r of P(x) itself consists of non degenerate ones which form a smooth manifold of codimension r(r + l)/2. Proof: We first show that the proof is reduced to the case that P and P are r x r matrix valued function. Without restrictions we may assume that x = 0. As in the previous section, we take T so that one has T-i 1 * w " ~ ' 0 G where G is non singular. Denote T~l P(x]T and T~1 PT by P(x) and P(x) again. Writing Vz) PK(X] we have (3.3) PH(Z) = X! + £ AjXj + 0(|x|2 ) = P0(x) + 0( x2 ). From the assumption PO(X) is diagonalizable for every x and {/, A^,..., An} span a r(r -h l)/2 dimensional subspace in M(r;R). By Lemma 2.1 all eigenvalues of PO(X) are real then one can apply Theorem 3.4 in [9]and conclude that: there is a constant matrix S such that where Aj are symmetric and {I,Ai,..., An} span Ms (r; R), the space of all r x r real symmetric matrices. We still denote o-i n / c A / c-i A - / c 0 I P(x I I I I P(x) I 0 / ^(x) 0 / ' 0 I }n j 0 /
- 35. Hyperbolic systems with nondegenerate characteristics 15 by P(x) and P(x) again so that writing , M*) P22(*) we may assume that (3.4) Pn(x) = xJ +£AM +O(|x|2 ) 3=2 where (3.5) {/, Ai,...,An} spanMs (r;R). Let {Fi, F2,..., FJ, FI = / be a basis for Ms (r; R) where k = r(r + l)/2. Writing we make a linear change of coordinates Xj = £,-(x), j = 1,..., n so that denoting Xj 1 < j < k again and (xk+,...,xn) = (yi, ...,yj) wehave (*} fi} P,,(r 11} — > ^ FT- -4- n((r -4- ii2 } O.D) rii^x, yj — 2_j ^jXj -f v((x -f ij/i; j- Note that the coefficient of Xi in Pu(x,y) is the identity matrix /. We now prepare the next lemma. Lemma 3.3 Let P(x) be a m x m matrix valued C°° function defined near x = 0. With a blocking A2i we assume thatAH andA22 has no common eigenvalue. Then there is e = e(Au.A22) > 0 such that if A2i + ||-A12|| < e then one can find a smooth matrix T(x) defined in |x| < e such that T(x)~1 P(x)T(x) = where T(x) = I + Ti(x) and Ti(Q} -+ 0 as A21 + A12 -> 0. Proof: We first show that there are G2, G2i such that (3.7) An A12( I G12 ( I Gl2(An+Xu I2i A22 JG2l I j ~ { G21 I l 0 provided m12|| + ||A21|| is small. The equation (3.7) is written as + A (~* A f1 _1_ A / A ! V /~* A i /~* V SilZ^Zl -^11^12 ~r /Il2 | _ I -^11+ -All O-i2/l22 + Cj-i2A22 + ^422^-r 2i ^42iCT i2 + -A22 I G2Ai + G2iX A22 + X22
- 36. 16 Nishitani This gives A^G^i = A'n, AiG-2 = X22- Plugging these relations into the remaining two equations, we have A12 + AnGi2 = Gl2^22 + Gi2^42lGi2 .421 + Ay^Gii —GiA + G^iAi^G^l- Let us set then the equations become { j'-* Fi(Gi2, G2i, A12, A-2i) = 0 F (G Go A A ) = 0 It is well known that */r ^(o,0,0,0) is non singular iA and ^422 have no common eigenvalue. Then by the implicitfunction theorem there exist smooth Gi2(^4i2, ^21) an d Gi{A-&, AH) defined in li^ull + H^bi i < 6 with Gi2(0, 0) = 0, G2i(0,0) = 0 verifying (3.8). This proves the assertion. We look for T(x) in the form T(x) =To + T!(X), T0(x) = ( / °f } , T!(0) =0. ^21 1 ) The equation which is verified by T(x] is: (3.9) (P0 H- Pi(x))(T0 + T^x)) = (T0 + T where P0 = P(0), P0T0 = r0P0 and O p f ~ i"£lX Recall that ' Pn(x) _ ° " ' ~ P21(x) Look for TI (x) in the form 0 T12(x) >i(x) 0 Equating the off diagonal entries of both sides of (3.9) we get (v in - V^iVi • 72)P22(x) V'-) -lu ^ A T_ . i D. .(~ _L r> _(~r<_ _L ^ _ _ . ~ , L / S~V t HH T-> / , r-r- / A , — (Gii •+• -ti2)P22(x) + ^i2(A22 + A22T21 + P21(x) + P22(x)G21 + P22(x)T2: = (G2i+T21)P11(x)+r21(A11 +
- 37. Hyperbolic systems with nondegenerate characteristics 17 On the other hand, equating the diagonal entries of both sides we have 22(z) =A21T12 + P22(x) + P21(x)(G12 + T12). Plugging (3.11) into (3.10) we obtain Fi(Tl2, x) = AuTiz - T12(A22 +A2lGi2) +Pu(x)Gi2 +Pl2(x) +Pn(x)T12 -(Gi2 +T12}(A21T12 + P21(x)[G12 + T12] + P22(x)) = 0 and F2(T2i, z) = A22T21 - T2l(An + A12G2i) + P22(x)G21 + P21(x) + P22(x)T12 ~(G2l +T21)(A12r21 + Pi2(x)[G21 + T21] + Pn(x)) = 0. Since *i(Ti2, 0) = AnTu - T12A22, F2(T21, 0) = A22T21 - T21AU when A2i = 0, Ai2 = 0, x = 0, it is clear that are non singular if ||^4i2|| + ||A21|| is small. Then by the implicit function theorem there exist smooth Ti2(x) and T21(x) with T12(0) = 0, T2i(0) = 0 such that P!(T12(x),x) = 0, F2(T21(x),x) = 0. This proves the assertion. We return to the proof of Proposition 3.1. Since P(x,y) is sufficiently close to P(x, y] and P ( 0 '°) = (o G detG ^° one can apply Lemma 3.3 to P(x, y] and find G(x, y) such that (3.12) G(Xtvr*P( Denote G(x,y)-l P(x,y)G(x,y) and G(x,y)-1 P(x,t/)G(x,7/) by P(x,y) and P(x,y) again. We summarize our arguments in Proposition 3.2 Assume that P^g and P0ng verify the assumption in Proposition 3.1. Then we may assume that P^ig and Parig have the form ~ 0 P22(x,y) ' > - P21(x,y)
- 38. 18 with Nishitani - R ( x , y ) , R(xty)=0((x + M)2 ) 3=1 3=1 where the following properties are verified: for any neighborhood U of the origin there is a neighborhood W c U of the origin such that for any e > 0 one can find e > 0 so that if Parig — Pong c2 (u) < ? then we have (3.13) (3.14) | Moreover one has Pn(x,y) - I k y}. j=l det(A -f Pn(x, j/)) = 0 => A is real. Proof: Since P(x, y] and P(x, y) are obtained from Porig and P^ig by a smooth change of basis and a linear change of coordinates then (3.13) is clear. Let us recall which verifies (3.12) where ||G12(0,0)|| + |G2i(0,0)|| becomes as small as we please if 1 is small. Hence G(x,y} is enough close to the identity and then (3.14) follows from (3.6). Note that det(A + Pang] = det(A + Pn(x,y))det(A + Pv(x,y}}. Then the last assertion follows immediately. We proceed to the next step. Write (3.15) Pu(x,y) = Pn(0,7/) + (^(x^y^Kij^r- Let us denote where Lemma 3.4 Assume that Pu(x,y) —Pu(x.y)ci(w) < e ana {(x,y) x ,y < e} C W. Then for x , y < e we have
- 39. Hyperbolic systems with nondegenerate characteristics 19 Proof: Write Pn(x,y} = Pu(Q,y} + f^Aj(y)xj + R(x,y)J R(x,y] = O(x2 ) j=i so that (3.16) T= (*j(z,y)) = £A,-(yto - E^ +*(*>*)• j=i j=i Noting dx,Pn(0,y) = Aj(y), dx,Pu(0,y) = ^ + dx,fl(0,y) and dXiR(Q,y)<Cy<Ce if |y| < e with C independent of P, one gets (3.17) Aj(y}-Aj<C€ if|y|<e. Now it is clear that (3-18) l^.^-^I^C'e i f | y | < € because of (3.14) and (3.17). On the other hand from 3-1 and (3.14) it follows that |Pii(0,y)| < Ce|y| +Cy2 < Cey if |y| < e. Moreover |Pn(0,y) - P.(Q,y)c*(w) < eshows (3.19) |Pn(0,y)|<c + Ce|y|<C'e if |y| < e. We now estimate T(x,y) = (t}(a:,y)) and dX ] T(x,y}. Note that |5XjJR(x,y)| < C|x| since dXiR(Q,y) = 0 and |^(x,y)| < C for |a| = 2 with C independent of P. Then by (3.16) and (3.18) one sees (3.20) |r(x, y)| < Ce|x| + C|x|2 < C'ex if |x| < e, |0,,r(z,y)| < Ce 4- Cx < C'c if |x|, |y| < e which proves the assertion. Let us study the map where Ba = {x € Rfc | |x| < a}. Let A : R* 3 x .-» 0
- 40. 20 Nishitani which is a lineax transformation on R^, Since <b](x}} i > j are linearly independent, A is non singular. Prom Lemma 3.4 one can choose a > 0 so that A-l $'x(x, y)-I < 1/2 if x , y < a. Let us write Pu(Q,y} = (^(y)) and we apply the implicit function theorem to where 9 6 R*; y G R*. Then there exist a: > 0. a2 > 0 and a smooth g(y,0) defined in 0 6 Bai , y e -Ba2 such that Note that (3-22) pfo,0) if t/l, |0| < e because of (3.19). Set (3-23) (q(y,9)) = then from (3.15) and (3.21) it follows that (3-24) ^(?/, £) = #}, z > j . Let us write tf5(y,0) = <5(j/) + x}M) where cj(y) = ^-(y, 0) and Xj-(y, 9} = O(9}. We show that c£(y) = 0 for p < 5. This, together with (3.24), implies (3.25) ' Ai(0(2/,0),i/) = 0. Let us put /i(A) = det(X+Pu(g(y, 9},y}}. From Proposition 3.2 it follows that h(] = 0 implies Ais real. Take #* = 0 for i > j unless (i,j] = (q,p). Then one has If c^(y) 7^ 0 then h(X] =0 has a non real root for sufficiently small 9 because Xq(Ui 0) = 0. This is a contradiction and hence the assertion. Here we extend Proposition 3.1 in [7]. Let where FI = I and {Fi, ..., Fk] be a basis for M5 (m; R) and hence k = m(m + l)/2. Proposition 3.3 Assume that P(x] is a m x m real matrix valued smooth function defined in a neighborhood of the origin o/Rn . Assume that all eigenvalues of P(x] are real and k ftp (3.26) £sr(°to j=l UX 3 is sufficiently close to F(x}. Then there is a 5 > 0 such that P(x] is diagonalizable for every x with x < 6.
- 41. Hyperbolic systems with nondegenerate characteristics 21 Proof: It is enough to repeat the proof of Proposition 3.1 in [7]with a slight modifica- tion. Let us write P(u +x} = P(u} + Q(x,u} so that Q(0,o;) = 0. For T € O(m), an orthogonal matrix of order m we consider + x)T = 'TP(u;)T Denoting QT (x,u) = (0j-(x,a;;T))i<iij<m we show that there exist a 6 > 0 and a neighborhood W of the origin of Rfe such that with x = (xa, x6), xa = (xi, ...,xfc), is a diffeomorphism from W into {?/ € Rfc | |y| < 6} for every T € O(m) and every xj,, w with |x&|, |c«;| < 6. To see this we write = E F i*i + E F)P E j=k+l then it is clear that for any e > 0 one can find 6' > 0 such that (3.27) ||^(x,w)||<e|x| if |x|, |u;| < 5' and (3.26) is sufficiently close to F(x). Let us study where lj(xa; T} are linear in xa. Since O(m) C Rm is compact it is clear that we have '(*!,..,**),_. > c>0 with some c > 0 for every T e O(m). In view of (3.27), taking e > 0 so small we conclude that '^M %^(0,0,0;T) with some d > 0 for every T G O(m). By the implicit function theorem and the compactness of O(m) there exists a smooth xa(ya,X6,u;;T) defined in |j/a|, |xb|, |w| < <5 and T e O(m) such that for
- 42. 22 Nishitani where we have set ya — (yl j]i>j € RA This proves the assertion. We now show that P(^J] is diagonalizable for every u £ Rn with u; < 6. Take T € 0(m) so that (3-28) 1=1 where {Aj are different from each other and Aij are Ti x TJ matrices such that Aij = 0 if i > j and A^ are upper triangular with zero diagonals. Let us set s-l - U((*> P=I < rp} where TO = 0. As observed above one can take ((y,-)i>j, £&) as a new system of local coordinates around the origin of Rn . Denote yn = y ( and, putting y/; = 0, 0:5 = 0. consider det(A = det(A + PT (w + x)) = [ det(A , u; T)) where with sl — J, s0 = 0. Note that we have if p>q, Hence applying the same arguments as in the proof of Lemma 3.4 in [7]we get AH = 0. Then from Lemma 3.5 in [7]it follows that PT (u] is diagonalizable. Since a;, u < 5 is arbitrary we get the assertion. We now prove that near (0,0) the characteristics of order r of P(x, y] form a smooth manifold given by x ~ g(y, 0). Let (x, y} be a characteristic of order r of P(x, y) close to (0,0). Then it is clear that (x, y) is a characteristic of the same order for Pu(x,y] because detP22(x, y} ^ 0 near (0,0). Recalling that Pu(x,y] has the form , y ) , x* = (x2,...,xfc) we see that detPn(xi,x# ,y) = (xi —Xi)r and hence Thus the zero is an eigenvalue of order r of Pi(x,y}. On the other hand Proposition 3.2 gives (3.29) •(0) - F3 , -(0) < Ce.
- 43. Hyperbolic systems with nondegenerate characteristics 23 Then one can apply Proposition 3.3 to conclude that P(x, y) is diagonalizable. This shows that Pii(x,y) = 0 and hence 0j = 0, i > j. Then one gets x = g(y, 0). Finally we show that the characteristics (g(y, 0), y) are non degenerate. From (3.25) we have P(<?(y,0),y)=(° p and hence (3.30) KerP(<7(y, 0), y) n ImP(5(y, 0),y) = {0}. It is also clear that P($(y,o),y)(x,y) is given by On the other hand since |Pn —Pii|c2 (w) < e it follows from Proposition 3.3 and (3.22) that *Pn (3.31) if |7/| < e. This clearly shows that r(r +1) (3.32) dimP(5(y>0),y) = . To finish the proof, taking P(p(y,o),y) (#) = ^ into account, it is enough to show that P(g(y,o),y)(x,y} is diagonalizable for every (x, y). Note that from Lemma 2.1 all eigen- values of P($(y,o),y)(x, y) a16 real. Then from Proposition 3.3 and (3.31) it follows that P(g(y,o)<y)(x,y) is diagonalizable for every (x,y) near (0,0) and hence for all (x,y). 4 Well posedness of the Cauchy problem Let us study a system (4.1) P(x, D) = E ^a(x)DQ , ^ = -^~ N<« z 9x ^ where >lQ(x) are m x m matrix valued smooth function defined in a neighborhood fi of the origin of Rn . We assume that xi = const, are non characteristic and without restrictions we may assume that (4-2) A(g)0,...,0)(x) = / the identity matrix of order m. We are concernedwith the following Cauchyproblem: (4 3) f P(x, D}u= /, supp/ c {xi >0} V ' ; suppn C {xi > 0}
- 44. 24 Nishitani Let P«j(x,£) be the principal symbol of P(x, D): a=q and we assume that (4.4) detP?(x,£) = 0 =Ki is real Vx 6 n,V£' - (&,...,£„) € R^1 . We prove the following result which extends those in [2], [4]. Theorem 4.1 Assume that every characteristic over (0,f), )£' = 1 of Pq(x,£) is at most double and non degenerate. Then the Cauchy problem for P(x, D] is C°° well posed near the origin for arbitrary lower order terms. Moreover if P(x, D} is another system of the form (4-1) verifying (4-4) with ^ne principal symbol Pq(x,£) = Ho=q AQ(x)£a of which AQ(x) are sufficiently close to Aa(x] in C2 (Q) for a = q then the Cauchy problem for P(x, D) is C°° well posed near the origin for arbitrary lower order terms. Assuming the analyticity of the coefficients we have Theorem 4.2 Assume that Aa(x), a —q are real analytic in fi and every character- istic of Pq(x,£) over (£,£'), 1 is non degenerate. Then the Cauchy problem for P(x, D) is C°° well posed near the origin for arbitrary lower order terms. The proof is very simple. We reduce the Cauchy problem for P(x, D} to that for a first order system "P(x, D). Taking the invariance of non degeneracy of characteristics proved in Proposition 3.1, it suffices to apply the previous results in [6] and [7]which assert the existence of a smooth symmetrizer <S(x, D'} for P(x, D) defined near the origin. Let us write (4.5) P(x, D}u = Dq lU + E A J(X ' D'}Dr°u = f. J=2 Put where (D'}2 = 1+ £"=2 D]. Then (4.5) is reduced to ^U + where U = 0 -/ -I Af(x,D') (D'}U = P , P = f (0, ..., 0,/) and
- 45. Hyperbolic systems with nondegenerate characteristics 25 Let us denote by A°(z,£') the principal symbol of Af(x,E>'} and set 0 -/ 0 0 -/ -/ (4.6) Fix (0,1"'), If I = 1- Let (0, A,,!"'), z = 1, ...,p, be characteristics of & + -4(z,f ) where (0, AJ,|> ) are at most double and non degenerate and A, are different from each other. Then there exists a smooth T(x, £') defined near (0,£'), homogeneous of degree 0, such that T(z, rrU(z,r)r(x,O = -4i(x,O e • • • e A>(^ f) where (0,Aj,f') is a non degenerate characteristic of "Pw (rr,£) = fi + -^(x, £')._From Proposition 3.1 it follows that the characteristic set of defP^ (z, £) near (0,At,<f') is a smooth manifold of codimension 3 through (0, Ai,£'). Then one can apply Lemma 3.1 in [6]to get a smooth <Sw (z, £') defined near (0,£'), homogeneous of degree 0 such that is symmetric. This proves that A(x,£') is smoothly symmetrizable near (0,1"') by <S(1) (z, f) 0 • •• © «S(p) (z, £'). By the usual argument of partition of unity one can prove that there is a smooth <S(z, £') which symmetrizes ,A(x, £')• Thus the Cauchy problem for P(x, D) is C°° well posed for arbitrary lower order terms and hence so is forP(x,p). Let Aa(x) be sufficiently close to Aa(x] for |a| = q in C2 (f2). Let ^4(x, ^') be defined by (4.6) from P9(x,£'). Then it is clear that A(x,g) is sufficiently close to -4(x,f') in C2 (fi x {1/2 < |^'| < 2}). To prove the last assertion it suffices to show that every characteristic^ P(x, £) =£1+ A(x, £') over (0,f') is non degenerate and at most double. Let (0, A, f ) be a characteristic of 7>(x, £)_ = Ci + ^(^, CO- If (0,A, f') is simple then the characteristic of P(x, £} close to (0,A,f ) is also simple. If (0, A, f) is a non degenerate double characteristic, then from Proposition 3.1 it follows that the characteristic of P(x,^} near (0, A,^') is simple except in a manifold of codimension 3 consisting of non degenerate double characteristics. This proves that the characteristic of P(x, £) close to (0, A, £') is non degenerate and at most double. This completes the proof of Theorem 4.1. To prove Theorem 4.2, it suffices to apply Theorem 1.1 in [7]: Proposition 4.1 ([7]) Assume that A(x,^'} is a m x m matrix valued real analytic function defined near (0,f ) and (0,A, f ) is a non degenerate characteristic for £j + .A(x,£') of order m. Then there exists a real analytic <S(x,£') defined near (0,<f') such that is symmetric.
- 46. 26 Nishitani 5 A remark As proved in Theorem 3.1, if 3=0 has a non degenerate multiple characteristic (x, £) then near P there is no strictly hyperbolic system of the form (5.1). When q = 1 more detailed facts are known (see [5], [1]). Let P be a first order (q = I) system with constant coefficients: (5.2) P($=ti-EA& j=2 where Aj are m x m constant matrices. We always assume that (5.3) all eigenvalues of Z^=2 AJ£J are real for every £' = (£2, -•-, £n)- Then from [5] P(£) can not be strictly hyperbohc if n > 3 and m = 2 modulo 4. Contrary to this if q —2 it is clear that for any n and m there exist strictly hyperbolic systems. In this section we study 2 x 2 first order systems with constant coefficients: from the result refered above if n > 3 then P(£) can not be strictly hyperbolic. On the other hand we have Proposition 5.1 Let m = 2 and P(£) verify (5.3). If n < 3 then P(£) can beap- proximated by strictly hyperbolic systems: there are strictly hyperbolic P€(£) of the form (5.2) converging to P(£) as e —* 0. Proof: Since the assertion is clear for n — 2 we may assume that n = 3. We first consider the case that P(£) is diagonalizable for every £. From Lemma 4.1 in [6]there exists a constant matrix T such that is symmetric for every £. Take a basis for Ms (2; R) and write with f ' = (^2, If 02 and 03 are linearly independent then P(£) itself is strictly hyperbolic because
- 47. Hyperbolic systems with nondegenerate characteristics 27 If 02 = 03 = 0 then taking ^2(£') and ^s(£') which are linearly independent we define P as Assume that 02 7^ 0 and 0s = A;02 with some constant k ^ 0. Take ^(f) so that 02 and -03 are linearly independent and define Pe(£) by Then one has and hence P€(£) is strictly hyperbolic which converges to P(f ) as e —> 0. If A: = 0 then it suffices to take P€ so that T~l PeT = & - &)/ - 02A - €^3B- The proof for the remaining case is just a repetition. We turn to the case that P(£) is not diagonalizable for some f = u, u ^ 0. Choose a constant matrix T so that With T-1 P(£)T - (£jj(0) we snow tnat ^n(C). ^22(0 and ^i(C) are linearly dependent. Suppose this were not true then we could choose £ so that ^n(£) = ^22(0> 1+^12(0 > 0 and £2i(C) < 0- This shows that T~1 P(u + £)T = r~1 P(a;)T + (£y(0) would have a non real eigenvalue which contradicts (5.3). Then one has (5-4) a^n(r) We first assume 7 7^ 0 so that with some a ^ 0 because ^21 contains no £1. Let us consider o W £n £12 W i o -a 1 £21 ^22 a 1 a , , -0^12 -^22 - ^1 Note that ^12 ^ 0 for (TS^P^TS is not diagonalizable at u. Let us write where (5-5)
- 48. 28 Nishitani by (5.3). Note that 0 ^ 0 because l2 ^ 0. Then one can write i-& — k<£> with some k because of (5.5) and hence where 1- a2 k2 > 0 by (5.5).If 1 k (5 '6) - -a't -1 Ct A/ J_ is diagonalizable then so is P(£) for every £ contradicting the assumption. If (5.6) is not diagonalizable and necessarily 1 —a2 /c2 = 0 one can take U so that r,_i / i K TT t/ I 2- , 1 U I k / O 1 a2jfe _i J^ - ^ o o;• Take 7/> so that ^ and 0 are linearly independent and define P£(£) by (5.7) It is clear that P€(^) is strictly hyperbolic and converges to P(£) as e —> 0 because If 7 = 0 then we have £n = i^i from (5.4). Hence one can write Since ^12(^)^21(^') > 0 by (5.3), assuming £12 7^ 0 we can write with some constant A; > 0 so that If k > 0 then P(^) is diagonalizable for every £ and contradicts the assumption. If k = 0 it suffices to define P£(£) by the right-hand side of (5.7) with t = £n and 0 = ^12- The proof for the case £2i 7^ 0 is just a repetition. Thus we complete the proof. References [1] S.FRIEDLAND, J.ROBBIN AND J.SYLVESTER, On the crossing rule, Comm. Pure Appl. Math. 37 (1984), 19-37.
- 49. Hyperbolic systems with nondegenerate characteristics 29 [2] L.HORMANDER, Hyperbolic systems with double characteristics, Comm. Pure Appl. Math. 46 (1993), 261-301. [3] F.JOHN, Algebraic conditions for hyperbolicity of systems of partial differential equations, Comm. Pure Appl. Math. 31 (1978), 89-106. [4] F.JOHN, Addendum to : Algebraic conditions for hyperbolicity of systems of partial differential equations, Comm. Pure Appl. Math. 31 (1978), 787-793. [5] P.D.LAX, The multiplicity of eigenvalues, Bull. Amer. Math. Soc. 6 (1982), 213- 214. [6] T.NlSHITANl, On strong hyperbolicity of systems, in Research Notes in Mathe- matics, 158, pp. 102-114, 1987. [7] T.NlSHITANl, Symmetrization of hyperbolic systems with non degenerate charac- teristics, J. Func. Anal. 132 (1995), 92-120. [8] T.NlSHITANl, Stability of symmetric systems under hyperbolic perturbations, Hokkaido Math. J. 26 (1997), 509-527. [9] T.NlSHITANl, Symmetrization of hyperbolic systems with real constant coefficients, Scuola Norm. Sup. Pisa 21 (1994), 97-130.
- 51. The Cauchy problemfor hyperbolicoperators dominated by the timefunction Kunihiko Kajitani Institute of Mathematics,University of Tsukuba, Tsukuba, Ibaraki 305,Japan In Honor of Jean Vaillant 0 Introduction This note is devoted to the C*30 well posedness of the Cauchy problem for a hyperbolic operator which are a generalization of the effectively hyperbolic operator. The result in this note is a joint work with S. Wakabayashi and K. Yagdjian and the detail of the proof will be appeared in [8]. Let j+a<m be a partial differential operator with smooth coefficients a^a(t,x) € Z?°°(Rn+1 ). We consider the Cauchy problem for P P(t,x,Dt,Ds)u(t,x) =/(*,*), i€[0,T], z 6 R n , . } D}u(Q,x) =Uj(x], j = 0,...,m-l, x 6 Rra . ( } For the principal symbol Pm(t,x, A, £) of the operator P defined by j+a=m we assume that for all t e R, x € Rn , A€ R, ^ € Rn , the following representation with the real-valued functions j(t, x,£), j ~ 1, . ..,m, and with an integer d > 2 and a nonnegative Lipschitz continuous function y>(t,x) and ipo(t,x) € B1 (Rn+1 ) satisfying dt<Po 7^ 0 if (/?o = 0, k, (0.2) 31
- 52. 32 Kajitani holds and v(t,x} does not vanish outside a compact set in FP+l . Thus the operator P(t, x,Dt, Dx] is a hyperbolic operator with the characteristics Xj(t, x,f), j = 1,..., m, of variable multiplicity in a compact set and strictly hyperbolic outside a compactset. It is well-known that the lower order terms ofthe operators with multiple characteristics play crucial role in the well-posedness of the Cauchyproblem (see, for example, [2], [9]). Therefore we make an assumption dtd%ajt0t(t,x} < Ck£<f>(t,x)d (3 --'XTn -J ')-fc -l/3 !. j + a=s, 1 < s < m, (0.3) for (t,x) € Rn+1 , where y>(t,x) is given in (0.2). These kind of conditions for the coefficients with 5 < m —1 are called Levi conditions. To describe a propagation phenomena in the Cauchy problem we denote A_ '^= sup A'(t x £) and define a hyperbolic cone of symbol Pm by T := { (A,£) € R"+1 ; A> A^KI } (0.4) while F* is dual cone of F that is r :={(t,x] 6 Rn+1 ; U 4- 1 • £ > 0 foraU (A,^) € T } (0.5) and is called a propagation cone of symbol Pm. For the Cauchy problem (0.1)we can prove the following theorem. Theorem 0. 1 Assume (0.2) and (0.3). TTien /or any Uj € #°°(Rn ) and for any f € C°°([0,r];Jff00 (Rn )) there exists a unique solution u € C°°([0,r];^00 (Rn )) o/ t/ie Cauchy problem (0.1). For t/ie support of the solution the following formula holds U *+ (o,y)lj U f f o y ) , (0-6) m-l Do = U supp Ui , fio = supp /, K+(r, y] = (r, y) + T* . (0.7) t=0 Thus according to this theorem the solution propagates along the propagation cone. Example 0. 2 The second-order operator P = JD? - (fc(t, x)24 satisfies the conditions (0.2), (0.3), if we take fa = x(^~£f^} and 02 € B( R2 ), where X € C°°(R) satisfing x(t) = t, t < 1 and x(^) = ±2, ±£ > 2 ,and (^ = ($ + <pi)5 does not vanish outside a compact set and PQ = fa. See [3] for more general hyperbplic operators with double characteristics.
- 53. Hyperbolic operators dominated by the time function 33 Example 0. 3 For the operator P = P3 + PI + PI, Ps = Ttj+a=saj,a(t)X: '^a , s = 1, 2, 3, with the principal symbol the condition (0.3) will be satisfied if oo.aMHEa^f-^"1 , N = 2, .7=1 ai,a(t, x) = £M^'1 , <*>,«(*, *) = £ c^-J $-2 , |a| = 1 , 3=1 3=2 where $,&i are the same as in Example 0.2 and a;-, bjCj are in BCC (R2 }. The microlocal version of Theorem 0.1 will be given in [7]. For the operators with the multiple characteristics a microlocal energy method developed in [4], [5]; allows to prove the well-posedness and gives a complete pictureofthe propagation ofsingularities. When wecan take tpo = t, the case that d in (0.2) depends on j is treated in [8]. When d = 1, Theorem is already proved in [6]. Therefore we consider here the case of d > 2. We shall describe the outline of the proof of Theorem 0.1 in the next section. 1 The outline of the proof In this section we shall give the outline of proof of an a priori estimate for the Cauchy problem (0.1) under the assumptions (0.2) and (0.3). The proof will be done in the following seven steps. Step 1: Reduction to the problem in usual Sobolev spaces The equation P(t, x, Dt, Dx}u(t,x) = f(t, x} obtained from (0.1) one can replace by the equation P7(t, x, Dt, Dx)u^(t, x) = /7(i, x) , /7(t, x] := e-^f(t, x) , (1.1) for new unknown function u^ (t,x) = e~^t u(t,x], where /7 € H°°(Rn+i }, while P7(t, x,A, Dx)v = P(t, Dt - fy, Dx)v = e~^P(t, Dt, Dx}(e^v] . (1.2) If we define v by ,x) := eA (t,x,Dx}v(t,x) = with weight function A(t,x,f) € C°°(R2n+1 ) denned by (1.8) below then the function v(t,x) satisfies P^A(t,x,Dt,Dx)v(t,x) = /7>A(U), A,A(*,*) :=eA (i,x,JDI)-1 /7(i,x), (1.3)
- 54. 34 Kajitani where P7iA(i, z, Dt, Dx) := eA (£, x, Dx}~l P(t, x, Dt - ry, D,)eA (t, or, I?) (1.4) and the inverse eA (t, x, Dx}~1 is assured by the ellipticity of the symbol eA(tjX ^. Our aim is to obtain a priori estimate for the solutions of the differential equation (1.3): ibl!^(R^i)<-^i|P7,A(^^A,^xX-)||^(R^i) for all v e H°°(Rn+l } . (1.5) v / This estimate holds for all 7 > 70, where 70 will be chosen and fixed, while C is inde- pendent of 7. Since P(t,x,Dt,Dx) = e^eA (t,x,Dx}P^(t,x,DtlDx}e-^lt eA (t1x,Dx)-1 1 from the last inequality we obtain ^^} (1.6) +l Let d > 2 be an integer given in (0.2) and ipi functions given in (0.9) and define 1 . 2 8 = -±-f <0, = V + l^i2 (1-7) (1-8) (1.9) x AQ(S, y, (1.12) where /i, 7, Af 6 R+ are parameters. The metric gt^,s(s, y,77, C) is a Riemann metric in T*(R7H "1 ) and slowly varying in the sense of Definition 18.4.1 [1] that follows from the fact that u(t,x,£) and r(t,x,£) are weights with respect to g. The constant M counts the loss of regularity, while 7 controls the support of solution and reflect the finite propagation speed. Step 2: Leray- Carding's method In this step we prove estimate (1.5). Denote HT£ = rd + £j=iCjC^ and (f, C) = - Set Q(i '*'A 'c) := (H *Pm}(t ' XjA ~''(7+Af) 'e "'Ai)for thesymbo1of the separating operator. To prove estimate (1.5) it is enough to show that there exists a constant CQ > 0 such that an estimate Im (P7^v, Qv)ff.(R«-n) > C0||v||^(R^i} (1-13) holds for all v € ^(R""1 ). Wenote Im (P^v, Qv)H,(nn+i} = — {(Q*P7,Av, u)H.(R-n) - (P7^Qv, v)H»(R-n)} • (1-14)
- 55. Other documents randomly have different content
- 56. Mr. Specter. Would you state your age, sir? Dr. Perry. 34. Mr. Specter. What is your profession? Dr. Perry. I am a physician and surgeon. Mr. Specter. Were you duly licensed to practice medicine by the State of Texas? Dr. Perry. Yes. Mr. Specter. Would you outline briefly your educational background, please? Dr. Perry. After graduation from Plano High School in 1947, I attended the University of Texas and was duly graduated there in January of 1951 with a degree of Bachelor of Arts. I subsequently graduated from the University of Texas Southwestern Medical School in 1956 with a degree of Doctor of Medicine. I served an internship of 12 months at Letterman Hospital in San Francisco, and after 2 more years in the Air Force I returned to Parkland for a 4-year residency in general surgery. I completed that in—— Mr. Dulles. Where did you serve in the Air Force, by the way? Dr. Perry. I was in Spokane, Wash., Geiger Field. At the completion of my surgery residency in June of 1962, I was appointed an instructor in surgery at the Southwestern Medical School. But in September 1962, I returned to the University of California at San Francisco to spend a year in vascular surgery. During that time, I took and passed my boards for the certification for the American Board of Surgery. I returned to Parkland Hospital and Southwestern in September of 1963, was appointed an assistant professor of surgery, attending
- 57. surgeon and vascular consultant for Parkland Hospital and John Smith Hospital in Fort Worth. Mr. Specter. What experience have you had, Dr. Perry, if any, in gunshot wounds? Dr. Perry. During my period in medical school and my residency, I have seen a large number, from 150 to 200. Mr. Specter. What were your duties at Parkland Memorial Hospital, if any, on November 22d, 1963? Dr. Perry. On that day I had come over from the medical school for the usual 1 o'clock rounds with the residents, and Dr. Ronald Jones and I, he being chief surgical resident, were having dinner in the main dining room there in the hospital. Mr. Specter. Will you describe how you happened to be called in to render assistance to President Kennedy? Dr. Perry. Somewhere around 12:30, and I cannot give you the time accurately since I did not look at my watch in that particular instant, an emergency page was put in for Dr. Tom Shires, who is chief of the emergency surgical service in Parkland. I knew he was in Galveston attending a meeting and giving a paper, and I asked Dr. Jones to pick up the page to see if he or I could be of assistance. The Chairman. Doctor, at this time I must leave for a session at the Supreme Court, and the hearing will continue. Congressman Ford, I am going to ask you if you will preside in my absence. If you are obliged to go to the Congress, Commissioner Dulles will preside, and I will be available as soon as the Court session is over to be here with you. (At this point, Mr. Warren withdrew from the hearing room.) Representative Ford. Will you proceed, please? Mr. Specter. What action did you take after learning of the emergency call, Dr. Perry?
- 58. Dr. Perry. The emergency room is one flight of stairs down from the main dining cafeteria, so Dr. Jones and I went immediately to the emergency room to render what assistance we could. Representative Ford. May I ask this: In the confirmation of the page call, was it told to you that the President was the patient involved? Dr. Perry. It was told to Dr. Jones, who picked up the page, that President Kennedy had been shot and was being brought to Parkland. We went down immediately to the emergency room to await his arrival. However, he was there when we reached it. Mr. Specter. Who else was present at the time you arrived on the scene with the President? Dr. Perry. When Dr. Jones and I entered the emergency room, the place was filled with people, most of them officers and, apparently, attendants to the Presidential procession. Dr. Carrico was in attendance with the President in trauma room No. 1 when I walked in. There were several other people there. Mrs. Kennedy was there with some gentleman whom I didn't know. I have the impression there was another physician in the room, but I cannot recall at this time who it was. There were several nurses there. Mr. Specter. Were any other doctors present besides Dr. Carrico? Dr. Perry. I think there was another doctor present, but I don't know who it was, I don't recall. Mr. Dulles. Can I ask a question here, Mr. Specter? Mr. Specter. Certainly. Mr. Dulles. What is the procedure for somebody taking command in a situation of this kind? Who takes over and who says who should do what? I realize it is an emergency situation. Maybe that is an improper question. Dr. Perry. No, sir. Mr. Dulles. But it would be very helpful to me——
- 59. Dr. Perry. No, sir; it is perfectly proper. Mr. Dulles. In reviewing the situation to see how you acted. In a military situation, you have somebody who takes command. Dr. Perry. We do, too. And it essentially is based on the same kind of thing. Mr. Dulles. I would like to hear about that. If it doesn't fit in here—— Mr. Specter. It is fine. Dr. Perry. It is based on rank and experience, essentially. For example, Dr. Carrico being the senior surgical resident in the area, at the time President Kennedy was brought in to the emergency suite, would have done what we felt was necessary and would have assumed control of the situation being as there were interns and probably medical students around the area, but being senior would take it. This, of course, catapulted me into this because I was the senior attending staff man when I arrived and at that time Dr. Carrico has noted I took over direction of the care since I was senior of all the people there and being as we are surgeons, the department of surgery operates that portion of the emergency room and directs the care of the patients. Mr. Dulles. Did you try to clear the room of unnecessary people? Dr. Perry. This was done, not by me, but by the nurse supervisor, I assume, but several of the people were asked to leave the room. Generally, this is not necessary. In an instance such as this, it is a little more difficult, as you can understand. Mr. Dulles. Yes. Dr. Perry. But this care of an acutely injured and acutely injured patients goes on quite rapidly. Over 90,000 a year go through that emergency room, and, as a result, people are well trained in the performance of their duties. There is generally no problem in asking
- 60. anyone to leave the room because everyone is quite busy and they know what they have to do and are proceeding to do it. Mr. Dulles. Thank you very much. Mr. Specter. Upon your arrival in the room, where President Kennedy was situated, what did you observe as to his condition? Dr. Perry. At the time I entered the door, Dr. Carrico was attending him. He was attaching the Bennett apparatus to an endotracheal tube in place to assist his respiration. The President was lying supine on the carriage, underneath the overhead lamp. His shirt, coat, had been removed. There was a sheet over his lower extremities and the lower portion of his trunk. He was unresponsive. There was no evidence of voluntary motion. His eyes were open, deviated up and outward, and the pupils were dilated and fixed. I did not detect a heart beat and was told there was no blood pressure obtainable. He was, however, having ineffective spasmodic respiratory efforts. There was blood on the carriage. Mr. Dulles. What does that mean to the amateur, to the unprofessional? Dr. Perry. Short, rather jerky contractions of his chest and diaphragm, pulling for air. Mr. Dulles. I see. Mr. Specter. Were those respiratory efforts on his part alone or was he being aided in his breathing at that time? Dr. Perry. He had just attached the machine and at this point it was not turned on. He was attempting to breathe. Mr. Specter. So that those efforts were being made at that juncture at least without mechanical aid?
- 61. Dr. Perry. Those were spontaneous efforts on the part of the President. Mr. Specter. Will you continue, then, Dr. Perry, as to what you observed of his condition? Dr. Perry. Yes, there was blood noted on the carriage and a large avulsive wound on the right posterior cranium. I cannot state the size, I did not examine it at all. I just noted the presence of lacerated brain tissue. In the lower part of the neck below the Adams apple was a small, roughly circular wound of perhaps 5 mm. in diameter from which blood was exuding slowly. I did not see any other wounds. I examined the chest briefly, and from the anterior portion did not see anything. I pushed up the brace on the left side very briefly to feel for his femoral pulse, but did not obtain any. I did no further examination because it was obvious that if any treatment were to be carried out with any success a secure effective airway must be obtained immediately. I asked Dr. Carrico if the wound on the neck was actually a wound or had he begun a tracheotomy and he replied in the negative, that it was a wound, and at that point—— Mr. Dulles. I am a little confused, I thought Dr. Carrico was absent. That was an earlier period. Dr. Perry. No, sir; he was present. Mr. Dulles. He was present? Dr. Perry. Yes; he was present when I walked in the room and, at that point, I asked someone to secure a tracheotomy tray but there was one already there. Apparently Dr. Carrico had already asked them to set up the tray.
- 62. Mr. Specter. Dr. Perry, backtracking just a bit from the context of the answer which you have just given, would you describe the quantity of blood which you observed on the carriage when you first came into the room where the President was located? Dr. Perry. Mr. Specter, this is an extremely difficult thing. The estimation of blood when it is either on the floor or on drapes or bandages is grossly inaccurate in almost every instance. As you know, many hospitals have studied this extensively to try to determine whether they were able to do it with any accuracy but they cannot. I can just tell you there was considerable blood present on the carriage and some on his head and some on the floor but how much, I would hesitate to estimate. Several hundred CC's would be the closest I could get but it could be from 200 to 1,500 and I know by experience you cannot estimate it more accurately. Mr. Specter. Would you characterize it as a very substantial or minor blood loss? Dr. Perry. A substantial blood loss. Mr. Specter. Now, you mentioned the President's brace. Could you describe that as specifically as possible? Dr. Perry. No, sir; I did not examine it. I noted its presence only in an effort to reach the femoral pulse and I pushed it up just slightly so that I might palpate for the femoral pulse, I did no more examination. Mr. Specter. In the course of seeking the femoral pulse, did you observe or note an Ace bandage? Dr. Perry. Yes, sir. Mr. Specter. In the brace area? Dr. Perry. Yes, sir. It was my impression, I saw a portion of an Ace Bandage, an elastic supporting bandage on the right thigh. I did not examine it at all but I just noted its presence.
- 63. Mr. Specter. Did the Ace Bandage cover any portion of the President's body that you were able to observe in addition to the right thigh? Dr. Perry. No, sir; I did not go any further. I just noted its presence right there at the junction at the hip. It could have been on the lower trunk or the upper thigh, I don't know. I didn't care any further. Mr. Specter. Would you continue to describe the resuscitative efforts that were undertaken at that time? Dr. Perry. At the beginning I had removed my coat and watch as I entered the room and dropped it off in the corner, and as I was talking to Dr. Carrico in regard to the neck wound, I glanced cursorily at the head wound and noted its severe character, and then proceeded with the tracheotomy after donning a pair of gloves. I asked that someone call Dr. Kemp Clark, of neurosurgery, Dr. Robert McClelland, Dr. Charles Baxter, assistant professors of surgery, to come and assist. There were several other people in the room by this time, none of which I can identify. I then began the tracheotomy making a transverse incision right through the wound in the neck. Mr. Specter. Why did you elect to make the tracheotomy incision through the wound in the neck, Dr. Perry? Dr. Perry. The area of the wound, as pointed out to you in the lower third of the neck anteriorly is customarily the spot one would electively perform the tracheotomy. This is one of the safest and easiest spots to reach the trachea. In addition the presence of the wound indicated to me there was possibly an underlaying wound to the neck muscles in the neck, the carotid artery or the jugular vein. If you are going to control these it is necessary that the incision be as low, that is toward the heart or lungs as the wound if you are going to obtain adequate control. Therefore, for expediency's sake I went directly to that level to obtain control of the airway.
- 64. Mr. Specter. Would you describe, in a general way and in lay terms, the purpose for the tracheotomy at that time? Dr. Perry. Dr. Carrico had very judicially placed an endotracheal tube but unfortunately due to the injury to the trachea, the cuff which is an inflatable balloon on the endotracheal tube was not below the tracheal injury and thus he could not secure the adequate airway that you would require to maintain respiration. (At this point, Mr. McCloy entered the hearing room.) Mr. Specter. Dr. Perry, you mentioned an injury to the trachea. Will you describe that as precisely as you can, please? Dr. Perry. Yes. Once the transverse incision through the skin and subcutaneous tissues was made, it was necessary to separate the strap muscles covering the anterior muscles of the windpipe and thyroid. At that point the trachea was noted to be deviated slightly to the left and I found it necessary to sever the exterior strap muscles on the other side to reach the trachea. I noticed a small ragged laceration of the trachea on the anterior lateral right side. I could see the endotracheal tube which had been placed by Dr. Carrico in the wound, but there was evidence of air and blood around the tube because I noted the cuff was just above the injury to the trachea. Mr. Specter. Will you now proceed to describe what efforts you made to save the President's life? Dr. Perry. At this point, I had entered the neck, and Dr. Baxter and Dr. McClelland arrived shortly thereafter. I cannot describe with accuracy their exact arrival. I only know I looked up and saw Dr. Baxter as I began the tracheotomy and he took a pair of gloves to assist me. Dr. McClelland's presence was known to me at the time he picked up an instrument and said, "Here, I will hand it to you."
- 65. At that point I was down in the trachea. Once the trachea had been exposed I took the knife and incised the windpipe at the point of the bullet injury. And asked that the endotracheal tube previously placed by Dr. Carrico be withdrawn slightly so I could insert a tracheotomy tube at this level. This was effected and attached to an anesthesia machine which had been brought down by Dr. Jenkins and Dr. Giesecke for better control of circulation. I noticed there was free air and blood in the right mediastinum and although I could not see any evidence, myself any evidence, of it in the pleura of the lung the presence of this blood in this area could be indicative of the underlying condition. I asked someone to put in a chest tube to allow sealed drainage of any blood or air which might be accumulated in the right hemothorax. This occurred while I was doing the tracheotomy. I did not know at the time when I inserted the tube but I was informed subsequently that Dr. Paul Peters, assistant professor of urology, and Dr. Charles Baxter, previously noted in this record, inserted the chest tube and attached it to underwater seal or drainage of the right pneumothorax. Mr. Dulles. How long did this tracheotomy take, approximately? Dr. Perry. I don't know that for sure, Mr. Dulles. However, I have —a matter of 3 to 5 minutes, perhaps even less. This was very—I didn't look at the watch, I have done them at those speeds and faster when I have had to. So I would estimate that. At this point also Dr. Carrico, having previously attached and assisting with the attaching of the anesthesia machine was doing another cut down on the right leg; Dr. Ronald Jones was doing an additional cut down, venous section on the left arm for the insertion of plastic cannula into veins so one may rapidly and effectively infuse blood and fluids. These were being done. It is to Dr. Carrico's credit, I think he ordered the hydrocortisone for the President having known he suffered from adrenal
- 66. insufficiency and in this particular instance being quite busy he had the presence of mind to recall this and order what could have been a lifesaving measure, I think. Mr. Specter. Would you identify who Dr. Baxter is? Dr. Perry. Yes. Dr. Charles Baxter is, when I noted when I asked for the call, is an assistant professor of surgery also and Dr. McClelland. Mr. Specter. And is Dr. McClelland occupying a similar position at Parkland Memorial Hospital as Dr. Baxter? Dr. Perry. That is correct. Mr. Specter. Would you identify Dr. Jenkins? Dr. Perry. Dr. M. T. Jenkins is professor and chairman of the department of anesthesiology and chief of the anesthesia service, and Dr. Giesecke is assistant professor of anesthesiology at Parkland. Mr. Specter. Have you now identified all of the medical personnel whom you can recollect who were present at the time the aid was being rendered to the President? Dr. Perry. No, sir; several other people entered the room. I recall seeing Dr. Bashour who is an associate professor of medicine and chief of the cardiology section at Parkland. Dr. Don W. Seldin, who is professor and chairman of the department of medicine, and I previously mentioned Dr. Paul Peters, assistant professor of urology, and I believe that Dr. Jackie Hunt of the department of anesthesiology was also there, and there were other people, I cannot identify them, several nurses and several others. Mr. Specter. Dr. William Kemp Clark arrived at about that time? Dr. Perry. Dr. Clark's arrival was first noted to me after the completion of the tracheotomy, and at this point, the cardiotachyscope had been attached to Mr. Kennedy to detect any electrical activity and although I did not note any, being occupied, it
- 67. was related to me there was initially evidence of a spontaneous electrical activity in the President's heart. However, at the completion of the tracheotomy and the institution of the sealed tube drainage of the chest, Dr. Clark and I began external cardiac massage. This was monitored by Dr. Jenkins and Dr. Giesecke who informed us we were obtaining a satisfactory carotid pulse in the neck, and someone whose name I do not know at this time, said they could also feel a femoral pulse in the leg. We continued external cardiac massage, I continued it as Dr. Clark examined the head wound and observed the cardiotachyscope. The exact time interval that this took I cannot tell you. I continued it until Dr. Jenkins and Dr. Clark informed me there was no activity at all, in the cardiotachyscope and that there had been no neurological or muscular response to our resuscitative effort at all and that the wound which the President sustained of his head was a mortal wound, and at that point we determined that he had expired and we abandoned efforts of resuscitation. Mr. Specter. Would you identify Dr. Clark's specialty for the record, please? Dr. Perry. Dr. Clark is professor and chairman of the department of neurosurgery at the University of Texas Southwestern Medical School, and chief of the neurosurgical services at Parkland Hospital. Mr. Specter. Now, you described a condition in the right mediastinum. Would you elaborate on what your views were of the condition at the time you were rendering this treatment? Dr. Perry. The condition of this area? Mr. Specter. Yes, sir. Dr. Perry. There was both blood, free blood and air in the right superior mediastinum. That is the space that is located between the lungs and the heart at that level. As I noted, I did not see any underlying injury of the pleura, the coverings of the lungs or of the lungs themselves. But in the
- 68. presence of this large amount of blood in this area, one would be unable to detect small injuries to the underlying structures. The air was indicated by the fact that there was some frothing of this blood present, bubbling which could have been due to the tracheal injury or an underlying injury to the lung. Since the morbidity attendant upon insertion of an anterior chest tube for sealed drainage is negligible and the morbidity which attends a pneumothorax is considerable, I elected to have the chest tube put in place because we were giving him positive pressure oxygen and the possibility of inducing a tension on pneumothorax would be quite high in such instances. Mr. Specter. What is pneumothorax? Dr. Perry. Hemothorax would be blood in the free chest cavity and pneumothorax would be air in the free chest cavity underlying collapse of the lungs. Mr. Specter. Would that have been caused by the injury which you noted to the President's trachea? Dr. Perry. There was no evidence of a hemothorax or a pneumothorax through my examination; only it is sufficient this could have been observed because of the free blood in the mediastinum. Mr. Specter. Were the symptoms which excited your suspicion causable by the injury to the trachea? Dr. Perry. They were. Mr. Specter. At what time was the pronouncement of death made? Dr. Perry. Approximately 1 o'clock. Mr. Specter. By whom was death announced? Dr. Perry. Dr. Kemp Clark.
- 69. Mr. Specter. Was there any special reason why it was Dr. Kemp Clark who pronounced the President had died? Dr. Perry. It was the opinion of those of us who had attended the President that the ultimate cause of his demise was a severe injury to his brain with subsequent loss of neurologic function and subsequent massive loss of blood, and thus Dr. Clark, being a neurosurgeon, signed the death certificate. Mr. Specter. In your opinion, would the President have survived the injury which he sustained to the neck which you have described? Dr. Perry. Barring the advent of complications this wound was tolerable, and I think he would have survived it. Mr. Specter. Have you now described all of the treatment which was rendered to the President by the medical team in attendance at Parkland Memorial Hospital. Dr. Perry. In essence I have, Mr. Specter. I do not know the exact quantities of balance salt solutions or blood that was given. I mentioned the 300 mg. of hydrocortisone Dr. Carrico ordered and, of course, he was given oxygen under pressure which has been previously recorded. The quantities of substances or any other drugs I have no knowledge of. Mr. Specter. In general you have recounted the treatment? Dr. Perry. That is correct. Mr. Specter. Have you now stated for the record all of the individuals who were in attendance in treating the President that you can recollect at this time? Dr. Perry. Yes, sir; I have. Mr. Specter. Will you now describe as specifically as you can, the injury which you noted in the President's head? Dr. Perry. As I mentioned previously in the record, I made only a cursory examination of the President's head. I noted a large avulsive wound of the right parietal occipital area, in which both scalp and
- 70. portions of skull were absent, and there was severe laceration of underlying brain tissue. My examination did not go any further than that. Mr. Specter. Did you, to be specific, observe a smaller wound below the large avulsed area which you have described? Dr. Perry. I did not. Mr. Specter. Was there blood in that area of the President's head? Dr. Perry. There was. Mr. Specter. Which might have obscured such a wound? Dr. Perry. There was a considerable amount of blood at the head of the cartilage. Mr. Specter. Would you now describe as particularly as possible the neck wound you observed? Dr. Perry. This was situated in the lower anterior one-third of the neck, approximately 5 mm. in diameter. It was exuding blood slowly which partially obscured it. Its edges were neither ragged nor were they punched out, but rather clean. Mr. Specter. Have you now described the neck wound as specifically as you can? Dr. Perry. I have. Mr. Specter. Based on your observations of the neck wound alone, do you have a sufficient basis to form an opinion as to whether it was an entrance wound or an exit wound, Dr. Perry. No, sir. I was unable to determine that since I did not ascertain the exact trajectory of the missile. The operative procedure which I performed was restricted to securing an adequate airway and insuring there was no injury to the carotid artery or jugular vein at that level and at that point I made the procedure. Mr. Specter. Based on the appearance of the neck wound alone, could it have been either an entrance or an exit wound?
- 71. Dr. Perry. It could have been either. Mr. Specter. Permit me to supply some additional facts, Dr. Perry, which I shall ask you to assume as being true for purposes of having you express an opinion. Assume first of all that the President was struck by a 6.5-mm. copper-jacketed bullet fired from a gun having a muzzle velocity of approximately 2,000 feet per second, with the weapon being approximately 160 to 250 feet from the President, with the bullet striking him at an angle of declination of approximately 45 degrees, striking the President on the upper right posterior thorax just above the upper border of the scapula, being 14 cm. from the tip of the right acromion process and 14 cm. below the tip of the right mastoid process, passing through the President's body striking no bones, traversing the neck and sliding between the large muscles in the posterior portion of the President's body through a fascia channel without violating the pleural cavity but bruising the apex of the right pleural cavity, and bruising the most apical portion of the right lung inflicting a hematoma to the right side of the larynx, which you have just described, and striking the trachea causing the injury which you described, and then exiting from the hole that you have described in the midline of the neck. Now, assuming those facts to be true, would the hole which you observed in the neck of the President be consistent with an exit wound under those circumstances? Dr. Perry. Certainly would be consistent with an exit wound. Mr. Specter. Now, assuming one additional fact that there was no bullet found in the body of the President, and assuming the facts which I have just set forth to be true, do you have an opinion as to whether the wound which you observed in the President's neck was an entrance or an exit wound? Dr. Perry. A full jacketed bullet without deformation passing through skin would leave a similar wound for an exit and entrance
- 72. wound and with the facts which you have made available and with these assumptions, I believe that it was an exit wound. Mr. Specter. Do you have sufficient facts available to you to render an opinion as to the cause of the injury which you observed in the President's head? Dr. Perry. No, sir. Mr. Specter. Have you had an opportunity to examine the autopsy report? Dr. Perry. I have. Mr. Specter. And are the facts set forth in the autopsy report consistent with your observations and views or are they inconsistent in any way with your findings and opinions? Dr. Perry. They are quite consistent and I noted initially that they explained very nicely the circumstances as we observed them at the time. Mr. Specter. Could you elaborate on that last answer, Dr. Perry? Dr. Perry. Yes. There was some considerable speculation, as you will recall, as to whether there were one or two bullets and as to from whence they came. Dr. Clark and I were queried extensively in respect to this and in addition Dr. Carrico could not determine whether there were one or two bullets from our initial examination. I say that because we did what was necessary in the emergency procedure, and abandoned any efforts of examination at the termination. I did not ascertain the trajectory of any of the missiles. As a result I did not know whether there was evidence for 1 or 2 or even 3 bullets entering and at the particular time it was of no importance. Mr. Specter. But based on the additional factors provided in the autopsy report, do you have an opinion at this time as to the number of bullets there were?
- 73. Dr. Perry. The wounds as described from the autopsy report and coupled with the wounds I have observed it would appear there were two missiles that struck the President. Mr. Specter. And based on the additional factors which I have provided to you by way of hypothetical assumption, and the factors present in the autopsy report from your examination of that report, what does the source of the bullets seem to have been to you? Dr. Perry. That I could not say. I can only determine their pathway, on the basis of these reports within the President's body. As to their ultimate source not knowing any of the circumstances surrounding it, I would not have any speculation. Mr. Specter. From what direction would the bullets have come based on all of those factors? Dr. Perry. The bullets would have come from behind the President based on these factors. Mr. Specter. And from the level, from below or above the President? Dr. Perry. Not having examined any of the wounds with the exception of the anterior neck wounds, I could not say. This wound, as I noted was about 5 mm., and roughly circular in shape. There is no way for me to determine. Mr. Specter. Based upon a point of entrance in the body of the President which I described to you as being 14 cm. from the right acromion process and 14 cm. below the tip of the right mastoid process and coupling that with your observation of the neck wound, would that provide a sufficient basis for you to form an opinion as to the path of the bullet, as to whether it was level, up or down? Dr. Perry. Yes, it would. In view of the fact there was an injury to the right lateral portion of the trachea and a wound in the neck if one were to extend a line roughly between these two, it would be going slightly superiorly, that
- 74. is cephalad toward the head, from anterior to posterior, which would indicate that the missile entered from slightly above and behind. Mr. Specter. Dr. Perry, have you been a part of or participated in any press conferences? Dr. Perry. Yes, sir; I have Mr. Specter. And by whom, if anyone, were the press conferences arranged? Dr. Perry. The initial press conference, to the best of my knowledge, was arranged by Mr. Hawkes who was identified to me as being of the White House Press, and Mr. Steve Landregan of the hospital administration there at Parkland, and Dr. Kemp Clark. They called me, I was in the operating suite at the time to assist with the care of the Governor, and they called and asked me if it would be possible for me to come down to a press conference. Mr. Specter. At about what time did that call come to you, Doctor? Dr. Perry. I am not real sure about that but probably around 2 o'clock. Mr. Specter. What action, if any, did you take in response to that call? Dr. Perry. I put in a page for Dr. Baxter and Dr. McClelland since they were also involved, and went down to the emergency room where I met Mr. Hawkes and Dr. Clark. And from there we went up to classrooms one and two which had been combined into a large press room, and was packed with gentlemen and ladies of the press. Mr. Specter. In what building was that located? Dr. Perry. This was in Parkland Hospital, in the classroom section. Mr. Specter. Are you able to identify which news media were present at that time?
- 75. Dr. Perry. No, sir; there were numerous people in the room. I would estimate maybe a hundred. Mr. Specter. What doctors spoke at that press conference? Dr. Perry. Dr. Clark and I answered the questions. Mr. Specter. Who spoke first as between you and Dr. Clark? Dr. Perry. I did. Mr. Specter. Would you state as specifically as you can the questions which were asked of you at that time and the answers which you gave? Dr. Perry. Mr. Specter, I would preface this by saying that, as you know, I have been interviewed on numerous occasions subsequent to that time, and I cannot recall with accuracy the questions that were asked. They, in general, were similar to the questions that were asked here. The press were given essentially the same, but in no detail such as have been given here. I was asked, for example, what I felt caused the President's death, the nature of the wound, from whence they came, what measures were taken for resuscitation, who were the people in attendance, at what time was it determined that he was beyond our help. Mr. Specter. What responses did you give to questions relating to the source of the bullets, if such questions were asked? Dr. Perry. I could not. I pointed out that both Dr. Clark and I had no way of knowing from whence the bullets came. Mr. Specter. Were you asked how many bullets there were? Dr. Perry. We were, and our reply was it was impossible with the knowledge we had at hand to ascertain if there were 1 or 2 bullets, or more. We were given, similarly, to the discussion here today, hypothetical situations. "Is it possible that such could have been the case, or such and such?" If it was possible that there was one bullet. To this, I replied in the affirmative, it was possible and conceivable that it was only one bullet, but I did not know.
- 76. Mr. Specter. What would the trajectory, or conceivable course of one bullet have been, Dr. Perry, to account for the injuries which you observed in the President, as you stated it? Dr. Perry. Since I observed only two wounds in my cursory examination, it would have necessitated the missile striking probably a bony structure and being deviated in its course in order to account for these two wounds. Mr. Specter. What bony structure was it conceivably? Dr. Perry. It required striking the spine. Mr. Specter. Did you express a professional opinion that that did, in fact, happen or it was a matter of speculation that it could have happened? Dr. Perry. I expressed it as a matter of speculation that this was conceivable. But, again, Dr. Clark and I emphasized that we had no way of knowing. Mr. Specter. Have you now recounted as specifically as you can recollect what occurred at that first press conference or is it practical for you to give any further detail to the contents of that press conference? Dr. Perry. I do not recall any specific details any further than that. Representative Ford. Mr. Specter—was there ever a recording kept of the questions and answers at that interview, Dr. Perry? Dr. Perry. This was one of the things I was mad about, Mr. Ford. There were microphones, and cameras, and the whole bit, as you know, and during the course of it a lot of these hypothetical situations and questions that were asked to us would often be asked by someone on this side and recorded by some one on this, and I don't know who was recorded and whether they were broadcasting it directly. There were tape recorders there and there were television cameras with their microphones. I know there were recordings made but who made them I don't know and, of course, portions of it
- 77. would be given to this group and questions answered here and, as a result, considerable questions were not answered in their entirety and even some of them that were asked, I am sure were misunderstood. It was bedlam. Representative Ford. I was thinking, was there an official recording either made by the hospital officials or by the White House people or by any government agency? Dr. Perry. Not to my knowledge. Representative Ford. A true recording of everything that was said, the questions asked, and the answers given? Dr. Perry. Not to my knowledge. Mr. Dulles. Was there any reasonably good account in any of the press of this interview? Dr. Perry. No, sir. Representative Ford. May I ask—— Dr. Perry. I have failed to see one that was asked. Representative Ford. In other words, you subsequently read or heard what was allegedly said by you and by Dr. Clark and Dr. Carrico. Were those reportings by the news media accurate or inaccurate as to what you and others said? Dr. Perry. In general, they were inaccurate. There were some that were fairly close, but I, as you will probably surmise, was pretty full after both Friday and Sunday, and after the interviews again, following the operation of which I was a member on Sunday, I left town, and I did not read a lot of them, but of those which I saw I found none that portrayed it exactly as it happened. Nor did I find any that reported our statements exactly as they were given. They were frequently taken out of context. They were frequently mixed up as to who said what or identification as to which person was who. Representative Ford. This interview took place on Sunday, the 24th, did you say?
- 78. Dr. Perry. No, there were several interviews, Mr. Ford. We had one in the afternoon, Friday afternoon, and then I spent almost the entire day Saturday in the administrative suite at the hospital answering questions to people of the press, and some medical people of the American Medical Association. And then, of course, Sunday, following the operation on Oswald, I again attended the press conference since I was the first in attendance with him. And, subsequently, there was another conference on Monday conducted by the American Medical Association, and a couple of more interviews with some people whom I don't even recall. Representative Ford. Would you say that these errors that were reported were because of a lack of technical knowledge as to what you as a physician were saying, or others were saying? Dr. Perry. Certainly that could be it in part, but it was not all. Certainly a part of it was lack of attention. A question would be asked and you would incompletely answer it and another question would be asked and they had gotten what they wanted without really understanding, and they would go on and it would go out of context. For example, on the speculation on the ultimate source of bullets, I obviously knew less about it than most people because I was in the hospital at the time and didn't know the circumstances surrounding it until it was over. I was much too busy and yet I was quoted as saying that the bullet, there was probably one bullet, which struck and deviated upward which came from the front, and what I had replied was to a question, was it conceivable that this could have happened, and I said yes, it is conceivable. I have subsequently learned that to use a straight affirmative word like "yes" is not good relations; that one should say it is conceivable and not give a straight yes or no answer. "It is conceivable" was dropped and the "yes" was used, and this was happening over and over again. Of course, Dr. Shires, for example, who was the professor and chairman of the department was identified in one press release as chief resident.
- 79. Mr. Dulles. As what? I didn't get it. Dr. Perry. As chief resident. And myself, as his being my superior, whereas Dr. Ronald Jones was chief resident of course, nothing could be further from the truth in identifying Dr. Shires as chief resident. I was identified as a resident surgeon in the Dallas paper. And I am not impressed with the accuracy of the press reports. Mr. McCloy. I don't know whether you have covered this very well. Let me ask you about the wound, the wound that you examined in the President's neck. You said that it would have been tolerable. Would his speech have been impaired? Dr. Perry. No, sir; I don't think so. The injury was below the larynx, and certainly barring the advent of any complication would have healed without any difficulty. Mr. McCloy. He would have had a relatively normal life? Dr. Perry. Yes, sir. Mr. McCloy. Did you, any other time, or other than the press conference or any other period, say that you thought this was an exit wound? Dr. Perry. No, sir; I did not. Mr. McCloy. When the President was brought, when you first saw the President, was he fully clothed, or did you cut the clothing away? Dr. Perry. Not at the time I saw him. Dr. Carrico and the nurses were all in attendance, they had removed his coat and his shirt, which is standard procedure, while we were proceeding about the examination, for them to do so. Mr. McCloy. But you didn't actually remove his shirt? Dr. Perry. No, sir; I did not. Mr. McCloy. Did you get the doctor's experience with regard to gunshot wounds?
- 80. Mr. Specter. Yes, sir; I did. Mr. McCloy. You said something to the effect that, of knowing the President had an adrenalin insufficiency, is that something you could observe? Dr. Perry. This is common medical knowledge, sir, that he had had in the past necessarily taken adrenalin steroids to support this insufficiency. Dr. Carrico, at this moment of great stress, recalled this, and requested this be given to him at that time, this is extremely important because people who have adrenalin insufficiency are unable to mobilize this hormone at the time of any great stress and it may be fatal without support from exogenous drugs. Mr. McCloy. In other words, you had a general medical history of the President before he was—common knowledge. Dr. Perry. No more so than anyone else, sir, except this would have stuck with us, sir, since they were already in that line. Mr. McCloy. Did you discuss with any of the other doctors present, and you named quite a number of them, as to whether this was an exit wound or an entrance wound? Dr. Perry. Yes, sir; we did at the time. But our discussion was necessarily limited by the fact that none of us knew, someone asked me now—you must remember that actually the only people who saw this wound for sure were Dr. Carrico and myself, and some of the other doctors were quoted as saying something about the wound which actually they never said at all because they never saw it, because on their arrival I had already made the incision through the wound, and despite what the press releases may have said neither Dr. Carrico nor myself could say whether it was an entrance or an exit wound from the nature of the wound itself and Dr. McClelland was quoted, for example, as saying he thought it was an exit wound, but that was not what he said at all because he didn't even see it. Mr. McCloy. And it is a fact, is it not, that you did not see what we now are supposed to believe was the entrance wound?
- 81. Dr. Perry. No, sir; we did not examine him. At that time, we attended to the matters of expediency that were life-saving and the securing of an adequate airway and the stanching of massive hemorrhage are really the two medical emergencies; most everything else can wait, but those must be attended to in a matter of minutes and consequently to termination of treatment I had no morbid curiosity, my work was done, and actually I was rather anxious to leave. Mr. McCloy. That is all. Mr. Specter. Yes. (Discussion off the record.) Mr. Dulles. I suggest, Mr. Specter, if you feel it is feasible, you send to the doctor the accounts of his press conference or conferences. And possibly, if you are willing, sir, you could send us a letter, send to the Commission a letter, pointing out the various points in these press conferences where you are inaccurately quoted, so we can have that as a matter of record. Is that feasible? Dr. Perry. That is, sir. Would you prefer that each clipping be edited individually or a general statement? Mr. Dulles. Well, I think it would be better to have each clipping dealt with separately. Obviously, if you have answered one point in one clipping it won't be necessary to answer that point if it is repeated in another clipping. Dr. Perry. Yes, sir. Mr. Dulles. Just deal with the new points. Dr. Perry. I can and will do this.
- 82. Representative Ford. This would be where Dr. Perry is quoted himself, or Dr. Carrico, or anyone else, they would only pass judgment on the quotes concerning themselves. Mr. Dulles. That would be correct. Dr. Perry. Yes, because some of the other circumstances in some of the press releases which have come to my attention have not been entirely accurate either, regarding sequence of events, and although I would not have knowledge about those you would not want those added necessarily, just any statement alluded to have been made by me. Mr. Dulles. I think that would be better. Don't you think so, Mr. Chairman? Representative Ford. I think it would be the proper procedure. Is this a monumental job, Mr. Specter? Mr. Specter. No, I think it is one which can be managed, Congressman Ford. I might say we have done that with some of the clippings. There was an article, as the deposition records will show when you have an opportunity to review them, they have not been transcribed, as to an article which appeared in La Expres, statements were attributed to Dr. McClelland—— Mr. Dulles. Which paper? Mr. Specter. A French paper, La Expres. And I questioned the doctors quoted therein and developed for the record what was true and what was false on the statements attributed to them, so we have undertaken that in some circles but not as extensively as you suggest as to Dr. Perry, because we have been trying diligently to get the tape records of the television interviews, and we were unsuccessful. I discussed this with Dr. Perry in Dallas last Wednesday, and he expressed an interest in seeing them, and I told him we would make them available to him prior to his appearance,
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