NISHITANI, TATSUO

Osaka City University Repository

Nishitani, T.
Osaka J. Math.
21 (1984), 843-850
A NOTE ON REDUCED FORMS OF EFFECTIVELY
HYPERBOLIC OPERATORS AND
ENERGY INTEGRALS
TATSUO NISHITANI
(Received May 2, 1983)
1. Introduction
Let P be the principal symbol of a hyperbolic differential operator. At
a double characteristic point of P, its Taylor expansion begins with the quad-
ratic form in the cotangent bundle. The coefficient matrix of the Hamiltonian
system associated with this quadratic form is called the fundamental (or Hamil-
ton) matrix. If the fundamental matrix has non-zero real eigenvalues, P is
said to be effectively hyperbolic operator ([1], [2]).
Ivrii and Petkov conjectured in [2] that C°° Cauchy problem for effectively
hyperbolic operators is well posed for any lower order term; that is effectively
hyperbolic operator is strongly hyperbolic.
In this note, in §2, we reduce effectively hyperbolic operators of second
order to certain standard forms by homogeneous canonical transformations.
Since we are concerned with the Cauchy problem, we shall use only homo-
geneous canonical transformations which do not depend on the time and its
dual variables. In §3, for some simple but essential examples, we indicate
how the standard forms relate to the energy integrals which assure the strong
hyperbolicity.
The detailed proofs of deriving the energy estimates for effectively hyper-
bolic operators of the standard forms will be appear elsewhere.
Denote *<»=(*» -, *ά ?»=&> -, &), *=*(0)> ξ=ξ™> 0<p<d9 and
consider
P(χ, ξ) = &-Q(χ, l(1)),
where Q(x, £(1)) is defined in a conic neighborhood of (0, i(1)), non-negative
and homogeneous of degree 2 in £(1).
Let (0, ξ) be a double characteristic point of P(x, ξ). That is dP(x, ξ)
vanishes at (0, ξ). This is the same thing as f=(0, l(1)), Q(0, ?(1))=0. De-
note by Fp(x9 ξ) the fundamental matrix evaluated at (#, ξ) (for the precise
definition, see [2]). In the following, {,} denotes the Poisson bracket. The
844 T. NISHITANI
standard forms are the followings.
Theorem 1.1. Assume that FP(0, i) has non-zero real eigenvalues. Then,
in a conic neighborhood of (0, l(1)), there exists a homogeneous canonical trans-
formation in T*R" taking (0, |(1)) to (0, l(1)) under which O(x, ξw) is transformed
to (1.1), with (1.1); or (1.2), «ώλ (1.2); and (1.2);'.
(1.1),
(i.i);
(1.2),
(1.2);
q{(x, ξ (1)), rf (Λr, f (1)) are positive, homogeneous of degree 2, 0 respectively,
ΨP> SP are non-negative, vanishing at (0, !(/>+1)), homogeneous of degree 0, 2 re-
spectίvely and φp is homogeneous of degree 0.
REMARK 1.1. The condition (1.2)" is closely related with the energy integ-
rals, see §3.
2. Proof of Theorem 1.1
First, we shall prove the following lemma which also will be useful to
study the standard forms for non effectively hyperbolic operators.
Lemma 2.1. Let 9gg(0, !(1))>0. Then Q(xy £(1)) is transformed to (1.1),
with (1.1); (Q^p^d— 1) or (1.2), fwϊλ (1.2); (!</><<*— 1), fry a local homogeneous
canonical transformation in T*Rd which takes (0, |(1)) to (0, l(1)).
Proof. From the Malgrange preparation theorem, we get
where φ0, v|r0 are homogeneous of degree 0 with ψ0>0 and qλ is positive,
homogeneous of degree 2. This is just (1.1)0
Now we assume that (l.l);_ι is not satisfied. Set Xp(x(p\ ξ(p})=φp-ι
(x(p\ ξ W). Then it follows that dφp^ and 23J., f y Λcy are linearly independent
at (0, |(ί)). In fact, if rfφ,_! were to be proportional to 2y-> %j dxj at (°» ^(ί))»
taking into account that J ψ^-^O, |(ί))=0, the Euler's identity would give (since
ψp-i is homogeneous of degree 0),
EFFECTIVELY HYPERBOLIC OPERATORS 845
This contradicts to our assumption. Thus following proposition 3.11 in Mel-
rose [3], we can construct a homogeneous canonical transformation [Xj(x(p\
£<»),
 Bχ*o>, f («)]'_, so that
(2.1) Xj(0, <«) = 0,p<j<d, B/0, o>) = 0, p<j<d-l, B,(0, ^>)ΦO .
After having done this transformation, remarking that
(9V,-ι/9 fl) (0» HΦO, £<» = (0, -.., 0, &),
the Malgiange preparation theorem gives that
where ^ is positive, homogeneous of degree —2, kp is non-negative, homo-
geneous of degree 2 and /^ is homogeneous of degree 1. Take
It is clear that {B ,^ XP} = 1. Moreover the differentials Σy-/» ξj dx^ dBp> dXp
are linearly independent at (0, !(/0). Indeed, if there were to be a dependence
relation
Σί-, ζj dxj = adBp+βdXp at (0, |^ ) ,
then applying this to HXp, the Hamilton vector field of Xp, would give α=0,
hence
Σy-> f y dxj = βdXp .
But this gives a contradiction because |(Λ=(0, •••, 0, ^
rf), dXp=dxp and/><ίί— 1.
Again, from proposition 3.11 in [3], one can extend H ,^ -X^ to a homogeneous
canonical transformation [Xj(x(p\ |(ί)), HX^(j>), £(ί))]y-/, satisfying (2.1). Since
d&p, p<j<d, it follows that ξj(X<*\ Bw) (/)+l<;<^) and (^
(p<j<d) do not depend on Bp. Thus we get (1.2)^ with gp(x^\ |(/>+1))>0,
r^+1>0 being homogeneous degree 2, 0 respectively.
Finally, assume that (1.2)J does not hold. Then one can write
with φpy typ which are homogeneous of degree 0 and -ψ^O, where ap is posi-
tive, homogeneous of degree 2. This yields (1.1) .^ Therefore, the induction
on p proves this lemma.
We proceed to the proof of theorem 1.1. If Ql Q(Q, !(1))=0, it is easily
seen that FP(0, I) has only pure imaginary eigenvalues (cf. [1], [2]). Then
846 T. NISHITANI
we may suppose that dl Q(Q, l(1))>0. Applying lemma 2.1, Q(x, £(1)) is re-
duced to (1.1), with (1.1); (0<p<d-l) or (1.2), with (1.2)J (\<ρ<d-ϊ).
We note that the fundamental matrix is transformed to a similar matrix
by a canonical transformation. Therefore to prove theorem 1.1, it suffices
to show that an operator P with Q of type (1.1), with (!.!)£ is in fact effectively
hyperbolic and an operator P with Q of type (1.2), with (1.2)£ is effectively
hyperbolic if and only if (1.2)J' holds. The following two propositions are
easily verified.
Proposition 2.1. Let
P = £o2-£ fcte-i-^'-ίί '< f f , ?, >0, r,>0 .
ί=l t=l
Then we have
Here P(xy ξ) is considered as a function of (XQ, •••, xp-u £0, •••, ?,-ι).
Proposition 2.2. Let
= β- ϊto-i-^- '< & ίι>0, r,>0 .
TTtew vie have
det (λ+ί» = Φ(λ, ?,, r, ) = λ2 ψ(λ, g, , r,), ψ(0, ?„
y=ι
JT(Λ;, |) ώ considered as a function of (XQ, •••, xp, ξQy •••, ,^).
First we consider the case when Q is of the form (1.1), with (l.l)J. We
denote by riy q{ the value of rfa ξ (1)), ?, (Λ?, ?(1)) at (0, ί(1)). Since φ,, ψ ,
depend only on (Λ;(^+I), |(^+1)) and {φ,, {φ,, ψ ,}> (0, |<*+1>)==0, it follows
that
det(λ+FP(0, 1)) =
where
is considered as a function of (#0, •••, Λ?,, £0, •••, f,) and £ is a non-negative
quadratic form in (x(p+1\ ξ(p+1)). By proposition 2.1, we get
Φ(λ) - det(λ+^) =
 λ
**«+...+φ(θ), Φ(0) = -(Π 43,) (Π r,)<0 .
EFFECTIVELY HYPERBOLIC OPERATORS 847
This shows that Φ(λ)=0 has non-zero real roots.
Next we consider the case when Q has the form (1.2)^ with (1.2)£. Put
-t-xtf- r, ξ1,, E = -gp(X«\ £(>+1)) rp+l(x,
From the non-negativity of gp, taking that (d2gpldxp) (0, |<*+1>)=0 into account,
it follows that
at (0, »
Then the same reasoning as before shows that
det(λ+FP(0, 1)) = det(λ+ί» det(λ+^(0, 1)) ,
where P, E are considered as functions of (x
ot •••, xp, £„, •••, ξp), (x(p+l\
respectively. From proposition 2.2, we get
det(λ+ J» = λV(λ, q,, r,) , ψ (λ, qh r,) = λ2>+ - +ψ(0, qt, r,) ,
(Π r,)
From [2], the equation ψ(χ, .^, r,)— 0 has only pure imaginary roots except
for at most one simple real root μ (ΦO) and for — μ, for any qi9 r, >0. Since
•^ (λ, qi9 Tf) depends continuously on qiy r{, in order that the equation ψ(λ,
5, , ri)=0 has a non-zero real root, it is necessary and sufficient that ι|r(0, qh rf )
is negative. Taking into account that det(λ+^)=0 has only pure imaginary
roots (since E is non-negative), -FP(0, ς) has a non-zero real eigen value if and
only if the condition (1.2)" holds.
These facts prove theorem 1.1.
3. Energy integrals
As a simple example, we shall indicate some connections between the
condition (1.2)" and the energy integrals. Let us consider the following oper-
ator,
P = 8S-
ί=0
where l<p<d, ί,>0 (0</</>— 1), r, >0 (!</</>— 1), ^>0. We assume
that P is effectively hyperbolic. If rp>Q, the effective hyperbolicity means
=1
 P
Here, we note that the condition 2J r7x>l (r,->0), is equivalent to the
«=ι
existence of real numbers {£,•}?„! such that
848 T. NISHITANI
(3.1) jSfi?r,<l,ij <•,= !.
Taking (3.1) into account, we use the following weight (or separating) function,
with {£,}?.! in (3.1) if rp>0 and £y=0, l<,j£p-l, Sf=l if r,=0. Then
the integration by parts gives that
(3.2) -2Re( Y(x)"Pu Y(x)"d^dx = 2n { Y(X)2"-1\dQ u\2dx+JQ~ Jo"
+2n JQ Y(Xγ"-1 {g qi(Xi-Xi+ιγ} I ddu I *dx+
+2n\ Y(xY"-1&ri\diu\*}dx+2q0\ Y(xγ*(x0-Xl)\ddu\2dx+Jo~ ί=ι JQ-
+2n( Y(xfn-l^£iri(QiJQ~ ί=ι
for u^Co(Rd+1)y wfiere Ω*^ {x^Rd+l; Y(x)^Q}, and w is a positive integer.
We shall estimate the last term of the right hand of (3.2). Let rp>0, then by
the Cauchy-Schwarz inequality, it follows that
2n 6, rtfi u 9^+9^ Q0u) <2nδ r, | dp \
i=l i=l
with δ>0. From (3.1), we can take δ so that
On the other hand, from 2 £i=l» we have | F(Λ?) (Λ:O— jcjl <c Σ <li(xi— xi+ιf>
with some c>0. Hence the right hand side of (3.2) is estimated from below
by
2n(\ - δ) ( Y(
x
γ*~i I QQu I *dx+(2n-c) \ Y(xf^ {Σ ϊK^—JQ~ J Q ~ ί=o
+2n(l-δ)jQ F -^1 φrΛdvW dx .
In the case r#=0, the last term of the right hand side of (3.2) is equal to
zero, and we have the following estimate from below,
2n ( Y(x)2^ \QQu\2 dx+(2n-c)
-
EFFECTIVELY HYPERBOLIC OPERATORS 849
Now let us consider the first order term BQdJ fieCV From the following
two inequalities
2Re Y(x)"Bddu Y(*)"&> dx<2^1 n~l\B\2 Y(x)*»+1 \ ddu \ 2 dx+JQ~ JQ~
SlM ί Y(x)2Λ-1\Q<iu\2dx+S1tf ( Y(x)2"-3\u\2 dx ,JQ~ JQ~
ιψι-2) ( Y(x)2'-3 1 u 1 2 dx<, ( Y(x)2"-1 1 a0« 1 2 dx ,JQ~ Jo~
it follows that
2Re ( Y(x)nBQdu Y(x)%* dx<2^1 n~l\B\2 \ Y(x)2»+l \ Qdu \ 2 dx+JQ~ JQ"
+38^ ( Y(x)2n-ίld
ίiu\
2
 dx, n>4.J Q ~
Then using the inequality Y(x)2<c 2! ?,•(#,-— Xi+ι)\ we get finally
1=0
(3.3) 2Re{ Y(x)nBddu Y(x)ndJQ~
Now we take δ^ n so that 2(1— S)— Sδ^O and n(2n—c)>28τ1 c\B\2y then
one can absorb the first order term BQd.
Proposition 3.1. Assume that P is effectively hyperbolic. Then we have
Y(x)2»+1 1 (P+Bdd)u \2dx> 2nS2 ( Y^)2*-1 1 d<μ \ 2 dx+
~ ~Q
+c2n Y(x)2*-1 { ί, (*,-^ +ι)2} 1 9,« 1 2 dx+
-
+2S2n , Y(x)2*-> { 'i 1 9, « \2\dx,
- ί=ι
By a similar way, one can obtain the energy estimate in Ω+ (cf. [4]).
Finally, we consider a simple example corresponding to operators of type
{(1.1),, (1.1);}. Let
where 9,>0, r, >0, ξ"=(ξt+ι, •••, ξΐ), p+I<,d. Taking the Fourier transform
with respect to x"=(xp+1, •••, XΛ), it suffices to consider
850 T. NlSHITANI
As a separating function, we take
(3.4) Y(X,
Denote by u(x', £") the partial Fourier transform with respect to x". Then
the same reasoning as before gives that
ί Y(X>ξγ'
Jω
[ Y(x, ξ)2'-1 fi qfa-Xi+tf+qfa-Wm I
Jω" ί=0
for n>c3\B\ , where ω±=ω±(£")= {*'; ^ -φίf'O^O}. (cf. [5]).
References
[1] L. Hδrmander: TΛe Cauchy problem for differential equations with double char-
acteristics, J. Analyse Math. 32 (1977), 118-196.
[2] V. Ja. Ivrii and V.M. Petkov: Necessary conditions for the Cauchy problem for
non-strictly hyperbolic equations to be well-posed, Russian Math. Surveys 29 (1974),
1-70.
[3] R.B. Melrose: Equivalence of glancing hyper surf aces, Invent. Math. 37 (1976),
165-191.
[4] T. Nishitani: The Cauchy problem for weakly hyperbolic equations of second order,
Comm. Partial Differential Equations 5 (1980), 1273-1296.
[5] T. Nishitani: Sur les operateurs fortement hyperboliques qui ne dependent que du
temps, Seminaire sur les equations aux derivees partielles hyperboliques et holo-
morphes, Univ. de Paris VI, (1981-82).
Department of Mathematics
College of General Education
Osaka University
Toyonaka, Osaka 560
Japan


A note on reduced forms of effectively hyperbolic operators and energy integrals

Nishitani, Tatsuo

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A note on reduced forms of effectively hyperbolic operators and energy integrals

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