Hyperbolic Differential Operators and Related Problems (Lecture Notes in Pure and Applied Mathematics)

0, (5 G R is defined in (2.2) fte/otu.

2.3

The Spaces JEP'J;A((0,T) X R n )

For T > 0, we set (0)T)xRT

(2.2)

and equip this space with its infimum norm. There is an alternative description of this space provided that s € N. Lemma 2.10. Let s € N, 6 € R and T > 0. Then the infimum norm of the space HS'5'X((Q,T) x R n ) is equivalent to the norm \\-\\SS.T} where

Here we have introduced the notation t% = (^) &, 0 = l/(i* + 1). For a proof of this and the following results, see [9]. Lemma 2.11. For 5, s' > 0, 6, 5' e R, and T > 0, H'^((Q,T) x R n ) C H3'>S'>X((Q,T) x R n ) only if s>s',

s + p6l*>s' + {36'l*.

(2.3)

Edge Sobolev spaces and branching of singularities

187

The two conditions in (2.3) are related to the fact that the elements of the edge Sobolev spaces have different smoothness for t > 0 and t — 0, respectively. The following two results provide a criterion when the superposition operators defined by the right-hand sides of the hyperbolic equations in (1.1) and (1.2) map an edge Sobolev space into itself. Proposition 2.12. Assume that s + 6 > 0. We suppose that s € N and inin{s, s + ££/,} > (n + 2)/2. Then #5'*;A((0,T) x R71) is an algebra under pointwise multiplication for any 0 < T < T0. In other words, we have IHUr < C |HUr IHW for u,v e £P'*;A((0,T) x If1). Moreover, the constant C is independent of 0 < T < TQ. The proof consists in a direct, but quite long calculation using the representation of the norm from Lemma 2.10. Then, an interpolation argument (see [2]) gives us the following result: Corollary 2.13. Let f = f(u) be an entire function with /(O) = 0, i.e., f ( u ) = J]°11 fjUj for allufR. Assume that [s\ + 6 > 0 and min{ [s\ , [s\ + /361*} > (n + 2)/2. Then there is, for each R > 0, a constant Ci(R) with the property that

provided that u,v € Ha>*'x((Q,T) x R") and \\u\\3tS.T < R, \\v\\S)S.T < R.

3

Linear and Semilinear Cauchy Problems

Our considerations start with the linear Cauchy problem Lw(t,x)=f(t,x),

(d}w)(Q,x) = wj(x),

We define

3=1

j = 0,l.

(3.1)

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Further we introduce the number

q

i+rop P

,

2

(3.2)

'

^

and fix Ao =

Theorem 3.1. Let s, Q € R, s > 1, Q > QQ. Further let WQ G wi € #5+A-^(Rn), and / e #5-1'c?;A((0, T) x R n ) to (3.1). Moreover, the solution w is unique in the space #5'Qo;A((0, T) x IP*). Remark 3. 2. The parameter A0 describes the loss of regularity. The explicit representations of the solutions for special model operators in [14] and [19] show that the statement of the Theorem becomes false if A < AQ. By interpolation, it suffices to prove Theorem 3.1 when s e N+. In this case, Theorem 3.1 will follow by standard functional-analytic arguments if the following a priori estimate is established. Proposition 3.3. For each s € N+, Q > QQ, there is a constant C0 — (7o(s, Q) with the property that \\W\\S,Q;T ^ °0

lKHff.+A(R«) + IK

for all Q
where u 6 C°°(R+) has support in [0,2], satisfies w(t) = 1 for 0 < t < 1, and takes values in [0,1]. Then we introduce the vector W(t,f) = £)w(t, £), Dtw(t, «f)) and obtain the first-order system DtW(t, 0 = A(*, 0^(4,0 + *U 0,

(3-3)

Edge Sobolev spaces and branching of singularities

189

It is clear that this first-order O.D.E system has a unique solution W for each fixed f E R n . Point-wise estimates of W(t, f) and its time derivatives will allow to establish the a priori estimate from Proposition 3.3. If X(t, t', f ) denotes the fundamental matrix, i.e.,

DtX(t,lf,£) = A ( t , ( ) X ( t t l f , Q ,

X(M,0 = J,

0 < f , t < T 0 , (3.4)

then W(t, 0 = X (t, f, ?)W(f , f ) + » Jj X(t, i", f )F(t", 0*"- Tms immediately gives estimates of |W(£, f)| if estimates of X(t, f , £) have been found. The following lemma is the crucial tool. Lemma 3.4. There is a constant C > 0 such that

holds for all £ E R n , where QQ is given by (3.2). Proof. This has been proved in [9] for the case of t% < if < t < TQ. Prom this, l|A(t,0|| < Cp(t,0, and \\X(t,t,{)\\ < exp(/; ||A(r,0|| O we derive the desired inequality for arbitrary 0 (n + 2)/2, where QQ be the number from (3.2). Suppose that f = f ( u ) is an entire function with /(O) = 0. Let Q > Qo and A = J3QI+. Then, for UQ 6 #3+A(Rn), m € fP+^-^IT), there is a number T > 0 with the property that a solution u E HS>Q'X((Q, T) x R71) to the Cauchy problem (1.1) exists. This solution u is unique in the space Ha>Qo'x((Q,T) x R n ). Theorem 3.6. Let s € N and assume that s — I > (n + 2)/2. Suppose that f = f(u,v,vi,...,vn) is entire with /(O, ...,0) = 0. Let Q > QQ and A = PQl*. Then, for UQ E tf'+^R"), t*i E tf'+^R71), there is a number T > 0 with the property that a solution u E fP'9;A((0,T) x R n ) to the Cauchy problem (1.2) ezi$is. TTiis solution u is unique in the space ) xR n ). Eventually, we state a result concerning the propagation of mild singularities.

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Theorem 3.7. Let s satisfy the assumptions of Theorem 3.5. Assume UQ E H'+W<**(1P), Ul E fp+^M'-^R"), where QQ is given by (3.2). Let v be the solution to I .7\JJ ——• —. I\jI JLJ

II /~r 1 * T* L/a 'T! U /I I\I \J «*/ I I ^T •""•" II Cv ••»I \ T* **/ 1I •

1 / ~"™ —""- I L/I % IX .

f I XO **\OI /

Then the solutions u,v E fP' 0o;A ((0,T) x R n ) to (1.1) and (3.5) saiw/y M - v E fr a+ * gojA ((0,r) x R n ). Proof. Corollary 2.13 implies f ( u ) E fP' Qo;A ((0, T) x R71). From Lemma 2.11 we deduce that /(u) E H3-1+W°+l'>x((Q,T) x R n ). The function w(t,x) = (it — v)(tt x) solves Lw = /(it) and has vanishing initial data. An application of Theorem 3.1 concludes the proof. D Example 3.8. Consider Qi Min-You's operator L from (1.4). Then 2« = 1, /? = 1/2, and Qo = 2m. Theorems 3.1, 3.5, and 3.7 state that the solutions it, v to (1.1), (3.5) satisfy it, i; E #s'2m;A((0, T) x R),

t/ - v E #3+1/2'2m;A((0, T) x R)

if 1*0 E #5+m(R), iti E #5+m-1/2(R). Proposition 2.7 then implies it, v E ^((O.T) x R),

t t - u E JJ^1/2((0,T) x R).

We find that the strongest singularities of u coincide with the singularities of i;. The latter can be looked up in (1.5) in case iti = 0.

4

Branching Phenomena for Solutions to Semilinear Equations

In this section, we consider the Cauchy problems

with L from (1.3), and we are interested in branching phenomena for singularities of the solution it. Our main result is Theorem 4.5. We know, e.g., from the example of Qi Min-You that we have to expect a loss of regularity when we pass from the Cauchy data at {t = 0} to the solution at {t ^ 0}. However, we also will observe a loss of smoothness if we prescribe Cauchy data at, say, t = —T0 and look at the solution for t = 0.

Edge Sobolev spaces and branching of singularities

191

This can be seen as follows. For simplicity, we only consider the operator of Taniguchi-Tozaki,

L = $-t2l*d2x-uJ*-ldx, ZER,

fceR.

(4.3)

In [19], it was shown that the Fourier transform of the solution v(t,x) to

can be written in the form v(t, 0 = exp(-iA(t)0 iFi (0(1 + 6)l./2, /?/., 2iA(t)0 £0(0 + 1 exp(-iA(*)0 iFi (0(1 + 6)l./2 + 0, 0(Z, + 2), 2iA(*)0 ^(0, where A(t) = f * X ( t ) d t f , and iFi(a,7, z) is the confluent hypergeometric function satisfying

for \z\ -> oo. This leads us to ) +exp(»A(t)00(|f |^))tio(0

(4.4)

for |f | —> oo provided that t ^ 0 has been fixed. Here we have set of = -0(1 + &)Z,/2, aj = -0(1 - 6)f./2, af = ao - 0, af = of - 0. In general, one of the exponents a^, aj and one of the exponents a^~, oif is positive, the other one is negative. We observe a loss of max{o^", aj} derivatives when we start from t = 0, and a loss of max{-ao , -aj} derivatives when we arrive at t = 0. Therefore, we have to expect both these phenomena when we prescribe Cauchy data at t = — T0 and cross the line t = 0. This leads us to the following definition. Definition 4.1. Let 5 > 0, 6 6 R We say that u € H5^A((-T,T) x R n ) if ti(t,x) 6 #'•**(((), T) x R n ), n(-t,x) 6 #a'*;A((0,T) x R71), and ti(t,x) -

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Dreher and Witt

Let 5_, s+ > 0,
s+ < 5_.

We say that u € #*->*+,*-,*+;A((_r,T) x Rn) if u e H3+>d+'>x((-T,T] x R n ) and ti(-t,x) 6 # a -'*- ;A ((0,r) x R71). We define the norm by \\u(ti x) lljr-' J +' 5 -' 5 + ;A ((-T,T)xR n ) ~^~ l l u ( ~ ^ x ) l J H" J -' <5 - ;A ((0,T)xR n ) •

This choice of the norm is possible, since # 5 -' J - ;A ((0,T) x R71) c fT a +'*+ ;A ((0,T) x R n ), compare Lemma 2.11. Let us consider the equation Lv — 0, again, where the operator L is from (4.3), and suppose that the initial data at t — — T0 satisfy WQ € jfif s -(R), Wi € fP— HR). We set £+. = Q0 = (l&l -l)/2, J_ = -1 -Q0 = (-|6| + l)/2, s+ = s_ + /36-.1* — (36+1*. The identity (4.4) leads, after some calculation, to a representation of v(t, £) in terms of iDo(£) and wi(0 which shows that the solution v belongs to #a-'a+'*-'*+;A((-r0,To) x R). Moreover, t;(0,aO e H and these statements about the smoothness are best possible. For the general operator L from (1.3), we can prove the following wellposedness result: Proposition 4.2. Let L be the operator from (1.3), Qo be the number defined in (3.2), and set Q+ = Qo, Q- = -1 - QQ,

s+ = s_ + 0Q J, - PQ+l*,

where 5_,5 + > 1. Suppose that WQ(X) € Hs-(Rn), Wl(x) € ff'— ^K1), /(t,x) e H3—l^-lA'+lA++liX((-TQlTQ) x R n ). T/ien there is a unique solution w e Ha-^+^-^+^((-T0,T0} x R n ) to Lw = /(t,o;),

(^)(-T0,x) = 1^(0;),

j = 0,1.

Moreover, (S?«;)(0,x) 6 ff'-+M-«--«(K*) /or j = 0, 1,

-, C

-^n) +

Edge Sobolev spaces and branching of singularities

193

The key of the proof is an estimate of the fundamental matrix. Lemma 4.3. Let X (£,£',£) be the solution to (3.4). Then it holds \\X(t,t'^)\\
,

0 < * < « * < T0.

Proof. Prom X(t,t,f) = X(t,t,€)~l and Lemma 3.4 we deduce that || JC (*,*',£) || < C (

f l )

| det X^,*, Or1,

0
Since dtX = iAX, we obtain dt det X(t, t7,0 = trace(iA(t, £)) det X(t, i7,0,

| det X(*, t;, 01 = exp f T ^f^ - co(r) dr) , \jf Plr5?; /

This completes the proof.

0 < t, f < T0.

D

Inverting the time direction, we will obtain an estimate of the fundamental matrix for negative times from this lemma. Proof of Proposition 4-2. The existence and uniqueness of w is clear since its Fourier transform solves an O.D.E. with parameter £. It remains to discuss the smoothness of w. By interpolation, we may assume s_ E N. Then we can make use of Lemma 2.10 to represent the norms of w ( _ To)0)xR n a^d /l(-T 0 ,o)xR«- We define 9(t,() = 9(-*,t) f°r (*,fl € (-T0,0) x R" and are going to show (4.5) and

(4.6) /=o J-T° 7R? ltf'-tRn) +

I k l l l f f ' — 1(R»))

i.e., the a priori estimate for t < 0. Proposition 3.3 then gives the a priori estimate for t > 0, which completes the proof.

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Dreher and Witt

For (t,£) e (-T0,0) x R n , we set above, and conclude that

*

U)»Aw(*,f)) as

for —T0 < t < 0, where X^.t1^) solves (3.4). Inverting the time direction in (3.3), we conclude from Lemma 4.3 that

-T Consequently, (4-7)

Taking squares, setting t = 0, and using
Multiplying with (£)2(s-~1) and integrating over 1RJ gives (4.5). Clearly, jm=0

for every I > 1. From ||L>^A(t,^)|| < Cp(t,0m+1 and (4.7) we then get 3_-2

/ ^

,t)->\D>tW(t,t)\ < C\W(t,®

i=0

r

^s_-2

Edge Sobolev spaces and branching of singularities

195

Squaring this relation and integration over (— T0, 0) x R" gives

which implies (4.6). The proof is complete.

D

Remark 4-4- Due to (3.2), QQ > —1/2, which is equivalent to s+ < s_. If s+ = s_, no loss of regularity occurs when we cross the line of degeneracy. The case of an hyperbolic operator with this property and countably many points of degeneracy (or singularity) accumulating at t = 0 has been discussed in [20]. The next theorem relates branching phenomena for the semilinear problem (4.1) with branching phenomena for the linear reference problem (4.2). This relation between a semilinear Cauchy problem and an associated linear reference problem has already been discussed in Example 3.8. Theorem 4.5. Let L be the operator from (1.3), QQ be the number from (3.2), and suppose that min{ |_s±J , |a±J + 0Q±Z.} > ^ - ,

|a±J + Q± > 0, s± > 1,

where Q+ = QQ, Q, = -1 - QQ, and s+ = s_ H- 0Q-1* Assume that w0 £ H3-(Rn), wi G F5-"^!71), and that / = /(«) is an entire function with /(O) = /'(O) = 0. Then there is an £Q > 0 such that for every 0 < e < £Q there are unique solutions u,v € ffa-'3+'0-g+;A((-T0,r0) x R") to (4.1) and (4.2), respectively, which, in addition, satisfy u-ve H3-+P'3++e>Q->Q+'>x((-T0, To) x R n ).

(4.8)

Proof. For the sake of simplicity, let us denote the Banach space The conditions on s± and Q± HS-,S+,Q-,Q+;\^_TQ^T^ x R n) by B imply that B is an algebra, see Corollary 2.13. Since /(O) = /'(O) = 0, we can conclude that U

\\>

\\f(u)-f(v)\\B
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Dreher and Witt

If we choose R small enough, and then choose e small enough, the usual iteration approach works, leading to a solution u to (4.1) with \\u\\B < R. From Lemma 2.11 we then deduce that f ( u ) G E C H3—l+0>3+-1+f}>Q-+l>Q++1*((-TQ,TQ)

x R n ).

The difference u — v solves L(u — v) — f ( u ) and has vanishing Cauchy data for t = -T0. Then Proposition 4.2 yields (4.8). D

References [1] K. Amano and G. Nakamura. Branching of singularities for degenerate hyperbolic operators. Publ. Res. Inst. Math. Sci., 20(2):225-275, 1984. [2] J. Bergh and J. Lofstrom. Interpolation Spaces. An Introduction, Grundlehren Math. Wiss.: Vol. 223, Springer, Berlin, 1976. [3] F. Colombini, E. Jannelli, and S. Spagnolo. Well-posedness in the Gevrey classes of the Cauchy problem for a non-strictly hyperbolic equation with coefficients depending on time. Ann. Scuola Norm. Sup. Pisa Cl Sci. (4), 10:291-312, 1983. [4] F. Colombini and S. Spagnolo. An example of a weakly hyperbolic Cauchy problem not well posed in C00. Ada Math., 148:243-253, 1982. [5] P. D'Ancona and M. Di Flaviano. On quasilinear hyperbolic equations with degenerate principal part. Tsukuba J. Math., 22(3):559-574, 1998. [6] M. Dreher. Weakly hyperbolic equations, Sobolev spaces of variable order, and propagation of singularities, submitted. [7] M. Dreher. Local solutions to quasilinear weakly hyperbolic differential equations. Ph.D. thesis, Technische Universitat Bergakademie Freiberg, Freiberg, 1999. [8] M. Dreher and M. Reissig. Propagation of mild singularities for semilinear weakly hyperbolic differential equations. J. Analyse Math., 82, 2000. [9] M. Dreher and I. Witt. Edge Sobolev spaces and weakly hyperbolic equations, submitted.

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[10] Y. Egorov and B.-W. Sdmlze. Pseudo-Differential Operators, Singularities, Applications, volume 93 of Oper. Theory Adv. Appl. Birkhauser, Basel, 1997. [11] V. Ivrii and V. Petkov. Necessary conditions for the Cauchy problem for non-strictly hyperbolic equations to be well-posed. Russian Math. Surveys, 29 (5): 1-70, 1974. [12] K. Kajitani and K. Yagdjian. Quasilinear hyperbolic operators with the characteristics of variable multiplicity. Tsukuba J. Math., 22(l):49-85, 1998. [13] O. Oleinik. On the Cauchy problem for weakly hyperbolic equations. Comm. Pure Appl. Math., 23:569-586, 1970. [14] M.-Y. Qi. On the Cauchy problem for a class of hyperbolic equations with initial data on the parabolic degenerating line. Acta Math. Sinica, 8:521-529, 1958. [15] M. Reissig. Weakly hyperbolic equations with time degeneracy in Sobolev spaces. Abstract Appl. Anal, 2(3,4):239-256, 1997. [16] M. Reissig and K. Yagdjian. Weakly hyperbolic equations with fast oscillating coefficients. Osaka J. Math., 36(2):437-464, 1999. [17] B.-W. Schulze. Boundary Value Problems and Singular Pseudodifferential Operators. Wiley Ser. Pure Appl. Math. J. Wiley, Chichester, 1998. [18] K. Shinkai. Stokes multipliers and a weakly hyperbolic operator. Comm. Partial Differential Equations, 16(4,5):667-682, 1991. [19] K. Taniguchi and Y. Tozaki. A hyperbolic equation with double characteristics which has a solution with branching singularities. Math. Japon., 25(3):279-300, 1980. [20] T. Yamazaki. Unique existence of evolution equations of hyperbolic type with countably many singular or degenerate points. J. Differential Equations, 77:38-72, 1989.

Sur les ondes superficieles de Peau et le developement de Friedrichs dans le systeme de coordonnees de Lagrange TADAYOSHI KANO Departement de Mathematiques, Universite d'Osaka, Japon SAE MIKI Departement de Mathematiques, Universite d'Osaka, Japon

1. LES ONDES SUPERFICIELLES DE L'EAU. Les ondes de surface de 1'eau ont tendance a s 'ecraser sur elles-memes et c'est ainsi non seulement sur la plage mais aussi en pleine mer. En effet, G.B.Airy a donne (non pas demontre mathematiquement) en 1848 [1] ses fameuses equations des ondes en eau peu profonde comme la seconde approximation pour les ondes de surface de 1'eau d'ampleur fmie apres la premiere approximation de Lagrange dans son livre [14] par 1'equation lineaire des ondes pour les oscillations infmitesimales : (1.1) (1.2)

u t + u u x + gy x =0 Y t +uy x +yu x = 0, y > 0 , (Airy, 1848),

ou u est la vitesse et y - y(t,x) = 0 represente le profil des ondes de surface de 1'eau. Elles s 'ecrasent, en effet, dans un temps flni; soient: (1.3)

P = u + 2(g7) 1/2 , 199

Q=u-2(gy)1/2,

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Kano and Miki

alors elles satisfont aux equations (1.4)

Pt + ((gy) 1/2+ u)P x = 0,

Qt - ((gy)l/2- u)Qx = 0,

representant des rnouvements de surface de 1'eau avec les vitesses dependantes d'apmleur des ondes, c'est-a-dire que la crete s'avance plus vite que le creux et ainsi la "tete" depasse "le corps" pour s 'ecraser sur ellememe. La decouverte de 1'onde solitaire de surface de 1'eau par J.Scott Russell en 1834[27] est done quelque chose de ties singulier et par la suite, un nombre considerable de mathematiciens eminents de 19e siecle se sont interesses aux ondes superficielles de 1'eau tels que, entre autres, G.G.Stokes, Lord Rayleigh... Airy s'est violemment oppose a 1'idee de 1'onde solitaire de Scott Russell, alors que G.G.Stokes donnait une seconde approximation de vitesse de propagation independante d'ampleur pour les ondes d'ampleur finie en 1848 [30]. L'idee de 1'onde solitaire est considerablement renforcee par la decouverte de 1 'equation de Boussinesq [2]: (1.5)

un-uxx-(3l'

et celle de Korteweg - de Vries[13]: (1.6)

u t + u x + 3uu x + (l/3)u xxx = 0,

qui admettent les ondes solitaires comme des solutions particulieres. Mais il est tres difficile de voir la structure des ondes superficielles de 1'eau a travers des equations "approchees" ainsi deduites, meme apres la demonstration mathematique de 1'existence des solutions analytiques periodiques d'ondes de surface de 1'eau par Levi-Civita en 1925[17] et ensuite par Struik egalement en 1925[31], et celle d'ondes solitaires par M.Lavrentiev en 1947[16] et Friedrichs-Hyers en 1954[6]. Or Friedrichs a propose (ou plutot decouvert) en 1948[5] une procedure systematique pour obtenir les equations d'Airy a partir d 'equations d'Euler originales pour les ondes superficielles de 1'eau en considerant le probleme

Les ondes superficieles de 1'eau

201

nondimensionnel pour le potentiel O et le profil de surface y = F(t,x) pour un ecoulement bidimensionnel: (1.7) (1.8)

(1.9) (1.10)

S 2 0 XX +O yy = 0

dans Q= |(x,y): x G R , 0 < y < F(t,x)| , Oy = 0, x G R , y = 0,

82(0t+ (1/2)0X2+ y) + (l/2)0y2 = 0, x G R , y = F(t,x) , 8 2 (F t +F x O x )- Oy = 0 , x G R , y = F(t,x).

En supposant un developpement de solutions et d 'equations en series entieres par rapport a un parametre non-dimensionnel 8 qui intervient d'une maniere naturelle dans la non-dimensionalisation du probleme representant le ratio de la profondeur moyenne h de 1'eau a la longuer des ondes A,: 5 = = hA,. Friedrichs a montre que les premiers termes de developpement satisfaisaient aux equations d'Airy. II s'est particulierement interesse a cette procedure, car les equations d'Airy sont exactement les memes que celles pour le mouvement de gaz polytropique [3]. Lui-meme n'a pas demontre la legitimite mathematique de son developpement, et c'est L.V.Ovsjannikov qui a donne une justification mathematique pour les premiers termes lorsque les ondes etaient spatialement periodiques [25]. Nous avons demontre que les solutions analytiques du probleme nondimensionnel existent et sont indefiniment differentiates par rapport a ce parametre nondimensionnel 5 et que par la suite le developpement de Friedrichs est mathematiquement legitime pour les solutions analytiques[9]. On a par consequent donne une justification mathematique pour les equations d'Airy egalement pour les solutions analytiques qui ne sont pas necessairement periodiques spatialement [8]. Le developpement de Friedrichs nous a permis de donner une justification mathematique pour 1'equation de Boussinesq et celle de Korteweg-de Vries comme des equations approchees decouvrant la structure des ondes o "longues"[10]; elles satisfont aux relations telles que les quantites e et 5 definies par

202

Kano and Miki

£ = (I'ampleur a)/(la profondeur A,), (#) 9

9

""= ((la profondeur h)/(la longuer A.))" sont de meme ordre comme infinitesimaux lorsque Ton fait tendre A vers 1'infini et a vers zero, alors que £ reste fini, 5 tendant vers zero pour les ondes en eau peu profonde d'Airy, (voir egalement F.Ursell [32]). Notons enfin que nous avons donne egalement une justification mathematique pour 1'equation de Kadomtsev-Petviashvili comme une equation approchee pour les ondes longues caracterisees en haut dans 1'ecoulement tridimensionnel [12]. Dans cet article, on donnera exactement la meme theorie que celle mentionnee plus haut, cette fois-ci dans le systeme de coordonnees de Lagrange. Les demonstrations detaillees seront publiees ulterieurement sous le titre: Ondes lagrangiennes de surface de 1'eau I, II.

2. LE PROBLEME DANS LE SYSTEME DE COORDONNEES DE LAGRANGE . Le probleme est regi par les equations suivantes: les equations dynamiques (2.1)

xttx^ + (ytt+ g)y^ + p^ /p - 0

(2-2)

XttV^tt+gVP^ 0 '

et 1' equation de continuite (2.3) dans

X

^n-V?=1>

Les ondes superficieles de 1'eau

(2.4)

203

Q 0 = Iferi): 0 < r, < h©| ,

avec les donnees initiales:

(2.5) (2.6)

xt (0£,T1) = X^TI), y t (0£,T1) = y1^);

ou p = p(t,^,T|) represente la pression, p constante et g la pesanteur.

la densite comme une

On s'explique, avant d'aborder le probleme, pourquoi on essaie de refaire les memes etudes que nous avons deja faites dans les annees 80 dans le systeme de coordonnees de Lagrange cette fois-ci. Dans le systeme de coordonnees d'Euler, il s'agissait du probleme aux limites pour 1 'equation elliptique (essentiellement 1 'equation de Laplace) avec les conditions aux bords defmies par les equations (1.9) - (1.10). Et tout le probleme etait concentre dans la fagon d'ecrire O sur la frontiere libre (ondes superficielle de 1'eau) par les valeurs de Ox sur la frontiere pour obtenir un systeme complet d 'equations sur la frontiere libre. La non-linearite est concentree sur cet operateur Dirichlet-Neumann de fonction du potentiel sur la frontiere libre. Pour une solution analytique, nous 1'avons explicite suivant Ovsjannikov par le noyau de Cauchy pour 1'ecoulement bidimensionnel et obtenu des estimations a priori uniformes par rapport au parametre 5 pour O par les valeurs de |O ,O I sur la frontiere libre y dans le cas de 1 'ecoulement tridimensionnel. On considere, dans cet expose, les ondes superficielles de 1'eau en ecoulement bidimensionnel dans le systeme de coordonnees de Lagrange ou la couche initiale est une bande, c'est-a-dire (2.4) avec h© = h: constante: (2.4)'

Q = |(£,T|): ^R, 0 < T ] < h : constante

Le cas plus general ou h = h(^) est une fonction de £ peut etre ramene au cas present si les ondes ne sont pas trop grandes, c'est-a-dire que d'apres

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Kano and Miki

1 'energie E(f) que Ton defmira ci-dessous h = h(£) est petite dans le sens qu'on ait E(h(q)-h) + E(D,(h©-h)), D, = 9 / 9 q, est petite. On va done considerer 1 'evolution temporaire de Q.Q dans (2.7)

Q t = i ( x , y ) : x G R , 0 < y < y ( t £ , h)|

d'apres les equations probleme sur une bande (2.4)"

(2.1), (2. 2)

et

(2.3). On etendra d'abord le

Q = |(5,Ti):£eR, 0 < |ri| < h : constante

en defmissant les inconnues pour T| < 0 par x(t£,ri) - x(t,^, -TI) (2.8)

y(t,^Tl) = - y(t£, -TI)

Le probleme nondimensionnel qu'on va resoudre ici s 'ecrit comme suit pour les quantites nondimensionnelles, avec prime, definies par (*)

(t,^,Ti) = ((Xyc)t', X51, hri'), (x,y,p) = (Kx\ hy', pghp'),

ou g, la pesanteur et c=(gh) 1/2 , la vitesse sonore; on ecrit les equations en laissant tomber le prime de quantites nondimensionnelles:

(2.9)

1' equation de continuite:

(2.1D

Les ondes superficieles de 1'eau

205

dans (2.12)

avec les donnees initiales

(2.13) (2.14)

x x t (0£,rj) = X^TI), yt(0£,r|) = y1^)

et les conditions aux bords (2.15)

p(t£,l)=0, y(t£,0)=0,

et enfm 1 'equation de surface sur T| = 1: (2-16) avec x l

xtt (l)x^(l) + (52ytt (1) + l)y^(l) = 0 =x t , , l

etc.

Pour obtenir les solutions analytiques du probleme (2.9)-(2.16) pour le systeme d 'equations elliptiques nonlineaires (2.9)-(2.11), on le transforme suivant M.Shinbrot [28] au probleme elliptique lineaire non-homogene pour les coefficients de developpement en series formelles de Taylor de supposees solutions. Soient en effet:

(2-17) 2

y=

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Kano and Miki

(2.17) 3

p = -(n-1) + S

() ^ n <

co ( pn(U;5) / n!) tn ,

alors on obtient les equations suivantes pour les coefficients de Taylor, d'ordre n, (X R , y n , p n )(£,Ti;8):

V - X n + w n ~ ! = 0 , n^2; D.x1 + D (5y ! ) = 0;

(2.19)

avec les conditions aux bords: p n (^,l;6) = 0, n^O; 6yn(5,0;5) - 0, n^

(2.20)

et enfm 1 'equation de la surface de 1'eau:

(2.21)

x n (l)+(l/5)D,(8y n ~ 2 (l)) + (j)n

1

(l) = 0, n^3;

x2(l)-0

ou n

(2.22)

= (x n , 5yn), V = ( D e , D ) = ( d / d ^ (1/8)9/911),

= I*2^n-! n -2 C v-2 ( x V V x n ~ V + (^X V5 y n "~ V ))> O 1 -0,

et

Les ondes superficieles de 1'eau

(2.23) w11-1

207

CPxv

En defmissant une certaine energie E(Xn) pour derivees par rapport au temps et E(DmXn), avec D m =2 0 ^ JI ^ m Dtm~^D ^ pour derivees par rapport aux variables spatiales, on obtient (voir § 3) une serie d'estimations a priori telles que:

PROPOSITION 2.1 II existe une constante C positive independante de (n,8) telle que Ton ait: E(X n )^c[E(D 2 X n ~ 2 ) + E(DXn~2) + + Z 2 ^ v ^ n _, n _ 2 C y _ 2 E(XV)(E(Xn-v) + E(DXn~v))+ (2.24) +E

+

^ v ^ j nCv E ( X ) ( E ( X ~ ) + E ( D X ~ ) ) + <;_

C

|E(DXV)E(DXn-v~2) + E(XV)(E(DXn-v~2)

PROPOSITION 2.2 II existe une constante C positive independante de (n,6) telle que Ton ait:

E(DmXn) ^ c[E(Dm+2Xn-2) + E(Dm+1Xn-2)

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(2.25)

Kano and Miki

+ 'O^u^rn mC^

m+l C ji

Pour obtenir ces estimations a priori uniformes par rapport au parametre 5 , supposons que le mouvement de 1'eau soit initialement irrotationnel: en effet, en supprimant la pression p de (2.9)-(2.10) et en integrant une fois par rapport au temps, on obtient pour la vitesse (u, 5v) = = (xt, 6yt), 1'equation suivante:

(2-26)

X

U

+ (SyCSv)- x

- (Syv

= 0,

qui donne

(2.27)

D xn - K

- r

D'ou, Ton obtient, compte tenu de 1'equation (2.19), les equations elliptiques lineaires non-homogenes pour (x n , 5yn), en supprimant pn :

Les ondes superficieles de 1'eau

(2.28)

209

D,2xn + D 2xn = f11"1 = -D (EM)11"1 - D.Vj/11"1) - D.w11"1,

(2.29)

D est a noter ainsi que les donnees initiales (x l , Sy1) doivent satisfaire aux equations (2.30)

DA1 + D^x1 = 0, D^2(5y]) + D^y1) = 0,

ce qui nous permet d'obtenir les estimations a priori uniformes par rapport a 5 par recurrence. On donnera les demonstrations dans le paragraphe suivant. On va ensuite demontrer dans § 5

la convergence de serie

2 (E(Xn)/n!)tn qui est la serie majorante pour les developpements de Taylor (2.17) 1''.9 3 Elle revient a la demonstration de la sommabilite de serie double d'estimations a priori des seconds membres dans les propositions 2.1-.2.2. Done toute la difficulte pour obtenir des estimations a priori uniformes par rapport au parametre 5 pour une solution d'equation elliptique nonlineaire a ete deplacee vers la demonstration de la sommabilite de serie double d'estimations a priori.

3. LES ESTIMATIONS PROPOSITIONS 2 . 1 - 2 . 2 .

A

PRIORI.

PREUVES

DES

Resumons d'abord ce qu'on a dit dans le precedent paragraphe: on voit que les vitesses (u,5v) = (xt, 5yt ) satisfont aux equations de continuite (2.11) et de sans-vorticite (2.26). L'equation (2.26) et une differentiation de (2.11) par rapport au temps nous donnent le systeme elliptique suivant d 'equations nonlineaires :

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(3.1)

+ x (Svt) = 0.

Et ainsi, les coefficients de series formelles de Taylor de solution (x,8y) satisfont aux equations (2.28) - (2.29).

On defmit des energies par

DEFINITION 3.1

E(DXn) et E(DmXn), m ^ 1, pour n

E((J>) pour <j>GC

ou

avec D = D* ou D 1

et oo

Et pour m ^ 1 , pour n ^ 1 :

E(Dmc{)) pour

Les ondes superficieles de 1'eau

211

+ DD<>i + ou

D'apres la transformee de Fourier, 1'equation (2.28)

se ramene a n

1 'equation integrale suivante pour la transformee de Fourier x (T|) avec la fonction de Green:

xnA(ri) = [ch(27c8kri)/ch(27i5k)]xnA(l) + 11

52/[(27i5k)sh(47i5k)][sh(27i:5k(r|-l))sh(27r5k(a+l))]fn~lA(a)da + 1 i

52/[(27i5k)sh(47i5k)][sh(2Tc5k(Ti+l))sh(27r5k(a-l))]fil~lA(o)da =

=[ch(Sri)/ch(S)]x nA (l)

52/[Ssh(2S)][

[sh(S(ri-l))sh(S(a+l))]f il ~ lA (a)da

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Kano and Miki

[sh(S(n+D)sh(S(a-l))]f n ~ lA (a)do] -

= [ch(Sr|)/ch(S)]x nA (l)-

S[5 2 /sh(2S)j[

sh(S(r|-l))sh(S(a+l))(t) n ~ lA (G)da+

+ / sh(S(ri+l))sh(S(a-l))({) n ~ lA (a)da]-

-is[5-/sh(2S)][ J shCSCri-l^chCSCa+l))^ 11 - 1 (a)da

l

sh(S(ri+l))ch(S(a-l))\|/ n - lA (a)da]-

i[8/sh(2S)][

sh(S(ri-l))sh(S(a+l))w n - lA (a)da

sh(S(r|+l))sh(S(a-l))w n - lA (a)da]

Les ondes superficieles de 1'eau

213

- <j> n ~ lA (r|) - [sh(STi)/shS](t) n ~ lA (l); ou

De meme pour 5yn, ordinaire.

S = 2:i5k, i = (-1)172

d'ou les lemmes 2.1-2.2

d'apres un calcul

4. LES ESTIMATIONS A PRIORI POUR LA PRESSION. Les coefficients equations (2.18):

pn

sont definis par

X n puisqu'ils satisfont aux

(2.18) X n + (l/8)V(8y n ~ 2 ) + Vp n ~ 2 +O n-1 = 0, n^3; X + V p ° =

c'est-a-dire:

(4.1)

D'ou, Ton obtient la proposition suivante comme les propositions 2.1-2.2:

PROPOSITION 4.1 II existe une constante positive independante de (n, 5 ) telle que Ton ait

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Kano and Miki

E(5p n ~~ 2 )^C[E(X n ) + E(X n ~ 2 ) + E(DX n ~ 2 ) + + Zo^n.! n _ 2 C v _ 2 E(X V )E(X n - v )J, E(Dm(5pn~2)) ^ C[E(DmXn)+E(Dm~' X n )+E(D m X n ~ 2 )+ +S

2^n-i n-2CV-2

5. LA CONVERGENCE DE LA SERIE

n (E(X

n

)/n!)t n .

Nous suivrons essentiellement les idees de Shinbrot dans [28]. Cette majorante pour (2.17) est convergente pour un temps suffisamment petit pour les donnees initiates X ! = (x 1 , 8y1)(^,T|) uniforrnement analytiques definies par

DEFINITION 5.1 sur la bande

<=>

X 1 = (x l , 8y l )(^,Ti) sont uniforrnement analytiques

II existe les constantes C et R positives telles que Ton ait < CR m m!, m = 0,l,2,3...

<=> etant donne un nombre q positif, il existe les constantes c et r positives telles que 1'on ait ^ cr m (m+ir q m!, m = 0,1,2,3....

Les ondes superficieles de 1'eau

215

On peut demontrer par recurrence le

THEOREME 5.2 Supposons que les donnees initiales Xl= (x 1 , Sy1)^,^) soient uniformement analytiques. Soit q > 3/2, il existe y > 0, r > 0 et 8 > 0, petite, telles que Ton ait:

(1)

(2) E(DmXn) ^ Y(m+n-l)!(n+ir q (m+n+ir q r m (r/e) n ~ 1 , m^l, n ^ l .

Ceci montre 1'existence de solution analytique pour notre probleme des ondes surperficielles de 1'eau en assurant la convergence de series (2.17)1'2'3. Soient en effet

yn!(n+l)

x

7 (m+n - l)!(n+l)~ q (m+n+l)~ q

m+n

avec

= (k- l)!

alors 1'estimation dans la Proposition 2. 1, par exemple, s 'ecrit comme:

SvSn-I n-2vc-2 -

c

c b - v + an-v n-vn-v+l vv(nn-v

""

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Kano and Miki

on voit qu'il existe des constantes Yj, y?, y^ ... etc telles que Ton ait les inegalites: ^

a

-

b

^7

£

" •etc"

qui nous donnent enfin

On a done E(DX n )^c n si Ton choisit e con venablement petit pour que Y3 + +....) soit plus petite que 1. Notons comme demiere remarque que toutes les estimations a priori sont uniformes par rapport au parametre 5, et que par consequent notre solution {x, 5y, 6p} analytique est indefiniment differentiate par rapport a 5. D'ou la legitimite mathematique du developpement de Friedrichs pour les ondes de surface de 1'eau.

REMARQUE 5.3 Plus haut, nous avons demontre qu'en fait notre solution {x, §y, 5p} satisfait aux equations comme les fonctions analytiques par rapport aux (^, §r|) qui sont indefiniment differentiables par rapport a 5 . Nous avons considere en effet le developpement de solution en series de Taylor par rapport aux (IX, D ) avec D = (1/5) d/dr\, ce qui revient a la derivation par rapport a 5r].

6. EQUATIONS APPROCHEES. Dans ce dernier paragraphe, nous donnons 1 'equation des ondes en eau peu profonde et 1'equation de Boussinesq via 1'equation de Korteweg-de Vries comme des equations approchees dans le systeme de coordonnees de Lasrange.

Les ondes superficieles de 1'eau

217

(1) L 'equation des ondes en eau peu profonde. On a d'abord le

THEOREME 6.1 La solution uniformement analytique du probleme (2.1) - (2.6) satisfait a 1 'equation x^^t. - Xtp = O(52) sur la surface de 1'eau rj = 1 , autrement dit, X(t,£) definie par x(t,£,l) = £+ X(t,£) satisfait a X t t -(l-hX,)- 1 X,,=O(5 2 ) ausensd'energie E(X).

C'est-a-dire que 1 'equation hyperbolique quasilineaire utt - (1+ UK) Wr = 0 est en fait 1 'equation d'Airy pour les solutions analytiques dans le systeme de coordonnees de Lagrange:

TTHEOREME 6.2 Le probleme de Cauchy pour utt - (1+ iit)"1^* = 0 avec les donnees initiales

u(0,^) = X(0£) = 0 et ut(0,^) = X t (0,^) =

x^O,^) admet une solution unique uniformement analytique satisfaisant a E ( U ) ^ C O < ! et qui approche X(t,^) satisfaisant a au sens que 1'on ait:

E(X(t,^)) ^C Q < 1

E(X t (t)- u t (t))-0(5), d'apres la continuite de solutions par rapport aux seconds membres.

REMARQUE 6.3

F.Ursell "approche" (1+X^r 1 par 1- X,

donne[32] "son" equation des ondes en eau peu profonde

comme 1 'equation d'Airy.

et il

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Kano and Miki

En effet, en partant de 1'equation (2.21) sur la surface, on obtient le theoreme 6.1 comme la consequence des deux propositions suivantes: d'abord

LEMME 6.4 5y n ~ l = O(5), O. (5y n ~ l ) - O(5), D (5y n ~ ] ) = O(5) D^x"'1) = 0(5), D r) (x n ~ 1 ) = 0(5).

LEMME 6.5 w n ~ 1 = O(52), n^2; (D (52) - D,\|/n~') = O(52),

On montre ces deux lemmes par recurrence en appliquant 1'un a 1'autre alternativement. Us impliquent la

PROPOSITION 6.6

2 Z nnx n ( l ) + ((f)n ^(1) = x^x, a ^ + O(5 - )

On a ensuite

LEMME 6.7

LEMME 6.8

/

D^w11 ldT\ = O(52).

(- 1/5)D (x n ~ 2 )(l) =

- (A5/5)DJxnn--22)(l) + (l/^ODcj) 11 - 1 - D - ' X I ) + 0(52),

ou Ag/5 est 1'operateur pseudo-differentiel defini par 1'image inverse de transformee de Fourier de i(th(2ic5s))/5 par rapport a s. D'ou la

Les ondes superficieles de 1'eau

PROPOSITION 6.9

219

1/2 / D,2xn~2dT] = D, 2 x n ~ 2 (l) + O(52). -l

PREUVE

On voit en effet que 1

1

DK 2 x n ~ 2 dTj = 1/2 / (-D 2xn~2 + f

1/2 / -1

-1 1

n 3 n 3 = -(1/5)DTj x n ~ 2 (l) - (l/8)(D^(j) ~ - D,\i/ ~ )(l) - 1/2 T| (^

-1

d'ou la proposition etant donnes les lemmes 6.7 - 6.8.

(2) L'equation de Boussinesq et de Korteweg-de Vries pour les ondes longues. Considerons maintenant le mouvement de deplacement moyen de surface de la surface de 1'eau en repos. C'est-a-dire qu'on etudie lafonction Y(t, ^, Tj) definie par y = T| + Y , ce qui revient a 1'etude de la surface libre de 1'eau Y = y - h, h etant la profondeur moyenne de 1'eau en repos. Soit, en outre, a 1'ampleur moyenne des ondes superficielles de 1'eau. Au moyent de variables non-dimensionnelles 1'ampleur non-dimensionnelle Y ' par

(*)

de § 2, on defmit

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Kano and Miki

y = n + Y = hy' =

= hTi' + Y = h ( T V + Y/h) = = h(n' + (cxY ')/h) = h(ri' + £¥'),

£=a/h.

D'ou y' = rj' + eY ' = ry + 52Y ' en defmissant les ondes longues (voir § 1) par le mouvement d'une surface Y '(I) defmie par

y'(l) = i + 6 2 Y'(l),

(##)

ou

y ' ( l ) = y'(t,C,,l)

etc, et la condition

(#)

"simplifiee" des ondes

longues : £ = 5~ . Correspondant a Y', on definit les autres quantites nondimensionnelles X ' et P ' par

x = A,x' - £ + 52X = _ £' T i O R^V — 1A,(^/ C, A »\J •,

p = pghp' = = pg(h-ri) + P = =

Pgh((l-TV)+§

2

P')

d'ou x'=^+82X'

et p'= (1-rO +8 2 P'.

Elles satisfont aux equations suivantes en laissant tomber le signe prime: les equations du mouvement,

Les ondes superficieles de 1'eau

(6.1)

221

X t t + (1/5)D^(8Y) + 62XttD^X + 62(5Y)ttD^(5Y) + D^P = 0,

(6.2) (6Y) tt + (1/5)D^(SY) 62XttDTiX + 522(5Y) (5Y)ttttDDTlT(( 6Y) + DP = 0; T(5Y) + S^DX

1' equation de continuite,

D^X + D^Y + 82(D^XDT1Y- D^XD^Y) = 1,

(6.3)

dans

(6.4)

Les series formelles de Taylor sont maintenant:

(6.5)2 (6.5)3

y = T 1 + 52Y = Ti + S 1 ^ n < 0 0 52(Yn(^,7i;5) / n!) tn, p = (1 - n) + 52P = -(Ti-1) + Z 0 ^ n < ^ 52(Pnft,ri;5) / n!) t11 .

D'ou Ton obtient, en minuscules comme plus haut, les equations suivantes:

(6.6)

x

(6.7) 8y n +(l/5)D (8yn""2>+-82V1"1+D^p11"^ 0,

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Kano and Miki

c'est-a-dire

D x n - Dv:(5y n ) + 52(Dn(|)n~1- D^n~{) = 0, T]

avec

(6.8)

S

M

S

(j)1 = \|/' = 0, et aussi

D.x11 + D (Sy11) + S^w11"1 = 0. n^2;

D^x 1 + D^Sy1) = 0,

avec les conditions aux bords:

p n (^, 1 ;6) = 0, n ^ 0; 5yn(^,0;5 ) = 0, n ^ 1

(6.9)

Enfm 1 'equation de la surface de 1'eau r\ - 1:

(6.10)

x n (l)+(l/5)D,(6y n ~ 2 (l))+5 2 (l) n ~ 1 (l) = ^^ ^3; x 2 (l) - 0

avec 1

(6.11) (l/5)D,(5y n ~ 2 (l)) =- (1/2) / D^xn~2dr| - (l/2)62 -1

c'est-a-dire

I

223

Les ondes superficieles de 1'eau

(1/2)8 2 J D,w n ~ 3 drj = 0.

x n (l)- (1/2)

-l

De meme que dans § 1, on a la

PROPOSITION 6.10 n

~2 - (l/3)S2DAn~2 (1) + O(54),

d'apres le

= O(52), D,wn~3(r|) = O(52).

LEMME 6.11

Cela implique alors le

THEOREME 6. 12

Sur la surface de 1'eau r| = 1, on a

= 0(54),

(6.12)

pour

x(t£) = X(t£,l).

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Kano and Miki

COROLLAIRE 6.13 Le theoreme revient a dire que u = -\r satisfait a u t t - u,p- S 2 ((l/2)u 2 + (l/3)u,,W = 0(54).

En effet, on voit que

x, K /(l + 5 2 x.) - (1 - 5 2 x.) x,t - 5 4 x, 2 x t t / ( l + SV) + O(84).

d'ou le corollaire. Mais il ne s'ensuit pas immediatement une justification mathematique pour 1'equation de Boussinesq comme une equation approchee des ondes longues superficielles de 1'eau. En effet, 1'equation de Boussinesq

Utt- U -

62((1/2)U2+ ( 1 / 3 ) U )

=0

a une " mauvaise" relation de dispersion. Nous avons en fait le

THEOREME 6.14

Les ondes longues de surface de 1'eau

approchees par les solutions F(t, ^) Korteweg - de Vries

et

G(t, £)

sont

des equations de

Ft - F,- 52(FF^+ (l/3)F^p) - 52R° , G t + G,+ 5 2 (GG,+ (l/3)Gtte) - S2Q° ,

satisfaiant aux conditions initiales

x

Les ondes superficieles de 1'eau

225

F(0) = x ( 0 , 5)+x(0, 5) et G(0) = x ( 0 , ^)-x(0, 5) avec

RO = -( u O v O +(1/ 2)(v°) 2 +(l/3)v 0 ,,),, Q°=-(u0v0-(l/2)(v°)2+(l/3)v0,,),,

ou

u° = x°t , v° = x° t , x° etant la premiere approximation

x°tt - x°tK = 0

pour

sur Tj = 1

(6.1)-(6.3), avec des donnees initiales

x°(0, £) = x(0, £) et x°t(0, Q = xt(0, ^)

satisfaisant a E((x°t - x t )(t)) - 0(52), E((x° - x ) ( t ) ) - 0(52)

PREUVE

Eneffet, (6.12)

s 'ecrit comme suit:

ut - vc= 0, v t - ut- 82(uu,+ ( l/3)ut t e) = O(54) avec

U = X K , v = xt

et des donnees initiales

u(0, 5) = x £ (0, 5) = X,(0, ^ 1), v(0, ^) = x t (0, ^) = X

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Kano and Miki

Alors, f = u+v, g = u-v

satisfait, respectivement, a 1 'equation suivante:

f t - ft- 5 2 (ff,+ (l/3f t pe) - 52R+ 0(54), g t + g,+ 62(gg^+ (l/3)g^) = 5 2 Q+ 0(54), avec R = -(uv + (l/2)v 2 +(l/3)v,K)K, Q = -(uv - (l/2)v 2 +(l/3)v, ,), .

ss s

ss s

On a done E(f(t) - F(t)) = 0(54) et E(g(t)-G(t)) =

d'apres la dependance continue de solutions des seconds membres, vu que R(t) - R°(t) = O(52), Q(t) - Q°(t) = O(52). D'ou le theoreme, puisque 1'on a des equations

u(t) = (l/2)(f+g)(t) = (l/2)(F+G)(t) + 0(54), v(t) = (l/2)(f-g)(t) - (l/2)(F-G)(t) + 0(54);

c'est-a-dire que Ton obtient

E(2x,(t) - (F + G)(t)) - 0(54) et E(2\ t (t) - (F - G)(t)) = O(54).

<;

Pour clore, remarquons que Ton retrouve le meme resultat que celui dans [10], page 405, la proposition 3.1: 1'equation de Koreteweg-de Vries est justifiee mathematiquement comme 1'equation approchee pour les ondes longues de surface de 1'eau dont les donnees initiales satisfont aux conditions E(f(0)) = O(l) et E(g(0)) = O(S2). C'est-a-dire que Ton a la

Les ondes superficieles de 1'eau

227

PROPOSITION 6.15 Supposons que les donnees initiales satisfassent aux conditions suivantes:

(##)

E(xJO, Q + xt(0, £)) = 0(1),

E(x,(0, 0 - xt(0, 5)) = O(52).

Alors les ondes longues de surface de l'eau sont approchees par les solutions des equations de Korteweg-de Vries suivantes:

Ft - Ft- 52((3/4)FF,+ (l/2)Fs «) = 0 , G t + GK+ 52((3/4)GG,+ (l/2)Gtst) = 52(-(3/4)FFK +

satisfaisant aux conditions initiales F(0) = F(0) et G(0) = G(0), au sens que Ton ait

E(f(t) - F(t)) = 0(54) et E(g(t) - G(t)) = O(54) .

PREUVE

D'apres les definitions de

§-Q s 'ecrivent comme

52R = 52(§2Q = 52

Par consequent, on obtient d'abord

f et g, on voit que 52R et

228

Kano and Miki

f t - f,- 5 2 ((3/4)ff,+ (l/2)f w ) - 0(54), puisque 1'on a g(t) = O(5~) d'apres la dependance continue des solutions des donnees initiates et des seconds membres d'equation pour g. Ainsi f(t) - F(t) satisfait a 1'equation linearisee non-homogene de KortewegdeVries avec un second membre d'ordre O(54) et la donnee initiale nulle. D'ou Ton obtient f(t) = F(t) + O(54). De meme, on obtient g(t) = G(t) + + O(54) puisque Ton a gt + g,+ 5 2 ((3/4)gg,+ (l/2)gttt) = 52(-(3/4)ff, + (l/6)f,,,) + O(54),

s

s

sss

compte tenu du fait que E(f(t) - F(t)) = O(54) D'ou la proposition, puisque Ton a

s

et

g(0) - G(0) = 0.

u(t) = x,(t, ^ = (l/2)(f + g)(t, fy = (l/2)(F(t, ^) + G(t, 0)+ 0(54), v(t) = x t (t, Q = (l/2)(f- g)(t, Q = (l/2)(F(t, Q - G(t, ^))+ 0(54).

Bibliographic

[1] G.B.Airy:Tides and wave, B.Fellowes, London, 1845, 241-396. [2] J.BoussinesqiTheorie des ondes, J.Math.Pure.Appl., 2e serie, t.17 (1872), 55-108. [3] R.Courant-K.-O.Friedrichs:Supersonic flow and shock waves, Springer, 1948 and 1976. [4] W.Craig:An existence theory for water waves and the Boussinesq and Korteweg-de Vnes scaling limits, Comm.in PDE, 10(1985), 787-1003.

Les ondes superficieles de 1'eau

229

[5] K.-O.Friedrichs:On the derivation of the shallow water theory, Appenndix to: "The formation of breakers and bores " by J.J.Stoker in Comm.PureAppl.Math., 1(1948), 1-87. [6] K.-O.Friedrichs-D.H.Hyers:The existence of the solitary waves, Comm. Pure Appl. Math., 7(1954), 517-550. [7] :Asymptotic phenomena in mathematical physics, Bull.AMS, 61(1955), 485-504. [8] T.Kano-T. Nishida:Sur les ondes de surface de 1'eau avec une justification mathematique des equations des ondes en eau peu profonde, J.Math.Kyoto Univ., 19(1979), 335-370. [9] :Water waves and Friedrichs expansion., Lect.Note. Num.Appl.Anal., Kinokuniya-North Holland, 6(1983), 39-57. [10] :A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves, Osaka J. Math., 23(1986), 389-413. [11] T.Kano:Une theorie trois-dimensionnelle des ondes de surface de 1'eau et le developpement de Friedrichs, J.Math. Kyoto Univ., 26(1986), 101-155 et 157-175. [12] T.Kano:L'equation de Kadomtsev-Petviashvili approchant des ondes longues de surface de 1'eau en ecoulement trois-dimensionnel., Studies Math.Appl., 18: "Patterns and Waves-Qualitative analysis of nonlinear differential equations ", North Holland, 1986, 431-444. [13] D.J.Korteweg-G.de Vries:On the change of the form of long waves advancing in a rectangular canal and a new type of water surface waves, Phil. Magazine, 39(1895), 422-443. [14] L.Lagrange:Mecanique analytique, t.2, Mme Ve Courcier, Paris, 1815. [15] H.Lamb:Hydrodynamics,6th ed., Dover, 1932. [16] M.Lavrentiev:On the theory of long waves, Recueil des Travaux de 1'Institut Mathematique de 1'Academie des Sciences de la RSS d'Ukraine, 1946, no. 8, 13-69(1947).

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[17] T.Levi-Civita:Determination rigoureuse des ondes permanentes d'ampleur fmie, Math.Annalen 93(1925), 264-314. [18] VI.Nalimov:Cauchy-Poisson problem, Continuum Dynamics, Hydrodynamics Institut, Siberian Section, URSS Akad. Nauk, 18(1974), 104210. [19] :A priori estimates of solutions of elliptic equations in the class of analytic functions and their applications to the Cauchy-Poisson problem, Soviet Math.Dokl., 10(1969), 1350-1354. [20] L.Nirenberg:An abstract form of the nonlinear Cauchy-Kowalevski theorem, J.Diff.Geometry, 6(1972), 561-576. [21] T.Nishida:Anote on a theorem of Nirenberg, J.Diff.Geometry, 12 (1977), 629-633. [22] T.Nishida:Nonlinear hyperbolic equations and related topics in fluid dynamics, Publications Mathematiques d'Orsay, Universite Paris-Sud, Departemerit de Mathematiques, Orsay, 1978. [23] T.Nishida:Equations of Fluid Dynamics - Free Surface Problems, Comm. Pure Appl.Math., 39(1986), 221-238. [24] L.V.Ovsjannikov:Probleme de Cauchy nonlineaire dans 1'echelle des espaces de Banach, Dokl.Akad.Nauk URSS, 200(1971), 789-792. [25] :To the shallow water theory foundation, Arch. Mech.,(Varsovie), 26(1974), 407-422. [26] :Cauchy problem in a scale of Banach spaces and its application to the shallow water theory justification, Lect.Note in Math., Springer-Verlag, 503(1976), 426-437. [27] J.Scott Russell:Report on waves, in "Report of the fourteenth meeting of the British Association for the Advancement of Science held at York in September 1844", John Murray, London, 1845, 311-390. [28] M.Shinbrot:The initial value problem for surface waves under gravity, I.The simplest case, Indiana Univ.Math.J., 25(1976), 281-300.

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231

[29] J.J.Stoker:Water waves, the mathematical theory with applications, Interscience, New York, 1957. [30] G.G.Stokes:On the theory of oscillatory waves, Trans.Camb. Phil. Soc., 8(1847), 441-455. [31] D.J.Struik:Determination rigoureuse des ondes irrotationnelles periodiques dans un canal a profondeur finie, Math.Annalen, 95(1925), 595-634. [32] F.UrselliThe long wave paradox in the theory of gravity waves, Proc. Phil. Soc. Cambridge, 49(1953), 685-694. [33] L.C.WoodsrThe theory of subsonic plane flow, Cambridge University Press, 1961. [34] H.Yosihara:Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ.RIMS, Kyoto Univ., 18(1982), 49-96. [35] :Gravity waves on the free surface of an incompressible perfect fluid of finite depth (Revised unpublished manuscript), april 1982. [36] :Capillary-gravity waves for an incompressible ideal fluid, J.Math.Kyoto Univ., 23(1983), 649-694.

Probleme de Cauchy pour certains systemes de Leray-Volevich du type de Schrodinger JIRO TAKEUCHI Faculty of Industrial Science and Technology, Science University of Tokyo, Campus Oshamanbe, Hokkaido, 049-3514, Japan e-mail : takeuchiOrs. kagu. sut . a c . j p

Dedie au Professeur Jean Vaillant

Resume. Nous donnons des conditions suffisantes et des conditions necessaires afin que le probleme de Cauchy pour certains operateurs du type de Schrodinger soit bien pose dans un espace de Frechet H°°. Les operateurs qui concernent sont certains systemes de Leray-Volevich d'operateurs differentiels lineaires. La methode que nous utilise dans cet article est une L2-symetrisation independante du temps d'operateurs a la perte de regularite; c'est une generalisation de la L 2 -symetrisation independante du temps d'operateurs sans perte de regularite obtenue par Takeuchi ([64]). 233

234

Takeuchi

Abstract. We give sufficient conditions and necessary conditions for the Cauchy problem for certain operators of Schrodinger type to be well posed in a Frechet space /f 00 . Operators of which we treat are certain Leray-Volevich systems of linear partial differential operators. The method that we use in this article is a time-independent L 2 -symmetrization of operators with loss of regularity which is a generalization of the time-independent L 2 -symmetrization of operators without loss of regularity obtained by Takeuchi ([64]). Mathematics Subject Classification 2000 : 35G10. 35Q40 Keywords and phrases : Cauchy problem, Schrodinger type equation, Leray-Volevich system of Schrodinger type. Time-independent L 2 -symmetrization of operators

1

INTRODUCTION ET ENONCE DES RESULTATS

1.1 Conditions suffisantes — On considere un systeme d'equations aux derivees partielles : P(x, Dx, Dt)u(x, t} = Dtu(x, t) - A(x, Dx)u(x, t) = 0, ou x = ( x ' i , . . . , xn] 6 M n , t € R 1 , Dj; — (1 < j < n), Dt — i~ld/dt, A(x, Dx] — (ajjt(x, Ac)) est une matrice m x m dont la (j, fc)-composante o^x, A) est un operateur differentiel lineaire d'order r]k a coefficients /3°°(Rn) et u(x, t) = f ( u i ( x , t ) , . . . , u m (x, t)) est un vecteur d'inconnues. On impose une hypothese a priori sur les ordres TJ^ des operateurs a,jk(x, Dx] Hypothese (L-V) — II existe une suite { , s i , . . . , sm} de m entiers non negatifs telle que Ton ait r]k < s^ — Sj + 2 (j, k — 1 , . . . , m). D'apres Volevich [66], il est toujours possible de trouver de telles suites {si,. . . , s m } si Ton suppose que max cr

—

r a(

< 2,

m ^

ou a parcourt toutes les permutations de {!, Cette formulation de (si, . . . , s m } est due a Mizohata [39] ; un tel systeme Dtl — A(x, Dx) est un des systemes de Leray-Volevich (voir Leray [34], Volevich [66], [67], Garding-Kotake-Leray [12], Hufford [18], Wagschal [68] et Miyake [37]).

Systemes de Leray-Volevich du type de Schrodinger

235

Si une suite (si, . . . , sm} de ra entiers satisfait a la condition rjk < Sfc — Sj+1 (j,k = 1, . . . , ra), alors, pour tout entier p, {si + p, . . . , sm + p} satisfait a la meme condition. Done, on peut choisir comme Sj (1 < j < ra) un entier non negatif et le plus petit satisfaisant a la condition r^ < Sk — Sj + 2 (j,A: = l,...,ra). Desormais, on fixe {si, . . . , sm} dans I'hypothese (L-V). Dans cette situation, 1'operateur 0^(0;, Dx) peut s'ecrire comme suit : (1.2)

ajk(x, Dx) =

]T

ajka(x)DS

(1 < j, k < ra),

— S+2

avec les coefficients djka(x) € jB°°(En), ou ajfc(x, Dx) = 0 si Sk - Sj + 2 < 0. Nous aliens donner des conditions afin que le probleme de Cauchy pour le futur et pour le passe en meme temps , . (*)

(Dtu(xtt)-A(x,Dx)u(x1t) = f ( x t t ) ^ I ii(x, 0) = UQ(X) dans Rn

sur Rn x [-T,T] (T > 0)

soit bien pose dans #°°(Rn) = DEFINITION 1.1 — (1) On dit que le probleme de Cauchy (*) pour le futur et pour le passe en meme temps est bien pose dans H00^71} si et seulement si pour tout UQ(X) 6 (#°°(Rn))m et tout f ( x , t ) G (C^([-T,T]; tf°°(Rn)))m, il existe une solution unique u(x,t} du probleme de Cauchy (*) telle que u(x,t) e

(C™([-T,T]-H°°(Rn)}r

et que de plus on ait 1'inegalite d'energie suivante : pour tout / £ R 1 , il existe /' E R1 et une constante C(/,/',T) qui ne dependent pas de u(t) tels que

ol

, Uj(t) = Uj(-,t) est la j-composante de u(t} = u(-,t] et | |itj (£)||({) est la J /"i(Rn)-norme de Uj(t). Si on peut prendre /' = /, on dit que le probleme de Cauchy (*) est bien pose dans Hl(Rn) pour tout / 6 R1. c

236

Takeuchi

(2) Soit KQ un nombre non negatif. On dit que le probleme de Cauchy pour le futur et pour le passe en meme temps (*) est bien pose dans H°°(W1)^ a la perte de regularite KQ pres, si et seulement si pour tout UQ(X) G (HCG(En})m et pour /(x, t) = 0, il existe une solution unique u(x,t) du probleme de Cauchy telle que

et que de plus, pour tout I G R, on ait 1'inegalite d'energie suivante :

ou C(Z,T) est une constante positive qui ne depend que des constantes I et T et de 1'operateur P(x,Dx,Dt). Dans le cas ou KQ — 0, on dit que le probleme de Cauchy (*) est bien pose dans # z (R n ) pour tout I € R 1 . Nos conditions s'expriment comme suit. La premiere condition pour la partie principale est : CONDITION (A.I) — On a ajka(x) — a-jka (Constante) pour a Sj + 2, 1 < j, k < m. On note a° fc (£) le symbole principal de 1'operateur ajk(x., Dx] : (1-3)

et on pose A2(£) = (a?A;(0) Qui s'appelle la matrice caracteristique. La deuxieme condition pour la partie principale est : CONDITION (A. 2) — Les valeurs propres de la matrice caracteristique A2(0 sont non nulles, reelles et distinctes pour ^ e Rn \ 0 :

A?(0 ^ Aj(0 (3 * k, ^ 0) ; Aj(0 ^ 0 (^ 0, 1 < j < m). N.B — La valeur propre A°(^) est positivement homogene de degre 2 en pour £ € Mn \ 0. La condition (A. 2) et 1'identite d'Euler impliquent que

Systemes de Leray-Volevich du type de Schrodinger

237

0) i.e., V^°(0 ^ 0 « ^ 0).

DEFINITION 1.2 — On dit que un operateur Dtl - A(x, Dx) satisfaisant a I'hypothese (L-V) est un systeme de Leray-Volevich du type de Schrodinger si et seulement si les valeurs propres de la matrice caracteristique A%(£) sont reelles pour tout £ € En. Afin que le probleme de Cauchy pour 1'operateur Dtl — A(x, Dx] pour le futur et pour le passe en meme temps soit bien pose dans #°°(]Rn) au sens de la definition 1.1, il est necessaire que les valeurs propres de la matrice caracteristique A%(x, f ) soient reelles pour tout (x, ^) G Rn xR n meme si les coefficients de ^2(0;, £) dependent de x (Takeuchi [51], voir aussi Petrowsky [44], Mizohata [38]). Pour caracteriser 1'operateur du type de Schrodinger, il faut considerer non seulement le probleme de Cauchy pour le futur ou pour le passe, mais aussi le probleme de Cauchy pour le futur et pour le passe en meme temps ; c'est pour exclure les systemes {{ paraboliques }} . Pour uj E. Sn~1, on note $(&} le zero-vecteur a gauche de (A°(cj)/— et r®(uj) le zero-vecteur a droite, c'est-a-dire que l^(uj) est une matrice 1 x m et r®(uj) une matrice m x 1 telles que )) = 0, (A ; M/ - A 2 (o;))r(a;) - 0. Pour £ € Rn \ 0, on definit ^(0 et r°(£) comme suit :

Et de plus, on impose la condition suivante (1.4)

#(0^(0 = 1, ^ €

d'ou (1.5) d'apres la condition (A. 2). On note oL.(z, £) le symbole sous-principal de 1'operateur d j k ( x , D x )

Takeuchi

238

a-k(x^) =

et on pose AI(X,£] = (a 1 fc (x,^)) qui s'appelle la martice sous-principals. Remarquons que /j(£)Ai(:c,£) r j(0 est une scalaire. DEFINITION 1.3 — Soit G(x) est une fonction continue dans Rn a valeurs reelles. On dit que G(x] est un poids modere si G(x) satisfait aux conditions suivantes : (1) 0 < G(x) < CQ(x) (Co > 0) ou (x) = (1 + xj 2 ) 1 / 2 ; (2) (sous-multiplicativite) G(x + y) SQ(X} £\ , ou si la definition 1.3, tels que Ton ait KJ(X, 0 < <$o < 1, et que !,w;

sup

Im

ds

<

+00,

ou Sn~l est la sphere unite dans Rn et n3(x,u) = CONDITION (B.2) — Pour tout j (1 < j < m) et pour tout multi-indice v (\v > 1), on a sup

ds <+00.

(x,u;)€M n x5 Tl

CONDITION (B.3) — Pour tout j (1 < j < m) et pour tout multi-indice v, on a

Systemes de Leray-Volevich du type de Schrodinger

sup

239

{{x>D;(lm/(o;)Ai(x,a;)r( W ))|} < +00.

CONDITION (B.4) — Pour tout j (1 < j < m) et pour tout multi-indice z/ (H > 1)? on & sup

{(x)\DZ (RelJ(e4Ai(*,w)r?M)|} < +00.

Les conditions (B.I) a (B.4) ne dependent pas du choix du zero-vecteur a gauche et du zero-vecteur a droite satisfaisant (1.4). Nos resultats s'enoncent ainsi : THEOREME 1.3 — Supposons les conditions (A.I), (A.2), (B.I) d (B.4) verifiees. Alors, le problems de Cauchy (*) pour le futur et pour le passe en meme temps est bien pose dans H"°°(Rn), d la perte de regularite KQ pres, ou « o ; ; u 6 S™"1,! < <m. COROLLAIRE 1.4 — Supposons que les conditions (A.I), (A.2), (B.I) avec KJ(U] = 0 (1 < j < m), (B.2) d (B.4) soient verifiees. Alors, le probleme de Cauchy (*) pour le futur et pour le passe en meme temps est bien pose dans -H"'(Rn) pour tout I G R1 au sens de la definition 1.1. Pour demontrer le theoreme 1.3, nous proposons une methode de (( L2symetrisation independante du temps d'operateurs a la perte de regularite}). Cette methode est comme suit : D'abord, on diagonalise ce systeme ; ensuite, on reduit ce systeme diagonal a un systeme diagonal et L2-symetrique par un operateur pseudo-differentiel independant du temps; nous le verrons a la section 2. Notre resultat principal est le theoreme 2.9. Ce resultat de I/2-symetrisation independante du temps est plus fort que celui du theoreme 1.3.

1.2 Conditions necessaires — Ann d'obtenir des conditions necessaires, on impose la condition suivante plus faible que la condition (A.2): CONDITION (A.2)' — Les valeurs propres A?(£) (1 < j < m} de la matrice caracteristique ^2(0 sont reelles et distinctes pour £ G Rn \ 0.

240

Takeuchi

N.B — La valeur propre A^(£) peut s'annuler pour certains £ E W1. Mais, d'apres 1'identite d'Euler, on a sur

La condition (B.I) implique la condition suivante : CONDITION (B.I)' — II existe une constante non negative KQ et une constante positive C telles que, pour tout p G M 1 , on ait sup <~]TJTl vx OTI— 1

Nos resultats s'enoncent ainsi: THEOREME 1.5 — Supposons les conditions (A.I) et (A.2)' verifiees. Alors, afin que le probleme de Cauchy (*) powr /e /w£ur et pour le passe en meme temps soit bien pose dans H°°(Wl) au sens de la definition 1.1, il est necessaire que la condition (B.I) 7 soit verifiee. COROLLAIRE 1.6 — Supposons les conditions (A.I) et (A.2)' verifiees. Afin que le probleme de Cauchy pour le futur et pour le passe en meme temps (*) soit bien pose dans H^W1} pour tout I G M 1 , il est necessaire que la condition (B.I)' avec KQ — 0 soit verifiee. Pour demontrer le theoreme 1.5, on construit des solutions asymptotiques du probleme de Cauchy (*) selon Takeuchi [52], [63] (voir aussi Birkhoff [2], Leray [35], Maslov [36] et Mizohata [40] a [43]). Nous donnerons la demonstration du theoreme 1.5 dans un autre article parce que la demonstration du theoreme 1.5 est assez longue. Le theoreme suivant est une consequence du theoreme 1.5. THEOREME 1.7 — Supposons les conditions (A.I) et (A.2)7 verifiees. De plus, supposons aussi qu'il existe j & {1,..., m}, u 6 S"71"1 et CQ > 0 tels que, pour tout x 6 W1, on ait

Systemes de Leray-Volevich du type de Schrodinger

241

Alors, une condition necessaire afin que le problems de Cauchy pour le futur et pour le passe en meme temps (*) soit bien pose dans H°°(W1} est que

COROLLAIRE 1.8 — Supposons les conditions (A.I) et (A.2)' verifiees. De plus, supposons aussi qu'il existe j G {!,••• ,m}, o> G 5n-1 et CQ > 0 tels que, pour tout x G R n , on ait

Alors, une condition necessaire afin que le probleme de Cauchy pour le futur et pour le passe en meme temps (*) soit bien pose dans Hl(1$in) pour tout I G R1 est que 6 > 1. 1.3. Notes historiques — Takeuchi [49], pour la premiere fois, a traite du probleme de Cauchy pour 1'operateur de Schrodinger avec un potentiel vectoriel a valeurs complexes dans le cas ou n = I (voir aussi Takeuchi [50], [51]). Mizohata ([40] a [42]) a donne des conditions suffisantes et des conditions necessaires, qui sont le point de depart de nos etudes, afin que le probleme de Cauchy (*) soit bien pose dans L2 (voir aussi Mizohata [43]). Apres Mizohata ([40] a [42]), Ichinose ([19] et [20]) a considere le probleme de Cauchy pour le meme operateur dans le cadre H°°. Takeuchi ([52], [55] a [57]) a aussi considere le probleme de Cauchy pour 1'operateur du type de Schrodinger dans le cadre H°°. Baba ([!]) a traite du probleme de Cauchy dans H°° par la methode un peu differente. Kara [13], qui generalise le resultat d'Ichinose [19], a donne une condition necessaire afin que le probleme de Cauchy pour 1'operateur du type de Schrodinger soit bien pose dans H°°. Tarama [65] a donne d'autres considerations sur le probleme de Cauchy pour 1'operateur du type de Schrodinger. Doi ([8], [9]), dans lequel il a traite d'un operateur plus general, a ajoute une nouvelle consideration (dite des ({ effets regularisants }) de solutions) importante pour quelques equations non lineaires dispersives et du type de Schrodinger (voir Constantin-Saut [4], [5], Hayashi-Nakamitsu-Tsutsumi [14], Kato [30], [31], Kenig, Ponce et Vega [32], Yajima [69]). Kajitani et Baba [29] a traite du probleme de Cauchy pour 1'operateur du type de Schrodinger dans la classe de Gevrey. Ichinose ([21] a [23]) a traite du probleme de Cauchy pour 1'operateur du type de Schrodinger dans une variete riemannienne (Voir aussi Ichinose [24] et [25]). Kajitani [28] a traite du probleme de Cauchy dans le cadre H°° pour un operateur plus general.

242

2

Takeuchi

L2-SYMETRISATTON INDEPENDANTE DU TEMPS

2.1 Diagonalisation du systeme — Soit A(x,Dx) = (ajk(x,Dx)) est la matrice m x m d'operateurs differentials lineaires definie dans la section 1. On considere un systeme : (2.1)

Dtu(x, t) - A(x, Dx)u(x, t) = /(z, t).

On note J(£) le symbole de 1'operateur pseudo-differentiel diagonal J(DX) : (2-2)

J(0

En not ant

u(x,t} — A(x,Dx] =

J(Dx}A(x,Dx)J(Dx}-\

on a un systeme suivant : (2.3)

Dtu(x, t) - A(x, Dx)u(x, t) = f ( x , t}.

On note (2.4)

A(x, Dx] = A2(DX] + Ai(x, Dx)

ou

A2(DX) = AifoA,) - J(Dx}Al(x,Dx}J(Dx}-1. Le symbole A
A 2 (0

Le symbole AI(X,£) de 1'opereteur Ai(x,Dx} s'exprime comme suit :

E ou

Systemes de Leray-Volevich du type de Schrodinger

243

Comme 1

€ Mat (m;S?,0(Rn)) ,

on a

(2.6)

A!(X,O 1

LEMME 2.1 — Soft a x

™ WO

eS R

(mod Mat(m;5? >0 (R n )))

n

.

Alors

°n a

€ C°°(R»), xoo(0 = 1 (lei > 1), Xoc(0 = 0 (|£| < 1/2).

Demonstration : — Voir le lemme 3.12 de Takeuchi [63].

D

D'apres (2.5), (2.6) et le lemme 2.1, on a

(modMat(m;S'? )0 (E n ))). Grace a la condition (A. 2), on peut diagonaliser le systeme (2.3) : PROPOSITION 2.2 (Diagonalisation parfaite) — Supposons que les conditions (A.I) et (A. 2) soient verifiees. Alors, il existe un operateur pseudodifferentiel diagonal D(x, Dx] £ Mat (m; OP52)0(Rn)) et un operateur pseudo-differentiel inversible N(x,Dx) G Mat (m; OP5f )0 (R n )) tels que (1) I'on ait I 'equation suivante : (x, DX}J(DX] (Dt - A(x, Dx})J(DxrlN(x, Ac)'1 = (A - T>(x, Dx))

(mod Mat (m; ou d est un entier plus grand que 2«o et KQ = max{|«- (cj)| ; a; G 5n~1, 1 < j < m}, (2) en notant ar(D)(x,£) on ait (2.7)

= (Sjk^k(x>£))

Aj-(x,fl = A ( 0 + A ( x , 0

IG symbole de I'opereteur V,

244

Takeuchi

OIL

(2.8) (2.9)

oo(0 € Mat (m; S^IT)) , /e zero-vecteur a gauche de

et

/e zero-vectew d

> 1),

- 0

< 1/2).

Demonstration : — En repetant le meme procede de la preuve de la proposition 3.13 de Takeuchi [63], on a la proposition 2.2. D 2.2 Z/2-symetrisation independante du temps du systeme — Dans cette sous-section, on toujours suppose les conditions (A.I), (A.2), (B.I) a (B.4) verifiees. On definit le symbole K(x,^} de 1'operateur pseudo-differentiel K(x,Dx) comme suit: K(x. £) = diag (k\(x, ^ ) , . . . , km(x, ^)),

ou X(0 G C°°(R"), x(0 = 1 (Kl > 2), X(0 - 0

< 1),

^(s) = 1 (s< 1), ^(s) = 0 (s > 2) et J? > 1. D'abord, on a les lemmes suivants. LEMME 2.3 — Supposons les conditions (A.I), (A. 2) et (B.I) verifiees. Alors, pour 1 < j < m, on a les estimations :

Re ^(1,

Systemes de Leray-Volevich du type de Schrodinger

245

ou KQ = max{|«t(u;)|, Kj(u)\ ; u G 5n-1, 1 < j < m}, la fonction est definie dans la condition (B.I) et C est une constante positive.

Kf(u)

LEMME 2.4 — Supposons les conditions (A.I), (A.2), (B.I) a (B.3) verifiees. Alors, pour 1 < j < m, \a\ + \(3\>letR>l,ona les estimations:

ou Ca@ est une constante positive independante de R. En combinant le lemme 2.3 et le lemme 2.4, on a la proposition suivante. PROPOSITION 2.5 — Supposons les conditions (A.I), (A.2), (B.I) A (B.3) verifiees. Alors, pour 1 < j < m et R > 1, on a les estimations:

ou KQ = max { | «^" (a;) |, |«~(a;)| ; a; € 5n-1,l < j < m}, et Cap est une constante positive independante de R, c'est-d-dire que k j ( x , £ , t ) G S^CM.71} uniformement pour R > 1 . PROPOSITION 2.6 — Supposons les conditions (A.I), (A.2), (B.I) A (B.3) verifiees. Alors, si R est suffisamment grand, I'operateur K(x,Dx] est inversible dans Mat (m; OPS^oO^71)); ou KQ = max{|«t(o;)|, K~(U)\;U £

D'apres le theoreme de Calderon-Vaillancourt [3], les operateurs K(x,Dx] et K(x,Dx)~1 sont continus de (fP(R n )) m dans (H s - /e °(R n )) m . L'equation du lemme suivant etait le point de depart du probleme. LEMME 2.7 — Supposons les conditions (A.I), (A.2), (B.I) A (B.3) verifiees. Alors, le symbole kj(x,£) de I'operateur kj(x, Dx] satisfait a I 'equation suivante : (V^A°(0 • Dx - ilm A)(x,0)^-(x,0 = 0 (mod S^IT)). La proposition suivante est au cosur du probleme. PROPOSITION 2.8 — Supposons les conditions (A.I), (A.2), (B.I) A (B.4) verifiees. Alors, on a K(x,Dxrl(K(x,Dx),V^(x,Dx)}

6 Mat (m; OPS£0(IRn)) ,

246

Takeuchi

ou [A, B] = AB — BA est le commutateur d'operateurs A et B,

le symbole A](x,£) est defini par (2.7) d (2.9). En resume, nous obtenons le theoreme suivant qui est le but de nos etudes : THEOREME 2.9 (L2-symetrisation) — Supposons les conditions (A.I), (A. 2), (B.I) d (B.4) verifiees. Alors, il existe un operateur pseudo-differentiel matriciel K(x, Dx] tels que, (1) K(x,Dx) appartienne d Mat (m; (2) K(x,Dx) soit inversible dans Mat (m;OPSo° 0 (R n )), (3) K(x,Dx} satisfasse a I' equation suivante: K(x, Dx}~lJ(Dx}(Dt - A(x, Dx)}J(Dx}-lK(x, Dx) (mod Mat (m- OP5j i0 (R n ))), ou KQ = max{j«:t(a;)j, |«J(o;)| ; uj 6 5 n-1 ,l < j < m}, le symbole J(£) de I' operateur J(DX] est defini par (2.2), D R (x, Dx) = diag (A* (x, D x ), • • • , A* (z, Dx))>

ei /e symbole Aj(x,^) esi defini par (2.7) d (2.9). Demonstration : — On prend H assez grand pour que les conclusions des propositions 2.6 et 2.8 sont satisfaites. En combinant les propositions 2.5, 2.6 et 2.8, on a la conclusion du theoreme 2.9. D 3

PREUVE DES THEOREMES 1.3 ET 1.7

3.1. Preuve du theoreme 1.3 — Dans cette sous-section, on toujours suppose les conditions (A.I), (A. 2), (B.I) a (B.4) verifiees. En appliquant JV(x, DX)J(DX) 2.2,

a (2.1) a gauche, on a, d'apres la proposition

{Dt - V(x, Dx) + £_ d (z, Dx)}v(x, t) = N(x, Dx)J(Dx)f(x, t),

247

Systemes de Leray-Volevich du type de Schrodinger

ou v(x,t) = N(x,Dx)J(Dx)u(x,t),

B.d(x,Dx) G Mat (m;

Le theoreme 2.9 implique K(x, Dx)~l(Dt - £>(x, DxM(z, Dx) = A - £>*(*, Dx) + B0(z, D x ), ou B0(z, Dx) 6 Mat (m; OP5gi0(Rn)). En posant v(x,t) — K(x,Dx}w(x,t] et vu que d > 2«o, on a un systeme suivant : (^, t) = K(x, Dx}J(Dx}f(x, t),

(A - £>*(x,

ou

= ^(x, Dx}-lB-d(x, Dx)K(x, Dx)+B0(x, Dx) € Mat (m; K(x, D x ) = tf(z, Dx)-1^^, Dx) 6 Mat (m; Le probleme de Cauchy pour 1'operateur Q(x, Dx, A) = A — T)R(x, Dx) + B(x,Dx] dans Rn x [-T, T] est bien pose dans (tf z (R n )) m pour tout I € R1 et on a Pinegalite suivante pour t G [— T, T] :

\\w(t)\\(l}
('\\K(r}J(Dx)f(r)\\(l)dr Jo

Vu le theoreme de Calderon-Vaillancourt [3] et vu que les normes | et | | | w ( t ) l l l Z sont equivalentes, on a 1'inegalite suivante

f

[

{ 111^(0)111(0 + / I Jo

dr

pour t G [—T, T]; d'ou la conclusion du theoreme 1.3. La demonstration du theoreme 1.3 est complete.

D

3.2. Preuve du theoreme 1.7 — Supposons les conditions (A.I) et (A.2)' verifiees. De plus, supposons aussi qu'il existe o> G Sn~l, j G {1,..., m} et Co > 0 tels que, pour tout x G R n , on ait (3.1)

Supposons que le probleme de Cauchy pour le futur et pour le passe en meme temps (*) est bien pose dans H°°. Alors, d'apres le theoreme 1.5,

248

Takeuchi

la condition (B.I)' est verifiee. La condition (B.I)' et (3.1) entrainent que, pour tout p > 0,

(3-2)

0 < Co

Jo

(x + ds

' Im | lj(uj)Ai(x + o

< KQ log (p) + a Si V^A°(cI;) — 0. la condition (B.I)' implique que, pour tout x G R n ,

qui est en contradiction avec (3.1). Done, la condition (B.I)' et (3.1) entrainent que (V^A°)(o>) ^ 0. En posant x = 0 dans (3.2), on a, pour tout p > 0,

0
< KQ log (p) + C •

d'ou 5 > 1. La demonstration du theoreme 1.7 est complete.

n

3.3. Preuve du corollaire 1.8 — Supposons les conditions (A.I) et (A. 2)' verifiees. De plus, supposons aussi qu'il existe tl> € 5n-1, j 6 (1, . . . , m} et Co > 0 tels que, pour tout x € R n , on ait (3.3)

Supposons que le probleme de Cauchy pour le futur et pour le passe en meme temps (*) est bien pose dans Hl. Alors, d'apres le corollaire 1.6, la

Systemes de Leray-Volevich du type de Schrodinger

249

condition (B.I)' avec KQ = 0 est verifiee. La condition (B.I)' avec KQ = 0 et (3.3) entrainent que, pour tout p > 0,

(3.4)

0 < Co

(z + a(V^

< I Jo

=

Jo

fPlm{l°j

Si V^A^(cj) = 0, (3.4) implique que, pour tout x 6 R n ,

qui est en contradiction avec (3.3). Done, la condition (B.I)' avec KQ = 0 et (3.3) entrainent que (V^)(c2;) ^ 0. En posant x = 0 dans (3.4), on a, pour tout p > 0, 0
ds

d'ou 5 > I. La demonstration du corollaire 1.8 est complete.

D

Remerciements — L'auteur tient a exprimer ici toute sa reconnaissance a Professeur Jean Vaillant qui a permis a 1'auteur de faire de la recherche dans son Laboratoire de 1'Universite de Paris-VI depuis 1989. Grace a son accueil et ses conseils, 1'auteur a pu avancer dans ses etudes et a pu realiser ce travail. L'auteur tient a remercier Professeur Jean Leray qui s'est interesse aux Notes de 1'auteur proposees aux Comptes Rendus de 1'Academie des Sciences de Paris ; il a donne beaucoup de conseils precieux pour les Notes de 1'auteur et il a fait 1'effort de les ameliorer. L'auteur souhaite egalement remercier Professeur Sigeru Mizohata qui a donne a 1'auteur 1'impact fort et qui a fait faire des progres sur le probleme de Cauchy pour 1'operateur du type de Schrodinger en proposant ses conditions importantes.

250

Takeuchi

REFERENCES [ 1 ] A. Baba, The H°°-wellposed Cauchy problem for Schrodinger type equations, Tsukuba J. Math., 18 (1994), 101-117. [2] G. D. BirkhofF, Quantum mechanics and asymptotic series, Bull. Amer. Math. Soc., 39 (1933), 681-700. [3] A. P. Calderon and R. Vaillancourt, A class of bounded pseudo-differential operators, Proc. Nat. Acad. Sci. U.S.A., 69 (1972), 1185-1187. [4] P. Constantin and J. C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439. [5] P. Constantin and J. C. Saut, Local smoothing properties of Schrodinger equations, Indiana Univ. Math. J., 38 (1989), 791-810. [6] W. Craig, T. Kappler and W. Strauss, Microlocal dispersive smoothing for the Schrodinger equation, Comm. Pure Appl. Math., 48 (1995), 769-860. [7] J. Derezihski et C. Gerard, Scattering Theory of Classical and Quantum NParticle Systems, Texts and Monographs in Physics, Springer-Verlag, Berlin Heidelberg, 1997. [8] S. Doi, On the Cauchy problem for Schrodinger type equations and the regularity of solutions, J. Math. Kyoto Univ., 34 (1994), 319-328. [9] S. Doi, Remarks on the Cauchy problem for Schrodinger type equations, Comm. Partial Differential Equations, 21 (1996), 163-178. [10] S. Doi, Smoothing effects of Schrodinger evolution groups on Riemannian manifolds, Duke Math. J., 82 (1996), 679-706. [11] V. Enss, Asymptotic completeness for quantum mechanical potential scattering, I. Short range potentials, Commun. Math. Phys., 61 (1978), 285-291. [12] L. Garding, T. Kotake et J. Leray, Uniformisation et developpement asymptotique de la solution du probleme de Cauchy lineaire a donnees holomorphes, Bull. Soc. Math. France, 92 (1964), 263-361. [13] S. Hara, A necessary condition for H°°-wellposed Cauchy problem of Schrodinger type equations with variable coefficients, J. Math. Kyoto Univ., 32 (1992), 287-305. [14] N. Hayashi, K. Nakamitsu and M. Tsutsurni, On solutions of the initial value problem for the nonlinear Schrodinger equations, J. Funct. Anal., 71 (1987), 218-245. [15] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc., Providence, 1957. [16] L. Hormander, The Analysis of Linear Partial Differential Operators, II, Springer-Verlag, Berlin, 1983. [17] L. Hormander, The Analysis of Linear Partial Differential Operators, III, Springer-Verlag, Berlin, 1985. [18] G. Hufford, On the characteristic matrix of a matrix of differential operators, J. Diff. Equations, 1 (1965), 27-38. [19] W. Ichinose, Some remarks on the Cauchy problem for Schrodinger type equations, Osaka J. Math., 21 (1984), 565-581.

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[20] W. Ichinose, Sufficient condition on HOC wellposedness for Schrodinger type equations, Comm. Partial Differential Equations, 9 (1984), 33-48. [21] W. Ichinose, The Cauchy problem for Schrodinger type equations with variables coefficients, Osaka J. Math., 24 (1987), 853-886. [22] W. Ichinose, On L2 well-posedness of the Cauchy problem for Schrodinger type equations on the Riemannian manifold and Maslov theory, Duke Math. J., 56 (1988), 549-588. [23] W. Ichinose, A note on the Cauchy problem for Schrodinger type equations on the Riemannian manifold, Math. Japonica, 35 (1990), 205-213. [24] W. Ichinose, On the Cauchy problem for Schrodinger type equations and Fourier integral operators, J. Math. Kyoto Univ., 33 (1993), 583-620. [25] W. Ichinose, On a necessary condition for L2 well-posedness of the Cauchy problem for some Schrodinger type equations with a potential term, J. Math. Kyoto Univ., 33 (1993), 647-663. [26] H. Isozaki and H. Kitada, Microlocal resolvent estimates for two-body Schrodinger operators, J. Funct. Anal., 57 (1984), 270-300. [27] H. Isozaki and H. Kitada, Modified wave operators with time-independent modifiers, J. Fac. Sci. Univ. Tokyo, SeclA., 32 (1985), 77-104. [28] K. Kajitani, The Cauchy problem for Schrodinger type equations with variable coefficients, J. Math. Soc. Japan, 50 (1998), 179-202. [29] K. Kajitani and A. Baba, The Cauchy problem for Schrodinger type equations, Bull. Sc. Math., 2e serie, 119 (1995), 459-473. [30] T. Kato, Non-linear Schrodinger equations, Schrodinger Operators, Springer Lecture Notes in Physics, 345 (1989), 218-263. [31] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in Appl. Math. Adv. in Math., Suppl. Studies., 8 (1983), 93-128. [32] C. E. Kenig, G. Ponce and L. Vega, Smoothing effects and local existence theory for the generalized nonlinear Schrodinger equations, Inventiones Math., 134 (1998), 489-545. [33] H. Kumano-go, Pseudo-Differential Operators, MIT Press, 1981. [34] J. Leray, Hyperbolic Differential Equations, Inst. Adv. Study, Princeton, 1953. [35] J. Leray, Lagrangian Analysis and Quantum Mechanics. A mathematical structure related to asymptotic expansions and the Maslov index, MIT Press, 1981. [36] V. P. Maslov, Theory of Perturbations and Asymptotic Methods, Moskow, 1965 (en russe); French translation from Russian, Dunod, Paris, 1970. [37] M. Miyake, On Cauchy-Kowalewski's theorem for general systems, Publ. Res. Inst. Math. Sci. Kyoto Univ., 15 (1979), 315-337. [38] S. Mizohata, Some remarks on the Cauchy problem, J. Math. Kyoto Univ., 1 (1961), 109-127. [39] S. Mizohata, On Kowalewskian systems, Russ. Math. Surveys, 29-7 (1974), 223-235. [40] S. Mizohata, On some Schrodinger type equations, Proc. Japan Acad., 57 (1981), 81-84.

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[41] S. Mizohata, Sur quelques equations du type Schodinger, Seminaire J. Vaillant 1980-1981, Univ. Paris-VI. [42] S. Mizohata, Sur quelques equations du type Schodinger, Journees {{ Equations aux Derivees Partielles )), Saint-Jean-de-Monts, Soc. Math. France, 1981. [43] S. Mizohata, On the Cauchy Problem, Notes and Reports in Math., 3, Academic Press, 1985. [44] I. G. Petrowsky, Uber das Cauchysche Problem fur ein System linearer partieller Differentialgleichungen im Gebiete der nicht-analytischen Funktionen, Bull. Univ. Etat Moscou, 1 (1938), 1-74. [45] M. Reed and B. Simon, Methods of Modern Mathematical Physics, III. Scattering Theory, (XL 17, p. 331-340), Academic Press, New York, 1979. [46] E. Schrodinger, Quantisierung als Eigenwertproblem (Vierte Mitteilung), Ann. der Physik, 81 (1926), 109-139. [47] E. Schrodinger, Gesammelte Abhandlungen, 3 : Beitrdge zur Quantentheorie, Herausgegeben von der Osterreichischen Akademie der Wissenschaften, Verlag der Osterreichischen Akademie der Wissenschaften, Wien, 1984. [48] B. Simon, Phase space analysis of simple scattering systems: Extensions of some work of Enss, Duke Math. J., 46 (1979), 119-168. [49] J. Takeuchi, A necessary condition for the well-posedness of the Cauchy problem for certain class of evolution equations, Proc. Japan Acad., 50 (1974), 133-137. [50] J. Takeuchi, Some remarks on my paper "On the Cauchy problem for some non-kowalewskian equations with distinct characteristic roots", J. Math. Kyoto Univ., 24 (1984), 741-754. [51] J. Takeuchi, On the Cauchy problem for systems of linear partial differential equations of Schrodinger type, Bull. Iron and Steel Technical College, 18 (1984), 25-34. [52] J. Takeuchi, A necessary condition for H°°-wellposedness of the Cauchy problem, for linear partial differential operators of Schrodinger type, J. Math. Kyoto Univ., 25 (1985), 459-472. [53] J. Takeuchi, Le probleme de Cauchy pour quelques equations aux derivees partielles du type de Schrodinger, C. R. Acad. Sci. Paris, Serie I, 310 (1990), 823-826. [54] J. Takeuchi, Le probleme de Cauchy pour quelques equations aux derivees partielles du type de Schrodinger, II, C. R. Acad. Sci. Paris, Serie I, 310 (1990), 855-858. [55] J. Takeuchi, Le probleme de Cauchy pour certaines equations aux derivees partielles du type de Schrodinger, III, C. R. Acad. Sci. Paris, Serie I, 312 (1991), 341-344. [56] J. Takeuchi, Le probleme de Cauchy pour certaines equations aux derivees partielles du type de Schrodinger, IV, C. R. Acad. Sci. Paris, Serie I, 312 (1991), 587-590. [57] J. Takeuchi, Le probleme de Cauchy pour certains systemes de Leray-Volevic du type de Schrodinger, V, C. R. Acad. Sci. Paris, Serie I, 312 (1991), 799802. [58] J. Takeuchi, Le probleme de Cauchy pour certaines equations aux derivees partielles du type de Schrodinger, VI, C. R. Acad. Sci. Paris, Serie I, 313

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(1991), 761-764. [59] J. Takeuchi, Le probleme de Cauchy pour certaines equations aux derivees partielles du type de Schrodinger, VII, C. R. Acad. Sci. Paris, Serie I, 314 (1992), 527-530. [60] J. Takeuchi, Le probleme de Cauchy pour certaines equations aux derivees partielles du type de Schrodinger, VIII ; symetrisations independantes du temps, C. R. Acad. Sci. Paris, Serie I, 315 (1992), 1055-1058. [61] J. Takeuchi, Le probleme de Cauchy pour certaines equations aux derivees partielles du type de Schrodinger, IX ; symetrisations independantes du temps, C. R. Acad. Sci. Paris, Serie I, 316 (1993), 1025-1028. [62] J. Takeuchi, Le probleme de Cauchy pour certaines equations aux derivees partielles du type de Schrodinger, X; symetrisations independantes du temps, Proc. Japan Acad., Ser. A, 69 (1993), 189-192. [63] J. Takeuchi, Le probleme de Cauchy pour certaines equations aux derivees partielles du type de Schrodinger, These de Doctorat de 1'Universite Paris-VI, Octobre 1995, pp. 115. [64] J. Takeuchi, Symetrisations independantes du temps pour certains operateurs du type de Schrodinger (I), Preprint, 1998, pp.50, Bollettino Unione Matematica Italiana (to appear). [65] S. Tarama, On the H°° -wellposed Cauchy problem for some Schrodinger type equations, Mem. Fac. Eng. Kyoto Univ., 55 (1993), 143-153. [66] L. R. Volevich, On general systems of differential equations, Soviet Math. Dokl., 1 (1960), 458-461. [67] L. R. Volevich, A problem of linear programming arising in differential equations, Uspehi Mat. Nauk., 18-3 (1963), 155-162. (en russe) [68] C. Wagschal, Diverses formulations du probleme de Cauchy pour un systeme d'equations aux derivees partielles, J. Math. Pures et Appl., 53 (1974), 5169. [69] K. Yajirna, On smoothing property of Schrodinger propagators, Functional Analytic Methods for partial differential equations, Proc. Intern. Conf. on Functional Analysis and its Applications, Springer Lecture Notes in Math., 1450 (1990), 20-35. [70] K. Yosida, Functional Analysis, Springer-Verlag, 1965.

Systemes du type de Schrodinger a raciness caracteristiques multiples DANIEL GOURDIN Universite de Paris 6, UFR 920 (Mathematiques) 4, place Jussieu, 75252 Paris Cedex 05, France e-mail :[email protected]

Dedie au Professeur Jean Vaillant

On considere un operateur scalaire de 2-evolution au sens de Petrowsky et on suppose que les racines (en r) du polynome principal a coefficients constants sont reelles de multiplicites constantes. On donne alors des conditions suffisantes pour que le probleme de Cauchy pour le futur et le passe soit bien pose en meme temps dans les espaces de Sobolev. Nos conditions sont en partie analogues aux conditions de Levi pour les systemes et aux conditions de bonne decomposition dans les cas hyperboliques introduites par J. Vaillant puis De Paris (respectivement en 1968 et en 1972). On generalise ensuite aux systemes 2 x 2, 3 x 3 et certains systemes N x N. On montre que ces conditions restent scalaires comme dans le cas hyperbolique. Le cas des systemes N x N du type de Schrodinger generaux a multiplicite deux n'est pas termine. La premiere partie de ce travail a fait 1'objet d'une Note aux CRAS en 1997 en collaboration avec S. Ngnosse et J. Takeuchi. La deuxieme partie est ecrite avec M. Ghedamsi, M. Mechab et J. Takeuchi. 255

256

Gourdin

— PREMIERE PARTIE —

I.

INTRODUCTION - ENONCE DES RESULTATS

On note (x, t) G Rn x R 1 , Dt = -id/dt, Dx = (D±, . . . , L> n ), D^ =

On considere un operateur de 2-evolution au sens de Petrowsky : (1)

P(x,DI,Dt) = D? + a1(x,Dx)D?-l + ...+am(x,

On suppose le symbol principal P2m a coefficients constants : P(£, r) = r m + ^(Or^ 1 + • • • + <&(£),

(2)

°?(o= E a«^a (i^^^)|a|=2j

On note aussi (3)

= E a^ |a|=2j-fc

P 2 m-i(^,^,^) est le symbole sous-principal deP(x,Dx,Dt) ; P 2 m(^ r ) et P 2 m-fc(^,C, r ) sont quasi-homogenes respectivement de degres 2m et 2m — fc de poids (1,2) : P2mK,r2r)=r2mP2m(£,r) P 2 m-fc(a:,^,r 2 r) = r 2 m - f c P 2 m _ f c (x,e,r)

(r e R 1 ), (r e R 1 , k > 1),

Nous allons donner des conditions suffisantes pour que le probleme de Cauchy, a la fois pour le futur et pour le passe, note

DJt~lu(x,Q) = g3(x]

dans R n (1 < j < m)

soit bien pose dans C m ([-T,T];fl roo (E n )) dans le cas des operateurs degeneres du type de Schrodinger. On impose la condition suivante : CONDITION (A.I) — Les racines caracteristiques en r de P 2 m (^,^) sont reelles pour C e R n . On dit alors que P(x,Dx,Dt) est du type de Schrodinger.

Systemes a raciness caracteristiques multiples

257

Remarquons que cette condition est necessaire pour que le probleme de Cauchy (*) soit bien pose dans Cm([— T, T]; ff°°(R n )). Lorsque les racines caracteristiques sont simples, Takeuchi a donne des conditions necessaires et des conditions suffisantes pour que le probleme de Cauchy (*) soit bien pose dans les espaces de Sobolev (cf. aussi les resultats de Mizohata, Ichinose, Tarama, Kajitani). Ici nous supposons les racines multiples et de multiplicites constantes et nous proposons des conditions suffisantes de resolubilite du probleme de Cauchy (*) analogues en partie aux conditions de Levi et aux conditions de bonne decomposition donnee dans le cas hyperbolique.

1°)

Cas general.

On impose les conditions supplementaires suivantes. CONDITION (A. 2) — Le symbole principal P2m(£,7") possede une decomposition en facteurs HS(£,T) premiers entre eux (deux a deux) dans C[£,r] de multiplicites constantes va.

5=1

CONDITION (A.3) — Toutes les racines caracteristiques A°(£) du radical R(£, T) 131=1 HS(£, T} sont non nulles et distinctes deux a deux pour ^ G Rn \ 0 : <5
0 = n0 < m < • • • < n

On adopte la nouvelle notation suivante des racines A°(£) :

3=1

j=li=l

< ki < • • • < kj < • • • < kp — sup {i/s ; 1 < s < a} = /^,

s

Soitent li = (J; kj >i}(l
258

Gourdin

r=l

Alors, on a P 2m (:r,0 = lEU^MCONDITION (A. 4) (Bonne decomposition) — II existe des operateurs differentiels en t et pseudo-differentiels en x notes Cj et TJ (1 < j < p,}

j rI ]\~L, - ( x J-^X) D -"^ty D*} = -"-^t Dmj +\^r D I / ^ ' (x fc\ > X

L/

fc=l

avec 4 € 5L 2/c (R n ), rjfc e 5L 2A: (R n ) tels que \m — fj, — k

r^ admettant RJ(£,T] pour symbole principal et d%+1 G 5I/ 2fc (IR n ). CONDITION (A.5) — Soit rj'1^) le symbole sous-principal de r J k ( x , D x ] (1 < k < rrij, 1 < j < /j,}. (i) il existe £ > 0, tel que, pour tout j3, on a ( x , O I ^C^z)- 1 -*-™ 1 (ii) on a, pour tout 7 > 1,

Cette conditions sont invariantes par passage a 1'adjoint. Nous obtenons le theoreme suivant. THEOREME 1 — Supposons les conditions (A.I) a (A.5) verifiees. Alors, pour tout 9l(x), ..., gm(x) dans //°°(Rn) et tout f ( x , t ) e Cl([-T,T}; H°°(Rn)), il existe une solution unique du problerne de Cauchy (*) a la fois pour le futur et pour le passe telle que u(t) =u(-,t) eC' m ([-T,T];F 00 (IR n )). De plus, pour tout v(t) e Cm([-T, T]; #°°(R n )), m-l


Systemes a raciness caracteristiques multiples

259

ou

\IDlu(t}\l(s} = sup

\\Diu(t)\\a+2(l_j}

et \u(t)\\s est la Hs(1R.n)-norme de u(t) =u(-,t} avec la convention |jD'~ 1 u(0)||( s ) = 0 si I = 0.

2°)

Cas particulier (multiplicite < 2).

CONDITION (A.2) — Solent ra0 et mi deux nombres entiers positifs tels que m = rriQ + mi et 1 < mo < m^ Le symbole principal .P2m(£> r ) admet dans C[£, r] la decomposition en facteurs premiers HS(£,T) (s = 0,1), unitaires en r, quasi-homogenes de degres respectifs 2m0 et 2(mi — mo) de poids (1, 2) notee

CONDITION (A.3)' — Les racines Aj(^) (1 < j < mi = m - m 0 ) en r du radical ,r) = HQ(^,T)HI(^,T} sont non nulles et distinctes deux a deux. On note

CONDITION (A.4)' — Le polynome H0(£,r) divise les polynomes (k = 1, 2, 3) dans B[£,r] avec 5 = B°°(Rn) et 2m- 1

/

v

HQ

'

1 -2

E *

-J

rrjr(a, 1^0

sy |

s

'

|a|=2

ou #°°(IRn) est 1'anneau des fonctions de classe C°°(Rn) a derivees bornees sur E n , et en notant -P^m-fc = -^o-f^m -k (^ — ^ — ^)> Qimo-k et R2mi-k sont respectivement le quotient et le reste de la division euclidienne de -P^mi-fc Par -^1CONDITION (A.5)' — Pour |a| = 2j - 1, 1 < j < ml = m - m 0 ,

260

Gourdin

(i) il existe s > 0, tel que, pour tout 13

(ii) on a, pour tout 7 ( 7 > 1),

PROPOSITION 1 — Supposons les conditions (A.I) et (A.2)' a (A.5)' venfiees. Alors, il existe des operateurs diffferentiels Ck(x,Dx) d'ordre Ik a coefficients dans B°°(Rn) (0 < k < m - 2) tels que m —2

P(x, D x , A) - Q(x, Dx, Dt)R(x, Dx, Dt) = Y Ck(x, Dx}D^~k ou 2m,0

Grace a la proposition 1, nous obtenons le theoreme suivant. THEOREME 1' — Supposons les conditions (A.I) et (A.2)' d (A.5)' verifiees. Alors, on a la meme conclusion que celle du theoreme 1 avec jj, = 2.

II. REMARQUES La demonstration repose sur des methodes utilisees en hyperbolicite a caracteristiques multiples de Gourdin et Mechab (1984) et des travaux de J. Takeuchi (1995). Dans le cas particulier 2°) (multiplicite < 2), la condition (A.4) equivaut a 1'existence de deux operateurs de 2-evolution : Q(x, Dx, Dt) = Dr + bi(x, DX)D?°-1 +•••+ 6 m o (x, Dx), R(x, Dx,Dt) = Drm° + ci(rc, D^D?-"*-1 + ••• + c m _ m o (x, Dx) d'ordres generalises 2mo et 2(m — mo) tels que P(x, Dx, Dt) - Q(x, Dx,Dt)R(x, DXJ Dt) = ]T dk(x, DX)D d'ordres generalises 2(m — 2).

Systemes a raciness caracteristiques multiples

261

En notant Q 2 m 0 = #o, Q2(m-m0) = #o#i, 3

k=i 3

la decomposition de leur symboles respectifs en symboles quasi-homogenes Qs et Rs de degres 5 en (£,r) de poids (1, 2), la condition (A. 4) equivaut a la divisivilite par HQ de trois polynomes P^m-ii P^m-zi -^m-s (et non d'un seu^ comme dans le cas hyperbolique) car P2m-k = Q2m 0 -fc#2(m-m 0 ) + ^2(m-mo)-fcQ2m 0 (1 < ^ < 3) et par consequent P^-k — Qirn0-kH\ + Rimi-k (1 < k < 3).

— DEUXIEME PARTIE — III. ETUDE D'UNE AUTRE CONDITION (A.4)" IMPLIQUANT LA CONDITION (A.4) POUR L'OPERATEUR (1) SCALAIRE II suffit de remplacer R^mi-i par sa valeur P^m^-i ~ Qim0-iHi donnee par la premiere relation de divisibilite P^m^-i — HoPzmi-ii dans les deux autres relations r>i

_ TT r>n

n/

_ tr p//

- r 2mi-2 — -"0-'2m 1 -2' i 2 m 1 - 3 —U0-r2ml-3-

La deuxieme relation P2mi-2 = ^-0^2^-2 donne 1'equation d'inconnue Q2m0-i '• <.Hi(Q2m0-l)

— P'Zm-L-lQ'Zrno-l + Plm-2

(1 < j < m 0 )

qui est une equation differentielle non lineaire ordinaire le long des bicaracteristiques associees a HQ. Le systeme differentiel equivalent est : en supposant

(2 ,2

, H

<,<

262

Gourdin

avec les conditions initiales

D'oii

2/2,

et 1'equation differentielle ordinaire

dz

ou

-1

On a F|| < \\A | 2 2 -f \B \\z + \\C \. Comme

d z\ \A

< 00,

cette equation differentielle non lineaire n'a pas de solution globale en general ([2] p.379-380). Pour qu'il y ait une solution globale, il faut supposer Q2m0-i constant par rapport a x et done P^m-i e^ -f)2m-2 aussi constants par rapport a x. Alors, la condition (A.5)' implique que Q2m 0 -i e^ R^mi-i doivent etre reels done que le discriminant de 1'equation algebrique (en X = Q2m0-i]

soit positif ou nul.

Systemes a raciness caracteristiques multiples

263

Done, on peut remplacer de la condition (A. 4)' par la condition suivante. CONDITION (A.4)" — (i) (divisibilite de P2rn_! - P^m_! par HO)

P^ = tfo^'mi_i,

(ii) on a 1'inegalite suivante : A°(0) > 0 (1 < j < m 0 ).

Dans ces conditions, on obtient Q2m 0 -i et par suite J? 2 mi-i que Ton reporte dans la troisieme relation de divisibilite -f^m-a = ^o-^mx-s 5 en tenant compte du fait que #2m! -1 = P^rm-Z ~ HlQlmo-Z,

cette troisieme relation de divisibilite s'ecrit sous forme d'une equation derivee partielle du premier ordre (equation differentielle lineaire ordinaire le long des bicaracteristiques associees a J^o) d'inconnue Q2m0-2- D'ou Q2m 0 -2 et R2mo-2- On remarque que la resolution est pseudo-differentiel en x pour Q et R. On a le theoreme suivant. THEOREME 2 — Supposons les conditions (A.I), (A.2)', (A.3)' et (A.4)" verifiees. Alors, le probleme de Cauchy (*) est bien pose dans

Cm([-T,T}-H°°(Rn)) et I'operateur de Schrodinger verifie la meme inegalite d'energie que celle du theoreme I avec j. — 2. IV. ETUDE D'UN SYSTEME DU PREMIER ORDRE DEGENERE DU TYPE DE SCHRODINGER Soit h(Dx,Dt) I'operateur matriciel 2 x 2 a coefficients constantes de 2-evolution

ou A(DX) est d'ordre 2 en x. On fait les trois hypotheses suivantes : (i) La partie principale H ( £ , r ) de I'operateur h(Dx,Dt) est triangularisee sous la forme : 0

T

avec AQ(£) et /^o(0 deux formes quadratiques en £ definies positives ou negatives a coefficients reels.

264

Gourdin

(ii) La partie sous-principale H*(£, T) de 1'operateur h(Dx, Dt) est une matrice 2 x 2 de formes lineaires en a coefficients reels :

(iii) La partie quasi homogene #**(£, T) de degre 0 de 1'operateur h(Dx, Dt) est une matrice 2 x 2 des constantes reels : , r) = H** =

a'

{3' 5'

Nous obtenons le theoreme suivant : THEOREME 3 — Si T(£) = 0 et (J(£) -a(0) 2 definie positive, alors le probleme de Cauchy

/orme quadratique

surRnx[-T,T]

(T > 0)

= ti0(x)

pose pour le futur et le passe au sens ou : pour tout UQ(X) G [#°°(IRn)]2 et tout f ( x , t ) G Cl([-T,T}; [H°°(Rn)]2), il existe une solution unique u(x,t] de plus on a

/' \\Dth(D^Ds}v(s}\\L,(^ Jo L

+ \h(Dx,Ds)v(s)

pour tout t G [-T,T] et pour tout v(t) G Cl([-T,T};

(H°°(Rn)f).

La preuve de ce theoreme repose sur la proposition suivante : PROPOSITION 2 — II existe -B*(£) matrice 2 x 2 de formes lineaires en £ et B*(£] matrice 2 x 2 de constante, Q(x,Dx,Dt) et R(x,Dx,Dt) deux opwrateurs de 2evolution differentiel en t et pseudo-differentiels en x de symbole principal comme #0/2, d0(x,Dx) G BL°(Rn) tels que P = h(Dx, D t ) [ B ( D x , A) + B*(DX) + B"(DX)\ = Q(x, Dx, Dt)R(x, Dx, Dt) + d0(x, Dx) ou

Systemes a raciness caracteristiques multiples

265

Q et R admettent des symboles sous-principaux reels independante de x. Par un raisonnement analogue a celui de [4] on deduit le theoreme 3. REMARQUE I — Le polynome sous-caracteristique

(5i, 2 (e, A 0 (0) ^ 0) du systeme (cf. [3]) est £(£, r) = P(r - A 0 ) 2 - fjLo(a + 5}(r - A 0 ) Par consequent, 5 = 0 4=> /C(^,r) est divisible par (r — A0 et ry

2

(5 - a) + 4//07' > 0 <#

r\

—

V. SYSTEME 3 x 3 DE SCHRODINGER A RACINES CARACTERISTIQUES DE MULTIPLICITY 1 ET 2 Soit h(Dx,Dt] 1'operateur aux derivees partielles (matriciel 3 x 3) a coefficients constantes de 2-evolution

ou A(DX} est d'ordre 2 en x. On fait les trois hypotheses suivantes : (i) La partie principale H ( ^ ^ r ] de 1'operateur h(Dx,Dt] est triangularisee sous la forme : /r-A0(0 )= 0 V 0

0

avec AO(<^), A!^), /^o(0> MiCO) ^2(0 cinq formes quadratiques en <^ a coefficients reels constants telles que A 0 (£), AI(^), ^o(0 et A 0 (0 — AI(£) soient definies, positives ou negatives. (ii) La partie sous-principale de 1'operateur h(Dx,Dt]

266

Gourdin

est formee de neuf formes lineaires en £ a coefficients reels. (iii) La partie quasi homogene H** de degre 0 de 1'operateur h(Dx, Dt) est formee de neuf constantes reelles :

Remarquons que les hypotheses sont invariantes par passage a 1'adjoint. La matrice des cofacteurs de H ( £ . T } est -/,2(r - A 0 (r-A0)2 Le polynome sous-caracteristique de 1'operateur h(Dx,Dt] est (cf. [13] /C(e,r)=

B^HB^,

En notant Hk — T — Afc (k = 0,1), nous obtenons le theoreme suivant : THEOREME 4 — Si 1C est divisible par HQ (<=> ^2i/> — (Ao — Ai)5 = 0) et

JC

x 2

a/ors /e probleme de Cauchy h(Dx,Dt)u(x,t)

= f(x,t)

surRnx{-T,T}

(T > 0)

u(z,0) ='u 0 (x)

lorsque u0(x) € [F°°(R a )] 3 e^ /(x,i) e C^f-T, T]; [tf°°(]R n )] 3 ) et on a la meme inegalite d'energie que celle dans le theoreme 3. La preuve de ce theoreme repose sur le meme type de proposition que dans le paragraphs precedent. Cette fois, on expose h(Dx,Dt] avec

b(Dx, Dt] = B(Dx,Dt) + B*(DX, Dt) + B** + B*** + 5**** ou les B*" * sont pseudo-differentiels independants de x en Dx et differentiels en Dt ; Q et R matriciels triangulaires 3 x 3 ont respectivement pour symboles principaux

267

Systemes a raciness caracteristiques multiples

HQI3 et H0HiIz et leurs symboles sous-principaux reels sont des matrices triangulaires dont les coefficients sont solutions d'equations algegriques du second degre pour les temps diagonaux et d'equations differentielles du premier ordre lineaire pour les temps non diagonaux (X = (x,i)).

VI.

REMARQUES

Par le meme type de demonstration qu'au §.V, on a un theoreme analogue au theoreme 2 du paragraphe 1 (en remplagant P par h) avec les operateurs differentiels matriciels (N + 2) x (N + 2) d'ordre m •\m—k

h(x, Dx, A) = D?IN+2 + ^ A fc (z, k=i telle que

1) Aj(x,Dx) soit un operateur differentiels d'ordre 2j (1 < j < m) : A3(x,Dx)= \a\<2j

2) La partie principale ( quasi- homogene de degre 2m)

k=l

\a\=2j

soit a coefficients reels constants et s'ecrive :

/HO 0

H=

K H0

0 0

•• ' ••

H!

'•

° \

0

0 n

n

rr

i

ou m-1

•^j (1 ^ J' < (A^+l)m) et /^j (1 < j < m — 1) sont des formes quadratiques telles que Aj (1 < j < (7V+l)m), A,,--A fc (j ^ k), fj,0, Xj-^k (1 < j < (N+l)m, l
l
268

Gourdin

soit divisible par HQ dans M[£, r\. 4) /C = Hole1 et pour / = 1 , . . . , ra,

(Les hypotheses sont invariantes par passage a 1'ajoint.)

BIBLIOGRAPHIE [I] J. Chazarain et A. Piriou, Introduction a la theorie des equations aux derivees partielles lineaires, Gauthier-Villars, Paris, 1981. [2] J. Dieudonne, Calcui infinitesimal, Collection Methodes, Hermann, Paris, 1968. [3] D. Gourdin, Systemes faiblement hyperboliques a caracteristiques multiples, C. R. Acad. Sci. Paris, 278 (1974), pp. 269-272. [4] D. Gourdin et M-, Mechab, Propagation des singularites et operateurs differentiels (ed. J. Vaillant), Travaux en Cours, Hermann, 1984, pp. 121-147. [5] D. Gourdin, S. Ngnosse et J. Takeuchi, Probleme de Cauchy pour certaines equations du type de Schrodinger, C. R. Acad. Sci. Paris, 324 (1997), pp. 1111-1116. [6] W. Ichinose, On the Cauchy problem for Schrodinger type equations and Fourier integral operators, J. Math. Kyoto Univ., 33 (1993), pp.583-620. [7] K. Kajitani, The Cauchy problem for Schrodinger type equations with variable coefficients, J. Math. Soc. Japan, 50 (1998), pp. 179-202. [8] K. Kajitani et A. Baba, The Cauchy problem for Schrodinger type equations, Bull. Sc. Math., 2emeserie, 119 (1995), pp.459-473. [9] S. Mizohata, Sur quelques equations du type de Schrodinger, Jean-de-Monts, Soc. Math. France, 1981.

Journees EDP, Saint-

[10] I. G. Petrowsky, Uber das Cauchysche Problem fur ein System linearer partieller Differentialgleichungen im Gebiete der nicht-analytischen Funktionen, Bull. Univ. Etat Moscou, 1 (1938), pp. 1-74. [II] J. Takeuchi, Le probleme de Cauchy pour certaines equations aux derivees partielles du type de Schrodinger, These de Doctorat de 1'Universite Paris 6, Octobre 1995, 115 pp. [12] S. Tarama, On the H°°-wellposed Cauchy problem for some Schrodinger type equations, Mem. Fac. Eng. Kyoto Univ., 55 (1993), pp.143-153. [13] L. Vaillant, Donnees de Cauchy portees par une caractenstique double, dans le cas d'un systeme lineaires d'equations aux derivees partielles, role des bicaracteristiques, J. Math. Pures et Appl. 47 (1968), pp.1-40.

Smoothing effect in Gevrey classes for Schrodinger equations Kunihiko Kajitani Institute of Mathematics University of Tsukuba 305 Tsukuba Ibaraki Japan

- to the memory of Nobuhisa Iwasaki -

Introduction We shall investigate Gevrey smoothing effects of the solutions to the Cauchy problem for Schrodinger type equations. Roughly speaking, we shall prove that if the initial data decay as e~c<x>K(Q < K < l,c > 0), then the solutions belong to Gevrey class 71/* with respect to the space variables. Let T > 0. We consider the following Cauchy problem, r\

(1)

— u(t, x) - i&u(t, x) - b(t, x, D}u(t, x) = 0, t E [-T, T], x e Rn,

u(Q,

(2)

where (3)

&(*,*, D)u =

and DJ = — i-^. We assume that the coefficients bj(t,x) satisfy (4)

for '(t,x) € [-T,r] x B?,a € Nn, where (x} = (1 + |a;|2)1/2. Moreover we assume that there is K 6 (0, 1] such that for j — 1, 2, • • • , n, (5)

lim Rebj(t, x}(x)l~* = 0, uniformly in t e [-T, T]. |l|-»00

269

270

Kajitani

For e € R denote e = z£ - i£x^(x}
Rn

where £(£) stands for a Fourier transform of it and d£ = (27r}~nd£. Then our main theorem is the following. Theorem. Assume (4)~(5) o,re valid and there is 60 > 0 such that I^UQ € ^{R1} for kl < £o- Then ifdKl, there exists a solution of (l)-(2) satisfying that there are C > 0, p > 0 and 8 > 0 such that l^t*(*,x)|",

(6)

for (t,x) € [-T,T]\0 x #",a € #n. Remark. (i) Kato T. and Yajima in [13] began to investigate the smoothing effect phenomena . A. Jensen in [6] and Hayashi,Nakainitsu &: Tsutsumi in [5] showed that if < x >k uo(x) e ^(K1), the solution u of (l)-(2) belongs to H^ for t ^ 0, Hayashi & Saitoh in [4] proved that if es<x>2uo (8 > 0) is in L2(En), the solution u is analytic in x for t ^ 0 and De Bouard, Hayashi & Kato in [1], Kato & Taniguti in [12] show that if UQ satisfies \\(x • V)JUQ\\ < Ci'Jrlj\d for j — 0, 1.2..., then the solution belongs to Gevrey 7d/2 with respect to x for t ^ 0. Theorem 1 is proved by Kajitani in [9] and [11], when a — K = \. In the forthcoming Part II, we shall discuss the Gevrey smoothing effect for Shrodinger equations with variable coefficients.

1

Weighted Sobolev spaces

We introduce some Sobolev spaces with weights. Let p, 6 be real numbers and K E (0, 1]. Define HI = {ue LL(#n);e5<x>*u(*) G L2(Rn}}. For p > 0 let define

where Fu stands for the Fourier transform of u. For p < 0 we define H* as the dual space of H!p. Then the Fourier transform F becomes bijective from H% to H*. We define the operator epK mapping continuously from H^ to H^_p as follows; eflKu(x] = F-l for u 6 Hpl and es<x>K maps continuously from #£ to H^_s. We define for 6 > 0 and

peR (1.1)

H^6 = {ue Hp- e'Ku e H?}.

Smoothing in Gevrey classes for Schrodinger equations

271

For 6 < 0 we define Hfo as the dual space of Hlp_s. We note that H*Q = £TJ, Hfo = H$ and HQ$ — L2(Rn). Furthermore we define for p > 0 and 6 € R (1.2)

HJjs = {ue Hf; es<x>\ € B«p}

and for p < 0 define H*tS as the dual spase of H^_6. Denote by H' the dual space of a topological space H . Then H^ = H^_6 and fij^ = H-p-$ hold for any p and 8 e R. We shall prove ff*tf = Hfy later on (see Proposition 3.8). Lemma 1.1. Let p, 6 e R. Then (i)

#;> =

(it)

Lemma 1.2 let 1 > p > 0,5 € # and t* G H^. Then (1.6)

for x £ I?1, a e Nn and 0 < £ < 1. We can prove these lemmas analogously to the case of K = 1 which is proved in [11].

2

Almost analytic extension of symbols

Following Honnander's notation we define the symbol classes of pseudo-differential operators. Let m(x, £), (x, £) be weights and g = v?~2dx2 -f tp~2d£2 a Riemann metric. We denote by S(m, 5) the set of symbols a(x, ^) satisfying

for (ar,^) € R2n,a,/3 € Nn, where a ^ = SfDja. Let d > 1. Moreover we say that a function a(ar,4) € S(m,g) belongs to 7d5(m,p), if a(z,£) satisfies that there are Ca > 0, pa > 0 such that (2-1)

|a$(*, 01 < Cam(x, Op^^'la

for (z,£) € R2n,a,/3 € ]Vn. We denote 50 = di2 + d£2 and 9l = (x}~2dx2 + (0~2^2- We remark that the symbol class ^1S(m,gi)(i = 0, 1) is introduced in [11] when d — I. Here

272

Kajitani

we consider the case of d > 1. Let x(t) € that x(t) = 0,t < 1/2, *(t) = M > 1, and

(2-2)

be a monotone function satsfying

|

for t £ R,k £ N. Then for a weight u»(x, £) € idS(m.gi} and a parameter 6 > 0 we can see easily that xC&wfoC)) € 7dS(l, #1) satisfying (2.3)

|JDfDfx(M*,0))| < 2n

for (x, 0 € # , a, 0 e Nn, 6 > 1. ^a

Lemma 2.1. Let d > l,di > 1 and {pk(x,£)}kLi (2-4)

of symbols satisfying

|pg)(*,OI < m(x,

/or (a:, f ) € .R2n, a,0€Nn and k>Q. Then there is p(x, f ) € 745(m, ^) suc/i • p(x,0 ~ £>(*,£) e 7d5(m((x)(OpP)-^!dl,5i),

(2.5)

fc=0

/or any integer N > 0. Proof. This lemma is essentially a result of [2]. The case of d = 1 is explained in [11]. Here we prove the lemma in the case of d > 1. Let 6* = p~lk\~kM and M > 2. Define

(2-6)

p(x, ib=0

Then we have from (2.3)

i /"\i = I V b(«)j (*. oi I

fc a',/?'

< E aE, " fc

K + ^^^(x

7

-a

for (a:, ^) 6 /?2n, a, (3 6 A"". Here we used the following inequality

(2.7)

Smoothing in Gevrey classes for Schrodinger equations

273

for PQ > pp. Moreover we can write N-l fc=0

1

) + X> fc=0

=:/ + //.

Noting that /£*Jfc!dl(M{z) (£})'* < 1 on suppxWfcXO)"1) for A; > ^ and p~kk\di(M(x}(£)}-N > 1/2 on supp(l - ^(^({^XO)"1)) for fc < N - 1 respectively, we can see that I and II belong to jdS(m((x)(^}pp)-NN\d,g). Q.E.D. Let a(z,f) € idS(m,gi), that is, a(x,£) satisfies (2.1) with ifr = <^} and (p = (or). d—i Denote ba(x) = Bp~14n(x)~1|a|!T^" for x 6 B?. We define an almost analytic extension of a(x,£) as follows, (2.8)

a(x + iy,t + n,) = E^^'^C-^W^^^I^Dx^a^)!^!)^^)-1, a,/3

for x , y , £ , r j € K1, where ajjj(ar,^) = d%(-idx}0a(x,£). to Lemma 2.1(see Proposition 5.2 in [7]).

Then we can prove analogously

Proposition 2.2. Leta(x, £) € 7^S(m,
sl

\<* +P +7 + *i!d,

t>+ ir>)\ > 0),

> 0), 1

n

/or x, y,^rje If , and a, 0, 7, 5 6 W . Let tf(x,f) be in jdS(6^(x}K + pXf}*,^), where d« < l,d > 1. Denote by 1^(2:, 0 defined by (2.8) the almost analytic extention of •&. Denote for x, f , z, C e (2.9)

V^(x7 2, 0 - f 1 V,^(z + t(x - z), C) A,

274

Kajitani

(2-10)

V^(ar, £, C) = j V€0(z, C + 1(£ - C)) A.

Then we can prove the following lemma by repeating the argument of Lemma 5.4 in [7]. Lemma 2.3. Let &(x,£) be in ^dS(6^(x)K + p#(£}*,gi), where d/c < 1 and d > 1. Then there exist $(ar, z,C) and $'(z,£, C) in C°°(C3n) satisfying respectively,

(2.11)

S-iV.tffo *,*) = £

(2.12) /or z, 2, £ G C7n

(2.13)

IflfflJflfW^O -01 <

(2.14) and (2.15) |

(2.16)

|a,,Z?f

/or 2, C, ^ € C11 anda,/3,6 € Nn. Moreover $(x, f ) = *(x, x, $) and $'(x, 0 = $'(x, f , 5aiis/i/ (2.17) I^War.O-OI^^Cp*^^^- 1 ^-^^- 1 )^)-^^-^ (2.18) |

for x, £ G fl" anrf a,0,6eNn. For simplicity denote ^/"S^W+P^}" }5o) by A~ 5 , where 50 = dx2 4- dg2. For Oi € >l^ Si(i = 1? 2) we define a product of ai and a% as follows, (2.19)

(aioa 2 )(x,0 = os = lim €-*0

where dr/ = (27r)~ndT?. Then we can show the proposition below. Proposition 2.4. (i) Let K < 1 and Oi € ^,^,i = 1,2. Then there is e0 > 0 i/ |/?i|, |<52| < e0, the product a: o a2 belongs to ^p1

Smoothing in Gevrey classes for Schrodinger equations

275

(ii) Let Oi € A^s.,i = 1,2,3. Then if \fr\(i = l,2),|$|(i = 2,3) < cb/2, we have (ai o 02) o 03 = a-i o (02 o 03) . Proof, (i) Put Si = e-rtV-kWaifaQ (2.19) implies

(2.20) V

^

~a(x C) = lim ^

S;

e-

and a = e-<«+«><0"-(Si+fc><*>-ai 0 03. Then

«

xai(x, C 4- 77)02 (^ + 2/> C)%*7 = £lim a e (ar, 0"^ +U We shall show that ae(x, C) converges in ^,0 tending e —>• 0. Put - (0")

3=1

and

Then Le^ = e**. Hence we get from (2.20) for any positive integer N, (2.21)

a£(x,t) = Jfe

Noting that [V^l2 4- JV^I 2 + 1 > \y\2 + \r)\2 + 1, Q.J satisfies (2.22)

and 6j and c also satisfy (2.12), we can see

where C^ > 0, eo > 0 are independent of e. If jp^ + l ^ j < e0, Re(i(p] —eo((y)* + (rj}K) < 0 and consequently we can see easily that oe(:r, £) converges in ^>0 tending £ —>• 0. Q.E.D. Proposition 2.5 Let d>l arid a* € 7d5'({ar)mi(^)^Ssri), i = 1, 2. Then 0,1 oa 2 belongs to S((x)mi+m2{£}*14"*2,
a 1 oa 2 (x,0=p(*,0+r(x,0,

276

Kajitani

where p(x, f) € 7dS((or)mi'HB2{f)/l4*:I,00 satisfies that there are C > 0 and £0 > such that p(x^}- 52 7!-M7

(2-24)

e for any non negative integer N, and r(x,£) belongs to

~

The proof of this proposition is given in [11] in the case of d = 1. We can prove analogously in the case of d > 1.

3

Pseudo-differential operators

Let 0 < K < 1. Now we want to define a pseudo differential operator a(x, D) for a symbol a(z,£) € A£g , which operates from -ff^g/to H%_pj,_6. When p and 8 are non positive, since A*6 is contained in the usual symbol class SQQ (denote by S£s the Hormander's class), we can define (3.1)

a(x, D)u(x) =

for u € L2(fT} and for a €. A*j. Moreover for a+ € A^.,1 = 1,2 (pi and Si non positive) the symbol cr(oi(ar,D)a2(x, D))(x,^) of the product of ai(z, D) and a2(x, £>) can be written as follows, (3.2)

c7(a1(x,I>)a2(x,D))(x,0 = ( fll o a2) (or, £)

and we have (3.3)

01(1, £>)(a2(:r, D)«)(x) = (ai o a2)(x, D)u(x)

for u € L2(f^}, where a!oa2 is defined by (2.9). Next we shall show that (3.2) and (3.3) are valid for any p,-, <5j. To do so, we need some preparations. Let a € A*^ and u € H*. Then we can define a(x, D)u(x] which belongs to H$ . In fact, put 0(2,17) = e~5(ar)'c~p'Xz,£). Then a(z, ^) € A50. Noting that ep^Ku(^) € Z2, we can define e~6{x}Ka(x, D)u(x) =

(3.4)

which is in L2, that is, a(x,D}u € H£. For e > 0 we denote Xe(a:) = e~€<x;|2 and Lemma 3.1. (i) Let a e A^6(p, 6 € R), u E L2 and e0 > 0 chosen in Proposition 2.3. Then for any e > 0

(3.5)

a(

Smoothing in Gevrey classes for Schrodinger equations

277

and

(3-6)

(a

(ii) Let u € L2 and CQ > 0 chosen in Proposition 2.3. Then there is £1 > 0 such that for any £ > 0 e-'K(e-6<x>Kxf(x)xt(D)u}(x)

(3.7)

= ac(z, D)u(x),

where

ae(x, 0 = e

(3.8)

/or jp| < e0 and \6\ < Ci.

Lemma 3.2. I/et u € #£$ and |p|, |<5| < eo/2 (€Q is ywen in Proposition 2.3). Then for any e > 0 there, is ue € (3.9)

Lemma 3.3. Let a € A*j,Q < e^o < e0(e0 is given in Proposition 2.3) and u € H^eo- Th-en there is e2 > 0 independent o f a , p and 6 such that a(x,D)v,(x)belongs to H5 -_6 ifQ<€/Q — p< min{e.Q, €2pa} and 0 < £Q — 8 < €Q. Lemma 3.4. Let a* e A*.j.(i = 1,2) and u € H^-G(^eQ > 0). Then if \PI\ < 0 < €Q — 52 — ^i < CQ are valid (e0 is given in Proposition 2.3), we have (3.10)

ai(x,D)(a2(x,D)u)(x)

=

is in

The above lemmas 3.1-3.4 are proved in [10] for d = 1. For d > I we can prove analogously. Let a 6 A^(\p\, \6\ < e 0 /4),u € ^/2)£0/2 and |Pl|, j^l < e0/4. Put w = which is in ^J/2-plieo/2-«r Since we can write u = e~pl<-D>'c'u;), we get by use of Lemma 3.4 with e'Q = e0/2 — PI,€Q = ^o/2 — <5i,ai = a(x,£)e~plK and a2 =

a(x,D)u(x) = a(x, Noting that ai(ar,0 := (e^-^^V^-'X^") o (a(x,0e-pl<4>") o e-5l<x>1< € A^Q, we obtain (3.11)

278

Kajitani

for any u € •ff£/2,eo/2- Since following theorem.

^ dense in

Lemma 3.2, we get the

Theorem 3.5 Let a 6 -A£$(|p|,|£| < eo/4), |pi|, \6i\ < e0/4, ly/iere e0 are given in Proposition 2.3. Then a(x, D) maps from H*l>Sl to H^_p^_6 and satisfies the following inequality

(3-12)

IMk-,*-. ^ CN

for any u € H«^. For a € A£5, we difine (3.13)

a*(ar, $ = 03- j j J^a^ + y,

and a*(z, ^) = a*(x, ^). Then we can prove the following lemma, by the same way as that of the proof (i) of Proposition 2.4. Lemma 3.6. Let a € A^6 and |p|, |<5| < e0.Then a*(x, ^) defined in (2.29) belongs to A*j. Moreover it holds (3.14)

(cf(x,D)u,
= (uta(

The relation (3.14) and the inequality (3.12) yield <

<

if |pj, |<5| < e0/4 and (p^, |$i| < eo/4. Therefore taking account that ^rf0/2)eo/2 is dense in f£ A , we get from (3.14) (3-15)

lla^lU^^^CJIttH^^^,

for any u G H£ljSl. Thus we get the following proposition. Propostion 3.7. Let a € ^4p, pseudo differential operators at(x,D] anda*(x,D] satisfy (3.15). Noting that (e6<x>K e?Ky = epK es<x>" , we have for u 6 H

eo/4.

i/ie

Smoothing in Gevrey classes for Schrodinger equations

279

= (esKeftKYo (

Moreover we can see from Proposition 2.4 that(es<x>KepKy o (e~6<x>"'e~pKy is in A%£. Hence we obtain the fact below. Proposition 3.8. Let |p|, |<5| < eo/4. Then u belongs to H*s if and only i f u € H*s. The following result on the multiple symbols of pseudodifferential operators is a special case of Lemma 2.2 of Chapter 7 in Kumanogo's book [12]. In [10] we gave the proof in the case of d = 1. We can show analogously in the case of d > 1. Lemma 3.9. Let TJ(X, C) € 7dS(l,£o)0' = 1, 2, ..., v) satisfying

and put qv(x, D} = n(ar, D)r2(x, D) • > • rv(x, D}. Then the symbol qv(x,Q belongs to 7d5(l,<7o) and satisfies (3-16)

!$>,(*, C)l < ^ fl CTj^a^\\a + 0\\, j=i

for (x, C) € R2n, a, 0 € ^Vn^ where C is independent ofv and ev = mmi<jOis sufficiently small, then there is the inverse (I + r(x, .D))"1 which is a pseudodifferential operator with its symbol contained in Lemma 3.11. Let j ( x , £ ) € 7d5(£i, #1). Then if ei > 0 is small enough, there are Mx,f) € T^M*)"1^"1^!).^ > 0 independent of EI and r 00 (ar,0 € Al^d0_£Q such that (I + j(x,D))-1 = k(x,D) + k1(x,D)+r00(x,D), where k(x,$ =

4

Fourier Integral Operators

For & € yS^O* + S#(xY,9i)(p#,bv > 0), where dn < 1, we denote

For a 6 AQ?O we define a Fourier integral operator with a phase function (x, £) as follows,

280

Kajitani

(4-1)

a+(x,D)u(x)=

for u € #€0,60- Putting p(z,f) = a(x,g)e*(x*\ we can see p(x,f) € A£^. Therefore we can regard a^(x, Z?) as a pseudo differential operator with its symbol p = ae^ defined in §2 and consequently it follows from Theorem 3.5 that a^(x, D) acts continuously from Hfy to Hp-p^s-6^- However in order to construct the inverse operator of p(x, D) it is better to regard p(x, D) as a Fourier integral operator. In paticular for a = 1 we denote (4.2)

/,(*, D)u(x) =

(4.3)

I*(x, D)v(x] =

Theorem 4.1. Let a € 1dS((x}m(^^9i}^ € ^dS(p^}K + 6*(x}K,gl) and(j) = x£iti(x, £) . Assume dn < 1 . Then ifp#, 6# are sufficiently small, a(x, D} = /^(ar, D)a(x, D)I$ and a'(x,D) — I<j>(x,D)~1a(x,D)I
(4.5)

where (4.6)

P(x^)-a(x

(4.7)

p'(z,0 -a(x

w/iere $ — $(x,x, ^) and $7 = ^'(x,^, ^) are ^iuen 6y (2.11) and (2.12) respectively and r,r' belong to >ydS(e~£0^)(x)) ~ ? ^ 0 ) for an EQ > Q independent of p#. This theorem is proved in [11] in the case of d = K = 1. We can prove it by the similar way to that of [11]. Taking account of Proposition 2.1 and Lemma 2.2 and repeating the argument in the section 6 in [7] we can obtain the following Lemma 4.2-4.5. Lemma 4.2. Let a(x,0 € 7d5((a:)m{C)£,5i) and $ € -fdS(p^}K + *«{*) 0, 64 > 0) and d/c < 1. Put
a(x,0=p(ar,0+r(x,0,

Smoothing in Gevrey classes for Schrodinger equations

281

forx,£ € /?*, and (4.9)

p(x,$- £ 7!-1 \~f\
for any N, where $(x,y,^) is a solution of the following equation, (4.10)

*(*, y, 0 - iVztf(z, », $(x, y, 0) = £,

(4.11)

V.tffo y, ^ =

J(x,y,& = °* 0 ane independent of p$. Lemma 4.3. Let a(or, ^) a?^ z? be satisfied with the same condition as one of Lemma 4.2. For <j> = x£- w0(z,f) put ct(x,£) = /^(x,D)o^(x,D). Then if p* and 6* are sufficiently small, a'(x,£) belongs to S((£)m(x)e,gi) and moreover satisfies (4.12) (4.13)

p'(x,$€

/or any non negative integer N, where $'(«/, ^,77) is a solution of the equation (4.14)

#(0,&»7) - ( ^ ( y . e . T ? ) , ^ * / ) = y

(4.15)

V€i?(y, e, 17) = f «/o

and J'(y^ri) = A?)-

fl

and r 7 ^,^) € 7 d 5 ( e - e o ( ^ ^ ) , ^ 0 ) ^ o > 0 is independent of

,

Lemma 4.4. Let i?(x, £) € 7dS(8j(x)K 4- A?(£) 5 ,<7i) and d« < l,p$ > 0,6-& > 0. i/iai i?(x, £) > A)^)* ~ ^(x)"- If OQ > 0 and p^ -4- ^ and /?o + l^ol are sufficently small, there is the inverse o//^(x, D), which maps continuously from Hpl^lto for |/9i|, \8\\ small enough and satisfies I\^x.i.\jj A \R\

r^fr } z_yj D^ i0^x

-1

=— iIR^v" Ar

282

Kajitani

= I^Gc, D)(k(x, D} + kfa D) + r(x,

^(x, D),

Lemma 4.5. Lei a(x, £) arwf $ 6e satisfied with the same condition as one of Lemma 3,3. Let fi = x£ — ifi. Then we have (4.17) (4.18)

ff(I+(x,

D)a(x,

(j(a(x, D)/^(x, D)(x, 0 = a o

w/iene r.r1 is in jdS(e~£°^^

~ ,go), if p# is sufficiently small, and q^tf

satisfies

(4.19)

^(x, 0 - E 7"l^^J{a(s, ^ + rj - *Vxi?(x, y,

(4.20)

/or any positive integer N, and C > 0 and £0 > 0 are independent ofp$, where V$ $(x, ^, 77) Jo1 V^(i, ^ + try) A and Vx&(x, y, 0 = /
5

Criterion to L 2 — well posed Cauchy problem

For T > 0 let consider the following Cauchy problem, (5.1)

dtu(t, x) - tAu(t, x) - 6(t, x, D)u(«, x) = 0,

(5.2) «(0,x) = woW, for (t,x) € (0,T) x /?». We assume that 6(t,x,^) is in (^([O^S^o). Moreover we suppose that there are C £ R, K > 0 such that (5.3)

Reb(t, x, 0 < C, 71

for x, f € -R with |x|, |^| > /f and t € [0, T]. Then we can prove the following theorem by use of the same method as that of [3] and [8]. Theorem 5.1. Assume that the above condition (5.3) is valid. For any UQ 6 I/2 and f € C°([0, T]; L 2 ) there exists a unique solution u 6 C°([0, T}\ L2} D C1 ([0, T]; ^~2) o/ the Cauchy problem (5.1)- (5. 2).

Smoothing in Gevrey classes for Schrodinger equations

6

283

Proof of Theorem

Assume that u(t, x) satisfies (l)-(2) in the introduction. We change the unknown function u to w as follows, (6.1)

w(t,x) = It(x,D)u(t,x),

where 0 = x£ - ipt(£)" - ied(t, x, f ) and i? is given by

and 0±(t) = x(±t),&(*) = 1 ~ #+(*) ~ 0-(*) ^d x(<) e 7rf(^) such that %(i) = 1 for £ > l,x(*) = 0 for i < l/2,x;(i) > 0 and 0 < x(0 < 1- Then we can see that tf(£,z,f) belongs to 7d5((x} 0, M > 0, /f > 0, CQ > 0 such that £ satisfies (6.2)

(dt + £ • Vx)*(i, a:, 0 >

for x,$ € ^ with |x|, |$| > K and , jt| < T. It follows from Lemma 4.4 that if |e| and \pt\ are sufi^ciently small, we have the inverse I
~w(t, x) = (dtW+fa D)-lw(t, x] + J^(«A + 6(t, x, £>))/,(*, ^'^(i, x),

(6.4)

w(0,*) = ^(x,D)tio(x).

Since p*{e>"+i?(t,ar,0 € 7d5«0*+
(6.5) =

where a, e ^({OW"1^),^ € S^x}2-2^}2* +
284

Kajitani

where 6t € S((x}*-l($s,gi} and c € S((x}-l(^2K-l + (x)ff-1(^5^-\gl). the equation of w from (6.3)-( 6.4), (6.6)

-

Thus we obtain

= (»A + 6(t, x, L>) + p))«;(*, x),

(6.7)

t»(0)

where a2 = 4 4- &! -f c 6 S ((£}** (X)*"-* + (£)(x)-1 + {x)*-1^)^)- Moreover taking account of the assumptions (5) in the introduction and of (6.2) , a -f 6 = K, we can choose conviniently A" > 0, e and p such that we have p(x}« + Reb(t, x, C) - e(dt + £ • V,)tf)(«, x, Q) + Beoa(t, x, 0 < 0, for a:, £ € -R" with jxj, |^| > /C,where K" > 0 is sufficiently large. Therefore we can solve the Cauchy problem (6.6)-(6.7) by use of Theorem 5.1, since w(0) = I^UQ belongs to L2, and cosequently we get the solution u = /^(i,D)~1iu(t,x), which satisfies (6) from Lemma 1.2. In fact, taking account of pt(£)K +e&(t, x, ^) > pt(£)K — 6Qpt(x)(* and pt > 0, we can see from Lemma 4.4 that /^(ar, D)~l maps from L2 to H^^. This completes the proof of Theorem.

References [1] De Bouard A. Hayashi N. & Kato K. Regularizing effect for the (generalized) Korteweg-de Vrie equations and nonlinear Schrodinger equations, Ann. Lost. Henri Poincare Analyse nonhnear vol. 12 pp. 673-725 (1995). [2] Boutet de Monvel L. & Kree P. Pseudo-differential operators and Gevrey classes, Ann. Inst. Fourier Grenoble vol.17 pp. 295-323 (1967). [3] Doi S. Remarks on the Cauchy problem for Schrodinger type equations, Cornm. P.D.E. vol. 21 pp. 163-178 (1996). [4] Hayashi N. & Saitoh S. Analyticity and smoothing effect for Schrodinger equation, Ann. Inst. Henri Poincare Math, vol 52 pp. 163-173 (1990). [5] Hayashi S., Nakamitsu K. & Tsutsumi M. On solutions of the initial value problem for the nonlinear Schrodinger equations in one space dimension, Math.Z. vol. 192 pp. 637-650 (1986). [6] Jensen A. Commutator method and a Smoothing property of the Schrodinger evolution group, Math. Z. vol. 191 pp. 53-59 (1986). [7] kajitani K. & Nishitani T. The hyperbolic Cauchy problem, Lecture Notes in Math. 1505,Springer-Verlag, Berlin, (1991). [8] Kajitani K. The Cauchy problem for Schrodinger type equations with variable coefficients , Jour. Math. Soc. Japan vol.50 pp. 179-202 (1998).

Smoothing in Gevrey classes for Schrodinger equations

285

[9] Kajitani K. Analytically smoothing effect for Schrodinger equations, Proceedings of the International Conference on Dynamical Systems & Differential Equations in Southwest Missouri State University (1996) .An added Volume I to Discrete and Continuous Dynamical Systems 1998, pp. 350-353 (1998) [10] Kajitani K. &; Baba A. The Cauchy problem for Schrodinger type equations , Bull. Sci. math. vol. 119 pp. 459-473 (1995) [11] Kajitani K. &; Wakabayashi S. Analytically smoothing effect for Schrodinger type equations with variable coefficients, Proceeding of Symposium of P.D.E. at University of Delaware 1997. [12] Kato K. &: Taniguti K. Gevrey regularizing effect fot nonlinear Schrodinger equations, Osaka J. Math, vol.33 pp. 863-880 (1996). [13] Kato T.&; Yajima K. Some examples of smoothing operators and the associated smoothing effect, Rev. Math. Phys. vol.1 pp. 481-496 (1989). [14] Kumanogo H. Pseudo-Differential

Operators, MIT Press (1981).

Semiclassical wavefunctions and Schrodinger equation MAURICE DE GOSSON BLEKINGE INSTITUTE OF TECHNOLOGY, SE 371 79 KARLSKRONA

AND UNIVERSITY OF COLORADO AT BOULDER, BOULDER CO 80 309

ABSTRACT. It is well-known that the metaplectic representation allows the explicit construction of the solutions of Shro'dinger's equation for all quadratic Hamiltonians. We generalise the metaplectic representation to the case of arbitrary physical Hamiltonians and obtain Feynman's formula as a particular case of our construction. Au Professeur Jean Vaillant, avec respect 1.

INTRODUCTION

Consider Schrodinger's equation d^f

h2

at

2m

, #(z,0) = *0

x

(1)

where x = (xi,...xn), m > 0. It is well-known that ([5, 6, 4]) when the potential U is a quadratic form in the variables Xj this equation can be explicitly solved in integral form. Let in fact (/ t ) be the flow determined by the Hamiltonian vector field XH = (VPH, -VXH) where

If U is quadratic (/ t ) is a continuous one-parameter subgroup of the symplectic group Sp(n) and can thus be lifted, in a unique way, to a continuous one-parameter subgroup (Ft) of the double covering Sp2(n) of Sp(n). Identifying the latter with the metaplectic group Mpn(n), which will be defined below, and setting, for ^0 € S(I£")

V(x,t) = FtV0(x)

(3)

the function ^ is a solution of Schrodinger's equation (1) with initial condition \I/o (see, e.g., [2, 5, 10] for a proof on the Lie-algebraic level; we will prove this by other means below). This solution \£ is given, except for at most isolated values tk o f t , by the formula r i ~ 1/2 r w x> t 7172 V(x,t) = (^bf) detHesst-VK) e^ ^ ' ^(x')fflx' (4) 287

288

de Gosson

where W is the generating function of the flow (/ t ); Hess ZiX '( — W) matrix of second derivatives of — W . In view of a famous result of Gronewold and van Hove (see [2, 6] for up-to-date expositions), this lifting procedure cannot be extended to solve Schrodinger's equation with an arbitrary potential. In fact, the construction of the solutions (3) relies on the following property of Mph(ri): for all polynomials H, K quadratic in (x3,pj] we have {H,K} = 2mh[H,K}

(5)

where H and K are the operators obtained from H and K by Schrodinger's quantization rule x i—>• x — x • , p i—>• p = —ihVx , px i—> \(px + xp). Gronewold and van Hove show that it is not possible to construct a linear mapping H i—>• H from the space Pk of polynomials in (XJ,PJ) with degree k > 2 in the space of Hermitian operators on I/ 2 (R") in such a way that (5) is preserved (see [6] for a discussion of this result). The purpose of this work is to study the properties of the functions (4) when Wt is the generating function determined by the flow (/ t ) associated with a Hamiltonian of the type (2) with U arbitrary of class C2. We will call these functions semiclassical wave/unctions (they are obtained from a classical object, the generating function W). We will see that for small values of t, these functions are close, "at the order t2" of the exact solution of (1), and this fact will allow us, using an iterative method, to find the exact solution. In particular, our approach will allow us to recover Feynman's formula. (Our method has some similarities with that used by J. Vaillant [11] in another context.) We will use the following standard notations: we denote by x2 the scalar square (x, x) of x = (#1, ...,:r n ); if A is a symmetric matrix we write Ax2 instead of (Ax,x). We will assume that the potential U is a (7°° function; most results will however hold under the weaker assumption that it is C2. 2. DESCRIPTION OF THE SOLUTIONS: THE QUADRATIC CASE Recall (see for example [1]) that s 6 Sp(n] is a free symplectic transformation if the equation (x,p] — s(x' ,p'} has, for given x and x' , a unique solution (p,p'). Writing s in the canonical basis as A B C D

(the blocks A, B, C, D being of order n), s is hence free if and only if detB ^ 0. In this case there exists a generating function W, defined on R£ x R" by

Semiclassical wavefunctions and Schrodinger equation

289

and this function is given by W(x, x1} = ±DB~lx2 -B~lx-x' + \B~lAxa. Conversely, if W(x,x') = \Px2 -Lx-x' + ±Qx'2.

(7)

is a quadratic form such that P = PT , Q = QT and det L ^ 0, it is a generating function for the symplectic matrix

PL~1Q-LT

PL

To every generating function (7) one associates the two generalized Fourier transforms:

where argz = ?r/2 and the quantity A(H^) is denned by = im /|detHess(-W)

y

x,x'

where Hess X)I '(—W) is the matrix of second derivatives of— W. The integer m (the "Maslov index") corresponds to a choice of the argument of the determinant of Hess XjX / ( — W ) : arg det Hess(—W) = myr x,x'

and there are of course only possible choices for m mod 2. By definition, the metaplectic group Mph (n) is the group of unitary operators generated by these generalized Fourier transforms; the projection II : S i—> s of Mph(n) on Sp(n) defined by (6) is then a covering mapping of order 2 (for a proof, see Leray [7]). The formula (3) can be interpreted in the following way: there exists £ > 0 such that ft G Sp(n) is free for 0 < \t\ < £ (see [4], p.137). Choosing a generating function W(x,x',t) for each ft (0 < |£| < £•) in such a way that the dependence t i—>• W ( x , x ' , t ) is C°°, the solution of Schrodinger's equation (1) is then given by formula ^(x,t) = Ft^/Q(x) where the operator Ft G Mp(n} is defined by ^w^x''t^Q(x') dnx'

(8)

290

de Gosson

the Maslov index m(t) of W(x, x' , t) being chosen so that

(see [7, 3]). It can happen that for large values of t formula (8) no longer makes sense, since W is not in general denned for all t. This is not, however, a limitation of the method. In fact, since Ft = (F t //v) jV for every integer N, we can write

allowing us to write the solution ^ as an iterated integral, choosing N large enough, so that .Fj/.v^o is given by formula (8) with t replaced by t/N. It is interesting to remark here that the integer N being arbitrary, we can write *(ar,t) = lim (Ft/N}NVQ(x). N— »oo

(9)

Replacing W by the (crude) approximation WFeyn = m (x ~ 2 ^ /)2 - U(x')t

(10)

we obtain the physicists Feynman's formula (see for instance [9]). We will generalize this construction to the case of an arbitrary potential without using Feynman's formula. Our approach, which involves a precise study of the Hessian of the generating function , allows in fact to obtain a much more precise result.

3.

VAN VLECK'S DETERMINANT

The flow (ft) determined by the Hamiltonian (2) has the following property: there exists E > 0 such that ft is a free symplectomorphism for 0 < \t\ < s ([4], p. 137). Hence there exists a C°° function W such that

( p = VxW(x,x'}t) (*,p) = /t(*',P')^

, _ , _ ( p' = -VXn/ 'W(x,x,t).

(11)

Moreover, for fixed x' , the generating function determined by H satisfies the HamiltonJacobi equation ?L+H(x,VxW)=Q (see [1, 4]). By definition, the determinant de van Vleck is the quantity p(x, *',*) = (^]

Kess(-W)

x,x

(12)

Semiclassical wavefunctions and Schrodinger equation

291

that is a dire, vu (11):

( Van Vleck's satisfies an important continuity equation: PROPOSITION 1. The function (x,t) i—>• p(x,x';t,t'} satisfies the equation

(13)

0

where v = p/m is the velocity vector at x at time t of the trajectory passing through x and emanating from x' at time t' . Before we prove this let us recall the following result from the theory of differential equations (see [4, 8]). Consider the system z(t) = /(z(t),t) , a r = ( x i , . . . , x n ) , / = (/i,...,/ n )

(14)

where the fj are real functions defined in an open set U C Rn. Each solution xi,...,xn depends on n parameters ai,...,a n . Set a = (ai,...,Q! n ) and write x = x(a,t). Suppose that

does not vanish for (a,£)in some open set D C Mn x R t . Then:

(16) (Tr: the trace). Set now p ( x , t ) = p(x,x';t] and consider the Hamilton equations f x = VpH(x,p,t] \

p=-VxH(x,p,t)

with initial conditions x(0) = x' and p(0) = p' (x' is fixed, and p' variable). The solutions x(t) of the first equation are parametrized by p' since x' is fixed, and we can hence apply (16) with / = Vpf/", Q.—p' and dx_

?P' 0

dx

at 1

292

de Gosson

Setting x = x ( t ) and x = x ( t ) , this function Y is given by

and formula (16) can hence be written

that is — p(x, t] + p(x, t] Tr —(V = 0. pH(x,p, t)} at [ vx \

(17)

Noting that the derivative of t \—>• p ( x ( t ) , t ) is

and that we have

we finally get dp

„

—— + Vzp • Vp.H" + /?Vx • ^7PH = 0 dt which is just (13). LEMMA 1. The following asymptotic expression of W holds for t —> 0: W(T T' , t\ K K I X, X L I — —

rH

2 3 (T IX — — T-M X I — — Tf(T L/ I X ? T'V X J t/ -In^ O Lyf ^f t/ " ) I

n { L Q^ cJ I

where we have set U(x,x'}= I U(sx + (l-s}x')ds Jo

(20)

(U(x,x') is thus the average value of the potential U on the segment [ x ' , x } ) . We moreover have the following estimate for van Vleck's density:

Semiclassical wavefunctions and Schrodinger equation

293

PROOF. (For details see [4], Chapter 7). Let us prove (19). When H — p2/2m the generating function W is given by (x - x'}2 W = Wf=m( 2 t } .

This suggests, in the general case, to write W = W/ 4- R. Inserting this expression in Hamilton-Jacobi's equation (12), we see that R has to be a regular of the following singular equation: 2

-'

R

+u_ _

=°n

(22)

/00,

Looking for an asymptotic solution of the type - t') + W2(t - £')2

of this equation, where the Wj (j = 0, 1, 2) only depend on x and x' one finds that WQ = 0, and that W\ , W2 must satisfy the conditions

(23)

The general solution of the first of these equations is JL

/-I

i (x, x'} = Jo

U(sx + (l- s}x'}ds + -x —x

where k is an arbitrary constant. Since we want a regular solution, we must take k = 0, and hence Wl = -U

where U is given by (20). Similarly, the only non-singular solution of the second equation (23) is W-2 = 0; the estimate 19 follows. Let us now prove (21). This can be done directly by using the continuity equation (13) satisfied by p, or alternatively by the following argument (see [4] for a rigorous justification). Since

294

de Gosson

we have °2"'

°2"'/

fU °-u

. +0(0. , nlt3, -t

Since d2Wf

_

f

det

J/nxn -

we have, by definition of p:

which is (21). For instance, for the harmonic oscillator with Hamiltonian

H = -— (p + m u x ) 2m (n = 1) one finds that W(x, x'; t, t'} = m(X~X'^ - —(x2 + xx' + x'2}t + O(t2}. Ziif

\j

Note that the approximation WFeyn(x, x'; t, t'} = m

(x — x'}2 —

1 -mu2x'2t.

of (10) used in the usual Feynman formula is a very rough approximation of W] in fact W - WFeyn = 0(t). 4. THE SEMICLASSICAL WAVEFUNCTION Assume, as above, that 0 < |i| < e, so that the generating function W determined by H exists, and set

x',t]

(24)

Semiclassical wavefunctions and Schrodinger equation

295

where ^fp is defined by

We then have: PROPOSITION 2. The function Gsci has the following properties: lim/Gsci(x,x';t} = 8(x-x'}

(25)

t—•+ 1

and for fixed x' the function (x,t) i—> G3Ci(x,x';t) satisfies

(26) where the function Q is given by

t?

(27)

*"1 V I P I PROOF. In view of Lemma 1 we have

and W(x,x'-t) = for t —> 0, and consequently /

m

x n/2

for t > 0. Hence •

/

777 ''*

\ 1/2

[777? Cr — r'")2 t

''

t

X

By a similar argument, one finds that lim Gsci(x,x'; t) = 6(x — x'}

X

296

de Gosson

hence (25). Let us now show that GscL verifies (26). Setting a = ^/\p\ we have ..dGacl

tw

3W

,...da

(29)

and also

and hence xa

+ Ua- 2ihVxa • VXW -

Since VXW = p = mv it follows that

dW

dt

H(x^xWJt)

+

^xa+(^+Vx(av}

+

-aVxv).

that is, since W satisfies the Hamilton-Jacobi equation (12): „

da

In view of the continuity equation (13) we have da — +

x(av)

1 + -a

and hence /I2

dt

J

2

2m

proving (26). Proposition 2 implies that Gsci is the Green function of a certain integro-differential equation: COROLLARY 1. For each ^0 e 5(R") the function V(x,t)=

[ Gscl(x,x';t)y0(x'}dnx' 7

(30)

Semiclassical wavefunctions and Schrodinger equation

297

is a solution of the Cauchy problem rtf

^

^ , \Er(z,0) = *<,(*)

ot

(31)

where Q is the operator <S(R") —>• <S(R£) defined by QV(x,t)= j Q(x,x'-t]Gscl(x,x'-t}^Q(x'}dnx'

(32)

*/

Q being the function (27). PROOF. Differentiating under the integration sign in (30) we get ih™

at

_ £* = [ (in J \

at

2™y vW hence

That ^(x,0) = ^?Q(X) immediately follows from formula (25). When the potential U is quadratic, one gets formula (8) yielding the solutions of Schrodinger's equation: COROLLARY 2. When U is a quadratic form in the coordinates Xj the function (30) is a solution of the Cauchy problem for Schrodinger's equation: r)\b

^

ih^ = HV , *(z,0) = *„(*).

(33)

PROOF. If U is quadratic, then the generating function W has the form (7), and Van Vleck's determinant is hence independent of the variables x, x''. It follows that in this case Q = 0, hence the corollary, using (31). 5. AN ALGORITHM FOR SOLVING SCHRODINGER'S EQUATION Denote by Gt the operator which to \&0 associates the function ^(-,£) denned by (30). In general (that is when Q ^ 0) we have GtGt>?Gt+t'.

(34)

de Gosson

298

This is hardly surprising, since equation (31) is then an integro-differential equation. The operators Gt allow us, however, to construct exact solutions to Schrodinger's equation (1). Here is how. Suppose, for example, that t > 0 and consider:

One can easily prove that when N —> oo the sequence of operators lit (TV) converges towards a limit Ft (see [4]) and that we have Ft Ft1 — F t t 1 -

In fact, the following essential result holds: PROPOSITION 3. For every <&Q
(35)

,t)= lira

N— >oo

is a solution of the Cauchy problem (36)

PROOF. Making a second-order Taylor expansion at t = 0 of ^ one gets, using the equation satisfied by $?: (37)

(see [9]). Next note that the integral Gsci(x, x'- At)*o(x x ) dnx'

G±tyQ(x) = satisfies the estimate

where W/ is defined by

Applying the method of the stationary phase for A£ —» 0 we have (A(2)

m

2m

"•

Semiclassical wavefunctions and Schrodinger equation

299

and

y

J

rn

= V( —r—r|m| )}

u x

( > *') + O (Ai2)

from which follows that

Comparing the estimates (37) and (38), we get

^!(T At^

r^A vir fxl = OffAi^ 2 )

(39)

Denoting by Ft the mapping which to ^o( x ) associates \3/ = ^(x,t) let us show that Ft = lim (G*t)N. N-*oo

(40)

In view of (39) we have FAt = GA* + O((At) 2 ) and hence

One immediately checks that

hence (40). REMARK. All the results above can be extended to the case where the potential appearing in the Hamiltonian depends on t, and even to the case where a vector potential is present, that is when H is of the type: H=~(p-A(x,t)}2

+ U(x,t}.

One must however in this case replace the flow (ft) determined by H by the family of symplectomorphisms (ft,r) where t \—> f t , t ' ( x ' , p ' ) is the solution of Hamilton's equations

300

de Gosson

with x(t'} = x' and p(t'} = p'. One also has to replace the Green function G = G(x, x'; t) by the function G(x,x';t,tf) defined by dt Ira x ' t-»t' Formula (35) then becomes the "ordered product" N-l

<S?(x,t)=

lirn^ "[[ Gt-jAt,, N ^°° j=o

where =

/Gsci(x,x';t,t')^(x')dnx'.

REFERENCES [1] V.I. Arnold, Mathematical Methods of Classical Mechanics, second edition, Graduate Texts in Mathematics, Springer-Verlag (1989) [2] G. Folland, Harmonic Analysis in Phase space, Annals of Mathematics studies (Princeton University Press, Princeton, N.J., 1989). [3] M. de Gosson, Maslov Indices on Mp(n), Ann. Inst. Fourier, Grenoble, 40(3) (1990), 537-555 [4] M. de Gosson, The Principles of Newtonian and Quantum Mechanics: The Need For Planck's Constant h, Imperial College Press/World Scientific, London (2001) [5] V. Guillemin and S. Sternberg, Geometric asymptotics, Math. Surveys Monographs 14, Amer. Math. Soc., Providence R. I. (1978) [6] V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, Cambridge, Mass. (1984) [7] J. Leray, Lagrangian Analysis and Quantum Mechanics, a mathematical structure related to asymptotic expansions and the Maslov index, MIT Press, Cambridge, Mass. (1981); [8] V.P. Maslov and M.V. Fedoriuk, Semi-Classical Approximations in Quantum Mechanics, Reidel, Boston (1981) [9] L.S. Schulman, Techniques and Applications of Path Integrals (Wiley, 1981). [10] J.-M. Souriau, Construction explicite de I'indice de Maslov, Group Theoretical Methods in Physics, Lecture Notes in Physics, 50 (Springer-Verlag, 1975), 17-148. [11] J. Vaillant, Polynome sous-caracteristique and derivee de Lie de la forme element de volume, Comptes Rendus Acad. Sci. Paris 268, 547-548 (1969)

Strong uniqueness in Gevrey spaces for some elliptic operators F. COLOMBINI

Dipartimento di Matematica-Universita di Pisa, Italy

C. GRAMMATICO Dipartimento di Matematica-Universita. di Bologna, Italy

A Jean Vaillant, en temoignage d'amitie.

I

INTRODUCTION

We consider here the problem of strong uniqueness from the origin for particular smooth elliptic operators. Let P be a second order, elliptic operator with smooth coefficients defined in an open neighborhood of the origin in R2. We denote by PI the principal part of P. Then if P has simple characteristic, as a consequence of the results in [I] and [2], we have the following: 1. if P2(Q, Dx] is real, then P has the strong uniqueness from the origin, 2. if p2(0, Dx) is not real, then there exists a C00 function a flat at zero (more exactly in a suitable Gevrey class) and a function u G C°° flat at zero, not identically zero, such that Pu + au = U.

(1)

We emphasize the problem of strong uniqueness at the origin for u satisfying the inequality of the type

>! + ••• -i 301

h \Wh (x}\ Vhu(x)

(2) x e fi ,

302

Colombini and Grammatico

where h 6 N and il is a neighborhood of the origin in R n . Let C^(£l) denote the space of functions in C'°°(O) which are flat at the origin; we recall that f (x} is said to be flat at zero if I)a f (0) = 0 for any a (E N" ; we say that the relation (2) has the property of strong unique continuation at the origin, if the only function u ( x ) £ C*b°°(Q) satisfying (2) is the zero function. We recall that, for a, differential operator P — P (x, y, t, Dx, Dy, D-t) of order m (m > 2) with smooth coefficients, defined in a neighborhood of the origin in R n (n > 2) and of principal symbol FTO (x, y, £,£, 77, r], Alinhac [1] has shown, in particular, the following result (we denote the coordinates in R" by (*,»/,/,) with /, e R ' 1 ™ 2 ) : If Pm (0, 0, 0; 1, ?/, 0) /i«s £«>o simple, non real and non conjugate roots, then there exist a neighborhood V of the origin, two functions a,u G C 1>J (V) both flat at zero, such that Pu — au = 0 in V , and suppu is a neighborhood of the origin. Hence, if we want to obta,in unique continuation results in C'°° for operators of order greater than two, it is not too restrictive to consider the case A'1 (//. <= N) . This problem for the case h = 1 i.e. for inequalities of the type Au(*)| < \ W » ( x } \ \ u ( x } \ + |H',(a;)||VT/.(.T)|

x €

ft,

(3)

where M'o(x) , ^(x) are singular at the origin, has been studied for a long time. The first results with singular H/o(-?') and V^i(x) were obtained by Aronszajn, Krzywicki and Szarski [4] who examined potentials of the form: \W0(x)\ < \x\~2+£ , |Wi(x)| < x "l+s with E > 0. Afterwards a number of authors considered the problem of unique continuation, with WO(.T) and Vl 7 i(rr) having singularities of type // (Hormander [10], Barcelo. Kenig, Ruiz, Sogge [5]). In 1993 Regbaoui [15] proved a result of strong uniqueness for WQ(X] and H^i(x) satisfying the following estimates: r2

(4)

p1

(5)

provided that C-2 < - . Already in 1992, Pan [14] had shown a similar result, in R2 , without any bound for the constant, C^ above, but for real valued functions u.

Strong uniqueness in Gevrey spaces for elliptic operators

303

In 1997 the result of Regbaoui was improved (see [9]), by proving the strong unique continuation respectively for WQ(X] and VFi(x) as in (4) and (5), but with C'2 < —j= . Later on Regbaoui in [16] extended this result to V2 the case of operators P (.x, D) = a,-,- (x) DjDj and P (0, D) = A . Alinhac and Baouendi [2] have shown, in particular, that the strong unique continuation property holds for the relation (2) when h = 2 and \Wi(x)\ 0 . In 1999 we proved in [6] that the strong unique continuation holds for the inequality (2) with |W/ (x)\ < d\xl~2h / — 0 , . . . , / ? , , with bounds only on CH constant. In our recent work [6] we have proved, in particular, that Given h G N positive integer, if u € C^° (R 2 ) verifies (r = x\]

x € R2 \ {0} ,

(6)

for a certain constant C < (2h — 1)!! , then u is the zero function. We have verified in [7] that the bounds for the constants in (6) are optimal. To this end we have the following result: Let h E N be a positive integer, then, for any e > 0 we can find two functions weC^(R2) with supp w = R2 and a e C°°(R2 \ {0}) with |L < (2h- !)!! + £• such that in R2 \ {0} .

More precisely 1) // h is even we can take ^s a < E , j ^ a < (2h — !)!! 2) if h is odd we can take < (2h — 1)!! -f e , |^a| < In [6] we have also proved that the strong unique continuation property holds for functions verifying

I2 < —

(8)

with Ai any positive constant and C < 9/4 . Whereas in [8] we have shown that For every C > 9/4 there exists a function wGCj^R 2 ) with supp w ~ 2 R satisfying the estimate (8) for a, suitable constant M. .

304

Colombini and Grammatico

Recently Le Borgne [12] has proved that for the inequality (8) it is possible to add derivatives of third order but with the potential |^j~ 2 + £ . In these notes, we study the case in which P has Gevrey coefficients. From now on we assume by strong unique continuation the following D E F I N I T I O N 1 .1 Let Q be an open neighborhood of the origin in R 2 . We say that P has the strong unique continuation property if whatever Pu — 0 in il and u£("^(Q] then n = Q in a neighborhood of zero. In 1981 Lerner [13] proved that if P is a, second order, elliptic operator with simple characteristic and with Gevrey coefficients of order s, defined in neighborhood of the origin in R 2 , then the equation Pu = 0 has the strong unique property at /ero if Gevrey 's index ,s is smaller than a quantity depending on the cone /?(0,R 2 ) , where p ( x , £ ) is the principal symbol of P . Similar results are t r u e in R" . We shall expose, briefly, Lerner's results in R 2 (see [13]). Then, we shall study particular operators P of fourth order with F\ (0, D] — AQ, and Q being a second order, elliptic, homogeneous operator with real coefficients. We shall obtain the strong uniqueness if P has Gevrey coefficients of order smaller t h a n a quantity depending on eigenvalues of the quadratic form associeted to Q (see theorem 3.1 below). We sha.ll obtain this result in two different way: in the first ca.se we shall give Carleman estimates for Q, in the second case as a consequence of the results in [13]. We hope that the first approach can be extended to R." . 2

SECOND ORDER CASE

Let Q be an open neighborhood of the origin in R 2 . We denote by :?R 2 , '3 z the real and imaginary part, respectively, of the complex number z . Finally we denote by G's the Gevrey space of order s. We state first a classical Sjostrand's lemma. L E M M A 2.1 ([13], pag. 1.174) Let P be a second, order, elliptic operator, defined in an open neighborhood of the origin in R2 and p its principal symbol,. Then, either p (0,R 2 \{()}) is a convex cone of C\{0), or it is C\{0). We state now the principal result in [13].

Strong uniqueness in Gevrey spaces for elliptic operators

305

THEOREM 2.2 ([13], pag. 1165) Let P be a second order, elliptic operator with Gevrey coefficients of order s > 1, defined in an open neighborhood of the origin in R 2 and p its principal symbol. 1. If p (0, R2\{0}) is a convex cone with angle 1
+ 2sm o • I

, , 9

then P has the strong unique continuation property at zero. 2. Tf p(0,R 2 \{0}) = C \ {0} and if P has simple characteristic, then there exists a real number 1 depending only on P ( Q , D X ) such that P has the strong unique continuation property ai zero for any S < (TQ.

We note that the operators as above are Gevrey hypoelliptic because their coefficients belongs to Gevrey class of order s, hence Pu — 0 implies r- f1S uE G .

We give two lemmas which we shall use later. L E M M A 2.3 ([13], pag. 1166) Let be: v > 0 and r 2 ( x ) a positive quadratic form in RTl; then the function u(x] = exp(— r~"} belongs to Gl+l/ (R n ). LEMMA 2.4 ([13], pag. 1166) Let Q be an open neighborhood of the origin in, Rn and u E Gs (Q). // u is flat at zero, then there exists a function v E flat at zero such that

provided 1 + v~l > s. The heart of the proof in theorem 2.2 is that if P is a second order, elliptic operator with smooth coefficients defined in an open neighborhood of the origin in R 2 and if it has simple characteristics then there esist two smooth, elliptic vector fields X\ , X% such that P = XVX2 + A where P\ is a first order differential operator with coefficients C°°. Mn this case, it is easy to see that P has simple characteristic

(10)

306

Colombini and Grammatico

Now, let X be a smooth, elliptic vector field, we set

where (/' = —v V " (v > 0) and

r — \x . If we choose v such that ,

.o

\x (0)r

(11

we can give a Carlernan estimate for X . Then if we iterate Carlernan estimates for X\ , A 2 fields in (10) we obtain a Carleman estimate for F by using P.,. — e-^'p^' .

Let (.i'i,,X2) be the coordinates in R 2 , we denote by (^1,^2) dual coordi nates. Setting po (£1,^2) = P (0,0; £1,62) w^ can write

j»o (6 , 6) = « (6 + *&) (6 + '"62)

«, 2:, w € c;

from ellipticity of po it follows that ^2, xr»; ^ 0We can suppose, now, that cv = 1; we note that

it is easy to see that if Css • 3ni > 0 then $ p$ vanishes and the imaginary axis is contained in the image of po • Hence p ((), R 2 \ {0}) = C \ {0}. If 3>z-^sw < 0 t h e real part of po can vanish or not vanish; in the second case the image of pG must be a convex cone with angle smaller than TT. Suppose that t h e image of p() is a convex cone with angle smaller than TT. We can suppose also t h a t 'Wpo is « positive quadratic form; then there exists a l i n e a r map (rotation) L : (*, t.) \—> (-?'i,-?'2) s "f'h that the principal symbol of p/, (,9.f;
(0, 0; a, T] = P(] (L (a, T}} = JW + *?-^r2

where 26 is the angle of the cone p (0, R2 \ {()}). When p verifies the cone property of angle 2<£ , 0 < 0 < — , we can find coordinates such t h a t the relation (11) can be written, for any vector field, as 2sin v > - -— - (12) 1 — sin 0

Strong uniqueness in Gevrey spaces for elliptic operators

307

REMARK 2.1 We note, taking into account lemma "2A, that the assumption vy = e 1^u for any 7 > 0 and t'0 £ C\°° is satisfied if s is as in the theorem. We stress that the result holds again for function u such that: 1. u — <°7ritv

for any 7 ^> 0 and ?;7 E C'°° flat at zero,

2. \Pu(x)\ < C x | g f ^ M +|X7(x)l) 2

x € fi neighborhood of

zero, for some constant C > 0 and for some s > 0. 3

FOURTH ORDER CASE

Now, we consider two second order, elliptic operators R, S not proportional with real, constant coefficients. After a change of coordinates, we can assume that = A 2 d 2 + n2d%, ,

A > LI > 0 .

(13)

Now, we can state our main result: THEOREM 3.1 Let P(x,D] a fourth order, elliptic operator with Gevrey coefficients of order s > 1, defined in an open neighborhood V of the origin in R 2 . Suppose that P(x,D) = L(x,D)Q(x,D) + a ( x )

(14)

where L and Q are second order differential operators. We denote by LI , Qi the principal part of L , Q respectively, and assume that L 2 (0, D) = A and Q2 (0, D) = T (D) with F (D) as in (13). If s <

A — //

then P has the strong unique continu-

ation property. REMARK 3.1 Taking into account the results in [1] the operator in (14) has not in general the strong unique continuation property, if its coefficients are not Gevrey enough. As annonced in introduction we give two different proofs.

308

3.1

Colombini and Grammatico

Proof I

We give Carleman estimates for Q operator and we note that similar estimates hold for L operator. Proof — In a neighborhood of the origin in R 2 , by the assumption, there exist two smooth, elliptic vector fields A i , X% such that g = A'1A'2 + Q1.

(15)

and Q\ is a first order differential operator with C'°° coefficients. Now, we consider the polar coordinates in the plane. We set, for x ^ 0 (see [9]) °

-

,

d

dxk-^fr •£ k

4+

J

r

xO

* '

.

kI- 1 1 1

- ' '

9

i

where ^vfe = -— and £}J;A, are suitable vector fields tangent to S . \x In polar coordinates, the vector fields in (15) becomes

Ar = 1 , 2 ,

(16)

with a-*. . 3k the coefficients of A' A- fields. Moreover from the hypothesis We give Carleman estimates for such vector fields, when applied to functions with support in r < Hy . Then, we can conclude in standard way. Thus, let

a smooth, elliptic vector field in t h e plain, with ^o- (0) — Sft$ (0) = 0 . For u£C£° having compact support in r < ^?o w^ consider

where i/'? — — i/~ 1 r~" (v > 0) and v^^C'^''. We can write

A- = v, + « v 2 , with Yi . Y'2 real smooth vector field defined in a neighborhood of the origin. In polar coordinates

Strong uniqueness in Gevrey spaces for elliptic operators

309

while Y2 - (wxi$a + u;X38/3) dr + ^ ttxi + ^ f^2 . Setting X7 = e-^Xe** , we have ^ = Yi + 7^-2 (Vi,x> + i (Y2 + 7r-"- 2 (Y 2 ,z» , o

where (F, a:) = ^ ofc (ar) a;*,

if Y = ]Pflfc(ar) ^— .

From now on we write v instead of u7; we have Y2:x)v+iY2v + jr^-(Yl:x)v 2

(17)

v 2

+ 2 » (Y! i; + * 7 r"^ {Y2 , ar> t; , » y2 v + 7 r" "
= 2» f where (-,-)L 2 (s 1 ) ^en°te the inner product in L 2 (S1) . Now taking into account the relations

we have - ( O ( r ) w , u) L a ( g l )

310

Colombia! and Grammatico

Thus,

2 5R (Yi v , 7 r^-1 (r, , *) v) = 7 ((/, 4- 2) (0(r)v,v) . Similarly it is easy to see that

+ ( O ( r ) v , v) .

We choose, now, // such that a2(())-/32(0)

+ 2> —

s

(19)

1

For z/ as above and R0 small enough, we can find EQ > 0 such that

where a is the scalar product in (17). Now with Z a smooth vector field, hence

'i^, ?'V'2?.') = 2?ft(Z?', v)

From now on, we denote by C any positive constant and by BR the ball which center is the origin and radius R. If we take a constant C < 7^ f / , from (17) we have -i

2$

+

So. if r < Rn we have 7 2/.2y1<;+n'2''' 2 — i

^-ir

•

I

_ £. _ I

/ , r

-

\

I

/y.2?_\r \ r/ I

+ C 7 ~ 2 r a ?:y'2?; + 7 J r ~ 2 - 1 /y 1?

(20)

Strong uniqueness in Gevrey spaces for elliptic operators

311

But, because X is an elliptic field, we deduce that there exists a constant > 0 such that 2

II f 1 I

<

€ C^° (#K O ) having its support in B Because r < R0, from (20), we have 1

/

V

1

X

X

l

i

'

z

l

(21)

r / Y l - i (\ Y<). 1 i ; -1 / ~2r2 Y1 ^ } — )/ /Y 2 ~ r ~ 2 ' t ?

rI

\

v

_i

_|_

i

l

X\

\ 'r/ i, _ K_Iy j lII 2 _ £o

So _

72

r,

a

2

+

° ° V'

|2

1 i _ji_j

2

7 2 r

2

/|| _I £ v

_»_!

7 — -2 r 2-Kv2 ?;

2 r 2 lt?

I _±l_! 2

0 — HQ 7 2r

2

v

Now, we note that ^ _ C 7 2

72 r

7

(22)

-

for any e > 0 and small enough. Thus, if we take e in (22) small enough and RQ such that Q-Q^O < -•> £ we obtain the following estimate 72

\\XyV\f >C

2"

(23)

for some positive constant C . Hence, we have proved that if RQ is small enough, then there exists C > 0 such that (23) holds for any v 6 C100, flat at zero and having its support in From (23), taking into account that X is an elliptic field, we have >C 7

r 2-

+7

-i r 2

7 r 2

i (24)

that is 3^ > 0 and 3 C > 0 such that (24) holds Vj > 0 and V v 6 C flat at zero with support in

Colotnbini and Grammatico

312

A similar estimate holds for ||raA'7i7 , that is 3RQ > 0 and 3C > 0 such that the estimate ,a v ,..I|2

r«A>r

>

(25)

C 7 12

holds for any 7 ^> 0 and for any v £ C°° flat at zero and having its support in BRQ • We return to Q operator and consider the operator Q\. We can write

with f,g,h smooth, complex- valued functions. Now, if v £ C'^1 (BRQ) with support in B/?0, we have

hence

Taking into account that r "

l

< r " 2 RQ , it follows that

^Qt (r^u) < C^Ro \r-v~2v 4- C P,t «|| + C ||^ and hence -

.

+ r'^

(26)

We evaluate, now. e.^^XiX^ (e^'in ; from (24) and (25) we have ~1 +r~d3:iv

4-

,

(27)

with C > 0 , provided that RQ is small enough and the function v is as above.

313

Strong uniqueness in Gevrey spaces for elliptic operators

Thus, from (26) and (27) we obtain that there exist C, RQ > 0 such that \\r~ldX2v\\) ,

(28)

holds for any 7 >> 0 and for any v G C™ (Bp^} having its support in BjR0. To obtain Carleman estimates for P operator we make the following change of variables X =

so, the operators d+d y and A2# •+ / /2c y become, respectively, and

Now, for both new transformed operators, (19) becomes v >

A—

Finally, if we iterate the above Carlernan estimate (28) we have

\

.-2t/-4,

(29)

for any 7 >• 0 and for any v (E C^° (Bp^} having its support in Let (p a real smooth function defined in R+ such that (T <

_ / 1 se r < T/2 ) se r > T. Let it be the function of the theorem. Considering lemma 2.4, for v as in (19) we can write tpu = e7^t>7 for some v 6 C^°(BR O ), hence ^(ftv-y). From (28) we have ,-2^-4,^., II

therefore, taking into account that P?/ — 0 in a neighborhood of zero, it follows that *

where

0f6)

— 2f—4

—-


'P(^)

A
From (30), we obtain

for any 7 large enough; if 7 goes to infinity we deduce that u = 0 in The proof is achieved.

(30)

314

Colombini and Grammatico

3.2

Proof II

Proof — The theorem is an easy consequence of theorem 2.2 and of the following remark: we write the principal symbol of P evalued at zero

where p\ and p2 are principal symbols of L = X\X-2 -f T\ and Q = X^X4 -{-T'2 respectively, while 7\ and T2 are first order operators; moreover from the assumption we have

A'I (o) = dxi + idx, , x2 (o) = dxi - idS2 and also A3 (0) = XdXl + indx.2

,

A4 (0) = XdX}

hence

that we can write as p (0. 0 - (Xtf + K22 + ^16 (A - //)) (Xtf + t£* - %& (A - p ) ) .

(31)

We note that both factors in (31) have as image a cone in C with angle . A - / / '2<£> where sin 0 = . A + // We consider, now, the operator P = A'2X3; applying the preceeding theorem to "P, we can choose new coordinates such that P becomes P — ~ ~ ~ ~ X — it A' 2 A'--5, and for A-?, A-^ the relation (12) becomes v > '- . V Return now to the operator P . In the new coordinates P becomes P=LQ+a where L = A] A 2 + T\ , while Q = X3A,} + T^ We note that for the vector fields A'i and A'4 the relation (12) becomes A - // v > —. V Taking into account (25), we can iterate the estimate (28) for operators L arid Q] from the remark 2.1, it follows that P has the strong unique continuation property if ,s is as in the theorem.

Strong uniqueness in Gevrey spaces for elliptic operators

315

References [1] S.ALINHAC: Non-unicite pour des operateurs differentiels a caracteristiques complexes simples, Ann. Sci. Ecole Norm. Sup. 13 (1980), 385-393. [2] S.ALINHAC, M.S.BAOUENDI: Uniqueness for the characteristic Cauchy problem and strong unique continuation for higher order partial differential inequalities, Amer. J. Math. 102 (1980), 179-217. [3] S.ALINHAC, M.S.BAOUENDI: A counterexample to strong uniqueness for partial differential equations of Schrodinger7s type, Comm. Partial Differential Equations 19 (1994), 1727 1733. [4] N.ARONSZAJN, A.KRZYWICKI, J.SZARSKI: A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark.Mat. 4 (1962), 417-453. [5] B.BARCELO, C.E.KENIG, A.Ruiz. C.D.SocGE: Weighted Sobolev inequalities for the Laplacian plus lower order terms, Illinois J. Math. 32 (1988), 230-245. [6] F.COLOMBINI, C.GRAMMATICO: Some remarks on strong unique continuation for the Laplace operator and its powers, Comm. Partial Differential Equations 24 (1999), 1079-1094. [7] F.COLOMBINI, C.GRAMMATICO: A counterexample to strong uniqueness for all powers of the Laplace operator, Comm. Partial Differential Equations 25 (2000), 585-600. [8] F.COLOMBINI, C.GRAMMATICO: Strong uniqueness for Laplace and biLaplace operators in limit, case, in "Carieman estimates and applications to uniqueness and control theory" (F. Colombini and C. Zuily editors), Progress in Nonlinear Differential Equations and their Applications, Birkhauser, Boston, 46 (2001), 49-60. [9] C.GRAMMATICO: A result on strong unique continuation for the Laplace operator, Comm. Partial Differential Equations 22 (1997), 1475-1491. [10] L.HoRMANDER: Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations 8 (1983), 21-64. [11] D.JERISON, C.E.KENIG: Unique continuation and absence of positive eigenvalues for Schrodinger operator, Ann. of Math. 121 (1985), 463494.

316

Colombia! and Grammatico

[12] P.LE BoRGNE: Unicite forte, pour Ic produit de deux operateurs elliptiques d'ordre 2, Indiana Univ. Math. J. 50 (2001). [13] N.LERNER: Resultats d'unicite forte pour des operateurs elliptiques a coefficients Gevrey, Cornrn. Partial Differential Equations 6 (1981), 1163-1177. [14] Y.PAN: Unique continuation for Schrodinger operators with singular potentials, Comm. Partial Differential Equations 17 (1992), 953-965. [15] R.REGBAOUT: Prolongement unique pour les operateurs de Schrodinger, These, Universite de Rennes, (1993). [16] R..REGBAOUI: Strong unique continuation for second order elliptic differential operators, ,]. Differential Equations 141 (1997), 201-217. [17] T.WOLFF: A counterexample in a unique continuation problem, Comm. Anal. Geom. 2 (1994), 79-102.

A remark on nonuniqueness in the Cauchy problem for elliptic operator having non-Lipschitz coefficients DANIELE DEL SANTO Dipartimento di Scienze Matematiche, Universita di Trieste, Via A. Valerio 12/1, 34127 Trieste, Italy

We are concerned here with the non-uniqueness of the solutions to the Cauchy problem for second order elliptic operators with real principal part. It is well known that if the coefficients of the principal part of a second order elliptic operator are real and Lipscbitz-continuous then a local uniqueness result holds for the solutions to the Cauchy problem belonging to C2 (see e. g. [1] or [2, Ch. XVII]). On the other hand a classical example due to Plis shows that the Lipschitzcontinuity assumption (on the coefficients of the principal part) cannot be very much weakened, since the uniqueness property is not verified by the solutions of a particular second order elliptic: operator which coefficients of the principal part are real and Holder-continuous of exponent a, for all a < 1 (see [3]). Recently a sharp result on this subject has been obtained by Tarama (see [4]). In order to recall this result we introduce some notations. Let /d be a modulus of continuity, i. e. /i is a real valued continuous function defined on [0,1], concave, strictly increasing and such that p,(0) = 0. Let / be a complex valued function denned on 17, open set of R n ; / is /^-continuous (we will write / E CM(17)) if for each K compact set in 0 there exists £ > 0 such that sup x,y£K, 0<\x-y\<£

It is easily seen that if for all 0 < a < 1, lini t-^- = 0 317

318

Del Santo

then C c u (ft) C C/'(ft) C f\ v < 1 C ( K ( V (ft) where C 0 ' 1 ^) and C°' a (ft) denote the space of Lipschitz-oontimioiis arid Holder-continuous functions of exponent cv respectively. THEOREM 1 ([4, Th. 1.1]) Let p. be a modulus of continuity satisfying (1). Suppose that o

-

Let ft be an open neighborhood of a point ZQ in R n ; let (p be a C2 real valued function on ft such that V
(3

There exists a function a e C M (R) and there exist four functions u, 61, /;2, c € C^(R3) such that 1/2 < a ( f ) < 3/2 for all t e R, supp (u) = {(*,:i;, :(/) G R3 : / > 0} and ()fu 4- rl^u + u,()fu + l>i()xu + bzdyU + cu = 0 on R3. (4)

Proof: we follow very closely the proof of the Theorem 1 in [3], outlining the main points. We consider four C00 functions A, B. C and J defined on R such that A(r) = 1 for T > 2/3. B(r) = 0

for r < 0 or r > 1,

C(T) = 0 J(r) = -2

for T > 1/2,

A(r] = 0

for r < 1/2,

D ( r ] = 1 for 1/6
for r > 1/6 or r > 5/6,

J(r) = 2 for 1/3 < r < 2/3.

We introduce two sequences of positive real numbers {rn}, { z n } such that + OC;

(5)

Cauchy problem for elliptic operator with non-Lipschitz coefficients

=0

319

and qn = ^zkr fc = 2 / __

^ \

Pn — \^n+i

^nj?m

and we suppose that

pn > I

for all n e N.

(7)

We set finally

and vn(t,x) = exp(-c/ n - zn(t - a n ))cosz n x, wn(t, ij) = exp(-c/ n - zn(t - an) + Jn(t)pn) cos zny. We define u(t,x,y) vi(t.x)

for t > a 1 ?

An(t)vn(t, x) + Bn(t)wn(t, y) + Cn(t)vn+i(t, x}

for a n + 1 < t < a n ,

0

for t<0.

It is possible to verify that supposing for all A ' i , . . . , K$ > 0

then the function u is in C°° (see [3, p. 97]). Moreover supp (it) = {(t,x,y] G R3 :

We set f ((-^(t}pn ~ zn}2 - J'n(t}pn}z-'2

1

for an+i
for t < 0 or t > a\.

Since there exists a constant C > 0, depending only on the £°° norms of J and its first and second derivatives, such that the condition sup C(r^pnz~l + r~'2p'nZ~2 + r~'2pnz~'2} < 1/2

(9)

implies that a(t] G [1/2,3/2] for all t G R. Moreover, for all t G [a n + i,a n ],

where again C depends only on J and its derivatives. Since pn > 1 for all n G N. we deduce from (10) that \a'(t)\ < C(rn-pnzn

320

Del Santo

for all t 6 [«7, + i, (i,n\- From the concavity of// we have that the function a i—> a///,(cr) is increasing and consequently

Then the //-continuity of a will be implied by sup H.EN

,.-2 n /n

_-l

,

-3 ' 2 _ - 2 Pn n

~ " ,+ ? " Mr n )

< +00.

(11)

If for all KI. K'2 and A;i > 0

'Vn-H'=°

(12)

then the functions b[. b^ and c: are in C!Xi and (4) holds. We refer to the cited paper of Plis for th( 5 details. We choose

From (1) and (2) the conditions (5), ( G ) , (7), (8). (9), (11) and (12) are verified if the integer A- is sufficiently large. The proof is complete. REFERENCES 1. L. Hormander. On the uniqueness of Cauchy problem II, Math. Scand., 7: 177190 (1959). 2. L. Hormander. "The Analysis of Linear Partial Differential Operators, III", Sprmgou--"Vorlatf. Berlin (1985). 3. A. Plis, On non-uniqueness in Cauchy problem for an elliptic second order differential equation. Dull, Acad. Pol. Sci., 11: 95-100 (1963). 4. S. Tarama, Local uniqueness in the Cauchy problem for second order elliptic equations with non -Lipschitzian coefficients, Publ. Res. Inst. Math. Sci., 33: 167188 (1997).

Sur le prolongement analytique de la solution du probleme de Cauchy YUSAKU HAMADA 61-36 Tatekura-cho, Shimogamo, Sakyo-Ku, Kyoto, 606-0806, Japon

Dedie a Jean Vaillant

Resume. J. Leray et L. Carding, T. Kotake et J. Leray ont etudie les singularites et des prolongements analytiques de la solution du probleme de Cauchy dans le domaine complexe. Nous etudions le prolongement analytique de la solution du probleme de Cauchy pour 1'operateur differentiel a coefficients de fonctions entieres. Dans cet expose, nous donnons une remarque sur le domaine d'holomorphie de la solution du probleme de Cauchy pour certains operateurs differentiels a coefficients polynomiaux. Ceci concerne les equations differentielles ordinaires de Darboux-Halphen, de J. Chazy et de la fonction modulaire.

1. INTRODUCTION ET RESULTATS J. Leray [L] et L. Carding, T. Kotake et J. Leray [GKL] ont etudie les singularites et un prolongement analytique de la solution du probleme de Cauchy dans le domaine complexe. [P], [PW] et [HLT] ont etudie des prolongements analytiques de la solution du probleme de Cauchy pour 1'operateur differentiel a coefficients de fonctions entieres ou polynomiaux. Soit x — (x 0 , x') [x' = (xi, • • • , xn)] un point de C71"1"1. On considere a(x,D) un operateur differentiel d'odre m, a coefficients de fonctions entieres sur C""1"1: a(x,D)= ^ aa(x)Da,Dl = d/dxl,Q
Sa partie principale est note g ( x , D ) . Nous supposons que

321

322

Hamada

Soit 5 Phyperplan XQ = 0, non caracteristique pour g. Etudions le probleme de Cauchy

(1.1)

a(z, D ) u ( x ) = v ( x ) , £>Ju(0, x'} = wh(x'), 0 < h < m - 1,

ou v(x)^Wfl(xf) sont des fonctions entieres sur C n+1 et Cn respectivement. D'apres le theoreme de Cauchy-Kowalewski, il existe une unique solution holomorphe au voisinage de 5 dans C n+1 . Jusqu'ou est-ce que cette solution locale peut se prolonger analytiquement? En general, il se passe des diverses phenomenes compliquees. Dans [HI], en appliquant un resultat de L. Bieberbach et P. Fatou, nous avons construit un exemple tel que le domaine d'holomorphie de la solution admet un point exterieur pour 1'operateur a coefficients de fonctions entieres. Dans [H2], nous avons donne un exemple tel qu' en disant rudement, le domaine d'holomorphie de la solution ramifiee admet un point exterieur pour un operateur differentiel a coefficients polynomiaux. Dans cet expose, nous donnons un complement de [H2]. D'abord, pour 1'expliquer, nous rappelons un resultat de [HLT]. THEOREME [HLT]. Dans le probleme (1.1), supposons que m

g ( x , D} = D% +

Lk(x, Dx>)D™-k, ou Lk(x, Dx>), I < k < mf est d'ordre

k en Dx> et un polynome en x' de degre /^A;, [i etant un entier > 0. Alors il existe une constante C(0 < C < 1) ne dependant que de M(R] telle que le probleme (1.1) possede une unique solution holomorphe sur {x e Cn+l-\xQ |< Cmin [(1+ || x' ||)-«™«(M-I,O) J ^j | ? ou M(R) est le module sur {XQ; \ XQ \< R} des coefficients des polynomes en x' dans g ( x , D) et || x' \\— maxi<;< n | X{ \. Done au cas ou p — 0, 1, la solution est une fonction entiere sur C n+1 . Ceci a ete deja demontre dans [P], [PW] et [HLT]. Dans cet expose, nous donnons des exemples tels que le domaine d'holomorphie de la solution soit univalent et admette un point exterieur pour 1'operateur differentiel a coefficients polynomiaux. En fait, J. Chazy [C] a etudie les equations differentielles ordinaires du troiseme ordre et le systeme d'equations de Darboux-Halphen. (Aussi voir [AF]). Nous employons ces resultats. Considerons les problemes de Cauchy

Prolongement analytique de la solution du probleme de Cauchy

323

3

(1.2)

{Do + ]£ #,•(*')A}tfi.j(*) = 0. Uij&x1) = X j , 1 < j < 3, 1=1 [x = (x0,x'),x' = (xi,x 2 ,x 3 )]

ou

HI(X') = -[(x2 + x 3 )xi - x 2 x 3 ], (1.3)

H2(x'} = -[(xi + x 3 )x 2 -xix 3 ], H3(x') - -[(xi + x 2 )x 3 - xix 2 ].

Ceci concerne le systeme d'equations differentielles ordinaires de DarbouxHalphen ([C], [AF]). Considerons les problemes de Cauchy (1.4) {Do + x 2 D x + x3D2 + (2xix 3 - 3xl)D3}U2jj(x) = 0, Ceci concerne 1'equation differentielle ordinaire de Chazy. ([C], [AF]). J. Leray [L] et L. Carding, T. Kotake et J. Leray [GKL] ont etudie le probleme de Cauchy, lorsque la surface initiale a des points caracteristiques. Dans [H2], nous avons etudie un cas exceptionel de [L] et [GKL]. Nous completons ici des resultats de [H2]. Considerons les problemes de Cauchy 3

(1.5)

{]T Ai(x')Di}U3j(x)

= 0, t/ 3l j(0,x') - x,, 1 < j < 3,

[x = (x 0 , x'), x' = (xi, x 2 , x 3 )] ou

A 0r-\ i\*^1* (1 o l

i ~~ / T* I 1 / ~~" **»*' i 1 1

mm nr* t I *** L )

i* *-* *^2l

^41 f2* } —— /I/-*13* 15*0

A2(x') = A 0 (x')x 3 , Ceci concerne 1'equation differentielle ordinaire de fonction modulaire. ([C], [Hi], [AF]). Nous avons alors PROPOSITION 1.1. Les domaines d'holomorphie £>;, 1 < ij < 3; des solutions U i j ( x ) , I < z, j' < 3, des problemes (1.2), (1.4) et (1.5) sont des domaines univalents dans C 4 . 7/5 admettent un point exterieur dans C4. En utilisant une application birationnelle et une application algebrique, les problemes (1.2) et (1.4) se transforment en les problemes (1.5) avec les donnees transformers.

324

Hamada

2. ESQUISSE DE LA PREUVE DE LA PROPOSITION 1.1 La fonction modulaire w = X(z) est holomorphe sur {z;5z > 0} et son inverse z — v(w] est holomorphe sur le revetement universel 7£[C \ {0,1}] du domaine C \ {0,1}. X(z) a la frontiere naturelle {z; ^sz = 0}. a W = X V( ct ,+; a) i ) ,' a.b.c.d ' ' ' etant des constantes,' ad — be = 1. satisfait 1'equation {W-t} = -R(W) ( ^§H , ou {W;£} est la derivee de Schwarz:

d d2W dW\ I d2W dW\ {W-1]J = — —— / —— —rr / —r2 2 dt I dt I dt ) 2 I dt I dt ) R(W) = (i-w + W2}/2W2(l - W}2. ([Hi]) Les £]_ = W, X'2 = dW/dt, x^ = d2W/dt2 satisfont done CL£^

CtX^

^**^3

O*Co

1-L

2/i ~f~ X-I ) J

Avec les donnees x'(0) = y', on a alors xi = X

c(y'}t

ou les fonctions

3/2

sont holomorphes au voisinage d'un point C x c ( \{°)) x Cetellesse prolongent y'(°) = (yj^^^yi^) ^ ( \{M}) analytiquement sur 7e[(C \ {0, 1}) x (C \ {0})] x C. Les solutions t/3,j(z), 1 < j < 3, des problemes (1.5) sont alors holomorphes au voisinage d'un point (0, x^ 0 )) de {x- xQ = 0, x' G (C \ {0, 1}) x (C \ {0}) x C} et nous obtenons

- M d(x '}

Prolongement analytique de la solution du probleme de Cauchy

325

En observant une ramification de v(w}, on voit que n/V \tg\JU

/s

lI — — °flfi(r^rfr'}'] -vJlttl X I C I X

f

f

A/fir'} pt II, 1VJL IX I — — n(r'\rl(r'\ til X l U I X I — — Ih(r M X }r(r ICI} X I CL

N(x'} = Ss[a(x')d(x')] sont des fonctions analytiques de variables reelles 3Rx;, ^sx' au point x'(°) et elles sont uniformes et analytiques en ftx'^x' sur (C\{0,l})x(C\{0})xC. P(x') = iM(x')/2Q(x'),R(xr) = 1/2 | Q(x') sont uniformes et analytiques en sRx', Sx' sur {x' € (C \ {0,1}) x (C \ {0}) x C; Q(x') ^ 0}. Definissons le domaine suivant: x = (x 0 , x') € C x (C \ {0,1}) x (C \ {0}) x C; | x0 - P(x') |> ^(x') pour Q(x') > 0, | x0 - P(x') |< R(x') pour Q(x') < 0, 9[M(x')z0] ~ ^V(^) < 0 pour Q(x') = 0 Le domaine T>3 est univalent et il admet un point exterieur dans C 4 . D'apres le theoreme de Cauchy-Kowalewski et les representations des solutions, en employant un technique des prolongements analytiques dans [HLT] (Proposition 7.1 dans [HLT]), on voit que les domaines d'holomorphie de C/ 3? j(x), 1 < j' < 3, sont ^3. Ceci prouve la Proposition 1.1 pour C7 3| j(x),l < j < 3. Ensuite, nous etudions [/i,j(x), 1 < j < 3. Considerons 1'application birationnelle de {x = (xi, x 2 , Xs) G C ; x\ ^ x 2 , x\ ^ xs, x 2 ^ x%} sur Xi = Xi(x') = (xi - x 3 )/(xi - x 2 ), X2 = X 2 (x 7 ) = (x 2 - x 3 )(xi - x 3 )/(xi - x 2 ) = (x 2 - x 3 )A"i(x ; ), ^3 = ^3(^0 = (Xl + X 2 - X 3 ) ( X 2 - X 3 ) ( X 1 - X 3 ) / ( X 1 - X 2 ) = (xi + x 2 - x 3 )X 2 (x ; ),

et done nous avons — XI^A I Y'\j — x\ — — -r^-

A2

-

326

Hamada

Par cette application, les problemes de Cauchy (1.2) se transfer-merit en les problemes de Cauchy suivants.

avec les donnees if)

V'\ _ -

3

.iW' A ) — T — A2

fr «\ A Y'\j = - —+ _i_ -- - -- —^3,3(0, A2

1 —A I

AI

OU

Nous avons alors

r3)31 j =

UB,I (xy

Posons Cl =

I I <

/ /\ — f-14 / / \ / / 2 = ( X o , X } G C , X — ( X i , X 2 , X 3 J ; X i 7= X 2 , X 2 7= X 3 , X 3 ' .

alors C/ij(x), 1 < ji < 3, sont holomorphes sur £"1. Designons par T>i = (^1)^°^ 1' intereur de 1'adherance ^i de £1. Nous obtenons alors D! \ {x^ — 2;, 1 < A; < / < 3} = S\. D'autre part, d'apres le theoreme de Cauchy-Kowalewsk, lesi C7ij(x), 1 < j' < 3, sont holomorphes au voisinage de 5 D {x^ — x/, 1 < k < / < 3}. D'apres le theoreme de Hartogs, C/I T J(X), 1 < j < 3, sont holomorphes sur PI. Nous pouvons voir facilement que les domaines d'holomorphie de C/i t j(x), 1 < j' < 3, sont D\. T>\ est un domaine univalent et il admet un poin exterieur dans C4. Ceci prouve la Proposition 1.1 pour C/ij(x), 1 < j' < 3. Finalement nous etudions les problemes (1.4).

Prolongement analytique de la solution du probleme de Cauchy

327

Considerons 1'application de C3 sur C3:

= x2(Xf) = 3 x3 =

Soit -Xj(x'), 1 < j' < 3, les determinations de la fonction algebrique definie par r3 - xir 2 + 2x 2 r - -x3 = 0, o a un point z'(0) de {xf = (xi,x.<2,x3) € C3; A(z') ^ 0}, ou A(x') est le discriminant de cette equation algebrique. Les Xj(x'), I < j• < 3, se prolongent analytiquement sur leur surfaces de Riemann 7^T, c'est - a - dire, le revetement du domaine {x/ = ( x 1 , x 2 , x 3 ) G C 3 ; A ( x / ) ^ 0 } . Xj = Xj(x'), x1 € 7^T, 1 < j < 3, applique 7£T sur {(Xi,X 2 ,X 3 ) € C3;X! / X 2 ,X 2 ^ X3,X3 ^ Xi}. L'application XQ = XQ,XJ = Xj(X'), 1 < j' < 3, transforment les problemes (1.4) en les problemes suivants 3

{DXo avec les donnees

,2(0, X') =

Nous avons alors , au voisinage d'un point de (0,x^ 0 ^) de

C7 2)J (x) =Wij(x 0 ,XV)).l < 3 < 3Nous obtenons done, au voisinage du point (0,x'(°)),

Hamada

Soient x' un point arbitraire de {x7; A(x 7 ) ^ 0} et 7 un chemin dans {x7; A(x 7 ) ^ 0} d'origine fixe x 7 (°) a x 7 . Prolongeons analytiquement tout definissons le domaine suivant:

=

x-

2

Designons par T>? = (£2}^ 1' interieur d'adherance de £*2 de £2. Nous avons P2 \ {x'\ A(x') = 0} = 82- Pour chaque point x de £*2 H (x 7 ; A(x') = 0}, il existe alors un voisinage W(x] in X>2 tel que les fonctions C/ 2 j, 1 < j' < 3, sont holomorphes, uniformes et bornees dans W(x] \ {x'; A(x') = 0}, et done d'apres le theoreme de Riemann, elles sont holomorphes sur X> 2 . Bien entendu , comme dans Uij, I < j < 3, nous pouvons le demontrer aussi, en utilisant les theoremes de Cauchy-Kowalewski et de Hartogs. Nous pouvons voir facilement que le domaine d'holomorphie de C/2,j, 1 < j < 3, sont £>2. T>2 est un domaine univalent et il admet un point exterieur dans C 4 . Ceci prouve la Poposition 1.1 pour t/ 2j , 1 < j < 3. On trouvera les demonstrations detaillees des resultats dans [H3], REFERENCES [AF] M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, Cambridge Texts in Applied Mathematics, Cambridge University Press, 1997. [C] J. Chazy, Sur les equations differentielles du troisieme ordre et d'ordre superieur dont 1'integrale generale a ses points critiques fixes, Acta Math. 34 (1911), 317-385.

Prolongement analytique de la solution du probleme de Cauchy

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[GKL] L. Carding, T. Kotake et J. Leray, Uniformisation et developpement asymptotique de la solution du probleme de Cauchy lineaire a donnees holomorphes; analogue avec la theorie des ondes asymptotiques et approchees, Bull. Soc. Math. France 92 (1964). 263-361. [G] R. C. Gunning, Introduction to Holomorphic Functions of Several Variables, Vol. I, Wadsworth & Brooks/Cole, 1991. [HLT] Y. Hamada, J. Leray et A. Takeuchi, Prolongements analytiques de la solution du probleme de Cauchy lineaire, J. Math. Pures Appl. 64 (1985), 257-319. [HI] Y. Hamada, Une remarque sur le domaine d'existence de la solution du probleme de Cauchy pour 1'operateur differentiel a coefficients des fonctions entieres, Tohoku Math. J. 50 (1998), 133-138. [H2] Y. Hamada, Une remarque sur le probleme de Cauchy pour 1'operateur differentiel de partie principale a coefficients polynomiaux, Tohoku Math. J. 52 (2000), 79-94. [H3] Y. Hamada, Une remarque sur le probleme de Cauchy pour 1'operateur differentiel de partie principale a coefficients polynomiaux II, a paraitre au Tohoku Math. J. [Hi] E. Hille, Ordinary Differential Equations in the Complex Domain, John Wiley, 1976. [L] J. Leray, Uniformisation de la solution du probleme lineaire analytique de Cauchy pres de la variete qui porte les donnees de Cauchy (Probleme de Cauchy I), Bull. Soc. Math. France 85 (1957) 389-429. [N] T. Nishino, Theory of Functions of Several Complex Variables [Tahensu Kansu Ron] (en japonais), Univ. of Tokyo Press, 1996. [P] J. Persson, On the local and global non-characteristic Cauchy problem when the solutions are holomorphic functions or analytic functionals in the space variables, Ark. Mat. 9 (1971), 171-180. [PW] P. Pongerard et C. Wagschal, J. Probleme de Cauchy dans des espaces de fonctions entieres, J. Math. Pures Appl. 75 (1996), 409-418.

On the projective descriptions of the space of holomorphic germs P. LAUBIN, University of Liege, Institute of Mathematics (B37), 4000 Liege, Belgium

Summary A natural topology on the set O(S) of holomorphic germs near a subset S of a Frechet space E is the locally convex inductive limit topology of the spaces O(£l} endowed with the compact open topology; here Q runs over all open subset of E containing 5. We discuss some known results concerning the projective characterization of this generally uncountable inductive limit. In particular, when 5 is a compact subset, Mujica gave a description of O(S) as the inductive limit of a suitable sequence of compact subsets. He used for this a set of intricate semi-norms. We give a projective characterization of this space using simpler semi-norms whose form is similar to the one used in the Whitney extension theorem for CQO functions. They are quite natural in a framework where extensions are involved. We also give a simple proof that this topology is strictly stronger than the topology of the projective limit of the non quasi-analytic spaces

Introduction In this paper, we present some known results and some new advances concerning the topology of spaces of holomorphic functions in an open subset of a Frechet space. The main problems which we will considered concern projective descriptions of spaces which appear as inductive limits. We restrict ourselves to the compact open topology but give a projective characterization of holomorphic germs using norms whose form is similar to the one used in the Whitney extension theorem for C^ functions. They are quite natural in a framework where extension properties are involved. 331

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1 NOTATIONS AND DEFINITIONS It is interesting to consider the problem in a general Frechet space. Let E be a complex Frechet space and 17 an open subset of E. There are many equivalent definitions of holomorphy in this framework [1]. Let us use the following one. DEFINITION 1 A function f : Q ^ C is holomorphic if it is continuous and the complex valued of one complex variable C 3 z »—» f ( x + zh) is holomorphic where it is defined for every fixed x,h € E. The space of holomorphic functions in Q, is denoted O(£t). If the E is finite dimensional, the assumption of continuity can be removed. In general, it can be replaced by the local boundedness. Let us remind some basic facts concerning polynomials in a Frechet space. A homogeneous polynomial p of degree A; is a function p(h) = u.h(k} = u(h, ..., where u : E x . . . x E -+ C : (hi, , . . . ,/ifc) H-» u(hi, , . . . , /ifc) is a /c-linear symmetric mapping. It is said continuous if u is continuous. If j < fc, we denote by u.h^ the (k — j)-linear form on E defined by

u.h^' .(v, . . . , v ) = u.(h, . . . , h, v, . . . , v) with j copies of h and k — j copies of v. If k = j, this is just an element of C. If A C E is bounded, let \\p\A = sup p(h) . It is well known [1] that if A is a bounded, convex and balanced subset of E then sup

kk u.(hi,... ,/i f c )| < yrlblU.

/c!

An holomorphic function has a local Taylor expansion + 00

k=0

The symmetric polynomials D k f ( x ) are defined by

If k > 0, we also use the notation x£K

if K is a compact subset of Q and A is a bounded, convex and balanced subset of E.

(1)

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333

We endow the space O(Q) of holomorphic functions in Q with the usual compact open topology TO defined by the semi-norms

p(f) = sup /| K where K runs over all compact subset of f2. We restrict ourselves to this topology although some interesting problems also occurs for the r^ and r§ topologies. Of course, Cauchy inequalities extend in this framework. If A is a convex and balanced subset of E such that x + rA C £1 then \\Dkf(x)\A<^

sup |/|.

T* x+rA

2 SOME KNOWN RESULTS At least two natural inductive limits occur. A) If E = Cn and 5 is an arbitrary subset of Rn we can consider

0(S) = i where f2 runs over all open subset of Cn containing the set 5. This inductive limit defines on O(S) the strongest locally convex topology for which the inclusions O(fi) —>• O(S) are continuous. In general, this is a strongly uncountable limit. This is the space of germs of real analytic functions on 5. It has be proved by Martineau [2] using abstract functional techniques that Here the topology is the weakest locally convex one such that the inclusions O(£l) —> O(S) are continuous. For an open subset uj of R n , this can also be proved using the representation of the analytic functionals on O(u) by harmonic functions in E \ {0} x ui presented in [3]. B) If E is a general Frechet space and K is any compact subset of E, we can also consider the inductive limit 0(K) = i where Q runs over all open subset of E containing the set K. Mujica in [12] gave a projective description of this space using explicit semi-norms. His result can be stated in the following way. The topology of O(K) is generated by semi-norms of the form +°° ,n n=0

= sup sup 2n

—'

m=Q

where • M is any compact subset of E,

m

*—'

771=0

ml

(3)

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• (e n )ra is any sequence of strictly positive numbers decreasing to 0, (xfc) and (yjt) are any sequences of K and (a^), (6 fc ) are any null sequences such that for any k

• (nfc) is any sequence of positive integers. The semi-norms of type (2) are the natural ones. It can easily been shown that the sum can be replaced by a sup. The semi-norms of type (3) are formed to be able in the proof to glue together several Taylor expansions at points where some ambiguity between Taylor expansions can occur. In some way, they look two restrictive: they involve estimates on limited Taylor expansions approximating the function outside the compact set K. Such restrictions do not occur in Whitney extension theorem for the C°° framework. Mujica also shows that for a locally connected compact subset K, the first type of semi-norms can be used alone. However if A' = (1/m : m € NO} U {0} C C

this is not the case. Indeed, consider the sequence fm € O{z e C : ftz ^ m+11,2 } with fm — 1 if 3R^ < m+11y2 and fm = 0 if ftz > m+11,2. It is bounded for the semi-norms of type (2) but not bounded in O(K}. The result of Mujica is also stated for a Riemann domain over a Frechet space. 3 NEW RESULTS 3.1 Theorem Consider the following semi-norms of Whitney type I/I

Dkf

K,M

k\ sup

sup

Here k > 0 and M is any compact subset of E. These semi-norms are simpler than the ones used by Mujica and only constrain limited Taylor expansions between two points of K. Note that ||/ K.M,k is increasing with M and that ||/ K,\M,k = ^k\\f\\K,M,k if A > 0. Using the Cauchy inequalities, one easily see that these semi-norms are continuous on 0(K). In a finite dimensional space, we can fix M equal to the closed unit ball and omit it in the notations. Then we obtain the semi-norms sup

Projective descriptions of the space of holomorphic germs

335

with \\Dkf\\K = sup^sup^,^ \Dkf(x).hW\. The main result of [4] is the following projective description of O(K). THEOREM 2 If K is a compact subset of a Frechet space E then the topology o f O ( K ) is defined by the semi-norms p(f)

=

SUp4\\f\\K,M,k

/CGN

where M runs over all compact subsets of E and (efc)fc>o over all sequences of real numbers decreasing to 0. Moreover, if K is locally connected then one can replace ||/||K,A/,A: by \\Dk f \ \ K , M / k l in the definition of the semi-norms p. An easy inspection of the proof of this theorem shows that in the definition of ||/||K-,M,A;» it is possible to replace t\ by any sequence at satisfying at > i\ This is not a surprise since the control on the growth of the derivatives is already performed by the first part. However, the previous semi-norms are the natural ones since they correspond to the usual estimation of the error in the Taylor expansion of a holomorphic function. It follows from the theorem 2 that, for any locally closed subset M of R n , the space O(M) is nuclear, complete, ultrabornological and a strictly webbed space. In fact, the first three properties are contained in the theorem 1.2 of [2]. Moreover, if Kj is a fundamental sequence of compact subsets of M, the sets eni,...,nr ={f e 0(M) : \f\\K.tk < n^+\k <= N, 1 < j < r} define a strict web of O(M). Using the projective description of theorem 2, this can be easily checked as in [5] where the case of an open subset of Rn is considered. 3.1 Another inductive limit The next result shows that the set of sequences (e/c)/c>o cannot easily be relaxed. Let L = (Lfe)fceN be an increasing sequence of real positive numbers. If £1 is an open subset of R n , we denote by C (L) (0) the set of all / € Coo(^) such that for any compact subset K of Q, we have \Dkf\\K < Al+kLkk for every k and some A > 0. This space is endowed with the semi-norms

fc€N

where K runs over all compact subsets of fi and (efc)fc>o over all sequences of real numbers decreasing to 0. L or C^ is said to be quasi-analytic if u G C^ L ^(Q) and Dau(x) = 0 for every Q € Nn and some x 6 £1 imply u = 0. A classical theorem of Denjoy-Carleman says that C( L )(Q) is non quasi-analytic if and only if X]fc^o V-^fc < +00. It is well known that, as a set, L

where the intersection runs over all non quasi-analytic sequences L. We have the following result.

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THEOREM 3 7/n is a non void open subset ofW1 then the topology of O(Q) is strictly stronger than the projective limit topology of the non quasi-analytic spaces Note that this projective limit topology is defined by the semi-norms

where K runs over all compact subsets of 0 and (^fc)fc>o °ver all sequences of real numbers decreasing to 0 and satisfying + 00 V—^

2_^rik < +00-

k-Q

This follows easily from the fact that, if the sequence % is summable, there is a sequence n/c converging to +00 such that n^k is still summable.

References [1] S. Dineen, "Holomorphic germs on compact subsets of locally convex spaces", Functional analysis, holomorphy and approximation theory, Springer-Verlag Lectures Notes in Math., Vol. 843, 1981, 247-263. [2] A. Martineau, "Sur la topologie des espaces de fonctions holomorphes", Math. Ann. 163,62-88, 1966. [3] L. Hormander, "The analysis of linear partial differential operators I", Grundl. der math. Wiss. 256, Springer, 1983. [4] P. Laubin, "A projective description of the space of holomorphic germs", Proc. of the Edinburgh Math. Soc., 2001, 44, 1-9. [5] H.G. Garnir, M. De Wilde and J. Schmets, "Analyse fonctionnelle III, Espaces fonctionnels usuels", Math. Reihe 45, Birkhauser, 1973. [6] K.D. Bierstedt, "An introduction to locally convex inductive limits", in Functional Analysis and Its Applications, 35-133, World Scientific, 1988. [7] S. Dineen, "Complex analysis in locally convex spaces", North-Holland Math. Studies, Vol. 57, 1981. [8] L. Ehrenpreis, "Solution of some problems of division. Part IV", Amer. J. of Math., 82, 522-588, 1960. [9] A. Hirschowitz, "Bornologie des espaces de fonctions analytiques en dimension infinie", Seminaire Pierre Lelong 1969/1970, Springer-Verlag Lectures Notes in Math., Vol.205, 1970,21-33. [10] L. Hormander, "An introduction to complex analysis in several variables", NorthHolland Publ. Co., 1973.

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[11] F. Mantovani and S. Spagnolo, " Funzionali analitici reali e funzioni armoniche", Ann. Sc. Norm. Sup. Pisa 18, 475-513, 1964. [12] J. Mujica, "A Banach-Dieudonne theorem for germs of holomorphic functions", J. Funct. Anal. 57, 31-48, 1984. [13] K. Rusek, "A new topology in the space of germs of holomorphic functions on a compact set in Cn", Bull. Acad. Polon. Sci. 25, 1227-1232, 1977. [14] P. Schapira, "Sur les ultra-distributions", Ann. Scient. EC. Norm. Sup. 1, 395-415, 1968.

Microlocal scaling and extension of distributions M.K. Venkatesha Murthy Dipartimento di Matematica Universita di Pisa Via Buonarroti 2 Pisa, Italy

We are concerned here with the following problem for extension of distributions defined in the complement of a submanifold to the whole manifold. Suppose X is a paracompact C°° manifold and Y is a C°° submanifold of X. We assume the following standard geometric condition regarding the position of the singularity set of the distribution with respect to the submanifold Y. Given a distribution u € V(X \ Y) such that the closure of WF(u) in T*(X) is orthogonal to the tangent bundle TY of Y", the question is concerned with determining conditions under which u admits an extension u G V(X). The extendability of a distribution naturally depends on its analytic behaviour as one approaches the submanifold Y. The analytic behaviour of u near Y is expressed by the notion of microlocal scaling degree. We shall introduce the concept of microlocal scaling degree of a distribution along the submanifold Y", which classifies the class of extendible distributions on X\Y. We obtain results analogous to those of Hormander for homogeneous distributions defined on lRn \ 0. 339

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Scaling degree in the Euclidean space We begin by recalling the notion of the scaling degree due to Steinman for distributions in the Euclidean space with respect to the origin. Roughly speaking the scaling degree of a distribution measures the strength of its singularity. The H+ - action (t,(p) -> (IRn) induces by the adjoint action an JR+ - action on the space of distributions D'(Rn) denned by (u^(p) = (u,
sd(u] = inf{k e H; V - lim tkut = 0} C' r U

Thus u —> sd(u) is a mapping on the space of distributions D'(IRn) into [—00, +00]. We shall need the following simple properties of sd(u): For C°° functions / we have sd(f) < 0 sd(d%u) < sd(u) + |a|, for any multi-index a sd(xQu) < sd(u] — a , for any multi-index a sd(fu) < sd(u) for any C°° function /

sd(u 0 v) = sd(u] + sd(v) For the function exp(^) which is not defined at the origin sd(exp(i)) = +00 For homogeneous distributions of degree / the scaling degree is —/ It is well known that all homogeneous distributions on Rn \ 0 are not extendible (see Hormander's book).

We have the following result on the extension of distributions from R \OtolRn: n

Theorem. — (a) If a distribution u £ X?'(Rn \ 0) has scaling degree sd(u) < n then there exists a unique u € X>'(Rn) which extends u and has the same scaling degree.

Microlocal scaling and extension of distributions

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(b) If a distribution u 6 X>7(lRn \ 0) has a finite scaling degree sd(u) > n then there exists a (non unique) extension u e ^'(IR/1) having the same scaling degree and it is uniquely determined by its values on the subspace complementary to the space of test functions (p E £>(IRn) which vanish at the origin upto order [sd(u)] — n. Thus in the second case the extension is determined uniquely modulo a finite linear combination of derivatives of the Dirac distribution. Definition. — The real number h = sd(u) — n is called the degree of singularity of the distribution u. If the degree of singularity is finite then the distribution can be extended to the whole space. Micro-local scaling degree along a submanifold In order to extend globally across a submanifold we need a microlocal version of the scaling degree which classifies the strength of the singularities of distributions defined in the complement of submanifolds of arbitrary codimension, which can be extended to the whole manifold. If Y is a submanifold of X and u G V(X \ Y) and if W is any submanifold of X which is transversal to Y then the restriction u\w of u to W is well defined. To classify the distributions which are extendible globally across Y we need a covariant extension of the scaling degree sd(u) introduced above, namely the following concept of microlocal scaling degree with respect to the conormal bundle N*Y \ 0 or more genrally with respect to a conic subset FQ of N*Y \ 0. First of all we recall the following standard facts on wave front sets of distributions (see Hormanders book for details). We denote by VT(X), for any closed conic set F of T*(X) \ 0, the space of all distributions v on X whose wave front set WF(v) is contained in F and VT(X) is provided with Hormander's (pseudo -) topology: A sequence Uj in VT(X) converges to u in T>'T(X) if

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(i) Uj -» u weakly in T>'(X], (ii) for any properly supported pseudo - differential operator A such that /^supp (A) fl F = 0, we have AUJ — >• Au in C°°(X). (where ^supp(^4) is the projection of the wave front set WF(kA) of the Schwartz kernel k& of A onto the second component. (i.e. IJL - supp(A) = 7T2(WF(KA)), where 7r2:T*X x T*X -» T*X is the projection onto to the second factor.) We also need the following property on the restriction: If M is a submanifold of X and u € V(X) are such that the wave front set WF(u) does not intersect the conormal bundle N*M of M then u admits a restriction JMU. Moreover, if F is any conic set such that F n N*M — 0, then the trace operator 7^ lifts to a sequentially continuous operator from VT(X) to V(M}. Suppose X is a C°° manifold and Y is a C°° submanifold of X. Given a distribution u G P'(J£ \ Y") such that the closure of WF(u] in T*(X] is orthogonal to the tangent bundle TY of Y", i.e. if (y, 0 € WF(u) with y € Y , £ € Ty*X then {£, 0) = 0 Vtf €

It follows, from the property of the trace operator recalled above, that under the hypothesis on u and Y , u admits a restriction to every submanifold C of X contained in a sufficiently small neighbourhood of Y in X intersecting Y transversally. We may assume that X is provided with a Riemannian structure and y is a totally geodesic submanifold of X. We take a star shaped neighbourhood of the zero section Z(TyX) in TX and suppose that / denotes the local diffeomorphism U — > f(U) C Y x X denned by /(ar,7?) = (or, expx'd). u^ denotes the distribution defined on U obtained as the pull back of (ly u) by / and let (uf)t(x, t?) = u*(x, t&) for * € R+. Consider a closed conic set FQ in the conormal bundle of y with the zero section removed, denoted by N*Y \ 0. Definition. — A distribution u € V(X) is said to have the microlocal scaling degree k along the submanifold Y with respect to the closed conic set FQ, denoted by p,sdy(u, ;Fo), if

Microlocal scaling and extension of distributions

343

(a) there exists a closed conic subset F of T*(TyX) \ 0 with the properties (i) FTy -L /* (Z(T*Y) x r 0 ) and (ii)WF((f*(lY®u))t)cr (b) nsdy (u; r 0 ) = k = mf{k € E; VT(TYX} - limt_»0 tkf*(lY <S> u) t = 0} in the sense of Hormander's topology on T>'r(TYX). Remark. - We may take for / in the above definition any local difTeomorphism satisfying certain obvious conditions: CY xY

, <W(y,0) = s Moreover, fj,sd(u: TO) is independent of the choice of such an /. This follows by making use of the stationary phase method. The microlocal scaling degree has properties analogous to the ones satisfied by the scaling degree on the Eucleidean space The other main properties of [isdy^ FQ) are the following: 1) The microlocal scaling degree fj,sd(u) behaves well with respect to product of distributions when they are defined: If u € V'T and v € V^(X] are such that the zero section Z(N*Y) of the conormal bundle of Y, is not contained in F 0 E then the product u.v is well defined in a neighbourhood of Y and LJ,sdy (w.u, A) < /isdy (u; F) -i- psdy (u; S) where A = FUSU(FeS) 2) Given a submanifold Y' of Y and the conic set FQ C N*Y \ 0 we have fisdy (u; FQ) < iJLsdy (u; FQ) where (T'Q = Fo|y)- This folllows using the sequential continuity of the restriction mapping. For the extension property of distributions it is more convenient

344

Venkatesha Murthy

to make use of an equivalent concept of transversal microlocal scaling degree. For this purpose we decompose TyX = TY 0 E and the subbundle E defines in a neighbourhood of Y in X transversal submanifolds C = {Cy;y € Y}. Let U be a neighbourhood of the zero section Z(TyX) in TX and ir<2 : Y x X -> X be the second projection. Then fs = ^2° f\Enu is & diffeomorphism onto a neighbourhood of Y in JC (where /(x, t?) = (x, expx$). The images of the fibers of E define submanifolds transversal to Y at each of its points. Setting (fgu}(x, fa?) = W£,JL(Z, i?) for (xj z?) E £ we define the transversal microscaling degree along Y by = inf{fc e E;limt.u < j ± = 0} the limit being taken in Hormander's topology in X>p£ where TE\Z(E) — /E(FO) is a closed conic set in T*(E) \ 0. (Here and else where in the following Z(E) stands for the zero - section of the bundle E.) As already mentioned and this can be proved by using local frames and explicitely writing down the Fourier transforms. Extension of distributions across submanifolds Suppose u € V(X \ Y) is such that the closure in T*(X) of the wave front set WF(u) is orthogonal to the tangent bundle TY. The microlocal scaling degree with respect to Y and the transversal microlocal scaling degrees of such distributions are defined in a natural way: If C € C°°(X) with suppC n Y = 0 then (/*(l y ® C)-/*(«))t is considered as a distribution in a neighbourhood of the zero section in TyX and the micro local scaling and the micro-local transversal scaling degrees are defined as before using these distributions. Choose a fibration C of the neighbourhood U of Y by transversal submanifolds Cy at y € Y defined by the subbundle E complementary to TY in TX. Then we have

Microlocal scaling and extension of distributions

345

with respect to the point {y} = Cy fl Y considered as the origin in Rd, d being the dimension of the transversal manifold Cy. In the following we restrict ourselves to the case of FQ = N*Y \ 0 and in which case we shall write //sdy(n) for /^5c?y(it;Fo) = Our main result is the following Theorem. — Suppose Y is a submanifold of X and u € V(X \ Y) satisfy the hypothesis that the closure in T*X of its wave front set WF(u) is orthogonal to TY and that jj,sdY(u;N*Y \ 0) < oo. (a) If nsdY(u) < codimY then u admits a unique extension u G V(X) such that nsdY(u) = fJ-sdy^). (b) If codimY < ^sdy(u\N*Y \0) < oo then u admits (nonunique) extensions u € V(X] with the same microlocal scaling degree as that of u such that u is uniquely determined by its values on the subspace T consisting of smooth functions in such that where Vy,h(X) denotes the subspace ofD(X) consisting of functions (p which vanish on Y upto order h = fj,sdy(u] N*Y \ 0) — codimY. Brief sketch of the proofs We shall only give a very rough idea of the proofs of the theorems. We first consider the case of the euclidean space, namely, X = Rn and Y" = the origin. (i) Suppose u € P;(lRn \ 0) and sd(u) < n. Let C € £>(IRn) be a cutoff function such that £(x) = 1 in \x\ < e for some fixed e > 0 and let £t(x) = C(to)> for * £ R+- Define the sequence of distributions {uj} on En by It is easily seen that {uj} is a Cauchy sequence in T>'(lRn) for the

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weak topology. By the weak sequential completeness of V(Rn) the sequence {uj} converges weakly to a limit u EX>'(lR n ) . Then one can check u is unique, also independent of the choice of the function C and defines the extension of u such that sd(u) = sd(u). (ii) In the case n < sd(u) < H-oo, let h = sd(u] — n. Consider the subspace Z> A (R n ) = {(^ € P(Rn); ^v>(0) = 0,V|a < [/i]} Then the subspace {v e X>'(IRn); (v^) = 0, W> € ^(Hn)} consists of distributions of the form v = 2|o|<[/i] °ad£5i where CQ are complex numbers. We can decompose any (p 6 P(lRn) uniquely as