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Weak Hölder Continuity of the Generalized Hamiltonian Flow
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
204, 164]182 Ž1996.
0430
Weak Holder Continuity of the Generalized ¨ Hamiltonian Flow Latchezar Stoyanov* Department of Mathematics, Uni¨ ersity of Western Australia, Nedlands, Perth 6907, Western Australia Submitted by Cathleen S. Morawetz Received September 25, 1995
The generalized Hamiltonian flow Ft generated by a smooth function on a symplectic manifold S with smooth boundary S is considered. It is proved that if Ft has no tangencies of infinite order to S, then given a metric d on S, an integral curve s Ž t . of Ft , a compact neighbourhood K of s Ž0. in S, and T ) 0, there exist a ) 0 and C ) 0 such that dŽ s Ž t ., r Ž t .. F CdŽ s Ž0., r Ž0.. a holds for < t < F T whenever r Ž t . is an integral curve of Ft with r Ž0. g K. Q 1996 Academic Press, Inc.
1. INTRODUCTION Given a symplectic manifold S with boundary S and a smooth ŽHamiltonian. function p on S, satisfying certain non-degeneracy conditions, the generalized Hamiltonian flow is a local flow Ft : S ª S. It was first defined w11, 12x in the case S s T *Ž M ., p being the by Melrose and Sjostrand ¨ principal symbol of a certain differential operator on the manifold with boundary M. Their study was motivated by investigations on propagations of singularities of differential operators and inspired by previous works of Melrose w8, 9x, Anderson and Melrose w1x, Ivrii w3x, Morawetz, Ralston, and w4, Sect. 24x and Strauss w13x, Taylor w16x, and others Žsee also Hormander ¨ the Notes there.. Later it was shown that the generalized Hamiltonian flow is encountered in many important problems in spectral and scattering theory Žcf., for example, w2, 10, 14x.. *Partially supported by Australian Research Council Grant 412r092. E-mail address: [email protected]. 164 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
GENERALIZED HAMILTONIAN FLOW
165
To give a rough idea what the generalized Hamiltonian flow is Žsee Section 2 for the precise definition., consider the classical case when S s T *Ž V = R., where V is a domain in R n with smooth boundary V, and p is the principal symbol of the wave operator. The projection of the generalized Hamiltonian flow on V determines a flow called the generalized geodesic flow in V. Roughly its integral curves can be described as the possible trajectories of a point moving with constant speed in the interior of V, reflecting from V following the usual law of the geometrical optic. When the point reaches V with direction of motion tangential to V, under certain circumstances, it may start to move on V along a geodesic line on V. Under similar circumstances the point might leave V entering the interior of V, etc. Clearly, this flow is connected with two very well known flows in the theory of dynamical systems: the billiard flow and the geodesic flow. For example, if V is a strictly convex bounded domain, then the generalized geodesic flow coincides with the billiard flow in the interior of V and with the standard geodesic flow on V. In general the behavior of the generalized Hamiltonian flow is rather complicated. In fact, as an example of M. Taylor w16x shows Žsee also w4, Sect. 24x., this is not always a flow in the usual sense of dynamical systems, since there may exist different integral curves issued from one and the same point of the phase space. In the present paper we make the assumption that the Hamiltonian vector field H p has no tangencies of infinite order to S which guarantees Žcf. w11x. that the generalized Hamiltonian flow Ft is well-defined as a local flow. However, even under this assumption the situation still looks rather complicated. The main difficulty comes from the fact that near the boundary S each integral curve consists of several pieces Žsegments. satisfying different systems of ŽHamiltonian. differential equations. In the case when only transversal reflections occur, each segment satisfies the same system of differential equations but then the condition of impact at the boundary must be taken into account and the fact that the number of segments Žpieces. of the curve tends to infinity as the curve gets closer to the boundary S. Having this in mind, one can see why the regularity properties of the generalized Hamiltonian flow cannot be very good. In fact, the flow is even discontinuous but making some standard identifications of points along the boundary S, one gets a continuous flow. This was w11x. It is known Žsee w9; 4, Sect. 24x. established by Melrose and Sjostrand ¨ that the strongest regularity property that the generalized Hamiltonian flow might have in general is Holder continuity of order 12 Žsee also Sec¨ tion 2 below.. In this paper we show that the generalized Hamiltonian flow has a property stronger than continuity}we call it weak Holder continuity. ¨ Given a metric space Ž X, d ., we say that a flow f t : X ª X Ž t g R. is
166
LATCHEZAR STOYANOV
weakly Holder continuous if for each x 0 g X and each T ) 0 there exist ¨ C s C Ž x 0 . ) 0 and a s a Ž x 0 . ) 0 such that a
d Ž f t Ž x 0 . , f t Ž x . . F Cd Ž x 0 , x . ,
Though this property has a lot to do with the geometry of S, it is quite easy to show that it is equivalent to its local version which has a purely analytic nature and concerns dependence of integral curves on their initial points. The situation here resembles that of an impulsive differential equation Žcf. w5x, for example.. The difference is that the times of impact at the boundary S and the number of these impacts on a given time interval < t < F T are different for different integral curves. It is worth mentioning that the result proved below is not trivial even in the special case of the generalized geodesic flow in a convex Žbut not strictly convex. domain V ; R 2 with smooth boundary V. Then of course the interesting case is when the initial integral curve is the boundary V Žor part of it.. If V is strictly convex, the weak Holder continuity follows ¨ easily from the remarkable results of Lazutkin Žsee, for example, w6x. and Melrose w9x. In this case the behaviour of the flow near V is rather simple because of the existence of invariant curves. However, if the curvature of V vanishes somewhere, invariant curves no longer exist Žsee Mather w7x. and the situation becomes much more complicated. The question remains whether in this special case the flow is Holder continu¨ ous. It seems very unlikely that this is so in the general situation considered in this paper.
2. PRELIMINARIES AND STATEMENT OF THE MAIN RESULT Let S be a symplectic manifold with boundary S and let p: S ª R be a smooth Ž C` . function with dp < S / 0. Following w11x Žsee also w4, Sect. 24x., one defines the generalized Hamiltonian flow of p as follows. Let w g C`Ž S . be a defining function of S, i.e., w ) 0 in S _ S and w s 0 on S Ž w might be only locally defined around S .. Assume that
w , w , p4 4 / 0. Here f, g 4 denotes the Poisson bracket of the functions f and g Žcf. w4, Sect. 21.1x, for example., i.e., f, g 4 s H f g s yHg f, where H f g s L H f g is the Lie deri¨ ati¨ e of g with respect to the Hamiltonian ¨ ector field H f determined by the function f. If x 1 , . . . , x n , j 1 , . . . , j n are Žlocal. symplec-
167
GENERALIZED HAMILTONIAN FLOW
tic coordinates in S, then Hf s
ž
f j 1
f
,...,
j n
n
f , g 4 s H f gs Ý js1
ž
,y
f
,...,y
x1
f g j j x j
f g
y
x j j j
f xn
/
/
,
.
The operators H fj are then defined by induction: H fj g s H f Ž H fjy1 g . for j ) 1. By definition H f0 g s g. We are going to determine the flow of p on the zero le¨ el set S s py1 Ž 0 . . Consider the following subsets of S: G s s g S : w Ž s . s Hp w Ž s . s 0 4 Gd s s g G : Hp2w Ž s . ) 0 4
Ž glancing set . ,
Ž diffractive set . ,
Gg s s g G : Hp2w Ž s . - 0 4
Ž gliding set . ,
G k s s g G : H pj w Ž s . s 0 ; j s 0, 1, . . . , k y 1 4 , G` s
`
F Gk.
ks2
The gliding ¨ ector field H pG on G is defined by H pG s H p q
Hp2w Hw2 p
Hw .
DEFINITION w11x. Let I ; R be an interval. A curve g : I ª S is called a generalized integral cur¨ e Ž bicharacteristic. of p if there exists a discrete subset B of I such that: Ži. if t g I _ B and g Ž t . g Ž S _ S . j Gd , then there exists
g 9 Ž t . s Hp Ž g Ž t . . ; Žii. if t g I _ B and g Ž t . g G _ Gd , then there exists
g 9 Ž t . s HpG Ž g Ž t . . ;
168
LATCHEZAR STOYANOV
Žiii. for each t g B, g Ž t q s . g S _ S for all small s / 0 and there exist the limits g Ž t y 0. / g Ž t q 0. which are points of one and the same integral curve of w on S. Clearly, such a curve g has discontinuities at the points of B. To get a continuous curve we have to identify some pairs of points on S. Consider the following equivalence relation on S: x ; y iff either x s y or x g S l S, y g S l S and x and y lie on one and the same integral curve of w on S. The quotient space ˜ S s Sr; , which carries a natural structure of a manifold with boundary, is called compressed characteristic set and the projection g ˜ of a generalized integral curve g on ˜S is a continuous curve called compressed integral cur¨ e of p. In what follows we assume that G` s B. In this case one can define a flow Ft : ˜ Sª˜ S, t g R, such that Ft : t g R4 is a compressed integral curve of p for each s g ˜ S Žcf. w11x.. It was shown in w11x that the maps Ft are continuous. Remark. It is clear from the definition that the maps Ft depend on w . In general w is only locally defined and so in such cases Ft 4 is a local flow defined for small < t <. However, the integral curves of p, disregarding their parametrization, are globally defined and do not depend on w . To avoid the inconvenience caused by the change of the parameter along integral curves, one may consider maps between cross-sections of a given integral curve Žthe same definition as that of a Poincare ´ map.. Since the problem we deal with below is of local nature, and locally the maps between cross-sections and Ft have equivalent behaviour, we consider the maps Ft as if they were globally defined. Note that in general the maps Ft are not smooth. This is easily seen for S s T * Ž V = R. ,
Ž 1.
V being a domain in R n with smooth boundary V, and p given by n
p Ž x, j . s
2 . Ý j i2 y j nq1
Ž 2.
is1
An elementary argument shows that if V is the interior or the exterior of a ball in R n, then the maps Ft are Holder continuous with Holder exponent 12 , ¨ ¨ 1 and 2 is the maximal number with this property.
GENERALIZED HAMILTONIAN FLOW
169
Given S and p as in the beginning of this section, fix an arbitrary metric d on ˜ S generating its topology. The following result has been announced in w15x. THEOREM 1. Let r 0 g ˜ S, K be a compact neighbourhood of r 0 in ˜ S, and T0 ) 0. There exist constants C ) 0 and a ) 0 such that d Ž Ft r 0 , Ft r . F C Ž d Ž r 0 , r . .
a
Ž 3.
for e¨ ery r g K and e¨ ery t with < t < F T0 . The rest of the paper is devoted to the proof of this theorem.
3. PROOF OF THEOREM 1 Consider the following local version of the statement of Theorem 1. LEMMA 1. Let r 0 g ˜ S be fixed. There exist a neighbourhood U0 of r 0 in ˜S and constants T ) 0, C ) 0, a ) 0 such that Ž3. holds for all r g U0 and t g w0, T x.
Let us first show that Theorem 1 follows from Lemma 1. Let r 0 g ˜ S, T0 ) 0, and let r t s Ft r 0 for t g R. It follows from Lemma 1 that for each t there exist Tt ) 0, a t ) 0, Ct ) 0, and a neighbourhood Ut of r t in ˜S such that at
d Ž Fs s , Fs r t . - Ct Ž d Ž s , r t . . ,
< s < - Tt .
There exist t 1 , . . . , t k g w0, T0 x such that k
D Ž t i y Ti , t i q Ti . > w0, T0 x , is1
where Ti s Tt i . We may assume that 0 s t 1 - ??? - t k s T0 and t iq1 t i q Ti for all i s 1, . . . , k y 1. Set k
U0 s
F Fty1 Ž Ut . l s : d Ž r 0 , s . - 14 , i
i
is1
ai a i a iy 1 C s max Ci Ciy1 Ciy2 ??? C1a i a iy 1 ??? a 2 : 1 F i F k 4 ,
a s min a 1 ??? a i : 1 F i F k 4 . Let r g U0 and t g Ž0, T0 x. We claim that Ž3. holds. There exists i with t i - t F t iq1. Set s s T y ti ,
s j s t jq1 y t j
170
LATCHEZAR STOYANOV
for j s 1, . . . , i y 1. Then we have s1 q ??? qs j s t jq1 ,
s1 q ??? qsiy1 q s s t.
Moreover 0 - s j - Tj for all j s 1, . . . , i y 1 and 0 - s - Ti . Denote r j s Fs j r 0 . The definition of U0 and r g U0 imply Fs1 r g Ut 2 . Therefore d Ž Ft 2 r , r 1 . s d Ž Fs1 r , r 1 . s d Ž Fs1 r , Fs1 r 0 . - C1 d Ž r , r 0 .
a1
.
In the same way d Ž Ft 3 r , r 3 . s d Ž Fs 2Ž Fs1 r . , Fs 2 r 1 . -C2 Ž C1 d Ž r , r 0 . s C2 C1a 2 d Ž r , r 0 .
a 1q a 2
a1 a 2
.
.
Applying this procedure i times, one gets ai a i a iy 1 d Ž Ft r , Ft r 0 . - Ci Ciy1 Ciy2 ??? C1a i a iy 1 ??? a 2 d Ž r , r 0 .
F Cd Ž r , r 0 .
a 1 a 2 ??? a i
a
which proves Ž3.. The rest of this section is devoted to the proof of Lemma 1. Denote again by r 0 an element of S the projection of which in ˜ S coincides with r 0 . It follows by w11x Žcf. also w4, Sect. 24x. that there exist local coordinates
Ž x, j . s Ž x 1 , . . . , x n ; j 1 , . . . , j n . around r 0 s Ž0, 0. in S such that w s x 1 , i.e., locally S s Ž x, j . : x 1 G 0 4 ,
S s Ž x, j . : x 1 s 0 4 , and p Ž x, j . s a Ž x, j . Ž j 12 y r Ž x, j 9 . . , where aŽ x, j . ) 0 and r Ž x, j 9. are smooth functions. Throughout we use the notation x9 s Ž x 2 , . . . , x n . ,
j 9 s Ž j 2 , . . . , jn . .
Introduce the function
ˆp Ž x, j . s j 12 y r Ž x, j 9 .
GENERALIZED HAMILTONIAN FLOW
171
and denote by Fˆt the generalized Hamiltonian flow of ˆ p on S. Clearly, Ft and Fˆt have the same integral curves, only the parametrization of these curves might be different. Hence Ft can be written in the form Ft s FˆlŽ t, x, j . , where lŽ t, x, j . is a smooth function with Ž drdt . l s aŽ Ft Ž x, j ... It is now clear that it is enough to prove the statements Ža. and Žb. replacing Ft by Fˆt . This means that it is sufficient to prove these statements in the special case when the function p has the form p Ž x, j . s j 12 y r Ž x, j 9 . .
Ž 4.
Define the metric d by d Ž Ž x, j . , Ž y, h . . s max max < x i y yi < , < j i y hi < 4 , 1FiFn
and set Ft Ž x, j . s Ž x Ž t . , j Ž t . . . There are several cases for r 0 . Case 1. r 0 g S _ S. In this case locally around r 0 the generalized integral curves of p coincide with the integral curves of the Hamiltonian vector field H p , so the assertion follows trivially with a s 1. Case 2. r 0 g Gd . This means that Ž rr x 1 .Ž r 0 . ) 0. Then there exists a neighbourhood V0 of r 0 in S and a constant c ) 0 with
r x1
Ž r . G c,
r g V0 .
Choose a neighbourhood U0 of r 0 and T ) 0 such that Ft ŽU0 . ; V0 for all t g w0, T x. It then follows by w4, Lemma 24.3.4x that for each r g U0 the generalized integral curve Ft r : t g w0, T x4 has at most one reflection. Using this one can easily derive that the assertion of the lemma holds with a s 12 . Case 3. r 0 g Gg . As in the previous case, we find neighbourhoods U0 ; V0 of r 0 and c ) 0 such that
r x1
Ž r . F yc,
r g V0 .
Using w4, Lemma 24.3.5x, we find a constant C9 ) 0 such that if Ft r : t g w0, T x4 is a reflecting bicharacteristic Žin this case it is equivalent to say that
172
LATCHEZAR STOYANOV
the bicharacteristic is not entirely contained in Gg ., then we have
h12 Ž t . q y 1 Ž t . F C9 Ž h12 Ž 0 . q y 1 Ž 0 . . for all t g w0, T x, where Ft Ž r . s Ž y Ž t . ; h Ž t . . .
Ž 5.
From this the assertion of the lemma follows easily with a s 12 . Case 4. r 0 g G k _ G kq1, k G 3. Let Ž ˜ x9Ž t ., j˜9Ž t .. be the integral curve of the vector field H pG on G with initial conditions ˜ x9Ž0. s x9Ž0., j˜9Ž0. s j 9Ž0.. Set r eŽ t . s Ž 0, ˜x9 Ž t . , j˜9 Ž t . . , x1 f Ž t . s < x9 Ž t . y ˜ x9 Ž t . < q < j 9 Ž t . y j˜9 Ž t . < . Given r g S, define er Ž t . and fr Ž t . as eŽ t . and f Ž t ., respectively, replacing r 0 with r . Choose neighbourhoods U0 ; V0 of r 0 and T with 0-TF
1 2
so small that H pk has a constant sign in V0 and Ft U0 ; V0 for all t g w0, T x. Later we will impose other conditions on U0 and T. In the case under consideration we have e Ž t . s at ky 2 q l Ž t . t ky 1 for some constant a / 0 and some smooth function lŽ t . Žcf. w11x or w4x.. Fix L ) 0 with < lŽ t . < F
L 2
,
< l9 Ž t . < F
L 2
; t g w 0, T x .
Using standard facts from the theory of differential equations, it follows that if U0 is small enough, then there exists a constant c ) 0 such that for every r g U0 we have the representation er Ž t . s a0 q a1 t q ??? qa ky2 t ky 2 q at ky 2 q m Ž t . t ky 1
Ž 6.
with < a i < F c d ; i s 0, 1, . . . , k y 2;
< m Ž t . < F L, < m9 Ž t . < F L ; t g w 0, T x ,
Ž 7.
173
GENERALIZED HAMILTONIAN FLOW
where
d s dŽ r0 , r . . Then Ž5. and Ž6. imply er Ž t . F at ky 2 q c d Ž 1 q t q ??? qt ky 2 . q Lt ky 1 - at ky 2 q c d
1 1yt
q Lt ky1 - at ky2 q 2 c d q Lt ky1 .
In the same way one gets a similar estimate from below for er Ž t . which implies at ky 2 y 2 c d y Lt ky1 F er Ž t . F at ky2 q 2 c d q Lt ky1 ,
t g w 0, T x .
Ž 8. Next, we distinguish two subcases. The first is the more difficult one. Subcase 4.1. a - 0. Fix an arbitrary b ) 0. The assertion of Lemma 1 follows immediately from the following LEMMA 2. U0 and T ) 0 can be chosen so small that there exists a constant A ) 0 with d Ž Ft r 0 , Ft r . F A Ž d Ž r 0 , r . .
Ž1y b .r2
for all r g U0 and all t g w0, T x. Proof of Lemma 2. Take r g U0 and set d s dŽ r 0 , r . as before. Then y1Ž 0. F d ,
and Ž7. and Ž8. yield
ž
1y
e 2
/
at ky 2 y 2 c d F er Ž t . F 1 q
ž
e 2
/
at ky2 q 2 c d
for all t g w0, T x. In particular, < er Ž t . < F 2 < a < t ky 2 q 2 c d ,
t g w 0, T x .
Ž 9.
In what follows we use the notation const to denote a positive constant which does not depend on the choice of U0 , T, r , and t.
174
LATCHEZAR STOYANOV
Applying the inequalities Ž24.3.7. in w4x, we get that
¡y Ž t . F const H Ž t y s . < e Ž s . < ds q d q d t t
ž
1
r
0
~
ž
1
r
/
/
,
,
Ž 10 .
¢f Ž t . F const ž H Ž t y s . < e Ž s . < ds q d t q d t t
r
2
r
0
2
/
.
Set hŽ t . s
r x1
Ž y Ž t . , h 9Ž t . . .
It follows from w4, p. 436x that < h Ž t . y er Ž t . < s
r
Ž y Ž t . , h 9Ž t . . y
x1
r x1
Ž 0, ˜y9 Ž t . , h˜ 9 Ž t . .
F const Ž fr Ž t . q y 1 Ž t . .
Ž 11 .
for all t g w0, T x. Our next aim is to estimate hŽ t . by means of d , a, and t. For this we will use Ž10. and Ž11.. First, for fr Ž t ., Ž9. and Ž10. imply fr Ž t . F const
t
t y s . Ž 2 < a < s ky2 q 2 c d . ds q d
žH Ž ž HŽ 0
t
s const t 2
0
2
2 < a < s ky2 q 2 c d . ds q d
s const d q 2 c d t 3 q
ž
2 < a< ky1
t kq1
/
/
/
F const Ž d q < a < t kq 1 . . Similarly, y 1 Ž t . F const
t
t y s . Ž 2 < a < s ky 2 q 2 c d . ds q 2 d
žH Ž ž HŽ
F const t
0
t
0
2 < a < s ky 2 q 2 c d . ds q 2 d
F const Ž d q < a < t k . .
/
/
175
GENERALIZED HAMILTONIAN FLOW
From these two estimates, Ž8. and Ž11. one gets h Ž t . F er Ž t . q const Ž fr Ž t . q y 1 Ž t . . F at ky 2 q const Ž d q t ky 1 . . In the same way, h Ž t . G at ky 2 y const Ž d q t ky 1 . . Thus, there exists a constant M ) 0 such that t g w 0, T x . Ž 12 .
at ky 2 y Md y Mt ky1 F h Ž t . F at ky2 q Md q Mt ky1 ,
Next, consider again the integral curve Ž ˜ y Ž t ., h ˜ Ž t .. of the vector field HpG such that ˜ y Ž0. s Ž0, y9Ž0.., h ˜ Ž0. s Ž0, h 9Ž0.. and recall that er Ž t . s
r x1
r
Ž ˜y Ž t . , h˜ 9 Ž t . . s
x1
Ž 0, ˜y9 Ž t . , 0, h˜ 9 Ž t . . .
For all t g w0, T x except finitely many points we have dh dt
2r
n
Ž t. s Ý is1
s
x1 x i
2r x 12
2r
n
Ž y Ž t . , h 9 Ž t . . ˙yi Ž t . q Ý is2
x 1 j i
Ž y Ž t . , h 9 Ž t . . h˙i Ž t .
Ž y Ž t . , h 9 Ž t . . 2h1 Ž t . 2r
n
yÝ is2
x1 x i 2r
n
qÝ is2
x 1 j i
r
Ž y Ž t . , h 9Ž t . . Ž y Ž t . , h 9Ž t . .
j i r xi
Ž y Ž t . , h 9Ž t . . Ž y Ž t . , h 9Ž t . . .
So, introducing the function R Ž x, j . s
2r x1
n
qÝ is2
2r
n
x, j 9 . 2 j 1 y 2 Ž
Ý is2
2r x 1 j i
Ž x, j 9 .
x1 x i r xi
Ž x, j 9 .
Ž x, j 9 . ,
we have dh dt
Ž t . s R Ž y Ž t . , h 9Ž t . .
r j i
Ž x, j 9 .
176
LATCHEZAR STOYANOV
for all t g w0, T x for which Ž dhrdt .Ž t . exists. Similarly one gets d dt
er Ž t . s R Ž ˜ yŽ t. , h ˜ 9Ž t . .
for all t g w0, T x. Using these expressions for Ž dhrdt .Ž t . and Ž drdt . er Ž t ., as in w4, pp. 435]436x one shows that dh dt
d
Ž t. y
dt
er Ž t . F const Ž fr Ž t . q y 1 Ž t . q
Ž 13 .
for all t g w0, T x. Using Ž9. and Ž10. one estimates
t
H0 Ž 2 cd q 2 < a< s
ky 2
. ds q const d F const Ž d q < a< t ky1 . . Ž 14 .
Combining this with Ž13. and the estimates for fr Ž t . and y 1Ž t ., we get dh dt
Ž t. y
d dt
er Ž t . F const Ž d q < a < t ky 1 . ,
t g w 0, T x .
Ž 15 .
On the other hand Ž6. gives d dt
ky2
er Ž t . s
Ý iai t iy1 q Ž k y 2. at ky3 q m9 Ž t . t ky1 q m Ž t . Ž k y 1. t ky2 is1
F cd
1 1yt
q Ž k y 2 . at ky 3 q Ž k y 1 . Lt ky2 q Lt ky1
F Ž k y 2 . at ky 3 q const Ž d q t ky 2 . . Similarly, d dt
er Ž t . F Ž k y 2 . at ky 3 y const Ž d q t ky2 . .
Combining the latter estimates for Ž drdt . er Ž t . with Ž15., one gets
Ž k y 2 . at ky 3 y const Ž d q t ky 2 . F
dh dt
Ž t . F Ž k y 2 . at ky 3 q const Ž d q t ky2 .
177
GENERALIZED HAMILTONIAN FLOW
for all t g w0, T x. That is, there exists a constant M ) 0, and we may assume this is the same constant as in Ž12., such that
Ž k y 2 . at ky 3 y Md y Mt ky 2 F
dh dt
Ž t . F Ž k y 2 . at ky 3 q Md q Mt ky2 ,
Fix M ) 0 with Ž12. and Ž16.. Define e by ky2ye
t g w 0, T x . Ž 16 .
s 1 q b.
Ž 1 y e . Ž k y 2.
Ž 17 .
We claim that 0 - e - 1. Indeed, Ž17. is equivalent to
e s Ž 1 q b . Ž 1 y e . Ž k y 2. y Ž k y 2. s Ž k y 2. Ž 1 q b . Ž 1 y e . y 1 s Ž k y 2. b y e Ž 1 q b . , which implies
es
b Ž k y 2. 1 q Ž 1 q b . Ž k y 2.
.
It is now clear that 0 - e - 1. Having fixed the constant M ) 0, choose T ) 0 such that TF
< a< 2M
,
Ž 18 .
and the neighbourhood U0 of r 0 so small that
d s dist Ž r 0 , r . -
e < a< T 2M
for every r g U0 . Given r g U0 , set td s
2 Md
1r Ž ky2 .
ž / e < a<
.
Ž 19 .
Clearly 0 - td - T. Moreover for each t g w td , T x we have t ky 2 Ž e < a < y Mt . G tdky2 Ž e < a < y MT . G tdky2 s
2 Md e < a <
e < a<
2
s Md .
e < a< 2
178
LATCHEZAR STOYANOV
Hence t g w td , T x .
M Ž d q t ky 1 . F e < a < t ky 2 ,
Ž 20 .
In the same way Žfor t g w td , T x. we get t ky 3 Ž e < a < y Mt . G tdky2 Ž e < a < y MT . G tdky2
e < a< 2
s Md ,
therefore t g w td , T x .
M Ž d q t ky 2 . F e < a < t ky 3 ,
Ž 21 .
We are going to estimate from above the ratio Ž dhrdt .Ž t .rhŽ t . on the interval w td , T x. Given t g w td , T x, use Ž12. and Ž20. to get yh Ž t . G < a < t ky 2 y M Ž d q t ky1 . G < a < t ky 2 y e < a < t ky 2 s < a < t ky 2 Ž 1 y e . ) 0.
Ž 22 .
In the same way, Ž16. and Ž21. imply y
dh dt
Ž t . F Ž k y 2 . < a< t ky 3 q M Ž d q t ky2 . F Ž k y 2 . < a < t ky 3 q e < a < t ky 3 s Ž k y 2 q e . < a < t ky 3 .
Consequently, y Ž dhrdt . Ž t . ky2qe Ž dhrdt . Ž t . Ž k y 2 q e . < a< t ky 3 s F s ky 2 < a< t e hŽ t . yh Ž t . Ž1 y e . Ž 23 . for all t g w td , T x. Consider the function g Ž t . s h12 Ž t . y y 1 Ž t . h Ž t . . It is clearly continuous and g 9Ž t . exists almost everywhere in w0, T x. For those t g w td , T x for which g 9Ž t . exists, we have dg dt
Ž t . s 2h1 Ž t . h˙1 Ž t . y ˙y 1 Ž t . h Ž t . y y 1 Ž t .
dh
s 2h1 Ž t . h Ž t . y 2h1 Ž t . h Ž t . y y 1 Ž t .
dt
Ž t.
dh dt
Ž t . s yy1 Ž t .
dh dt
Ž t. G 0
179
GENERALIZED HAMILTONIAN FLOW
for t g w td , T x. This and Ž23. imply yy 1 Ž t . Ž dhrdt . Ž t . ky2qe Ž dgrdt . Ž t . Ž dhrdt . Ž t . s F F . 2 gŽ t. hŽ t . Ž1 y e . t h1 Ž t . y y 1 Ž t . h Ž t . Integrating the latter inequality gives ln
gŽ t. g Ž td .
s
t
Ht
d
ky2qe t1 t Ž dgrdt . Ž s . ds F ds s ln H g Ž s. td Ž 1 y e . td s
Ž ky2q e .r Ž1y e .
ž /
.
According to Ž17. we have ky2qe
Ž1 y e .
s Ž 1 q b . Ž k y 2. ,
while Ž19. yields tdŽ1q b .Ž ky2. s
1q b
2 Md
ž / e < a<
.
Therefore g Ž t . F g Ž td .
t
Ž1q b .Ž ky2 .
ž /
FN
td
g Ž td .
d 1q b
,
Ž 24 .
where Ns
e < a<
ž / 2M
1q b
.
Thus, gŽ t. F N
g Ž td .
d 1q b
,
t g w td , T x .
Ž 25 .
On the interval w0, td x the function g Ž t . can be estimated in a trivial way. Indeed, for t g w0, td x, Ž14. implies
t g w 0, td x .
180
LATCHEZAR STOYANOV
Now for t g w td , T x the latter and Ž25. yield g Ž t . F const
d2 d 1q b
s const d 1y b .
Therefore g Ž t . F const d 1y b ,
t g w 0, T x .
Ž 26 .
Consequently,
t g w 0, T x .
Ž 27 .
For y 1Ž t . we already have y 1Ž t . F const d on w0, td x. Let t g w td , T x. Then from Ž22. we have < h Ž t . < s yh Ž t . F < a < t ky 2 Ž 1 y e . ) 0. This and Ž24. imply y 1 Ž t . < h Ž t . < F g Ž t . F g Ž td .
t
Ž1q b .Ž ky2 .
ž / td
Therefore y1Ž t . F
g Ž td .
t Ž1q b .Ž ky2.
tdŽ1q b .Ž ky2.
< hŽ t . <
F const d 1y b t b Ž ky2. F const d 1y b .
Thus, y 1 Ž t . F const d 1y b ,
t g w 0, T x .
Ž 28 .
Now, applying a standard argument from the theory of differential equations to the rest of coordinate functions, one derives that there exists a constant C ) 0 such that d Ž Ft r 0 , Ft r . F Cd Ž1y b .r2 for all t g w0, T x. This completes the proof of Lemma 2. Subcase 4.2. have
a ) 0. Take T such that T - arL. Then for t g Ž0, T x we
e Ž t . s at ky 2 q l Ž t . t ky 1 G t ky 2 Ž a y LT . ) 0. Therefore Ft r 0 : t g w0, T x4 is an integral curve of the vector field H p . Given r g U0 , set d s dŽ r , r 0 .. As in the proof of Lemma 2 we see that
181
GENERALIZED HAMILTONIAN FLOW
there exists a positive constant M with Ž12.. Fix m with this property and take T such that Ž18. holds. Determine td g Ž0, T . by td s
2 Md
1r Ž ky2 .
ž / a
.
Then for each t g w td , T . we have t ky 2 Ž a y Mt . ) tdky2 Ž a y MT . G tdky2
a 2
s
2 Md a a
2
s Md .
Hence M Ž d q t ky 1 . - at ky2 ,
t g w td , T x
and therefore h Ž t . G at ky 2 y M Ž d q t ky 1 . ) 0 for all t g w td , T .. Consequently, Ft r : t g w td , T x4 is an integral curve of Hp . Clearly, d Ž Ft r 0 , Ft r . F const d Ž Ft d r 0 , Ftd r . ,
t g w td , T x .
Ž 29 .
Since td s const d 1rŽ ky2. and along an integral curve Ft r of the generalized Hamiltonian flow, the functions y Ž t . and h 9Ž t . are Lipschitz while
'
This and Ž29. now imply d Ž Ft r 0 , Ft r . F const d 1r2Ž ky2. ,
t g w 0, T x ,
which completes the proof of Subcase 4.2.
REFERENCES 1. K. Anderson and R. Melrose, The propagation of singularities along gliding rays, In¨ ent. Math. 41 Ž1977., 197]232. 2. V. Guillemin and R. Melrose, The Poisson summation formula for manifolds with boundary, Ad¨ . Math. 32 Ž1979., 204]232. 3. V. Ivrii, Propagation of singularities of solutions to the wave equation near the boundary, Dokl. Acad. Nauk SSSR 235 Ž1977., 1013]1016. 4. L. Hormander, ‘‘The Analysis of Linear Partial Differential Operators,’’ Vol. III, ¨ Springer-Verlag, Berlin, 1985.
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LATCHEZAR STOYANOV
5. V. Lakshmikantham, D. Bainov, and P. Simeonov, ‘‘Theory of Impulsive Differential Equations,’’ World Scientific, Singapore, 1989. 6. V. Lazutkin, The existence of caustics for a billiard problem in a convex domain, Math. USSR}Iz¨ . 7 Ž1973., 185]214. 7. J. Mather, Glancing billiards, Ergodic Theory Dynamical Systems 2 Ž1982., 397]403. 8. R. Melrose, Singularities and energy decay in acoustical scattering, Duke Math. J. 46 Ž1979., 43]59. 9. R. Melrose, Equivalence of glancing hypersurfaces, In¨ ent. Math. 37 Ž1976., 165]191. 10. R. Melrose, ‘‘Geometric Scattering Theory,’’ Cambridge Univ. Press, Cambridge, 1995. 11. R. Melrose and J. Sjostrand, Singularities in boundary value problems, I, Comm. Pure ¨ Appl. Math. 31 Ž1978., 593]617. 12. R. Melrose and J. Sjostrand, Singularities in boundary value problems, II, Comm. Pure ¨ Appl. Math. 35 Ž1982., 129]168. 13. C. Morawetz, J. Ralston, and W. Strauss, Decay of solutions of the wave equation outside nontrapping obstacles, Comm. Pure Appl. Math. 30 Ž1977., 447]508. 14. V. Petkov and L. Stoyanov, ‘‘Geometry of Reflecting Rays and Inverse Spectral Problems,’’ Wiley, Chichester, 1992. 15. L. Stoyanov, Regularity properties of the generalized Hamiltonian flow, in ‘‘Seminaire ´ Equations aux Derivees Partielles,’’ Ecole Polytechnique, Centre de Math., Expose ´ VI, 1992]1993. 16. M. Taylor, Grazing rays and reflection of singularities to wave equations, Comm. Pure Appl. Math. 29 Ž1976., 1]38.
204, 164]182 Ž1996.
0430
Weak Holder Continuity of the Generalized ¨ Hamiltonian Flow Latchezar Stoyanov* Department of Mathematics, Uni¨ ersity of Western Australia, Nedlands, Perth 6907, Western Australia Submitted by Cathleen S. Morawetz Received September 25, 1995
The generalized Hamiltonian flow Ft generated by a smooth function on a symplectic manifold S with smooth boundary S is considered. It is proved that if Ft has no tangencies of infinite order to S, then given a metric d on S, an integral curve s Ž t . of Ft , a compact neighbourhood K of s Ž0. in S, and T ) 0, there exist a ) 0 and C ) 0 such that dŽ s Ž t ., r Ž t .. F CdŽ s Ž0., r Ž0.. a holds for < t < F T whenever r Ž t . is an integral curve of Ft with r Ž0. g K. Q 1996 Academic Press, Inc.
1. INTRODUCTION Given a symplectic manifold S with boundary S and a smooth ŽHamiltonian. function p on S, satisfying certain non-degeneracy conditions, the generalized Hamiltonian flow is a local flow Ft : S ª S. It was first defined w11, 12x in the case S s T *Ž M ., p being the by Melrose and Sjostrand ¨ principal symbol of a certain differential operator on the manifold with boundary M. Their study was motivated by investigations on propagations of singularities of differential operators and inspired by previous works of Melrose w8, 9x, Anderson and Melrose w1x, Ivrii w3x, Morawetz, Ralston, and w4, Sect. 24x and Strauss w13x, Taylor w16x, and others Žsee also Hormander ¨ the Notes there.. Later it was shown that the generalized Hamiltonian flow is encountered in many important problems in spectral and scattering theory Žcf., for example, w2, 10, 14x.. *Partially supported by Australian Research Council Grant 412r092. E-mail address: [email protected]. 164 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
GENERALIZED HAMILTONIAN FLOW
165
To give a rough idea what the generalized Hamiltonian flow is Žsee Section 2 for the precise definition., consider the classical case when S s T *Ž V = R., where V is a domain in R n with smooth boundary V, and p is the principal symbol of the wave operator. The projection of the generalized Hamiltonian flow on V determines a flow called the generalized geodesic flow in V. Roughly its integral curves can be described as the possible trajectories of a point moving with constant speed in the interior of V, reflecting from V following the usual law of the geometrical optic. When the point reaches V with direction of motion tangential to V, under certain circumstances, it may start to move on V along a geodesic line on V. Under similar circumstances the point might leave V entering the interior of V, etc. Clearly, this flow is connected with two very well known flows in the theory of dynamical systems: the billiard flow and the geodesic flow. For example, if V is a strictly convex bounded domain, then the generalized geodesic flow coincides with the billiard flow in the interior of V and with the standard geodesic flow on V. In general the behavior of the generalized Hamiltonian flow is rather complicated. In fact, as an example of M. Taylor w16x shows Žsee also w4, Sect. 24x., this is not always a flow in the usual sense of dynamical systems, since there may exist different integral curves issued from one and the same point of the phase space. In the present paper we make the assumption that the Hamiltonian vector field H p has no tangencies of infinite order to S which guarantees Žcf. w11x. that the generalized Hamiltonian flow Ft is well-defined as a local flow. However, even under this assumption the situation still looks rather complicated. The main difficulty comes from the fact that near the boundary S each integral curve consists of several pieces Žsegments. satisfying different systems of ŽHamiltonian. differential equations. In the case when only transversal reflections occur, each segment satisfies the same system of differential equations but then the condition of impact at the boundary must be taken into account and the fact that the number of segments Žpieces. of the curve tends to infinity as the curve gets closer to the boundary S. Having this in mind, one can see why the regularity properties of the generalized Hamiltonian flow cannot be very good. In fact, the flow is even discontinuous but making some standard identifications of points along the boundary S, one gets a continuous flow. This was w11x. It is known Žsee w9; 4, Sect. 24x. established by Melrose and Sjostrand ¨ that the strongest regularity property that the generalized Hamiltonian flow might have in general is Holder continuity of order 12 Žsee also Sec¨ tion 2 below.. In this paper we show that the generalized Hamiltonian flow has a property stronger than continuity}we call it weak Holder continuity. ¨ Given a metric space Ž X, d ., we say that a flow f t : X ª X Ž t g R. is
166
LATCHEZAR STOYANOV
weakly Holder continuous if for each x 0 g X and each T ) 0 there exist ¨ C s C Ž x 0 . ) 0 and a s a Ž x 0 . ) 0 such that a
d Ž f t Ž x 0 . , f t Ž x . . F Cd Ž x 0 , x . ,
Though this property has a lot to do with the geometry of S, it is quite easy to show that it is equivalent to its local version which has a purely analytic nature and concerns dependence of integral curves on their initial points. The situation here resembles that of an impulsive differential equation Žcf. w5x, for example.. The difference is that the times of impact at the boundary S and the number of these impacts on a given time interval < t < F T are different for different integral curves. It is worth mentioning that the result proved below is not trivial even in the special case of the generalized geodesic flow in a convex Žbut not strictly convex. domain V ; R 2 with smooth boundary V. Then of course the interesting case is when the initial integral curve is the boundary V Žor part of it.. If V is strictly convex, the weak Holder continuity follows ¨ easily from the remarkable results of Lazutkin Žsee, for example, w6x. and Melrose w9x. In this case the behaviour of the flow near V is rather simple because of the existence of invariant curves. However, if the curvature of V vanishes somewhere, invariant curves no longer exist Žsee Mather w7x. and the situation becomes much more complicated. The question remains whether in this special case the flow is Holder continu¨ ous. It seems very unlikely that this is so in the general situation considered in this paper.
2. PRELIMINARIES AND STATEMENT OF THE MAIN RESULT Let S be a symplectic manifold with boundary S and let p: S ª R be a smooth Ž C` . function with dp < S / 0. Following w11x Žsee also w4, Sect. 24x., one defines the generalized Hamiltonian flow of p as follows. Let w g C`Ž S . be a defining function of S, i.e., w ) 0 in S _ S and w s 0 on S Ž w might be only locally defined around S .. Assume that
w , w , p4 4 / 0. Here f, g 4 denotes the Poisson bracket of the functions f and g Žcf. w4, Sect. 21.1x, for example., i.e., f, g 4 s H f g s yHg f, where H f g s L H f g is the Lie deri¨ ati¨ e of g with respect to the Hamiltonian ¨ ector field H f determined by the function f. If x 1 , . . . , x n , j 1 , . . . , j n are Žlocal. symplec-
167
GENERALIZED HAMILTONIAN FLOW
tic coordinates in S, then Hf s
ž
f j 1
f
,...,
j n
n
f , g 4 s H f gs Ý js1
ž
,y
f
,...,y
x1
f g j j x j
f g
y
x j j j
f xn
/
/
,
.
The operators H fj are then defined by induction: H fj g s H f Ž H fjy1 g . for j ) 1. By definition H f0 g s g. We are going to determine the flow of p on the zero le¨ el set S s py1 Ž 0 . . Consider the following subsets of S: G s s g S : w Ž s . s Hp w Ž s . s 0 4 Gd s s g G : Hp2w Ž s . ) 0 4
Ž glancing set . ,
Ž diffractive set . ,
Gg s s g G : Hp2w Ž s . - 0 4
Ž gliding set . ,
G k s s g G : H pj w Ž s . s 0 ; j s 0, 1, . . . , k y 1 4 , G` s
`
F Gk.
ks2
The gliding ¨ ector field H pG on G is defined by H pG s H p q
Hp2w Hw2 p
Hw .
DEFINITION w11x. Let I ; R be an interval. A curve g : I ª S is called a generalized integral cur¨ e Ž bicharacteristic. of p if there exists a discrete subset B of I such that: Ži. if t g I _ B and g Ž t . g Ž S _ S . j Gd , then there exists
g 9 Ž t . s Hp Ž g Ž t . . ; Žii. if t g I _ B and g Ž t . g G _ Gd , then there exists
g 9 Ž t . s HpG Ž g Ž t . . ;
168
LATCHEZAR STOYANOV
Žiii. for each t g B, g Ž t q s . g S _ S for all small s / 0 and there exist the limits g Ž t y 0. / g Ž t q 0. which are points of one and the same integral curve of w on S. Clearly, such a curve g has discontinuities at the points of B. To get a continuous curve we have to identify some pairs of points on S. Consider the following equivalence relation on S: x ; y iff either x s y or x g S l S, y g S l S and x and y lie on one and the same integral curve of w on S. The quotient space ˜ S s Sr; , which carries a natural structure of a manifold with boundary, is called compressed characteristic set and the projection g ˜ of a generalized integral curve g on ˜S is a continuous curve called compressed integral cur¨ e of p. In what follows we assume that G` s B. In this case one can define a flow Ft : ˜ Sª˜ S, t g R, such that Ft : t g R4 is a compressed integral curve of p for each s g ˜ S Žcf. w11x.. It was shown in w11x that the maps Ft are continuous. Remark. It is clear from the definition that the maps Ft depend on w . In general w is only locally defined and so in such cases Ft 4 is a local flow defined for small < t <. However, the integral curves of p, disregarding their parametrization, are globally defined and do not depend on w . To avoid the inconvenience caused by the change of the parameter along integral curves, one may consider maps between cross-sections of a given integral curve Žthe same definition as that of a Poincare ´ map.. Since the problem we deal with below is of local nature, and locally the maps between cross-sections and Ft have equivalent behaviour, we consider the maps Ft as if they were globally defined. Note that in general the maps Ft are not smooth. This is easily seen for S s T * Ž V = R. ,
Ž 1.
V being a domain in R n with smooth boundary V, and p given by n
p Ž x, j . s
2 . Ý j i2 y j nq1
Ž 2.
is1
An elementary argument shows that if V is the interior or the exterior of a ball in R n, then the maps Ft are Holder continuous with Holder exponent 12 , ¨ ¨ 1 and 2 is the maximal number with this property.
GENERALIZED HAMILTONIAN FLOW
169
Given S and p as in the beginning of this section, fix an arbitrary metric d on ˜ S generating its topology. The following result has been announced in w15x. THEOREM 1. Let r 0 g ˜ S, K be a compact neighbourhood of r 0 in ˜ S, and T0 ) 0. There exist constants C ) 0 and a ) 0 such that d Ž Ft r 0 , Ft r . F C Ž d Ž r 0 , r . .
a
Ž 3.
for e¨ ery r g K and e¨ ery t with < t < F T0 . The rest of the paper is devoted to the proof of this theorem.
3. PROOF OF THEOREM 1 Consider the following local version of the statement of Theorem 1. LEMMA 1. Let r 0 g ˜ S be fixed. There exist a neighbourhood U0 of r 0 in ˜S and constants T ) 0, C ) 0, a ) 0 such that Ž3. holds for all r g U0 and t g w0, T x.
Let us first show that Theorem 1 follows from Lemma 1. Let r 0 g ˜ S, T0 ) 0, and let r t s Ft r 0 for t g R. It follows from Lemma 1 that for each t there exist Tt ) 0, a t ) 0, Ct ) 0, and a neighbourhood Ut of r t in ˜S such that at
d Ž Fs s , Fs r t . - Ct Ž d Ž s , r t . . ,
< s < - Tt .
There exist t 1 , . . . , t k g w0, T0 x such that k
D Ž t i y Ti , t i q Ti . > w0, T0 x , is1
where Ti s Tt i . We may assume that 0 s t 1 - ??? - t k s T0 and t iq1 t i q Ti for all i s 1, . . . , k y 1. Set k
U0 s
F Fty1 Ž Ut . l s : d Ž r 0 , s . - 14 , i
i
is1
ai a i a iy 1 C s max Ci Ciy1 Ciy2 ??? C1a i a iy 1 ??? a 2 : 1 F i F k 4 ,
a s min a 1 ??? a i : 1 F i F k 4 . Let r g U0 and t g Ž0, T0 x. We claim that Ž3. holds. There exists i with t i - t F t iq1. Set s s T y ti ,
s j s t jq1 y t j
170
LATCHEZAR STOYANOV
for j s 1, . . . , i y 1. Then we have s1 q ??? qs j s t jq1 ,
s1 q ??? qsiy1 q s s t.
Moreover 0 - s j - Tj for all j s 1, . . . , i y 1 and 0 - s - Ti . Denote r j s Fs j r 0 . The definition of U0 and r g U0 imply Fs1 r g Ut 2 . Therefore d Ž Ft 2 r , r 1 . s d Ž Fs1 r , r 1 . s d Ž Fs1 r , Fs1 r 0 . - C1 d Ž r , r 0 .
a1
.
In the same way d Ž Ft 3 r , r 3 . s d Ž Fs 2Ž Fs1 r . , Fs 2 r 1 . -C2 Ž C1 d Ž r , r 0 . s C2 C1a 2 d Ž r , r 0 .
a 1q a 2
a1 a 2
.
.
Applying this procedure i times, one gets ai a i a iy 1 d Ž Ft r , Ft r 0 . - Ci Ciy1 Ciy2 ??? C1a i a iy 1 ??? a 2 d Ž r , r 0 .
F Cd Ž r , r 0 .
a 1 a 2 ??? a i
a
which proves Ž3.. The rest of this section is devoted to the proof of Lemma 1. Denote again by r 0 an element of S the projection of which in ˜ S coincides with r 0 . It follows by w11x Žcf. also w4, Sect. 24x. that there exist local coordinates
Ž x, j . s Ž x 1 , . . . , x n ; j 1 , . . . , j n . around r 0 s Ž0, 0. in S such that w s x 1 , i.e., locally S s Ž x, j . : x 1 G 0 4 ,
S s Ž x, j . : x 1 s 0 4 , and p Ž x, j . s a Ž x, j . Ž j 12 y r Ž x, j 9 . . , where aŽ x, j . ) 0 and r Ž x, j 9. are smooth functions. Throughout we use the notation x9 s Ž x 2 , . . . , x n . ,
j 9 s Ž j 2 , . . . , jn . .
Introduce the function
ˆp Ž x, j . s j 12 y r Ž x, j 9 .
GENERALIZED HAMILTONIAN FLOW
171
and denote by Fˆt the generalized Hamiltonian flow of ˆ p on S. Clearly, Ft and Fˆt have the same integral curves, only the parametrization of these curves might be different. Hence Ft can be written in the form Ft s FˆlŽ t, x, j . , where lŽ t, x, j . is a smooth function with Ž drdt . l s aŽ Ft Ž x, j ... It is now clear that it is enough to prove the statements Ža. and Žb. replacing Ft by Fˆt . This means that it is sufficient to prove these statements in the special case when the function p has the form p Ž x, j . s j 12 y r Ž x, j 9 . .
Ž 4.
Define the metric d by d Ž Ž x, j . , Ž y, h . . s max max < x i y yi < , < j i y hi < 4 , 1FiFn
and set Ft Ž x, j . s Ž x Ž t . , j Ž t . . . There are several cases for r 0 . Case 1. r 0 g S _ S. In this case locally around r 0 the generalized integral curves of p coincide with the integral curves of the Hamiltonian vector field H p , so the assertion follows trivially with a s 1. Case 2. r 0 g Gd . This means that Ž rr x 1 .Ž r 0 . ) 0. Then there exists a neighbourhood V0 of r 0 in S and a constant c ) 0 with
r x1
Ž r . G c,
r g V0 .
Choose a neighbourhood U0 of r 0 and T ) 0 such that Ft ŽU0 . ; V0 for all t g w0, T x. It then follows by w4, Lemma 24.3.4x that for each r g U0 the generalized integral curve Ft r : t g w0, T x4 has at most one reflection. Using this one can easily derive that the assertion of the lemma holds with a s 12 . Case 3. r 0 g Gg . As in the previous case, we find neighbourhoods U0 ; V0 of r 0 and c ) 0 such that
r x1
Ž r . F yc,
r g V0 .
Using w4, Lemma 24.3.5x, we find a constant C9 ) 0 such that if Ft r : t g w0, T x4 is a reflecting bicharacteristic Žin this case it is equivalent to say that
172
LATCHEZAR STOYANOV
the bicharacteristic is not entirely contained in Gg ., then we have
h12 Ž t . q y 1 Ž t . F C9 Ž h12 Ž 0 . q y 1 Ž 0 . . for all t g w0, T x, where Ft Ž r . s Ž y Ž t . ; h Ž t . . .
Ž 5.
From this the assertion of the lemma follows easily with a s 12 . Case 4. r 0 g G k _ G kq1, k G 3. Let Ž ˜ x9Ž t ., j˜9Ž t .. be the integral curve of the vector field H pG on G with initial conditions ˜ x9Ž0. s x9Ž0., j˜9Ž0. s j 9Ž0.. Set r eŽ t . s Ž 0, ˜x9 Ž t . , j˜9 Ž t . . , x1 f Ž t . s < x9 Ž t . y ˜ x9 Ž t . < q < j 9 Ž t . y j˜9 Ž t . < . Given r g S, define er Ž t . and fr Ž t . as eŽ t . and f Ž t ., respectively, replacing r 0 with r . Choose neighbourhoods U0 ; V0 of r 0 and T with 0-TF
1 2
so small that H pk has a constant sign in V0 and Ft U0 ; V0 for all t g w0, T x. Later we will impose other conditions on U0 and T. In the case under consideration we have e Ž t . s at ky 2 q l Ž t . t ky 1 for some constant a / 0 and some smooth function lŽ t . Žcf. w11x or w4x.. Fix L ) 0 with < lŽ t . < F
L 2
,
< l9 Ž t . < F
L 2
; t g w 0, T x .
Using standard facts from the theory of differential equations, it follows that if U0 is small enough, then there exists a constant c ) 0 such that for every r g U0 we have the representation er Ž t . s a0 q a1 t q ??? qa ky2 t ky 2 q at ky 2 q m Ž t . t ky 1
Ž 6.
with < a i < F c d ; i s 0, 1, . . . , k y 2;
< m Ž t . < F L, < m9 Ž t . < F L ; t g w 0, T x ,
Ž 7.
173
GENERALIZED HAMILTONIAN FLOW
where
d s dŽ r0 , r . . Then Ž5. and Ž6. imply er Ž t . F at ky 2 q c d Ž 1 q t q ??? qt ky 2 . q Lt ky 1 - at ky 2 q c d
1 1yt
q Lt ky1 - at ky2 q 2 c d q Lt ky1 .
In the same way one gets a similar estimate from below for er Ž t . which implies at ky 2 y 2 c d y Lt ky1 F er Ž t . F at ky2 q 2 c d q Lt ky1 ,
t g w 0, T x .
Ž 8. Next, we distinguish two subcases. The first is the more difficult one. Subcase 4.1. a - 0. Fix an arbitrary b ) 0. The assertion of Lemma 1 follows immediately from the following LEMMA 2. U0 and T ) 0 can be chosen so small that there exists a constant A ) 0 with d Ž Ft r 0 , Ft r . F A Ž d Ž r 0 , r . .
Ž1y b .r2
for all r g U0 and all t g w0, T x. Proof of Lemma 2. Take r g U0 and set d s dŽ r 0 , r . as before. Then y1Ž 0. F d ,
and Ž7. and Ž8. yield
ž
1y
e 2
/
at ky 2 y 2 c d F er Ž t . F 1 q
ž
e 2
/
at ky2 q 2 c d
for all t g w0, T x. In particular, < er Ž t . < F 2 < a < t ky 2 q 2 c d ,
t g w 0, T x .
Ž 9.
In what follows we use the notation const to denote a positive constant which does not depend on the choice of U0 , T, r , and t.
174
LATCHEZAR STOYANOV
Applying the inequalities Ž24.3.7. in w4x, we get that
¡y Ž t . F const H Ž t y s . < e Ž s . < ds q d q d t t
ž
1
r
0
~
ž
1
r
/
/
,
,
Ž 10 .
¢f Ž t . F const ž H Ž t y s . < e Ž s . < ds q d t q d t t
r
2
r
0
2
/
.
Set hŽ t . s
r x1
Ž y Ž t . , h 9Ž t . . .
It follows from w4, p. 436x that < h Ž t . y er Ž t . < s
r
Ž y Ž t . , h 9Ž t . . y
x1
r x1
Ž 0, ˜y9 Ž t . , h˜ 9 Ž t . .
F const Ž fr Ž t . q y 1 Ž t . .
Ž 11 .
for all t g w0, T x. Our next aim is to estimate hŽ t . by means of d , a, and t. For this we will use Ž10. and Ž11.. First, for fr Ž t ., Ž9. and Ž10. imply fr Ž t . F const
t
t y s . Ž 2 < a < s ky2 q 2 c d . ds q d
žH Ž ž HŽ 0
t
s const t 2
0
2
2 < a < s ky2 q 2 c d . ds q d
s const d q 2 c d t 3 q
ž
2 < a< ky1
t kq1
/
/
/
F const Ž d q < a < t kq 1 . . Similarly, y 1 Ž t . F const
t
t y s . Ž 2 < a < s ky 2 q 2 c d . ds q 2 d
žH Ž ž HŽ
F const t
0
t
0
2 < a < s ky 2 q 2 c d . ds q 2 d
F const Ž d q < a < t k . .
/
/
175
GENERALIZED HAMILTONIAN FLOW
From these two estimates, Ž8. and Ž11. one gets h Ž t . F er Ž t . q const Ž fr Ž t . q y 1 Ž t . . F at ky 2 q const Ž d q t ky 1 . . In the same way, h Ž t . G at ky 2 y const Ž d q t ky 1 . . Thus, there exists a constant M ) 0 such that t g w 0, T x . Ž 12 .
at ky 2 y Md y Mt ky1 F h Ž t . F at ky2 q Md q Mt ky1 ,
Next, consider again the integral curve Ž ˜ y Ž t ., h ˜ Ž t .. of the vector field HpG such that ˜ y Ž0. s Ž0, y9Ž0.., h ˜ Ž0. s Ž0, h 9Ž0.. and recall that er Ž t . s
r x1
r
Ž ˜y Ž t . , h˜ 9 Ž t . . s
x1
Ž 0, ˜y9 Ž t . , 0, h˜ 9 Ž t . . .
For all t g w0, T x except finitely many points we have dh dt
2r
n
Ž t. s Ý is1
s
x1 x i
2r x 12
2r
n
Ž y Ž t . , h 9 Ž t . . ˙yi Ž t . q Ý is2
x 1 j i
Ž y Ž t . , h 9 Ž t . . h˙i Ž t .
Ž y Ž t . , h 9 Ž t . . 2h1 Ž t . 2r
n
yÝ is2
x1 x i 2r
n
qÝ is2
x 1 j i
r
Ž y Ž t . , h 9Ž t . . Ž y Ž t . , h 9Ž t . .
j i r xi
Ž y Ž t . , h 9Ž t . . Ž y Ž t . , h 9Ž t . . .
So, introducing the function R Ž x, j . s
2r x1
n
qÝ is2
2r
n
x, j 9 . 2 j 1 y 2 Ž
Ý is2
2r x 1 j i
Ž x, j 9 .
x1 x i r xi
Ž x, j 9 .
Ž x, j 9 . ,
we have dh dt
Ž t . s R Ž y Ž t . , h 9Ž t . .
r j i
Ž x, j 9 .
176
LATCHEZAR STOYANOV
for all t g w0, T x for which Ž dhrdt .Ž t . exists. Similarly one gets d dt
er Ž t . s R Ž ˜ yŽ t. , h ˜ 9Ž t . .
for all t g w0, T x. Using these expressions for Ž dhrdt .Ž t . and Ž drdt . er Ž t ., as in w4, pp. 435]436x one shows that dh dt
d
Ž t. y
dt
er Ž t . F const Ž fr Ž t . q y 1 Ž t . q
Ž 13 .
for all t g w0, T x. Using Ž9. and Ž10. one estimates
t
H0 Ž 2 cd q 2 < a< s
ky 2
. ds q const d F const Ž d q < a< t ky1 . . Ž 14 .
Combining this with Ž13. and the estimates for fr Ž t . and y 1Ž t ., we get dh dt
Ž t. y
d dt
er Ž t . F const Ž d q < a < t ky 1 . ,
t g w 0, T x .
Ž 15 .
On the other hand Ž6. gives d dt
ky2
er Ž t . s
Ý iai t iy1 q Ž k y 2. at ky3 q m9 Ž t . t ky1 q m Ž t . Ž k y 1. t ky2 is1
F cd
1 1yt
q Ž k y 2 . at ky 3 q Ž k y 1 . Lt ky2 q Lt ky1
F Ž k y 2 . at ky 3 q const Ž d q t ky 2 . . Similarly, d dt
er Ž t . F Ž k y 2 . at ky 3 y const Ž d q t ky2 . .
Combining the latter estimates for Ž drdt . er Ž t . with Ž15., one gets
Ž k y 2 . at ky 3 y const Ž d q t ky 2 . F
dh dt
Ž t . F Ž k y 2 . at ky 3 q const Ž d q t ky2 .
177
GENERALIZED HAMILTONIAN FLOW
for all t g w0, T x. That is, there exists a constant M ) 0, and we may assume this is the same constant as in Ž12., such that
Ž k y 2 . at ky 3 y Md y Mt ky 2 F
dh dt
Ž t . F Ž k y 2 . at ky 3 q Md q Mt ky2 ,
Fix M ) 0 with Ž12. and Ž16.. Define e by ky2ye
t g w 0, T x . Ž 16 .
s 1 q b.
Ž 1 y e . Ž k y 2.
Ž 17 .
We claim that 0 - e - 1. Indeed, Ž17. is equivalent to
e s Ž 1 q b . Ž 1 y e . Ž k y 2. y Ž k y 2. s Ž k y 2. Ž 1 q b . Ž 1 y e . y 1 s Ž k y 2. b y e Ž 1 q b . , which implies
es
b Ž k y 2. 1 q Ž 1 q b . Ž k y 2.
.
It is now clear that 0 - e - 1. Having fixed the constant M ) 0, choose T ) 0 such that TF
< a< 2M
,
Ž 18 .
and the neighbourhood U0 of r 0 so small that
d s dist Ž r 0 , r . -
e < a< T 2M
for every r g U0 . Given r g U0 , set td s
2 Md
1r Ž ky2 .
ž / e < a<
.
Ž 19 .
Clearly 0 - td - T. Moreover for each t g w td , T x we have t ky 2 Ž e < a < y Mt . G tdky2 Ž e < a < y MT . G tdky2 s
2 Md e < a <
e < a<
2
s Md .
e < a< 2
178
LATCHEZAR STOYANOV
Hence t g w td , T x .
M Ž d q t ky 1 . F e < a < t ky 2 ,
Ž 20 .
In the same way Žfor t g w td , T x. we get t ky 3 Ž e < a < y Mt . G tdky2 Ž e < a < y MT . G tdky2
e < a< 2
s Md ,
therefore t g w td , T x .
M Ž d q t ky 2 . F e < a < t ky 3 ,
Ž 21 .
We are going to estimate from above the ratio Ž dhrdt .Ž t .rhŽ t . on the interval w td , T x. Given t g w td , T x, use Ž12. and Ž20. to get yh Ž t . G < a < t ky 2 y M Ž d q t ky1 . G < a < t ky 2 y e < a < t ky 2 s < a < t ky 2 Ž 1 y e . ) 0.
Ž 22 .
In the same way, Ž16. and Ž21. imply y
dh dt
Ž t . F Ž k y 2 . < a< t ky 3 q M Ž d q t ky2 . F Ž k y 2 . < a < t ky 3 q e < a < t ky 3 s Ž k y 2 q e . < a < t ky 3 .
Consequently, y Ž dhrdt . Ž t . ky2qe Ž dhrdt . Ž t . Ž k y 2 q e . < a< t ky 3 s F s ky 2 < a< t e hŽ t . yh Ž t . Ž1 y e . Ž 23 . for all t g w td , T x. Consider the function g Ž t . s h12 Ž t . y y 1 Ž t . h Ž t . . It is clearly continuous and g 9Ž t . exists almost everywhere in w0, T x. For those t g w td , T x for which g 9Ž t . exists, we have dg dt
Ž t . s 2h1 Ž t . h˙1 Ž t . y ˙y 1 Ž t . h Ž t . y y 1 Ž t .
dh
s 2h1 Ž t . h Ž t . y 2h1 Ž t . h Ž t . y y 1 Ž t .
dt
Ž t.
dh dt
Ž t . s yy1 Ž t .
dh dt
Ž t. G 0
179
GENERALIZED HAMILTONIAN FLOW
for t g w td , T x. This and Ž23. imply yy 1 Ž t . Ž dhrdt . Ž t . ky2qe Ž dgrdt . Ž t . Ž dhrdt . Ž t . s F F . 2 gŽ t. hŽ t . Ž1 y e . t h1 Ž t . y y 1 Ž t . h Ž t . Integrating the latter inequality gives ln
gŽ t. g Ž td .
s
t
Ht
d
ky2qe t1 t Ž dgrdt . Ž s . ds F ds s ln H g Ž s. td Ž 1 y e . td s
Ž ky2q e .r Ž1y e .
ž /
.
According to Ž17. we have ky2qe
Ž1 y e .
s Ž 1 q b . Ž k y 2. ,
while Ž19. yields tdŽ1q b .Ž ky2. s
1q b
2 Md
ž / e < a<
.
Therefore g Ž t . F g Ž td .
t
Ž1q b .Ž ky2 .
ž /
FN
td
g Ž td .
d 1q b
,
Ž 24 .
where Ns
e < a<
ž / 2M
1q b
.
Thus, gŽ t. F N
g Ž td .
d 1q b
,
t g w td , T x .
Ž 25 .
On the interval w0, td x the function g Ž t . can be estimated in a trivial way. Indeed, for t g w0, td x, Ž14. implies
t g w 0, td x .
180
LATCHEZAR STOYANOV
Now for t g w td , T x the latter and Ž25. yield g Ž t . F const
d2 d 1q b
s const d 1y b .
Therefore g Ž t . F const d 1y b ,
t g w 0, T x .
Ž 26 .
Consequently,
t g w 0, T x .
Ž 27 .
For y 1Ž t . we already have y 1Ž t . F const d on w0, td x. Let t g w td , T x. Then from Ž22. we have < h Ž t . < s yh Ž t . F < a < t ky 2 Ž 1 y e . ) 0. This and Ž24. imply y 1 Ž t . < h Ž t . < F g Ž t . F g Ž td .
t
Ž1q b .Ž ky2 .
ž / td
Therefore y1Ž t . F
g Ž td .
t Ž1q b .Ž ky2.
tdŽ1q b .Ž ky2.
< hŽ t . <
F const d 1y b t b Ž ky2. F const d 1y b .
Thus, y 1 Ž t . F const d 1y b ,
t g w 0, T x .
Ž 28 .
Now, applying a standard argument from the theory of differential equations to the rest of coordinate functions, one derives that there exists a constant C ) 0 such that d Ž Ft r 0 , Ft r . F Cd Ž1y b .r2 for all t g w0, T x. This completes the proof of Lemma 2. Subcase 4.2. have
a ) 0. Take T such that T - arL. Then for t g Ž0, T x we
e Ž t . s at ky 2 q l Ž t . t ky 1 G t ky 2 Ž a y LT . ) 0. Therefore Ft r 0 : t g w0, T x4 is an integral curve of the vector field H p . Given r g U0 , set d s dŽ r , r 0 .. As in the proof of Lemma 2 we see that
181
GENERALIZED HAMILTONIAN FLOW
there exists a positive constant M with Ž12.. Fix m with this property and take T such that Ž18. holds. Determine td g Ž0, T . by td s
2 Md
1r Ž ky2 .
ž / a
.
Then for each t g w td , T . we have t ky 2 Ž a y Mt . ) tdky2 Ž a y MT . G tdky2
a 2
s
2 Md a a
2
s Md .
Hence M Ž d q t ky 1 . - at ky2 ,
t g w td , T x
and therefore h Ž t . G at ky 2 y M Ž d q t ky 1 . ) 0 for all t g w td , T .. Consequently, Ft r : t g w td , T x4 is an integral curve of Hp . Clearly, d Ž Ft r 0 , Ft r . F const d Ž Ft d r 0 , Ftd r . ,
t g w td , T x .
Ž 29 .
Since td s const d 1rŽ ky2. and along an integral curve Ft r of the generalized Hamiltonian flow, the functions y Ž t . and h 9Ž t . are Lipschitz while
'
This and Ž29. now imply d Ž Ft r 0 , Ft r . F const d 1r2Ž ky2. ,
t g w 0, T x ,
which completes the proof of Subcase 4.2.
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182
LATCHEZAR STOYANOV
5. V. Lakshmikantham, D. Bainov, and P. Simeonov, ‘‘Theory of Impulsive Differential Equations,’’ World Scientific, Singapore, 1989. 6. V. Lazutkin, The existence of caustics for a billiard problem in a convex domain, Math. USSR}Iz¨ . 7 Ž1973., 185]214. 7. J. Mather, Glancing billiards, Ergodic Theory Dynamical Systems 2 Ž1982., 397]403. 8. R. Melrose, Singularities and energy decay in acoustical scattering, Duke Math. J. 46 Ž1979., 43]59. 9. R. Melrose, Equivalence of glancing hypersurfaces, In¨ ent. Math. 37 Ž1976., 165]191. 10. R. Melrose, ‘‘Geometric Scattering Theory,’’ Cambridge Univ. Press, Cambridge, 1995. 11. R. Melrose and J. Sjostrand, Singularities in boundary value problems, I, Comm. Pure ¨ Appl. Math. 31 Ž1978., 593]617. 12. R. Melrose and J. Sjostrand, Singularities in boundary value problems, II, Comm. Pure ¨ Appl. Math. 35 Ž1982., 129]168. 13. C. Morawetz, J. Ralston, and W. Strauss, Decay of solutions of the wave equation outside nontrapping obstacles, Comm. Pure Appl. Math. 30 Ž1977., 447]508. 14. V. Petkov and L. Stoyanov, ‘‘Geometry of Reflecting Rays and Inverse Spectral Problems,’’ Wiley, Chichester, 1992. 15. L. Stoyanov, Regularity properties of the generalized Hamiltonian flow, in ‘‘Seminaire ´ Equations aux Derivees Partielles,’’ Ecole Polytechnique, Centre de Math., Expose ´ VI, 1992]1993. 16. M. Taylor, Grazing rays and reflection of singularities to wave equations, Comm. Pure Appl. Math. 29 Ž1976., 1]38.