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Periodic solutions of Lagrangian systems on manifolds with boundary
Nonlinear AnslysiF. Theory, Methods Printed in Great Britain.
& Applications,
PERIODIC
Vol. 16, No. 6, pp. 567-586,
1991. 0
SOLUTIONS OF LAGRANGIAN SYSTEMS MANIFOLDS WITH BOUNDARY
ANNAMARIA CANINO Dipartimento di Matematica, Universita della Calabria, 87036-Arcavacata (Received
15 January
1990; received
Key words and phrases:
0362-546X/91 $3.00+ .OO 1991 Pergamon Press plc
in revised form
ON
di Rende CS, Italy
10 August 1990; received for publication
28 August
1990)
Lagrangian systems, manifolds with boundary, periodic solutions, Lusternik-
Schnirelmann category.
INTRODUCTION
a compact submanifold of R” without boundary and let N,M denote the normal subspace to M at x. The study of periodic solutions of the Lagrangian system
LET M BE
(0.1) has been carried out in [ 11, where the existence of infinitely many periodic solutions to (0.1) is proved under quite general assumptions. The proof is based on well-known results on the cohomology of the free loop space A(M) combined with techniques of critical point theory for smooth functionals. As usual for unilateral constraints [2-5, 12-161, the corresponding problem on manifolds with boundary involves a typical difficulty: variational techniques have to be applied to a nonsmooth functional. In the recent years a critical point theory for nonsmooth functionals has been developed [6-9, 121, which allows one to deal with such a problem. The aim of this paper is to extend the result of [l] to manifolds with boundary. The main difficulty is to verify that the referred techniques of nonsmooth analysis can be applied to this situation; this is the aim of Section 2. As regards the topological aspects, instead of using cohomology as in [l], we take advantage of the recent paper [lo], which allows us to evaluate the Lusternik-Schnirelmann category of A(M). In this way simpler techniques of critical point theory can be applied. Let us point out that the existence of a periodic solution to (0.1) on manifolds with boundary was proved in [16] for a wide class of Lagrangians L. However, a particular condition is imposed in [16]. For instance, if M is the closure of a smooth open set and u(x) is the exterior unit vector to M at x E dM, then, in [ 161, it is supposed that VUE n?“. (u, v(x))(D,W, 4, v(x)) 2 0 The above condition is not required in the present paper. 1. THE GENERAL
FRAMEWORK
Let M c IR” be a compact C2-submanifold
AND THE MAIN
RESULT
(possibly with boundary dM). 567
A. CANINO
568
Definition
1.1. If x E A4, let us denote by N,M the set of cys such that CYE IR”and limsup(~‘Y-X)
We will call N,M the outward normal cone to M at x. Let us point out that if x $ aM, N,M is the usual normal subspace to M at x; otherwise it is the outward normal halfspace to M at x. Now, let us consider a Cz-function L:lRxlR”xlR”~m and let us make the following assumptions: t/(t,q,u)~ R x IR” x R” and VWE R" (I&W,
there exist two constants c, v > 0 such that:
q, u)w, w) 2 vbd2
(1.1)
b?J4t,q,
@I 5 41 + id)
(1.2)
iD;,W,q,
dl 5 41 + bi2)
(1.3)
&JAG
4, u)l 5 ~(1 + 1~1)
(1.4)
lQ?J(t,
4, u)l 5 c.
(1.5)
Moreover, let us suppose that L is l-periodic in the first variable, that is -Ut + 1, 4, u) = L(t, q, u).
(1.6)
Let us remark that modifying the constants c and v, the following estimations are true: V t E R, qEM,
UE R” b,Ut,
q, u)l 5 41 + bi2)
(1.7)
ID&t,
4, u)l 5 ~(1 + InI)
(1.8)
vlu(2 I L(t, q, u) + c.
(1.9)
Now, we can state the main result of this paper. 1.2. Let A4 c IR"be a compact, connected, noncontractible in itself, C2-submanifold (possibly with boundary). Let us suppose that either (a) zi(M) has infinitely many conjugacy classes or (b) n,(M) has a finite number of elements. Then there exists a sequence (yhJh C W2,“(lR, IR”) such that for every h E ~PJ: (i) y,, is l-periodic and y(t) E M; (ii) d/dt(D,L(t, y,,(t), yp))) - D,L(t, yhWr v#N E N,,&4 ax. in IO, 11; and (iii) lim, _ mS;L(t, Y,Af), v;(t)) dt = +a.
THEOREM
Let us fix some notations which we will use throughout Definition
the paper.
1.3. If x E A4, let us define T,M=(uEIR”:(u,~)~OVWEN,M]
T,M is said the tangent cone to A4 at x. Moreover, let us set c;,M=
(u - W:U,WE TIM).
569
Periodic solutions of Lagrangian systems
1.4. If x $ aM, T,M = FxM is the usual tangent subspace to M at x. Now, let V:M + IF? be a Cl-map such that
Remark
v x E M : v(x) E FxM vxEaM:IV(X)I
= 1
and
Let us denote with px the orthogonal l-c,
w =
N,M
and
Iv(x)1 5 1
= (w + Av(x): w E (CM)‘,
A L 01.
projection on EM and let us set
P*(w - (w, v(x))v(x)) = PI w - (Fxw, v(x))v(x)
t/WEIR”
Let us list some properties of these two maps. 1.5. Let us take x, y E M: u, w E If?“; u E N,M. Then (v,y - x) 1 -constIu) Iy - x12. IFxu - GUI 5 constlul (x - yl. (%U, w) = (K r&w). 17r,uI I const(ul. 171,~- 7rYuJI constlu) Ix - yJ. If y, 6 E W1*2(0, 1; R”) and y(t) E M, then rr,,S E W1P2(0, 1; R”) and l(Q)‘(f) - q(t)s’(t)1s constld(t)) Iy’(t)l a.e.
PROPOSITION
(1) (2) (3) (4) (5) (6)
Proof. Since M is compact and of class C2, (l), (2), (4) and (5) are trivial. Let us prove the other ones (r&u, w) = (Q - (u, v(x))v(x)), w) = (u - (u, y(x))y(x), P+)
= (u, E’,w) - (u, v(x))(E w, v(x)) = (U, Fx w - (Fx w, v(x))v(x) = (U, 7t, w). Now, let us take t E [0, l] and h > 0 such that t + h E [0, 11. We have
Inr~t+~~~(~ + h) - q&(0 5 l~t+hj4t + h) - ~y~t+~~40 l + l~y~t+~~W) - ~,~,~W)l I const I&t + h) - b(t)1 + const Is(t)1 Iy(t + h) - y(t)J. Since, 6 E L”, then x,6 E W’*2(0, 1; Ii?“). Moreover
q(t+&f
+ h) - qcr)&f) - ~t)&t
+ h) -
h =
q(r+,+W
5 constl&t
and the estimate follows.
n,(t)
WI
h + h) - q(t) W + h) h
+ h)l
Y(t + “i-
Y(t)
n 2. THE VARIATIONAL
STRUCTURE
In this section, our aim is to prove that periodic orbits of the considered Lagrangian system can be characterized as lower critical points of a certain functional on a suitable functional space.
570
A. CANINO
First of all, let us recall some notions of nonsmooth analysis ([4-7, 91). Let us denote by H a real Hilbert space, and by I- 1 and (- , -) its norm and scalar product, respectively. Definition 2.1. (See also [4, 5, 71.) Let f: H -+ IR U (+mj be a map. We set D(f) = [u E H : f(u) < +a~]. Let u belong to D(f). The functionf is said to be subdifferentiable at u if there exists (YE H such that lim inf f(u) - f(u) - (a, 0 - u) > O U-U IV-U/ -. We denote by a-f(u) the (possibly empty) set of such (YSand we set D(a-f)
= {u E D(f):a-f(u)
# 0).
It is easy to check that a-f(u) is convex and closed v u E D(f); if u E D(d-f), denote the element of minimal norm of a-f(u). Moreover, let E be a subset of H. We denote by ZE the function O,
Z,(u) =
grad-f(u)
will
UEE
u E H/E. c +a, It is easy to check that a-Z,(U) is a cone v u E E. We will call (outward) normal cone to E at u the set a-Z,(U). Remark 2.2. It is readily seen that N&Z = a-Z,(x). Definition 2.3. A point u E D(f) is said to be a lower critical point for f if 0 E a-f(u); c E F? is said to be a lower critical value for f if there exists u E D(f) such that 0 E a-f(u) and f(u) = c. Definition 2.4. (See also [6, 91.) A function f: H + IR U [ +a) is said to have a p-monotone subdifferential of order 2 if there exists a continuous function x: D(f)2 x IR2 -+ FR+such that (a! - P, 24- u) 2 -X(U, v,f(u),f(u))(l whenever U, u E D(a-f),
Q!E a-f(u),
+ bl2 + IP12)lu - u12
and p E a-f(u).
Now, in order to formulate the desired characterization, admissible paths, that is
let us denote by X the space of the
X = (y E W’*2(0, 1; IR”): y(t) E M, v t, y(0) = y(1)) and let us define the functional f:L2(0, in the following way f(Y) =
1; R”) + mu (+a)
1
0
Uf,
i’ +*,
~(0,
Y’(O)
dt,
YEX y E L2(0, 1; fR”)\X.
Moreover, let us set C, = (t E [0, l] : y(t) E 634]. Let us state the following theorem.
Periodic
solutions
THEOREM 2.5. Let us take y E X. If a-f(y)
of Lagrangian
571
systems
# 0 then
y E W2,2(0, 1; R”),
Y:(O) = Y’(1)
and lY”WI 5 WM)(l4)I
a.e. in IO, l[,
+ 1 + IYV)12)
where 8: R + R is a continuous function. To prove this theorem, we need some lemmas. LEMMA 2.6.Let Q c R be a bounded open set and g E L ‘(Cl; II?“). Let us suppose that there exists a, E L’(Q) such that
IS a
(g(t), a’(t)) dt
5 I
li n
vU)]W]
v 6 E C,“(sz; R”).
dt,
Then g E WIS1(Q R”) and ]g’(t)] 5 v(t) a.e. in a.
Proof. Let t,, f2 (tl < t2) be two Lebesgue points of g with [ti, t2]C Q. Let [oh, be a sequence of mollifiers and let Bh E C,“(sZ, R”) be such that
4%) = (%# - f2) - Vh(f -
fd)(g(f2)
-
g(t1))
for every h E IN. If we pass to the limit as h -+ 00 in the inequality Ii’ (g(t), &At)) dt I 5 c ~(0]&(0] IJn I Jn
we get
M2>
hence the thesis.
-
dt,
g(4)12 5
n
LEMMA 2.7. Let s1 c R be a bounded open set and g E L’(Q). Let us suppose that there exists p E L’ (Cl) such that
s n
Then, we have
MO, 6 ‘(0) dt 5
n
v6
dWW dt,
E
C,“(sz), 6 L 0.
t+h g(t
+
h)
-
g(t)
1
-
ds)
st for almost every t, t + h E n with [t, t + h] c iI. Proof.
It is a simple variant of the previous proof.
W
d.s
A. CANINO
512
LEMMA2.8. Let Q C fRbe a bounded open set and g E L’(Q; IR”). Let us suppose that there exist g, E W’3’(Q), g, E w1S2(a), q~ E L’(Q) and a72E L2(Q) such that b ItAt
+
h)
-
g(Oi
dt
5
s LI
bhdt s a
+
A)
-
g,Wl
+
(IV)2(f
+
41
+
lfP2Wl)k2(~
+
(lvdt
+
Wl
+
bd0lPl
+
h)
-
g2wl
dt
whenever h 2 0, [a, b + h] c 0. Then g E W1S1(s2;R”) and
IdWl 5 Proof.
Ig;W
+
h72(0
I&WI
+
a.e. in Q.
sP,(0
It is readily seen that g E SV(G; R”) and that
whenever [a, b] c Q, where iu = g’ as measure. Therefore, if K is a compact subset of Sz with zero Lebesgue measure, we have /P](K) = 0. Since 1,uI is absolutely continuous with respect to Lebesgue measure, the thesis follows. H LEMMA2.9. Let us take y E X and a E a-f(y).
Then
1 W&(f,
Y,
04
+
P,W,
Y,
~‘1,
S’N
df
.r0
2
’ 6~
4
dt,
s 0
V 6 E Wlv2(0, 1; R”) with 6(O) = 6(l) and s(t) E T,&4,
v t.
Proof. Let m be the dimension of M. For every x E M, let i.J, be an open neighbourhood x in R”, g: U, + lR”+’ and V: U, + R two maps of class C2 such that
of
v y E U, : dg(y) is onto; vy E U,:grad
V(y) # 0;
M f7 U, = (y E U, : g(y) = 0, l’(y) s 0);
either
V(x) = 0
or
KY) < 0
Moreover, let us set N = (y E U, : g(y) = 0). By substituting of x, we can assume that there exist an open set A containing class C2. Now, let x E M and let 6 E W’.2(0, 1; R”) be such that compact support in y-‘(U,) and, if x E IBM, there exists E > (a(t), grad V(Y@)))5 4 If s is sufficiently P(u,).
&t)l Igrad
VMNI,
small, we have (y + sS) E A on y-‘(U,).
vy E u,. U, with a smaller neighbourhood N and a retraction r: A + N of
6(O) = 6(l), s(t) E l?$,M, 0 such that v c E [O,
6 has
11.
We claim that r(y + .sd) E M on
Periodic
solutions
of Lagrangian
573
systems
Actually, if x $ &VZthe fact is obvious. If x E aM, we have V(r(v + ~6)) I IV(v)) + (grad I+-(y)), dr(y)(s8)) + const s21612 = V(/(y)+ (grad V(y), dr(y)(s@) + const s21612. Since sS E F_M, we have dr(y)(sa) = ~8, hence ~(r(y + sS)) 5 s(grad V(Y),6) + const s21612 I -sslgrad
Al
161 + const s2j6j2
= -sj6l(elgrad
v(y)1 - const s161).
Therefore, r(y + ~8) E M foras sufficiently small. If we set, r(y + ~6) = y outside of y-‘(U,), we have, by definition of subdifferential 1
1 L(t,
r(y
+
d),
(r(y
+
dt -
~3))‘)
s0
L(t, Y, Y’)dt 1 0
2
:(a,
4~
+
~8)
-
Y)
dt
-
NY
+
~4
-
y)b(r
+
~4
-
YIILZ
I
where limIlWllL2 -o E(W) = 0. Thus, dividing by s, if 0 < z < 1 we have
s I
0
D,W,
+
L
Y +
zw + s4
s
lD”L(t, 0 r(Y
Y +
+
m
- Y), Y’ + z((eJ + ml’ - Y’))
zw + m -
Y
s
r(Y +
m-
Y), Y’ + z(W + @I - v’)) 4Y
dt - E(T(Y+ sS) - y) >
+
m
-
dt
(mJ + s&Y - Y’
dl
s
Y
s
II
Y
s
(2.9.1) /I 2’
Passing to the limit in (2.9.1) as s -+ 0 and recalling that dr(y)(b) = 6, hence d2r(y)(r’, 6) + dr(y)(b’) = 8, we have 1 MD&(&
Y,
Y’),
6)
+
(RW,
Y,
Y’),
WI
df
2
.r0
’ (a, s 0
4
(2.9.2)
dt.
Using partition of unity, it is readily seen that 1 ND&(&
Y,
Y’),
4
+
(DJ(f,
Y,
Y’),
W,
s0
01
dt
1
‘((~3 1 0
4
dt
whenever 6 E W132(0, 1; IR”), 6(O) = 6(l), s(t) E FY&4 and there exists E > 0 such that (s(t), v(W)) 5 --Elwl Iwml in a neighbourhood of C,. Finally, if 6 E W’*2(0, 1; R”), 6(O) = 6(l), s(t) E T&M, let us consider the sequence W)
= s(t) - ; W(0).
A.CANINO
514
Obviously,
lim,&
= 6 in W’S2(0, 1; R”) and
(W),
W0))
= (d(f), W(0))
- ;
IWO) I2 5 - ;
iftEC,.
By continuity (M), if t belongs
W0))
to a neighbourhood
5 _/.(l
I40 Iww)l
+l,lall,-,
of C,. So, we can apply (2.9.2) to Bh v h and we get
1 IP,Ut,
i Passing
Y, v'h4A
+
(D,Ut,
Y, y'), WI
lb, 0
dt 2
0
(2.9.3)
dddt.
n
to the limit in (2.9.3) as h + COwe have the thesis.
LEMMA 2.10.Let y E X, let t E IO, l[ and let h > 0 be such that t + h E IO, l[. Let us assume that y is differentiable at t and t + h. Then
Ir’U f h) - y’(t) - &,,ow + h) - v’W)l < Proof. First s E]O, l[. so
of all, let us observe
Iv’0 + h) - Y’(f)- &t,w
that pYo,(y’(s))
const]y’(t
+ h)l Iy(t + h) - y(t)l.
= y’(s), whenever
y is differentiable
at
+ h) - W)l = Iw + h) - &)(YV + h))l = I&t+/&4 + h)) - ~~,,,W + h))l.
Now,
we can apply (2) of proposition
1.5. to obtain
the thesis.
W
LEMMA 2.11. Let us take y E X and (II E a-f(y). Then (i) D,L(t, y, y’) E W’,‘(]O, l[\C,; I?); (ii) y’ E W’,‘(]O, l[\C,; I?“); and, (iii) Jr”(t)1 5 la(t)1 + const(1 + ly’(t)12) a.e. in IO, l[\C,. Proof. Let us consider q E C,“(O, 1; R”) with supt(q) By applying lemma 2.9 to 6 and -6, we get
C IO, l[\C,
and let us set 6 = p?‘;lr.
1 I(D,W,
i
and by (2) of proposition
so
+
~~~:,D,Ut,
Y, ~'1, ~11 dt =
1.5, it is readily
bp, i0
v)dt
seen that
l(~yv)’- @7’)l 5 codYI Id. 1
Ii 0 By lemma
Y, ~'1, (@T)
0
(p;,D,L(t,
y, y’), q’) dt
i
2.6 and (1.7) and (1.8), pTD,L $(~y(D,L(t,
y, y’)
‘(la - D,LI
I 1
I
+ constly’l
]D,L()]~l
dt.
(2.11.1)
0
E W’*‘(]O, l[\C,;
R”) and
Ia(t)( + const(1
+ Iy’(t)12)
a.e.
(2.11.2)
Periodic
solutions
of Lagrangian
if [t, t + h] c IO, l[\C,,
On the other hand,
systems
using the following
notations
e,(t) = t + sh r,(t) = y(t) + s(y(t + h) - y(t)) c,(t) = y’(t) + s(lJ’(t + h) - y’(t)) we have by (1.1) and (1.5) vly’(t + h) - ?m12 5
l(~F,Ue,(t), r0
+
r,(t),
S,(0)(y’(t
+ h) - v’(t)), y’(t + h) - Y’(0) d..s
+ h) - y’(t)1 Iy’(t + h) - y’(t) - &,(y’(t
constly’(t
By (1.2), (1.4) and (2) of proposition
+ h) - y’(t))l.
(2.11.3)
1.5, we have
1 (~,2"W,W,
i
<,(0, i,(ow(t
+
h) -
Y'(0), &)ow
+
h) -
Y'(0)) d.s
0 =
Py(t+h) D"W
+
A, Y(f +
h), y'(t +
h)) -
+
+
h, ?J(t+
h), y'(t +
h))
-s
-
5
(&f)D"Uf r(t+h)D"ut
+
lvCww~, s0
h, Y(t +
A), y'(t +
r,(t), wwh
IPy(l+hp,ut
+
k
y(t +
hh
Y(t), Y'(0), Y'@
h)), y'(t +
&tjw(t
y'(t +
&p,u4
+
h)) -
w
-
h) -
00))
&tp"ut,
+
h) -
Y'(0)
Y'(0)
b
y(t),
Y’(t))llY’u+ h) - Y’WI
+ conO,Ut + h, y(t+ 4, v’(t+ WI IvU+ h) - WI Iv’@+ h) - r’(O + const(1
+ ly’(t)l
+ const(l
+
5
+ Iy’(t + h)l)h(y’(t
+ h) - y’(t)1
Iv’Wl + Ir’tt + Nl)lv(t + W - WI Ir’tt + W - y’(Ol.
tl~-y(t+,,~“w+ fh YG+ h),y’(t + h)) - &)D”W, y(t),rwl + const(l+ Iv’W + Ir’(t + fOl)lv(t+ h) - rWl + const(1
+ If(t)1
+ Iy’(t + h)l)h)ly’(t
+ h) - y’(t)l.
(2.11.4)
A.
516
Combining
(2.11.4)
with lemma
CANINO
(2. lo), we deduce
that
vly’(t + h) - W)l 5 l&,,h$“W
By lemma
+ A, V(t + h), y’(t + h)) - &,D”WV
+ const(1
+ Iv’(0
+ Iv’@ + h)l)lv(t
+ const(1
+ Ir’(t)l
+ ]v’(t + h)l)h.
2.8 and (2.11.2),
y’ E W’,‘(]O,
l[\C,)
By composition,
we deduce and
+ h) -
y(t), Y’(0)l
~(0
that
Iv”(t)1 I
la(t)/
+ const(1
+ ]~‘(t)]~)
a.e. in IO, l[\C,.
we also have D”L(I,
y, y’)
E
W19’(]0,l[\C,;
H
R”).
LEMMA 2.12. Let us take y E X and Q! E a-f(y). Then: (i) rc,D,L(t, y, y’) E W’S1(O, 1; F?“); and, (ii) ](n,D,L(t, y, y’))‘l 5 const(lcx] + 1 + lv’12) a.e. in IO, l[. Proof. Let us take r;l E W’S2(0, 1; I?“) with q(O) = q(l) and let us set s(t) = q,,(q(t)). Obviously, s(t) E T,,,,M tl (-r,,,,M) tl t. So, we can apply lemma 2.9 to 6 and -6 to obtain 1
I(D,W,
Y,
~‘1,
nn,l;l)
+
(D&t,
Y,
~‘),0q,Wl
dt
‘b,
=
.i 0
7q)
dt
.I 0
and thus by (3) of proposition
1.5
1 ((Q&Q,
Y,
~‘1,
v)
+
(QW,
Y,
Y’),
(qd’N
dt
s0
rl) dt.
(2.12.1)
i 0
By using (4) and (6) of proposition
1.5 in (2.12.1)
we obtain
1
IS (D&t,
1 Y,
~‘1,
~,WN
(n&y
dt
0
-
and then by (3) and (4) of proposition Y,
Y’),
rl’)
i
1.5
1 (n,&W,
D&Q,
Y,
~‘1,
u?) dt
0
+ const
: 1~1Ir’] I&L]
dt
‘tb - D&I + IY’I I~,~OIvIdt.
dt
0
By lemma
’ b,w
=
0
2.6 and hypotheses
(1.7) and (1.8), we have from (2.12.2) qD,L(t,
y, y’) E W’vl(O, 1; R”)
and I(rQ&W,
Y, v’))‘l 5 const(]a]
+ 1 + ]r’12)
a.e. in IO, l[.
n
(2.12.2)
511
Periodic solutions of Lagrangian systems LEMMA 2.13.
Let us take y E X and (YE a-f(y).
Then for almost every t, t + h E IO, l[ (h > 0)
we have (D”Ut + A, Y(t + h), y’(t + h)), VW + h))) - (D”L(t, v(t), y’(t)), v@(t)))
s t+h
2
(ICYI+ 1 + l#)dr.
-const
t
Proof.
1) with q 2 0 and let d(t) = -q(t)v(y(t)).
Let q E C,“(O,
-
l(DJK i0
Y, ~'1, (~?W)')dt
1
-
'(01 - Q&V, 1 ,O
By lemma (2.9), we have Y, ~'1, mW)dt.
Therefore -
b,L(t, s0 2
Y, Y'), W)rl'dt
-
jb
- D&Q,
Y, ~'1, V(Y)) -
(D,Ut,
Y, ~'1, (WY)lv
dt.
1
By (1.7) and (1.8), we have
s 1
(D,L(t,
y, y’), v(y))q’ dt 5 const
‘(lcvi + 1 + Iy’j’)q dt s0
0
n
and by lemma 2.7 the thesis follows.
LEMMA 2.14. Let us take y E X and cx E d-f(y). Let t E IO, l[ and h > 0 be such that t + h E IO, l[ and such that y is differentiable at t and t + h. Then we have (y’(t + h) - y’(t), v(y(t)))
2s const ( 1 + jjWt
dr) i:‘*(lal
+ 1 + lY’?)dT t+h
+
consWWl+ Iv’@+ WI) f ly’lcit. s
Proof.
First of all let tl , t, E IO, l[ be such that there exist yL(tl) and y:(t2). Then we have rl(t,)
E Qt,,M
(rr_(t,), vbJ(t1))) 2 09
Y:m
E Qtz,M
(YX2h
W~z)N
5 0.
Therefore oC(&) - rr_(t,), VWl))) 5 W(fA
VW,)))
= (Y1(f*), WW) 5 const
- VW*))) + (Wz),
IY;(h)I IY&) - r(tl)I.
MM)) (2.14.1)
Now, let t E IO, l[, h > 0 with t + h E 10, l[ and y differentiable at t and t + h. If [t, t + h] fl C, # 0, let t, , t2 E C, be such that t I t, I t2 I t + h, (It, tl[ U ]t2, t + h[) fl C, = 0.
A. CANINO
578
By lemma 2.11, there exist r_(tr) and y:(fJ.
Therefore
(y’(t + h) - v’(t)* MO)) 5 (Y’([ + h) - V;(tz), VW))) + W(&) - y’(tA WO))
+ (rr_(t,) - y’(t), W(O))
I constly’(t + h) - Y:(&)[ + constlyl.(t,) + (Y:(G) - rr_(t,), Wtl)))
+ (YX)
- y’(t)1
- r’(tr), WO)
- VW,))).
Now, by applying lemma 2.11 and (2.14.1), we obtain (y’(t + h) - Y’(f), W(O)) 5 const
:’ IY(r), dr + .I;” /Y”(r), dr]
[I
+ constIr:&)l
IV(&)- r(tJ +consdMd- YWA Iv(O- Wdl
1
t+h 5
const
+
(/a(r)1 + 1 + lv’(r)12)dr
ii’.t
constly’(t + h)l Ir(t + h) - y(t)1 + constly’(t + h) - ~:(fJl
x
(Ir(f + h) - Y&>l+ Ir(f,) - ?+)I)
+ constly’(t+ MtlvU + h) - Y&)I + Ir(tJ - r(r)l) + constlf(t + 4 - Y:&)I IrG + f0 - WI + constlf@+ 4 - v’WlIv(t) - Y(~~)I + cona[Ir’@ + 4 - Y:WI + IvYtd - ~‘(0l1l~W- v(tJ t+h
t+h 5
const
[i
(la(r)1 + 1 + IY’(r)l’)dr t
+ const)y’(t + h)l f
const [.r t
(Ia(r
+ 1 + lr’(r)?)dr t+h
t-
dr
II’ t+h
t+h +
IY’(~
i;
1
constly’(t + h)( f i
f
Iv’(d
dz t+h
IY’(@I dt + consdy’@ + h) - v’(t)1t IY’(~dt i
t+h 5
const
[i
t
(la(r)1 + 1 + Iv’(r)?) dr ] ( 1 + 1: IfCr)l di) t+h
+
const(lVO + 41 + Iv’Wl) t ~Y’(G dr. i
Finally, it is readily seen that the same inequality holds if [t, t + h] fl C, = 0. Now, we come back to the following proof.
n
Periodic solutions of Lagrangian systems
519
Proof of theorem 2.5. Let us take 0 < t < t + h < 1 and let us assume that y is differentiable at t and t + h. If we keep the notations of lemma 2.11, by (1. l), (1.5) and lemma 2.10, we have vly’(t + h) - ~‘(t)~2 5
l(Q%&(O, s0
T,(t), W))(r’(t
+ h) - y’(t)), y’(t + h) - y’(0) ti
=
b,2,W,(t), 5 0
W),
+
W)W(t
h)
-
y’(t)),
E’,,,W
v’(t
5
+
+
h)
-
h)
y’(t)
-
y’(0))
-
ds.~
&W
+
h)
-
y’(t)))
’U’,2,W,W,L(t), S,(tM”(t + h) - y’(t)), ~,~,~(y’(t+ h) - y’(t))) ds ’V’2,,WW),t-,(t), LWW(t + h) - y’(O), WOM&t~W + h) - y’(t)), WO))’(%W,W, t-,(t), tXOW(t + h) - y’(t)), W(t)))
i0
s s
d.~
d.s
0
+
0
x (y’(t + h) - y’(t), &p’oW)+ CLS + constly’(t+ h) - y’(t)1Ir(t + 41ly(t + h) - WI. By (1.2), (1.4), (1.8), (3) and (5) of proposition
s
1.5 we obtain
1
0
U’:J@,W,t-,(t), M)W’(t + h) - y’(t)), q&‘(t =
+ h) - y’(t))) ds
(DJ(t + h, y(t + h), v’(t + h)) - D,W, v(t), y’(O), xr&‘(t
-s 1(@&V4W, 0
5
t-,(t), WNW + h) - y(O), ~rttjW(t + h) - y’(O)) d.~
(n,~t+,,RUt + h, Y(t + h), y’(t + h)) - q&Ut, + (+$‘,Ut
+ h) - y’(t)))
y(t), v’(t)), 7’0 + h) - y’(t))
+ h, Y(t + h), v’(t + h)) - nyct+@uL(t + h, y(t + h), y’(t + h)), v’(t + h) - y’(t))
+ const(1 + (y’(t + h)l + (y’(t)()h(y’(t + h) - y’(t)( + const(l + Iv’(t + h)l + b’(t)l)lv(t 5
+ h) - y(t)1 iv’(t + h) - r’(t)1
(bry~t+&L(f + h, y(t + h), y’(t + h)) - q#,Ut,
+
const(1 + (y’(t + h)l + (y’(t)0
y(t), y’W)l
t+h t IY’(~/ dr
5
+ const(1 + [y’(t + h)( + Iy’(t)l)h)ly’(t
+ h) - y’(t)l.
(2.5.2)
580
A. CANINO
By lemma 2.13, we have
5
-(~&w
+
x
W”W
+
(D”Jw
h)
+ +
h, k
-
Y’(0),
Y(t
+
Y(t
+
W0)-h),
y’(t
A),
y’(t
ii
h)),
+
constly’(t + h) - y’(t)1
5
+
t
h)),
V(Y(f
+
W(0)
h))) -
-
(D”W,
VW
+
YW,
Y’(0),
W(0))
h)))
(I~(~)/ + 1 + lr’(r)1*)d~ t+h
+
(1
+
Iv’@
+
Ml
+
Iv’@)l) t bwl dr + (1 + IYV + WI + l~~00~ . (2.5.3) 1
.r
Finally, since v@(t)) E ~Y;rct+4,by lemma 2.14 we have
sl 0
+ h) - Y’Wt &tj VW)))+d.s
vz,w,w9 MO, L(mJ’(~ + h) - Y’(O),wo))(Y’(t 5
constly’(t + h) - y’(t)/ {( 1 + j: Iv’(t)1 d7) j:‘“(k)1
+ 1 + lr’(.r)I*) ds
/yh b’td dT]. + (Ir’(t + 41 + IY’WO Combining (2.5.1)-(2.5.4), v(y’(t + h, - f(t)/
S
(2.5.4)
we get D”W
b++h)
+ const(l
+
h,
+ j:ly(r)[
Y(t
+
A),
Y’@
+
h))
-
~,(,)~“W,
Y(0,
Y’W)l
dr) jtt+h(lcx(r)l + 1 + lf(t)[*)dr t+h
+
const(1 + Jy’(t + h)l + If(t)/) j
+ (1 +
IY’U + WI + ly’m~.
t
Iv’(d
dt
(2.5.5)
581
Periodic solutions of Lagrangian systems
By applying lemmas 2.12 and 2.8, we can conclude that y E W2,2(0, 1; R”)
and
where 13:R + IRis a continuous function. Now, it remains to prove that y:(O) = y:(l).
Let us define p E X in such a way Oltl+
YU + *>3 HO =
*std.
i YU - t>v We can easily see that a-f(p) THEOREM 2.15.
a.e. in IO, l[,
]~“(t)l 5 ~V(Y))(]C#)] + 1 + ]~‘(t)]~)
# 0. Then p E W2V2(0,1; R”). Thus, y:(O) = y’(l).
Let us take y E Xn W2,2(0, 1; R”). If CYE a-f(y),
o(t) + it (D”JV, y(t), Y’(0)) - (Q&4,
y(t), Y’(0) E &,,M
n
then a.e. in IO, l[.
Proof. Let t,, E IO, l[ be a Lebesgue point for c(t)
(
o(t) + -$D”LU, Y(f), Y’(0) - Q&t,
y(t), Y’(0) >
Let us consider w E 7&$4 with (w, v(y(t,)) < 0 if to E C,. Let q,,(t) = ph(t - to) v h E N where 1~~)~is a sequence of mollifiers. Then, by continuity we have (p,,:rtow, v(y(t))) < 0 in a neighbourhood of to, hence (l?&(qh(t)w), v(y(t))) I 0 eventually as h + co. Since PJq,, w) E Hd and ff&(qh(t)w) E 7&M, we have, by lemma 2.9 1 6 0 SC
a
+
&“Uf,
(
(2.15.1)
Y, Y’) 3 W q,, dt I 0. > >
Y, Y’) - QJ(C
Passing to the limit in (2.15.1) as h + 00 we get
(
4fo) + $D,L(f,,
YGO),Y’GO))- Q&to,
YGO),Y’GO)),w 50 >
so that o(t0) + $“Ut,,
YQO),y’(t0)) - D,Uo,
YGO),Y’VO))E &,,,,M.
n
As a consequence of theorems 2.5 and 2.15, we have the following corollary. COROLLARY 2.16.
Let us take y E X. If 0 E a-f(y), y E W2.“(0, 1; V),
then Y:(O) = Y’(I)
and -&(f,
y(t), Y’(0) - Q$(t,
Finally, let us prove the cp-convexity off.
y(t), Y’(0) E N,,,M
a.e. in IO, l[.
582
A. CANINO
2.17. The functional f: L’(0, 1; R”) + R U (+a~) is 1.s.c. and there continuous function e+,; R2 -+ R such that
THEOREM
f(Y + 4 2 f(r)
+ (a, 42
whenever y, y + 6 E X and CYE d-f(y). order two.
- %(f(Y)t
f(Y + 4)(1
+
exists a
II&wllt2
In particular, f has a p-monotone
subdifferential
of
Proof. First, we will prove that f is I.s.c. Let us consider a sequence (Y~)~C X such that lim, yh = y in L2(0, 1; R”) and f(yh) I b. By (1.9) we have
and then:
that is, 1~~)~converges weakly to y in WiS2(0, 1; R”). On the other hand, L is continuous in the three variables and convex in the third one, so it is weakly 1.s.c. in WiS2(OI1; R”). Then j: L(t, y, y’) dt I b. Finally, it is enough to observe that 1~~)~converges uniformly to y and recall that A4 is compact to prove that y E X. Thus SAL(t, y, y’) = f(y) and f is 1.s.c. Now, let us take y E X fl W2S2(0, 1; R”), CYE d-f(y) and let 6 E W1*2(0, 1; R”) with 6(O) = 6(l) be such that y + 6 E X. By Taylor’s formula, we have Ut, Y + 6, Y’ + a’) - Ut, Y, Y’) = (D&V, Y, Y’), 6) + (D”L(t, Y, v’), 6’) + *(D&L@,
y + sd, y’ + ssy,
+ #g”L(t,
y + sd, y’ + sS’)S, S’)
+ +(D;“L(t,
y + sd, y’ + sS’)&, 6’)
for some s = s(t) E IO, I[. BY (l.l), (1.3), (1.4), theorem 2.14 and (1) of proposition f(Y + 6) -f(y)
2
-
6)
1.5 we have
iwW I’0
b&Y,
Y'> -
-$“L&
Y, Y’) - a, 6) dt
s
-‘const(l
+ ll~~‘@ + (Iy’ + S’ll~z)llSllfm
- constll
+
Ily’IIL~+ lly’ + ~‘Il~~)ll~ll~-ll.116’ll~~ + 5 ll~‘ll3
5 -constllD,L(t,Y, Y’)- $+D,,Ut, Y, Y’)- ~Il~~ll%= - const(l + ll~‘lltz + lly’ + S’ll~2)11Sll~~ - const(1 +
IIY’II~~ + IIY’+ ~‘ll~~~ll~ll~-ll~‘ll~~ + f IlfJ’llLz.
Periodic solutions of Lagrangian systems
583
By (1.7), (1.2), (1.4) and (1.5) we have f(r + 6) - f(v) -
- const(1 + Ily’l@ + lly’ + fJ’ll~~)llf?ll$ - const(l +
IIy’lLz+ lly’+ ~‘ll~~)llG4l~‘ll~~ + i IlS’II3.
By theorem 2.5 and (1.9) we conclude that f(v + 6) - f(y) -
’ (~6)
dt
I 2
-conNllV’ll~~ + Il&)(ll~ll~~+ 11~11~41~‘11~2) - const(l+ Ilv’llt~ + Ilr’ + ~‘ll3>ClldlL~+ Il&4l~‘ll~~) - const(l + Ily’lL2 + lly’ + ~‘llL~)(ll~ll~41~‘lI~2 + 11~112~211~‘11~4’> + i IlS’ll2
2 -%m%f(Y
+ a)(1 + Il~llm4lz~
for some continuous function cpO:R2 -+ R. Remark
n
2.18. From the previous proof, it turns out that if
y Ex
n w~,~(o,1;~“1,
Y:(o)= Y’(l),
ct!E L2(0,1; rn”)
and a(t) + $ (D”Uf, y(t), y’(0)) - D,JW, y(t), y’(0) E &,,M
a.e. in]O, l[,
then (11E a-f(r). Therefore, the converse of theorem 2.15 also holds. 3. TOPOLOGICAL
RESULTS
Since we proved the p-monotonicity off (see theorem 2.17), we are able to evaluate the number of lower critical points for f by means of a technique such as the LusternikSchnirelmann one (see, Section 4). Thus, in this section, we investigate the category of the space of the admissible paths X (as defined in Section 2). The following result, proved in a recent paper [lo], is crucial for our aim. THEOREM 3.1. Let F A E s B denote a Hurewicz fibration with E path-connected. Let us assume that there exists a section o: B + E. Then cat F I cat E. In [lo], theorem 3.1 is applied in order to estimate cat(A( Y)) wherte Y is a simply connected manifold without boundary. Our aim is to show that an analogous argument works on manifolds with boundary. If Y is a topological space, let us denote by A(Y), the free loop space of Y and if y0 E Y by Q(Y, yO) the based loop space of Y with base point yO.
A. CAMNO
584
LEMMA3.2. Let A be an open subset of R”, connected, noncontractible n,(A) is finite. Let us take x,, E A and let us set
in itself and such that
Q&4, x0) = (y : y E CJ(A, x0) and y is contractible in A]. Then, cat &,(A, x0) = +a. Proof. Let us consider the universal covering 71:A’ -+ A. Let y0 E A be such that n(yO) = x0, then &(A, x0) is homeomorphic to Q(A, uO). By the extension of [17] given in [l 1, corollary 3.21, we have catQ,(A,x,,) = catQ(A,y,) = +a~. n
By theorem 3.1 and lemma 3.2, we can state the following theorem. THEOREM 3.3. Let A be an open subset of IT?“,connected Moreover, let us suppose that either (i) x1(A) has infinitely many conjugacy classes or (ii) nl(A) has a finite number of elements. Then cat A(A) = +a.
and noncontractible
in itself.
Proof. If (i) is true, the thesis is trivial because A(A) has infinitely many path-components. If (ii) is true, let us set A,,(A) = {y : y E A(A) and y is contractible
in A).
Let us point out that, since the map p: A(A) + A defined by p(y) = y(O) gives a fibration and A,,(A) is a path-component of A(A) then also the restriction pO: A,(A) + A gives a fibration, of course with A&A) path-connected. We can define a section 0: A -+ A,(A) in such a way: v x E A, a(x)(s) = x v s E [0, 11. The fiber is p; ‘(x0) = Q,(A) x,,). Then, we can apply theorem 3.1 and we obtain: cat O,(A, x,,) I cat A,(A). Obviously, cat A,,(A) I cat A(A) and by lemma 3.2 we have the thesis. n Now, in order to evaluate the category of X (endowed with W’,2-topology), recall a result contained in [ 151. THEOREM3.4. The inclusion map i: X -+ A(M) is a homotopy
it is enough to
equivalence.
Finally, this gives the following theorem. THEOREM3.5. Let A4 c R”, be a C2 submanifold (possibly with boundary), connected and noncontractible in itself. Moreover, let us suppose that either (i) z,(M) has infinitely many conjugacy classes or (ii) n,(M) has a finite number of elements. Then cat(X) = +oo. Proof. Let A be an open subset of IR”such that M is a deformation retract of it. Then, A(A) is homotopically equivalent to A(M). Now, it is enough to apply theorems 3.3 and 3.4 to have the thesis. n
Periodic solutions of Lagrangian systems 4. PROOF
OF THE MAIN
585
RESULT
To prove the existence of infinitely many periodic orbits to by corollary 2.16 and remark 2.18, the existence of infinitely on the space of the admissible paths X, we will use a theorem with the number of the critical points of a functional defined
our Lagrangian system, that is, many lower critical points for f linking the category of a space on it (cf. [12 and 81).
4.1 (see [S]). Let H be a real Hilbert space, g: H + L?U (+a) a p-monotone subdifferential of order two. We set
THEOREM
d*(u, u) = Iu - VI + I&) - g(n)1
a 1.s.c. function with
v U, V E D(g).
Let us suppose that: (i) inf g > -00; (ii) /(u& c D(d-g) with s;p g(uh) < +m and lip grad- g(uh) = 0, (u&, has a subsequence converging in H. Then, g has at least cat@(g), d*) lower critical points. Moreover, if cat@(g), d*) = +oo, then sup g(D(g)) = SUPk(U) : 2.lE ma-g), 0 E a-&II. Finally, this leads to the following proof. Proof of theorem 1.2. Let us consider the functional f defined in Section 2. By hypothesis (1.9) and theorem 2.17, f is a 1.s.c. function, bounded below and it has a v-monotone subdifferential of order two. Moreover, by corollary 2.16 we want to prove the existence of a sequence (yhJh E X such that 0 E a-f&) and lim, f(yh) = + co. Let us remark that D(f) = X and that d* induces the W’*‘(O, 1, IR”)-topology on X. Therefore, cat@(f), d*) = +W by theorem 3.5. Now, we will consider a sequence 1~~)~C D(d-f)with s;P f(Yh) < + *
lihmgrad- f(y,,) = 0.
and
By hypothesis (1.9), (Y;)~ is bounded in L2(0, 1; IR”)and, on the other hand, since the manifold A4 is compact, (yhJh is bounded in L’(O, 1; IR”). By Rellich’s theorem, we deduce that lyhlh has a subsequence converging in L2(0, 1; R”). To complete the proof, it remains to show that sup f(y) = +oo. 7eX Let y E X and let y be nonconstant. by (1.9)
If we set ~&Y) = I
f(%l) + c 1 v
r1
30
v
h E N, we have (q,,)* E X and
lllltlZ~.
If by contradiction {f (q*),, is bounded, then up to a subsequence (q,, jh converges uniformly to the constant y(O). But y is nonconstant by hypothesis, so y(s) # y(O) for some s E IO, l[ and there exists s,, such that qh(sh) = I. This is an absurdity. n
586
A. CANINO
Acknowledgement-The author wishes to thank E. Fade11 who pointed out that a previous version of the proof of lemma 3.2 was incomplete. REFERENCES 1. BENCI V., Periodic solutions of Lagrangian systems on a compact manifold, J. DgJ Eqns 63, 135-161 (1986). 2. CANINOA., On p-convex sets and geodesics, J. DI~J Eqns 75, 118-157 (1988). 3. CANINOA., Existence of a closed geodesic on p-convex sets. Ann. Inst. H. Poincard, AnaIyse non LinPaire 5,
501-518 (1988). 4. COBANOVG., MARINOA. & SCOLOZZID., Evolution equation for the eigenvalue problem for the Laplace operator with respect to an obstacle, preprint no. 214, Dip. Mat. Pisa (1987). 5. COBANOV,G., MARINO A. & SCOLOZZID., Multiplicity of eigenvalues for the Laplace operator with respect to an obstacle and nontangency conditions, Nonlinear Analysis 15, 199-215 (1990). 6. DE GIORCI E., DEGIOVANNIM., MARINOA & TOSQUESM., Evolution equations for a class of nonlinear operators,
Atti Accad. Naz. Lincei Rend. Cl. Sri. Fix Mat. Natur. 75, l-8 (1983). 7. DE GIORCI E., MARINO A. & TOSQUESM., Problemi di evoluzione in spazi metrici e curve di massima pendenza, Atti Accad. Naz. Lincei Rend. Cl. Sri. Fis. Mat. Natur. 68, 180-187 (1980). 8. DEGIOVANNIM., Homotopical properties of a class of nonsmooth functions, Ann. Mat. pura appl. (in press). 9. DEGIOVANNIM., MARINO A. & TOSQUESM., Evolution equations with lack of convexity, Nonlinear Analysis 9, 1401-1443 (1985). 10. FADELLE. & HUSSEINIS., A note on the category of the free loop space, Proc. Am. math. Sot. 107,527-536 (1989). 11. FADELL E & HUSSEINI S., Extending Serre’s theorem on the category of loop spaces, preprint (1990). 12. MARINO A. & SCOLOZZID., Geodetiche con ostacolo, BON. Un. Mat. Ital. B(6) 2, 1-31 (1983). 13. SCOLOZZID., Esistenza e molteplicita di geodetiche con ostacolo e con estremi vairabili, RicercheMar. 33, 171-201
(1984). 14. SCOLOZZID., Un teorema di esistenza di una geodetica chiusa su varieta con bordo, Bolt. Un. Mat. ftal. A(6) 4, 451-547 (1985). 15. SCOLOZZID., Molteplicita di curve con ostacolo e stazionarie per una classe di funzionali non regolari, preprint no. 69, Dip. Mat. Pisa (1984). 16. SCOLOZZID., Esistenza di una curva chiusa stazionaria e con ostacolo, preprint no. 70, Dip. Mat. Pisa (1984). 17. SERRE J. P., Homologie singuliere des espaces fib&, Ann. Math. 54, 425-505 (1951).
& Applications,
PERIODIC
Vol. 16, No. 6, pp. 567-586,
1991. 0
SOLUTIONS OF LAGRANGIAN SYSTEMS MANIFOLDS WITH BOUNDARY
ANNAMARIA CANINO Dipartimento di Matematica, Universita della Calabria, 87036-Arcavacata (Received
15 January
1990; received
Key words and phrases:
0362-546X/91 $3.00+ .OO 1991 Pergamon Press plc
in revised form
ON
di Rende CS, Italy
10 August 1990; received for publication
28 August
1990)
Lagrangian systems, manifolds with boundary, periodic solutions, Lusternik-
Schnirelmann category.
INTRODUCTION
a compact submanifold of R” without boundary and let N,M denote the normal subspace to M at x. The study of periodic solutions of the Lagrangian system
LET M BE
(0.1) has been carried out in [ 11, where the existence of infinitely many periodic solutions to (0.1) is proved under quite general assumptions. The proof is based on well-known results on the cohomology of the free loop space A(M) combined with techniques of critical point theory for smooth functionals. As usual for unilateral constraints [2-5, 12-161, the corresponding problem on manifolds with boundary involves a typical difficulty: variational techniques have to be applied to a nonsmooth functional. In the recent years a critical point theory for nonsmooth functionals has been developed [6-9, 121, which allows one to deal with such a problem. The aim of this paper is to extend the result of [l] to manifolds with boundary. The main difficulty is to verify that the referred techniques of nonsmooth analysis can be applied to this situation; this is the aim of Section 2. As regards the topological aspects, instead of using cohomology as in [l], we take advantage of the recent paper [lo], which allows us to evaluate the Lusternik-Schnirelmann category of A(M). In this way simpler techniques of critical point theory can be applied. Let us point out that the existence of a periodic solution to (0.1) on manifolds with boundary was proved in [16] for a wide class of Lagrangians L. However, a particular condition is imposed in [16]. For instance, if M is the closure of a smooth open set and u(x) is the exterior unit vector to M at x E dM, then, in [ 161, it is supposed that VUE n?“. (u, v(x))(D,W, 4, v(x)) 2 0 The above condition is not required in the present paper. 1. THE GENERAL
FRAMEWORK
Let M c IR” be a compact C2-submanifold
AND THE MAIN
RESULT
(possibly with boundary dM). 567
A. CANINO
568
Definition
1.1. If x E A4, let us denote by N,M the set of cys such that CYE IR”and limsup(~‘Y-X)
We will call N,M the outward normal cone to M at x. Let us point out that if x $ aM, N,M is the usual normal subspace to M at x; otherwise it is the outward normal halfspace to M at x. Now, let us consider a Cz-function L:lRxlR”xlR”~m and let us make the following assumptions: t/(t,q,u)~ R x IR” x R” and VWE R" (I&W,
there exist two constants c, v > 0 such that:
q, u)w, w) 2 vbd2
(1.1)
b?J4t,q,
@I 5 41 + id)
(1.2)
iD;,W,q,
dl 5 41 + bi2)
(1.3)
&JAG
4, u)l 5 ~(1 + 1~1)
(1.4)
lQ?J(t,
4, u)l 5 c.
(1.5)
Moreover, let us suppose that L is l-periodic in the first variable, that is -Ut + 1, 4, u) = L(t, q, u).
(1.6)
Let us remark that modifying the constants c and v, the following estimations are true: V t E R, qEM,
UE R” b,Ut,
q, u)l 5 41 + bi2)
(1.7)
ID&t,
4, u)l 5 ~(1 + InI)
(1.8)
vlu(2 I L(t, q, u) + c.
(1.9)
Now, we can state the main result of this paper. 1.2. Let A4 c IR"be a compact, connected, noncontractible in itself, C2-submanifold (possibly with boundary). Let us suppose that either (a) zi(M) has infinitely many conjugacy classes or (b) n,(M) has a finite number of elements. Then there exists a sequence (yhJh C W2,“(lR, IR”) such that for every h E ~PJ: (i) y,, is l-periodic and y(t) E M; (ii) d/dt(D,L(t, y,,(t), yp))) - D,L(t, yhWr v#N E N,,&4 ax. in IO, 11; and (iii) lim, _ mS;L(t, Y,Af), v;(t)) dt = +a.
THEOREM
Let us fix some notations which we will use throughout Definition
the paper.
1.3. If x E A4, let us define T,M=(uEIR”:(u,~)~OVWEN,M]
T,M is said the tangent cone to A4 at x. Moreover, let us set c;,M=
(u - W:U,WE TIM).
569
Periodic solutions of Lagrangian systems
1.4. If x $ aM, T,M = FxM is the usual tangent subspace to M at x. Now, let V:M + IF? be a Cl-map such that
Remark
v x E M : v(x) E FxM vxEaM:IV(X)I
= 1
and
Let us denote with px the orthogonal l-c,
w =
N,M
and
Iv(x)1 5 1
= (w + Av(x): w E (CM)‘,
A L 01.
projection on EM and let us set
P*(w - (w, v(x))v(x)) = PI w - (Fxw, v(x))v(x)
t/WEIR”
Let us list some properties of these two maps. 1.5. Let us take x, y E M: u, w E If?“; u E N,M. Then (v,y - x) 1 -constIu) Iy - x12. IFxu - GUI 5 constlul (x - yl. (%U, w) = (K r&w). 17r,uI I const(ul. 171,~- 7rYuJI constlu) Ix - yJ. If y, 6 E W1*2(0, 1; R”) and y(t) E M, then rr,,S E W1P2(0, 1; R”) and l(Q)‘(f) - q(t)s’(t)1s constld(t)) Iy’(t)l a.e.
PROPOSITION
(1) (2) (3) (4) (5) (6)
Proof. Since M is compact and of class C2, (l), (2), (4) and (5) are trivial. Let us prove the other ones (r&u, w) = (Q - (u, v(x))v(x)), w) = (u - (u, y(x))y(x), P+)
= (u, E’,w) - (u, v(x))(E w, v(x)) = (U, Fx w - (Fx w, v(x))v(x) = (U, 7t, w). Now, let us take t E [0, l] and h > 0 such that t + h E [0, 11. We have
Inr~t+~~~(~ + h) - q&(0 5 l~t+hj4t + h) - ~y~t+~~40 l + l~y~t+~~W) - ~,~,~W)l I const I&t + h) - b(t)1 + const Is(t)1 Iy(t + h) - y(t)J. Since, 6 E L”, then x,6 E W’*2(0, 1; Ii?“). Moreover
q(t+&f
+ h) - qcr)&f) - ~t)&t
+ h) -
h =
q(r+,+W
5 constl&t
and the estimate follows.
n,(t)
WI
h + h) - q(t) W + h) h
+ h)l
Y(t + “i-
Y(t)
n 2. THE VARIATIONAL
STRUCTURE
In this section, our aim is to prove that periodic orbits of the considered Lagrangian system can be characterized as lower critical points of a certain functional on a suitable functional space.
570
A. CANINO
First of all, let us recall some notions of nonsmooth analysis ([4-7, 91). Let us denote by H a real Hilbert space, and by I- 1 and (- , -) its norm and scalar product, respectively. Definition 2.1. (See also [4, 5, 71.) Let f: H -+ IR U (+mj be a map. We set D(f) = [u E H : f(u) < +a~]. Let u belong to D(f). The functionf is said to be subdifferentiable at u if there exists (YE H such that lim inf f(u) - f(u) - (a, 0 - u) > O U-U IV-U/ -. We denote by a-f(u) the (possibly empty) set of such (YSand we set D(a-f)
= {u E D(f):a-f(u)
# 0).
It is easy to check that a-f(u) is convex and closed v u E D(f); if u E D(d-f), denote the element of minimal norm of a-f(u). Moreover, let E be a subset of H. We denote by ZE the function O,
Z,(u) =
grad-f(u)
will
UEE
u E H/E. c +a, It is easy to check that a-Z,(U) is a cone v u E E. We will call (outward) normal cone to E at u the set a-Z,(U). Remark 2.2. It is readily seen that N&Z = a-Z,(x). Definition 2.3. A point u E D(f) is said to be a lower critical point for f if 0 E a-f(u); c E F? is said to be a lower critical value for f if there exists u E D(f) such that 0 E a-f(u) and f(u) = c. Definition 2.4. (See also [6, 91.) A function f: H + IR U [ +a) is said to have a p-monotone subdifferential of order 2 if there exists a continuous function x: D(f)2 x IR2 -+ FR+such that (a! - P, 24- u) 2 -X(U, v,f(u),f(u))(l whenever U, u E D(a-f),
Q!E a-f(u),
+ bl2 + IP12)lu - u12
and p E a-f(u).
Now, in order to formulate the desired characterization, admissible paths, that is
let us denote by X the space of the
X = (y E W’*2(0, 1; IR”): y(t) E M, v t, y(0) = y(1)) and let us define the functional f:L2(0, in the following way f(Y) =
1; R”) + mu (+a)
1
0
Uf,
i’ +*,
~(0,
Y’(O)
dt,
YEX y E L2(0, 1; fR”)\X.
Moreover, let us set C, = (t E [0, l] : y(t) E 634]. Let us state the following theorem.
Periodic
solutions
THEOREM 2.5. Let us take y E X. If a-f(y)
of Lagrangian
571
systems
# 0 then
y E W2,2(0, 1; R”),
Y:(O) = Y’(1)
and lY”WI 5 WM)(l4)I
a.e. in IO, l[,
+ 1 + IYV)12)
where 8: R + R is a continuous function. To prove this theorem, we need some lemmas. LEMMA 2.6.Let Q c R be a bounded open set and g E L ‘(Cl; II?“). Let us suppose that there exists a, E L’(Q) such that
IS a
(g(t), a’(t)) dt
5 I
li n
vU)]W]
v 6 E C,“(sz; R”).
dt,
Then g E WIS1(Q R”) and ]g’(t)] 5 v(t) a.e. in a.
Proof. Let t,, f2 (tl < t2) be two Lebesgue points of g with [ti, t2]C Q. Let [oh, be a sequence of mollifiers and let Bh E C,“(sZ, R”) be such that
4%) = (%# - f2) - Vh(f -
fd)(g(f2)
-
g(t1))
for every h E IN. If we pass to the limit as h -+ 00 in the inequality Ii’ (g(t), &At)) dt I 5 c ~(0]&(0] IJn I Jn
we get
M2>
hence the thesis.
-
dt,
g(4)12 5
n
LEMMA 2.7. Let s1 c R be a bounded open set and g E L’(Q). Let us suppose that there exists p E L’ (Cl) such that
s n
Then, we have
MO, 6 ‘(0) dt 5
n
v6
dWW dt,
E
C,“(sz), 6 L 0.
t+h g(t
+
h)
-
g(t)
1
-
ds)
st for almost every t, t + h E n with [t, t + h] c iI. Proof.
It is a simple variant of the previous proof.
W
d.s
A. CANINO
512
LEMMA2.8. Let Q C fRbe a bounded open set and g E L’(Q; IR”). Let us suppose that there exist g, E W’3’(Q), g, E w1S2(a), q~ E L’(Q) and a72E L2(Q) such that b ItAt
+
h)
-
g(Oi
dt
5
s LI
bhdt s a
+
A)
-
g,Wl
+
(IV)2(f
+
41
+
lfP2Wl)k2(~
+
(lvdt
+
Wl
+
bd0lPl
+
h)
-
g2wl
dt
whenever h 2 0, [a, b + h] c 0. Then g E W1S1(s2;R”) and
IdWl 5 Proof.
Ig;W
+
h72(0
I&WI
+
a.e. in Q.
sP,(0
It is readily seen that g E SV(G; R”) and that
whenever [a, b] c Q, where iu = g’ as measure. Therefore, if K is a compact subset of Sz with zero Lebesgue measure, we have /P](K) = 0. Since 1,uI is absolutely continuous with respect to Lebesgue measure, the thesis follows. H LEMMA2.9. Let us take y E X and a E a-f(y).
Then
1 W&(f,
Y,
04
+
P,W,
Y,
~‘1,
S’N
df
.r0
2
’ 6~
4
dt,
s 0
V 6 E Wlv2(0, 1; R”) with 6(O) = 6(l) and s(t) E T,&4,
v t.
Proof. Let m be the dimension of M. For every x E M, let i.J, be an open neighbourhood x in R”, g: U, + lR”+’ and V: U, + R two maps of class C2 such that
of
v y E U, : dg(y) is onto; vy E U,:grad
V(y) # 0;
M f7 U, = (y E U, : g(y) = 0, l’(y) s 0);
either
V(x) = 0
or
KY) < 0
Moreover, let us set N = (y E U, : g(y) = 0). By substituting of x, we can assume that there exist an open set A containing class C2. Now, let x E M and let 6 E W’.2(0, 1; R”) be such that compact support in y-‘(U,) and, if x E IBM, there exists E > (a(t), grad V(Y@)))5 4 If s is sufficiently P(u,).
&t)l Igrad
VMNI,
small, we have (y + sS) E A on y-‘(U,).
vy E u,. U, with a smaller neighbourhood N and a retraction r: A + N of
6(O) = 6(l), s(t) E l?$,M, 0 such that v c E [O,
6 has
11.
We claim that r(y + .sd) E M on
Periodic
solutions
of Lagrangian
573
systems
Actually, if x $ &VZthe fact is obvious. If x E aM, we have V(r(v + ~6)) I IV(v)) + (grad I+-(y)), dr(y)(s8)) + const s21612 = V(/(y)+ (grad V(y), dr(y)(s@) + const s21612. Since sS E F_M, we have dr(y)(sa) = ~8, hence ~(r(y + sS)) 5 s(grad V(Y),6) + const s21612 I -sslgrad
Al
161 + const s2j6j2
= -sj6l(elgrad
v(y)1 - const s161).
Therefore, r(y + ~8) E M foras sufficiently small. If we set, r(y + ~6) = y outside of y-‘(U,), we have, by definition of subdifferential 1
1 L(t,
r(y
+
d),
(r(y
+
dt -
~3))‘)
s0
L(t, Y, Y’)dt 1 0
2
:(a,
4~
+
~8)
-
Y)
dt
-
NY
+
~4
-
y)b(r
+
~4
-
YIILZ
I
where limIlWllL2 -o E(W) = 0. Thus, dividing by s, if 0 < z < 1 we have
s I
0
D,W,
+
L
Y +
zw + s4
s
lD”L(t, 0 r(Y
Y +
+
m
- Y), Y’ + z((eJ + ml’ - Y’))
zw + m -
Y
s
r(Y +
m-
Y), Y’ + z(W + @I - v’)) 4Y
dt - E(T(Y+ sS) - y) >
+
m
-
dt
(mJ + s&Y - Y’
dl
s
Y
s
II
Y
s
(2.9.1) /I 2’
Passing to the limit in (2.9.1) as s -+ 0 and recalling that dr(y)(b) = 6, hence d2r(y)(r’, 6) + dr(y)(b’) = 8, we have 1 MD&(&
Y,
Y’),
6)
+
(RW,
Y,
Y’),
WI
df
2
.r0
’ (a, s 0
4
(2.9.2)
dt.
Using partition of unity, it is readily seen that 1 ND&(&
Y,
Y’),
4
+
(DJ(f,
Y,
Y’),
W,
s0
01
dt
1
‘((~3 1 0
4
dt
whenever 6 E W132(0, 1; IR”), 6(O) = 6(l), s(t) E FY&4 and there exists E > 0 such that (s(t), v(W)) 5 --Elwl Iwml in a neighbourhood of C,. Finally, if 6 E W’*2(0, 1; R”), 6(O) = 6(l), s(t) E T&M, let us consider the sequence W)
= s(t) - ; W(0).
A.CANINO
514
Obviously,
lim,&
= 6 in W’S2(0, 1; R”) and
(W),
W0))
= (d(f), W(0))
- ;
IWO) I2 5 - ;
iftEC,.
By continuity (M), if t belongs
W0))
to a neighbourhood
5 _/.(l
I40 Iww)l
+l,lall,-,
of C,. So, we can apply (2.9.2) to Bh v h and we get
1 IP,Ut,
i Passing
Y, v'h4A
+
(D,Ut,
Y, y'), WI
lb, 0
dt 2
0
(2.9.3)
dddt.
n
to the limit in (2.9.3) as h + COwe have the thesis.
LEMMA 2.10.Let y E X, let t E IO, l[ and let h > 0 be such that t + h E IO, l[. Let us assume that y is differentiable at t and t + h. Then
Ir’U f h) - y’(t) - &,,ow + h) - v’W)l < Proof. First s E]O, l[. so
of all, let us observe
Iv’0 + h) - Y’(f)- &t,w
that pYo,(y’(s))
const]y’(t
+ h)l Iy(t + h) - y(t)l.
= y’(s), whenever
y is differentiable
at
+ h) - W)l = Iw + h) - &)(YV + h))l = I&t+/&4 + h)) - ~~,,,W + h))l.
Now,
we can apply (2) of proposition
1.5. to obtain
the thesis.
W
LEMMA 2.11. Let us take y E X and (II E a-f(y). Then (i) D,L(t, y, y’) E W’,‘(]O, l[\C,; I?); (ii) y’ E W’,‘(]O, l[\C,; I?“); and, (iii) Jr”(t)1 5 la(t)1 + const(1 + ly’(t)12) a.e. in IO, l[\C,. Proof. Let us consider q E C,“(O, 1; R”) with supt(q) By applying lemma 2.9 to 6 and -6, we get
C IO, l[\C,
and let us set 6 = p?‘;lr.
1 I(D,W,
i
and by (2) of proposition
so
+
~~~:,D,Ut,
Y, ~'1, ~11 dt =
1.5, it is readily
bp, i0
v)dt
seen that
l(~yv)’- @7’)l 5 codYI Id. 1
Ii 0 By lemma
Y, ~'1, (@T)
0
(p;,D,L(t,
y, y’), q’) dt
i
2.6 and (1.7) and (1.8), pTD,L $(~y(D,L(t,
y, y’)
‘(la - D,LI
I 1
I
+ constly’l
]D,L()]~l
dt.
(2.11.1)
0
E W’*‘(]O, l[\C,;
R”) and
Ia(t)( + const(1
+ Iy’(t)12)
a.e.
(2.11.2)
Periodic
solutions
of Lagrangian
if [t, t + h] c IO, l[\C,,
On the other hand,
systems
using the following
notations
e,(t) = t + sh r,(t) = y(t) + s(y(t + h) - y(t)) c,(t) = y’(t) + s(lJ’(t + h) - y’(t)) we have by (1.1) and (1.5) vly’(t + h) - ?m12 5
l(~F,Ue,(t), r0
+
r,(t),
S,(0)(y’(t
+ h) - v’(t)), y’(t + h) - Y’(0) d..s
+ h) - y’(t)1 Iy’(t + h) - y’(t) - &,(y’(t
constly’(t
By (1.2), (1.4) and (2) of proposition
+ h) - y’(t))l.
(2.11.3)
1.5, we have
1 (~,2"W,W,
i
<,(0, i,(ow(t
+
h) -
Y'(0), &)ow
+
h) -
Y'(0)) d.s
0 =
Py(t+h) D"W
+
A, Y(f +
h), y'(t +
h)) -
+
+
h, ?J(t+
h), y'(t +
h))
-s
-
5
(&f)D"Uf r(t+h)D"ut
+
lvCww~, s0
h, Y(t +
A), y'(t +
r,(t), wwh
IPy(l+hp,ut
+
k
y(t +
hh
Y(t), Y'(0), Y'@
h)), y'(t +
&tjw(t
y'(t +
&p,u4
+
h)) -
w
-
h) -
00))
&tp"ut,
+
h) -
Y'(0)
Y'(0)
b
y(t),
Y’(t))llY’u+ h) - Y’WI
+ conO,Ut + h, y(t+ 4, v’(t+ WI IvU+ h) - WI Iv’@+ h) - r’(O + const(1
+ ly’(t)l
+ const(l
+
5
+ Iy’(t + h)l)h(y’(t
+ h) - y’(t)1
Iv’Wl + Ir’tt + Nl)lv(t + W - WI Ir’tt + W - y’(Ol.
tl~-y(t+,,~“w+ fh YG+ h),y’(t + h)) - &)D”W, y(t),rwl + const(l+ Iv’W + Ir’(t + fOl)lv(t+ h) - rWl + const(1
+ If(t)1
+ Iy’(t + h)l)h)ly’(t
+ h) - y’(t)l.
(2.11.4)
A.
516
Combining
(2.11.4)
with lemma
CANINO
(2. lo), we deduce
that
vly’(t + h) - W)l 5 l&,,h$“W
By lemma
+ A, V(t + h), y’(t + h)) - &,D”WV
+ const(1
+ Iv’(0
+ Iv’@ + h)l)lv(t
+ const(1
+ Ir’(t)l
+ ]v’(t + h)l)h.
2.8 and (2.11.2),
y’ E W’,‘(]O,
l[\C,)
By composition,
we deduce and
+ h) -
y(t), Y’(0)l
~(0
that
Iv”(t)1 I
la(t)/
+ const(1
+ ]~‘(t)]~)
a.e. in IO, l[\C,.
we also have D”L(I,
y, y’)
E
W19’(]0,l[\C,;
H
R”).
LEMMA 2.12. Let us take y E X and Q! E a-f(y). Then: (i) rc,D,L(t, y, y’) E W’S1(O, 1; F?“); and, (ii) ](n,D,L(t, y, y’))‘l 5 const(lcx] + 1 + lv’12) a.e. in IO, l[. Proof. Let us take r;l E W’S2(0, 1; I?“) with q(O) = q(l) and let us set s(t) = q,,(q(t)). Obviously, s(t) E T,,,,M tl (-r,,,,M) tl t. So, we can apply lemma 2.9 to 6 and -6 to obtain 1
I(D,W,
Y,
~‘1,
nn,l;l)
+
(D&t,
Y,
~‘),0q,Wl
dt
‘b,
=
.i 0
7q)
dt
.I 0
and thus by (3) of proposition
1.5
1 ((Q&Q,
Y,
~‘1,
v)
+
(QW,
Y,
Y’),
(qd’N
dt
s0
rl) dt.
(2.12.1)
i 0
By using (4) and (6) of proposition
1.5 in (2.12.1)
we obtain
1
IS (D&t,
1 Y,
~‘1,
~,WN
(n&y
dt
0
-
and then by (3) and (4) of proposition Y,
Y’),
rl’)
i
1.5
1 (n,&W,
D&Q,
Y,
~‘1,
u?) dt
0
+ const
: 1~1Ir’] I&L]
dt
‘tb - D&I + IY’I I~,~OIvIdt.
dt
0
By lemma
’ b,w
=
0
2.6 and hypotheses
(1.7) and (1.8), we have from (2.12.2) qD,L(t,
y, y’) E W’vl(O, 1; R”)
and I(rQ&W,
Y, v’))‘l 5 const(]a]
+ 1 + ]r’12)
a.e. in IO, l[.
n
(2.12.2)
511
Periodic solutions of Lagrangian systems LEMMA 2.13.
Let us take y E X and (YE a-f(y).
Then for almost every t, t + h E IO, l[ (h > 0)
we have (D”Ut + A, Y(t + h), y’(t + h)), VW + h))) - (D”L(t, v(t), y’(t)), v@(t)))
s t+h
2
(ICYI+ 1 + l#)dr.
-const
t
Proof.
1) with q 2 0 and let d(t) = -q(t)v(y(t)).
Let q E C,“(O,
-
l(DJK i0
Y, ~'1, (~?W)')dt
1
-
'(01 - Q&V, 1 ,O
By lemma (2.9), we have Y, ~'1, mW)dt.
Therefore -
b,L(t, s0 2
Y, Y'), W)rl'dt
-
jb
- D&Q,
Y, ~'1, V(Y)) -
(D,Ut,
Y, ~'1, (WY)lv
dt.
1
By (1.7) and (1.8), we have
s 1
(D,L(t,
y, y’), v(y))q’ dt 5 const
‘(lcvi + 1 + Iy’j’)q dt s0
0
n
and by lemma 2.7 the thesis follows.
LEMMA 2.14. Let us take y E X and cx E d-f(y). Let t E IO, l[ and h > 0 be such that t + h E IO, l[ and such that y is differentiable at t and t + h. Then we have (y’(t + h) - y’(t), v(y(t)))
2s const ( 1 + jjWt
dr) i:‘*(lal
+ 1 + lY’?)dT t+h
+
consWWl+ Iv’@+ WI) f ly’lcit. s
Proof.
First of all let tl , t, E IO, l[ be such that there exist yL(tl) and y:(t2). Then we have rl(t,)
E Qt,,M
(rr_(t,), vbJ(t1))) 2 09
Y:m
E Qtz,M
(YX2h
W~z)N
5 0.
Therefore oC(&) - rr_(t,), VWl))) 5 W(fA
VW,)))
= (Y1(f*), WW) 5 const
- VW*))) + (Wz),
IY;(h)I IY&) - r(tl)I.
MM)) (2.14.1)
Now, let t E IO, l[, h > 0 with t + h E 10, l[ and y differentiable at t and t + h. If [t, t + h] fl C, # 0, let t, , t2 E C, be such that t I t, I t2 I t + h, (It, tl[ U ]t2, t + h[) fl C, = 0.
A. CANINO
578
By lemma 2.11, there exist r_(tr) and y:(fJ.
Therefore
(y’(t + h) - v’(t)* MO)) 5 (Y’([ + h) - V;(tz), VW))) + W(&) - y’(tA WO))
+ (rr_(t,) - y’(t), W(O))
I constly’(t + h) - Y:(&)[ + constlyl.(t,) + (Y:(G) - rr_(t,), Wtl)))
+ (YX)
- y’(t)1
- r’(tr), WO)
- VW,))).
Now, by applying lemma 2.11 and (2.14.1), we obtain (y’(t + h) - Y’(f), W(O)) 5 const
:’ IY(r), dr + .I;” /Y”(r), dr]
[I
+ constIr:&)l
IV(&)- r(tJ +consdMd- YWA Iv(O- Wdl
1
t+h 5
const
+
(/a(r)1 + 1 + lv’(r)12)dr
ii’.t
constly’(t + h)l Ir(t + h) - y(t)1 + constly’(t + h) - ~:(fJl
x
(Ir(f + h) - Y&>l+ Ir(f,) - ?+)I)
+ constly’(t+ MtlvU + h) - Y&)I + Ir(tJ - r(r)l) + constlf(t + 4 - Y:&)I IrG + f0 - WI + constlf@+ 4 - v’WlIv(t) - Y(~~)I + cona[Ir’@ + 4 - Y:WI + IvYtd - ~‘(0l1l~W- v(tJ t+h
t+h 5
const
[i
(la(r)1 + 1 + IY’(r)l’)dr t
+ const)y’(t + h)l f
const [.r t
(Ia(r
+ 1 + lr’(r)?)dr t+h
t-
dr
II’ t+h
t+h +
IY’(~
i;
1
constly’(t + h)( f i
f
Iv’(d
dz t+h
IY’(@I dt + consdy’@ + h) - v’(t)1t IY’(~dt i
t+h 5
const
[i
t
(la(r)1 + 1 + Iv’(r)?) dr ] ( 1 + 1: IfCr)l di) t+h
+
const(lVO + 41 + Iv’Wl) t ~Y’(G dr. i
Finally, it is readily seen that the same inequality holds if [t, t + h] fl C, = 0. Now, we come back to the following proof.
n
Periodic solutions of Lagrangian systems
519
Proof of theorem 2.5. Let us take 0 < t < t + h < 1 and let us assume that y is differentiable at t and t + h. If we keep the notations of lemma 2.11, by (1. l), (1.5) and lemma 2.10, we have vly’(t + h) - ~‘(t)~2 5
l(Q%&(O, s0
T,(t), W))(r’(t
+ h) - y’(t)), y’(t + h) - y’(0) ti
=
b,2,W,(t), 5 0
W),
+
W)W(t
h)
-
y’(t)),
E’,,,W
v’(t
5
+
+
h)
-
h)
y’(t)
-
y’(0))
-
ds.~
&W
+
h)
-
y’(t)))
’U’,2,W,W,L(t), S,(tM”(t + h) - y’(t)), ~,~,~(y’(t+ h) - y’(t))) ds ’V’2,,WW),t-,(t), LWW(t + h) - y’(O), WOM&t~W + h) - y’(t)), WO))’(%W,W, t-,(t), tXOW(t + h) - y’(t)), W(t)))
i0
s s
d.~
d.s
0
+
0
x (y’(t + h) - y’(t), &p’oW)+ CLS + constly’(t+ h) - y’(t)1Ir(t + 41ly(t + h) - WI. By (1.2), (1.4), (1.8), (3) and (5) of proposition
s
1.5 we obtain
1
0
U’:J@,W,t-,(t), M)W’(t + h) - y’(t)), q&‘(t =
+ h) - y’(t))) ds
(DJ(t + h, y(t + h), v’(t + h)) - D,W, v(t), y’(O), xr&‘(t
-s 1(@&V4W, 0
5
t-,(t), WNW + h) - y(O), ~rttjW(t + h) - y’(O)) d.~
(n,~t+,,RUt + h, Y(t + h), y’(t + h)) - q&Ut, + (+$‘,Ut
+ h) - y’(t)))
y(t), v’(t)), 7’0 + h) - y’(t))
+ h, Y(t + h), v’(t + h)) - nyct+@uL(t + h, y(t + h), y’(t + h)), v’(t + h) - y’(t))
+ const(1 + (y’(t + h)l + (y’(t)()h(y’(t + h) - y’(t)( + const(l + Iv’(t + h)l + b’(t)l)lv(t 5
+ h) - y(t)1 iv’(t + h) - r’(t)1
(bry~t+&L(f + h, y(t + h), y’(t + h)) - q#,Ut,
+
const(1 + (y’(t + h)l + (y’(t)0
y(t), y’W)l
t+h t IY’(~/ dr
5
+ const(1 + [y’(t + h)( + Iy’(t)l)h)ly’(t
+ h) - y’(t)l.
(2.5.2)
580
A. CANINO
By lemma 2.13, we have
5
-(~&w
+
x
W”W
+
(D”Jw
h)
+ +
h, k
-
Y’(0),
Y(t
+
Y(t
+
W0)-h),
y’(t
A),
y’(t
ii
h)),
+
constly’(t + h) - y’(t)1
5
+
t
h)),
V(Y(f
+
W(0)
h))) -
-
(D”W,
VW
+
YW,
Y’(0),
W(0))
h)))
(I~(~)/ + 1 + lr’(r)1*)d~ t+h
+
(1
+
Iv’@
+
Ml
+
Iv’@)l) t bwl dr + (1 + IYV + WI + l~~00~ . (2.5.3) 1
.r
Finally, since v@(t)) E ~Y;rct+4,by lemma 2.14 we have
sl 0
+ h) - Y’Wt &tj VW)))+d.s
vz,w,w9 MO, L(mJ’(~ + h) - Y’(O),wo))(Y’(t 5
constly’(t + h) - y’(t)/ {( 1 + j: Iv’(t)1 d7) j:‘“(k)1
+ 1 + lr’(.r)I*) ds
/yh b’td dT]. + (Ir’(t + 41 + IY’WO Combining (2.5.1)-(2.5.4), v(y’(t + h, - f(t)/
S
(2.5.4)
we get D”W
b++h)
+ const(l
+
h,
+ j:ly(r)[
Y(t
+
A),
Y’@
+
h))
-
~,(,)~“W,
Y(0,
Y’W)l
dr) jtt+h(lcx(r)l + 1 + lf(t)[*)dr t+h
+
const(1 + Jy’(t + h)l + If(t)/) j
+ (1 +
IY’U + WI + ly’m~.
t
Iv’(d
dt
(2.5.5)
581
Periodic solutions of Lagrangian systems
By applying lemmas 2.12 and 2.8, we can conclude that y E W2,2(0, 1; R”)
and
where 13:R + IRis a continuous function. Now, it remains to prove that y:(O) = y:(l).
Let us define p E X in such a way Oltl+
YU + *>3 HO =
*std.
i YU - t>v We can easily see that a-f(p) THEOREM 2.15.
a.e. in IO, l[,
]~“(t)l 5 ~V(Y))(]C#)] + 1 + ]~‘(t)]~)
# 0. Then p E W2V2(0,1; R”). Thus, y:(O) = y’(l).
Let us take y E Xn W2,2(0, 1; R”). If CYE a-f(y),
o(t) + it (D”JV, y(t), Y’(0)) - (Q&4,
y(t), Y’(0) E &,,M
n
then a.e. in IO, l[.
Proof. Let t,, E IO, l[ be a Lebesgue point for c(t)
(
o(t) + -$D”LU, Y(f), Y’(0) - Q&t,
y(t), Y’(0) >
Let us consider w E 7&$4 with (w, v(y(t,)) < 0 if to E C,. Let q,,(t) = ph(t - to) v h E N where 1~~)~is a sequence of mollifiers. Then, by continuity we have (p,,:rtow, v(y(t))) < 0 in a neighbourhood of to, hence (l?&(qh(t)w), v(y(t))) I 0 eventually as h + co. Since PJq,, w) E Hd and ff&(qh(t)w) E 7&M, we have, by lemma 2.9 1 6 0 SC
a
+
&“Uf,
(
(2.15.1)
Y, Y’) 3 W q,, dt I 0. > >
Y, Y’) - QJ(C
Passing to the limit in (2.15.1) as h + 00 we get
(
4fo) + $D,L(f,,
YGO),Y’GO))- Q&to,
YGO),Y’GO)),w 50 >
so that o(t0) + $“Ut,,
YQO),y’(t0)) - D,Uo,
YGO),Y’VO))E &,,,,M.
n
As a consequence of theorems 2.5 and 2.15, we have the following corollary. COROLLARY 2.16.
Let us take y E X. If 0 E a-f(y), y E W2.“(0, 1; V),
then Y:(O) = Y’(I)
and -&(f,
y(t), Y’(0) - Q$(t,
Finally, let us prove the cp-convexity off.
y(t), Y’(0) E N,,,M
a.e. in IO, l[.
582
A. CANINO
2.17. The functional f: L’(0, 1; R”) + R U (+a~) is 1.s.c. and there continuous function e+,; R2 -+ R such that
THEOREM
f(Y + 4 2 f(r)
+ (a, 42
whenever y, y + 6 E X and CYE d-f(y). order two.
- %(f(Y)t
f(Y + 4)(1
+
exists a
II&wllt2
In particular, f has a p-monotone
subdifferential
of
Proof. First, we will prove that f is I.s.c. Let us consider a sequence (Y~)~C X such that lim, yh = y in L2(0, 1; R”) and f(yh) I b. By (1.9) we have
and then:
that is, 1~~)~converges weakly to y in WiS2(0, 1; R”). On the other hand, L is continuous in the three variables and convex in the third one, so it is weakly 1.s.c. in WiS2(OI1; R”). Then j: L(t, y, y’) dt I b. Finally, it is enough to observe that 1~~)~converges uniformly to y and recall that A4 is compact to prove that y E X. Thus SAL(t, y, y’) = f(y) and f is 1.s.c. Now, let us take y E X fl W2S2(0, 1; R”), CYE d-f(y) and let 6 E W1*2(0, 1; R”) with 6(O) = 6(l) be such that y + 6 E X. By Taylor’s formula, we have Ut, Y + 6, Y’ + a’) - Ut, Y, Y’) = (D&V, Y, Y’), 6) + (D”L(t, Y, v’), 6’) + *(D&L@,
y + sd, y’ + ssy,
+ #g”L(t,
y + sd, y’ + sS’)S, S’)
+ +(D;“L(t,
y + sd, y’ + sS’)&, 6’)
for some s = s(t) E IO, I[. BY (l.l), (1.3), (1.4), theorem 2.14 and (1) of proposition f(Y + 6) -f(y)
2
-
6)
1.5 we have
iwW I’0
b&Y,
Y'> -
-$“L&
Y, Y’) - a, 6) dt
s
-‘const(l
+ ll~~‘@ + (Iy’ + S’ll~z)llSllfm
- constll
+
Ily’IIL~+ lly’ + ~‘Il~~)ll~ll~-ll.116’ll~~ + 5 ll~‘ll3
5 -constllD,L(t,Y, Y’)- $+D,,Ut, Y, Y’)- ~Il~~ll%= - const(l + ll~‘lltz + lly’ + S’ll~2)11Sll~~ - const(1 +
IIY’II~~ + IIY’+ ~‘ll~~~ll~ll~-ll~‘ll~~ + f IlfJ’llLz.
Periodic solutions of Lagrangian systems
583
By (1.7), (1.2), (1.4) and (1.5) we have f(r + 6) - f(v) -
- const(1 + Ily’l@ + lly’ + fJ’ll~~)llf?ll$ - const(l +
IIy’lLz+ lly’+ ~‘ll~~)llG4l~‘ll~~ + i IlS’II3.
By theorem 2.5 and (1.9) we conclude that f(v + 6) - f(y) -
’ (~6)
dt
I 2
-conNllV’ll~~ + Il&)(ll~ll~~+ 11~11~41~‘11~2) - const(l+ Ilv’llt~ + Ilr’ + ~‘ll3>ClldlL~+ Il&4l~‘ll~~) - const(l + Ily’lL2 + lly’ + ~‘llL~)(ll~ll~41~‘lI~2 + 11~112~211~‘11~4’> + i IlS’ll2
2 -%m%f(Y
+ a)(1 + Il~llm4lz~
for some continuous function cpO:R2 -+ R. Remark
n
2.18. From the previous proof, it turns out that if
y Ex
n w~,~(o,1;~“1,
Y:(o)= Y’(l),
ct!E L2(0,1; rn”)
and a(t) + $ (D”Uf, y(t), y’(0)) - D,JW, y(t), y’(0) E &,,M
a.e. in]O, l[,
then (11E a-f(r). Therefore, the converse of theorem 2.15 also holds. 3. TOPOLOGICAL
RESULTS
Since we proved the p-monotonicity off (see theorem 2.17), we are able to evaluate the number of lower critical points for f by means of a technique such as the LusternikSchnirelmann one (see, Section 4). Thus, in this section, we investigate the category of the space of the admissible paths X (as defined in Section 2). The following result, proved in a recent paper [lo], is crucial for our aim. THEOREM 3.1. Let F A E s B denote a Hurewicz fibration with E path-connected. Let us assume that there exists a section o: B + E. Then cat F I cat E. In [lo], theorem 3.1 is applied in order to estimate cat(A( Y)) wherte Y is a simply connected manifold without boundary. Our aim is to show that an analogous argument works on manifolds with boundary. If Y is a topological space, let us denote by A(Y), the free loop space of Y and if y0 E Y by Q(Y, yO) the based loop space of Y with base point yO.
A. CAMNO
584
LEMMA3.2. Let A be an open subset of R”, connected, noncontractible n,(A) is finite. Let us take x,, E A and let us set
in itself and such that
Q&4, x0) = (y : y E CJ(A, x0) and y is contractible in A]. Then, cat &,(A, x0) = +a. Proof. Let us consider the universal covering 71:A’ -+ A. Let y0 E A be such that n(yO) = x0, then &(A, x0) is homeomorphic to Q(A, uO). By the extension of [17] given in [l 1, corollary 3.21, we have catQ,(A,x,,) = catQ(A,y,) = +a~. n
By theorem 3.1 and lemma 3.2, we can state the following theorem. THEOREM 3.3. Let A be an open subset of IT?“,connected Moreover, let us suppose that either (i) x1(A) has infinitely many conjugacy classes or (ii) nl(A) has a finite number of elements. Then cat A(A) = +a.
and noncontractible
in itself.
Proof. If (i) is true, the thesis is trivial because A(A) has infinitely many path-components. If (ii) is true, let us set A,,(A) = {y : y E A(A) and y is contractible
in A).
Let us point out that, since the map p: A(A) + A defined by p(y) = y(O) gives a fibration and A,,(A) is a path-component of A(A) then also the restriction pO: A,(A) + A gives a fibration, of course with A&A) path-connected. We can define a section 0: A -+ A,(A) in such a way: v x E A, a(x)(s) = x v s E [0, 11. The fiber is p; ‘(x0) = Q,(A) x,,). Then, we can apply theorem 3.1 and we obtain: cat O,(A, x,,) I cat A,(A). Obviously, cat A,,(A) I cat A(A) and by lemma 3.2 we have the thesis. n Now, in order to evaluate the category of X (endowed with W’,2-topology), recall a result contained in [ 151. THEOREM3.4. The inclusion map i: X -+ A(M) is a homotopy
it is enough to
equivalence.
Finally, this gives the following theorem. THEOREM3.5. Let A4 c R”, be a C2 submanifold (possibly with boundary), connected and noncontractible in itself. Moreover, let us suppose that either (i) z,(M) has infinitely many conjugacy classes or (ii) n,(M) has a finite number of elements. Then cat(X) = +oo. Proof. Let A be an open subset of IR”such that M is a deformation retract of it. Then, A(A) is homotopically equivalent to A(M). Now, it is enough to apply theorems 3.3 and 3.4 to have the thesis. n
Periodic solutions of Lagrangian systems 4. PROOF
OF THE MAIN
585
RESULT
To prove the existence of infinitely many periodic orbits to by corollary 2.16 and remark 2.18, the existence of infinitely on the space of the admissible paths X, we will use a theorem with the number of the critical points of a functional defined
our Lagrangian system, that is, many lower critical points for f linking the category of a space on it (cf. [12 and 81).
4.1 (see [S]). Let H be a real Hilbert space, g: H + L?U (+a) a p-monotone subdifferential of order two. We set
THEOREM
d*(u, u) = Iu - VI + I&) - g(n)1
a 1.s.c. function with
v U, V E D(g).
Let us suppose that: (i) inf g > -00; (ii) /(u& c D(d-g) with s;p g(uh) < +m and lip grad- g(uh) = 0, (u&, has a subsequence converging in H. Then, g has at least cat@(g), d*) lower critical points. Moreover, if cat@(g), d*) = +oo, then sup g(D(g)) = SUPk(U) : 2.lE ma-g), 0 E a-&II. Finally, this leads to the following proof. Proof of theorem 1.2. Let us consider the functional f defined in Section 2. By hypothesis (1.9) and theorem 2.17, f is a 1.s.c. function, bounded below and it has a v-monotone subdifferential of order two. Moreover, by corollary 2.16 we want to prove the existence of a sequence (yhJh E X such that 0 E a-f&) and lim, f(yh) = + co. Let us remark that D(f) = X and that d* induces the W’*‘(O, 1, IR”)-topology on X. Therefore, cat@(f), d*) = +W by theorem 3.5. Now, we will consider a sequence 1~~)~C D(d-f)with s;P f(Yh) < + *
lihmgrad- f(y,,) = 0.
and
By hypothesis (1.9), (Y;)~ is bounded in L2(0, 1; IR”)and, on the other hand, since the manifold A4 is compact, (yhJh is bounded in L’(O, 1; IR”). By Rellich’s theorem, we deduce that lyhlh has a subsequence converging in L2(0, 1; R”). To complete the proof, it remains to show that sup f(y) = +oo. 7eX Let y E X and let y be nonconstant. by (1.9)
If we set ~&Y) = I
f(%l) + c 1 v
r1
30
v
h E N, we have (q,,)* E X and
lllltlZ~.
If by contradiction {f (q*),, is bounded, then up to a subsequence (q,, jh converges uniformly to the constant y(O). But y is nonconstant by hypothesis, so y(s) # y(O) for some s E IO, l[ and there exists s,, such that qh(sh) = I. This is an absurdity. n
586
A. CANINO
Acknowledgement-The author wishes to thank E. Fade11 who pointed out that a previous version of the proof of lemma 3.2 was incomplete. REFERENCES 1. BENCI V., Periodic solutions of Lagrangian systems on a compact manifold, J. DgJ Eqns 63, 135-161 (1986). 2. CANINOA., On p-convex sets and geodesics, J. DI~J Eqns 75, 118-157 (1988). 3. CANINOA., Existence of a closed geodesic on p-convex sets. Ann. Inst. H. Poincard, AnaIyse non LinPaire 5,
501-518 (1988). 4. COBANOVG., MARINOA. & SCOLOZZID., Evolution equation for the eigenvalue problem for the Laplace operator with respect to an obstacle, preprint no. 214, Dip. Mat. Pisa (1987). 5. COBANOV,G., MARINO A. & SCOLOZZID., Multiplicity of eigenvalues for the Laplace operator with respect to an obstacle and nontangency conditions, Nonlinear Analysis 15, 199-215 (1990). 6. DE GIORCI E., DEGIOVANNIM., MARINOA & TOSQUESM., Evolution equations for a class of nonlinear operators,
Atti Accad. Naz. Lincei Rend. Cl. Sri. Fix Mat. Natur. 75, l-8 (1983). 7. DE GIORCI E., MARINO A. & TOSQUESM., Problemi di evoluzione in spazi metrici e curve di massima pendenza, Atti Accad. Naz. Lincei Rend. Cl. Sri. Fis. Mat. Natur. 68, 180-187 (1980). 8. DEGIOVANNIM., Homotopical properties of a class of nonsmooth functions, Ann. Mat. pura appl. (in press). 9. DEGIOVANNIM., MARINO A. & TOSQUESM., Evolution equations with lack of convexity, Nonlinear Analysis 9, 1401-1443 (1985). 10. FADELLE. & HUSSEINIS., A note on the category of the free loop space, Proc. Am. math. Sot. 107,527-536 (1989). 11. FADELL E & HUSSEINI S., Extending Serre’s theorem on the category of loop spaces, preprint (1990). 12. MARINO A. & SCOLOZZID., Geodetiche con ostacolo, BON. Un. Mat. Ital. B(6) 2, 1-31 (1983). 13. SCOLOZZID., Esistenza e molteplicita di geodetiche con ostacolo e con estremi vairabili, RicercheMar. 33, 171-201
(1984). 14. SCOLOZZID., Un teorema di esistenza di una geodetica chiusa su varieta con bordo, Bolt. Un. Mat. ftal. A(6) 4, 451-547 (1985). 15. SCOLOZZID., Molteplicita di curve con ostacolo e stazionarie per una classe di funzionali non regolari, preprint no. 69, Dip. Mat. Pisa (1984). 16. SCOLOZZID., Esistenza di una curva chiusa stazionaria e con ostacolo, preprint no. 70, Dip. Mat. Pisa (1984). 17. SERRE J. P., Homologie singuliere des espaces fib&, Ann. Math. 54, 425-505 (1951).