Periodic Solutions of Nonlinear Wave Equations and Hamiltonian ...
Periodic Solutions of Nonlinear Wave Equations and Hamiltonian Systems Author(s): Haim Brezis and Jean-Michel Coron Source: American Journal of Mathematics, Vol. 103, No. 3 (Jun., 1981), pp. 559-570 Published by: The Johns Hopkins University Press Stable URL: http://www.jstor.org/stable/2374104 . Accessed: 21/02/2011 10:06 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=jhup. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].
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PERIODIC SOLUTIONS OF NONLINEAR WAVE EQUATIONS AND HAMILTONIAN SYSTEMS By HAIMBREZIsand JEAN-MICHEL CORON
Abstract. We consider the nonlinear vibrating string equation u,,-uxx + h(u) = 0 under Dirichiet boundary conditions on a finite interval. We assume that h is nondecreasing, h(O) = 0 and limjuj_I[h(u)/u] = 0. We prove that for T sufficiently large, there is a nontrivial T-periodic solution. A similar result holds for Hamiltonian systems. 0. Introduction. u,,-u,,,,
Consider the following nonlinear wave equation:
+ h(u)=O
O < x < 7r,tE R.
(1)
under the boundary conditions: u(O, t) = u( 7r,t) = 0,
(2)
where h: R - R is a continuous nondecreasing function such that h(O) = 0. We assume: lim IuI-o
h(u) =0 U
(3)
There exists a constant R such that h(u) ? 0 for Iu ? R. (4) We seek nontrivial solutions of (1), (2) which are T-periodic (in t). By "nontrivial" we mean that h(u(x, t)) ? 0 on a set (x, t) of positive measure; in particular, u(x, t) ? 0 on that set. In Section 1 we prove the following THEOREM 1. There exists To > 0 such that for every T : To, with T/7r rational, Problem (1), (2) admits a nontrivial T-periodic (weak) solution u ELw.
Manuscript received May 15, 1980. AniericaniJournailof Mathemiiatics,Vol. 103, No. 3, pp. 559-570 0002-9327/81/1033-0559 $01.50 Copyright ? 1981 by The Johns Hopkins University Press
559
560
HAIM BREZIS AND JEAN-MICHEL CORON
By a result of [4], weak solutions are in fact smooth if h is smooth and strictly increasing. The existence of nontrivial solutions for (1), (2) has been considered by several authors under assumptions which differ from ours (see [1, 4, 5, 7, 8]). In Section 2 we discuss a comparable result for Hamiltonian systems. Our investigation has been stimulated by the results of [3] (Section 4). Our technique relies on a duality device used in [6] for Hamiltonian systems and subsequently in [5] for the wave equation. We thank P. Rabinowitz for helpful discussions. Proof of Theorem 1. The proof is divided into five steps. Step 1 Step 2 Step 3
Generalities about Au = ut - uxx Determination of To. Existence of a nontrivial solution for Au + h(u) + cu = 0
Step 4 Step 5
(c > 0 small).
Estimates. Passage to the limit as c - 0.
Step 1 Generalities about Au = ut - ux Since T/7r E Q we may write T = 27rb/a where a and b are coprime integers. Let H = L2(Q) with Q (0, 7r) X (0, T). In H we consider the operator Al = U,,acting on functions satisfying (2) and which are T-periodic in t. We summarize some of the main properties of A which we shall use (see e.g. [4] and the references in [4]): i) A* =A ii) N(A) consists of functions of the form N(A) = {p(t + x)-p(t-x),
wherep has period
2ir
T
a
b
-= -and
T/b
)
P= ) Oi
iii) R(A) is closed and R(A) = N(A)'; whenever u EH we shall write U = III + u2with uI E R(A), u2 E N(A).
PERIODIC SOLUTIONS AND HAMILTONIAN SYSTIEMS
iv) The eigenvalues of A arej2 - [(27r/T)k]2, j = 1, 2, 3, ... k = 0, 1, 2, .... The corresponding eigenfunctions are sin jx sin(2J
and sin jx cos2(
kt)
561 and
kt).
We denote by X 1(T) the first negative eigenvalue. Note that X-1(T) - 0 as T - oo. Indeed, let ,u = j2 - [(2 7r/T)k]2 with j = 1 and k = [T/2 7r]+ 1. We have 1- [1 + (2 7r/T)]2 < ,u < 0 and so
X1l(T)l
sH
?y1s4 j
1+ r
v) Givenf ER(A), there exists a unique u ER(A) n C(Q) such that
Au =1. We set i/ =Kf
-(A
')J
We have KfHLOO s CIHIIILIVf E R(A), IKf HI s CIHIfL2 Vf E R(A).
K is a compact self-adjoint operator in R(A). Step 2 Determination of To We set H(u) =
h(s)ds 0
He(u) = H(u) + 2-I
1U2
e > 0
so that He is convex and we denote by H.* its conjugate convex function (He* is C1 and (HE*)' is the inverse function of h(u) + cu). We shall use the same "duality" approach as in [5].
562
HAIM BREZIS AND JEAN-MICHEL CORON
On R(A) we define: ?+
Fe(v) = 2 tKv.v 2 1
*(v).
The following lemma plays a crucial role: LEMMA
1.
There exists To > 0 such that if T > To and T/r is ra-
tional, then Inf Fe
R(A)
Proof of Lemma 1.
-1
v c>
0.
By (4) we may assume that
H(u)2
pIuI -C
Vu
for some constants p > 0 and C. Hence He(u)
2 pIuI
-
C
Vu
and HE*(v) < C
forI v I
p.
As a testing function for evaluating InfR(A) Fe we choose an eigenfunction of A corresponding to the eigenvalue X_1(T). More precisely, let = X I(T). Thus, v = p sinjx sin[(2ir/T)kt] with j2 -[(2ir/T)k]2 2
Fe(v)
IX(T)
V12 1 +
1
7rTp2 -
81 X (T) ?CI?
provided T - To for some large To. In what follows we fix T - T(. Step 3 Existence of a nontrivial solution for Au + h(u) + cu = 0
(c > 0 small)
PERIODIC SOLUTIONS AND HAMILTONIAN SYSTEMS
563
We start with LEMMA
2.
There exists constants
a >
0 and C (independent of
e)
such that Fe(v) 2
CZ V |22 -
Proof of Lemma 2. such that
Vv ER(A), E c l/41X1 I
C
Let 6 =
1/4 IX_ I.
By (3) there is a constant C
Vu.
H(u) s 2 JU12+ C Thus
_ J + C I_lU12
s He(u)~ HIE
Vu e
and He*(v) 21
2v2-C
Vv.
On the other hand, iKvv v -
V
1v 2
VvER(A)
and the conclusion follows. It is now clear that for c s 1/4 X1 X , MinR(A)Fe is achieved at some vE. Indeed if v,1is a minimizing sequence, then by Lemma 2, v,, is bounded in L2 and we may assume that v,, converges weakly to some v in L2. Then lim | Kv,1 v,= Kv v and lim |HE*(v,,) 2 H,*(v) (by the convexity of H.*). Clearly, we have KvE + (He*)' (v,)
-
X E N(A).
Set ue
=
(He*)
(VE)
so that v. = h(u,) + sue and Au, + h(u,) + sue = 0. Note that v, 0 0 since FE(vE) s - 1.
564
HAIM BREZIS AND JEAN-MICHEL CORON
Step 4.
(c
S
Estimates
In what follows we denote by C various constants independent of c 1/4X1 1- ). By Lemma 2 we already know that |VE C. Thus L2 S
IlAuE I|L2
C and so IIU1EHLO I
S
C.
S
We shall now prove
3.
LEMMA
C
S
11UE||L??
Proof of Lemma 3. We follow the same technique as in [2]. We first prove that IIUE I ILI S C. Indeed h(u)u >H(u)>pIul
C
Vu.
Therefore
p
= 0. Finally we prove that u is a nontrivial solution. Indeed we have FE(vE)
KvE.vE+
2
HE*(vE)< -1
and in particular
2
KvE Ve
-1.
566
HAIM BREZIS AND JEAN-MICHEL CORON
On the other hand, v,= Therefore, v 0 0.
h(u,,) +
EUe,,
v and so
-
1/I
Kv*v
?
-1.
2. Nontrivial periodic solutions of Hamiltonian systems. Let - R be a Cl convex function such that H(0) = H`(0) = 0. Consider the Hamiltonian system
H:R2,,
Ji;
Jr
HI,(1) = O
ER
(S)
where
u
(pandq are n-tuples) andJJ
L
We assume lim
H(u)
-
0
(6)
lim H(ii) = oo.
(7)
1tt1-??
We seek nonconstant solutions of (5) which are T-periodic. OUr main result is the following THEOREM 2. There exists To > 0 such that for every T > To, Problem (5) possesses a solution with minimal period T.
Remark. Theorem 2 is closely related to Theorem 4.7 in [3]. In [3] there is no convexity assumption; however, they assume (7) and Hu(u)I > 0, H(u) > O IHu(u)I 5M
VuER2'1\{O} VuER2n.
Theorem 2 is also related to the main result of [6] and our technique has been inspired by the duality device of [6]. Note, however, that we make no assumption about the behavior of H near 0; while the result of [6] requires the additional assumption
PERIODIC SOLUTIONS AND HAMILTONIAN SYSTEMS
567
H(u) >0.
lim
ii7V-o lU 12 Proof of Theorem 2. The proof follows essentially the same pattern as the proof of Theorem 1 and we shall omit some details; it is somewhat simpler since dim N(A) < oo. In H = L2(0, T)2",we consider the operator Au =Ju acting on functions which are T-periodic. We summarize some properties of A: i) A* = A. ii) N(A) consists of constants. iii) R(A) is closed and R(A) = N(A)'; whenever u EH we shall write u = uI + u2 with uI ER(A), u2 EN(A). iv) The eigenvalues of A are (2 7r/T)k, k E Z, and the corresponding eigenfunctions are kt) + Ja cos( 2 kt)
u(t) = a sin(
where a E R2",is arbitrary (a ? 0). Note that X_1 = -2ir/T. v) Given f ER(A) there exists a unique u ER(A) such that Au = We set
f.
u = Kf = (A-If).
K is a compact self-adjoint operator in R(A). Given c > 0 we set
He(u) = H(u) + 22
U 12
and we denote by He*(v) its conjugate convex function. Note that He* is Cl and that (Hf*)v is the inverse mapping of (Hu + JI). On R(A) we define 1 Fe(V = 2
~TT
Kvv
2 oo
H?*(v) +
568
HAIM BREZIS AND JEAN-MICHEL CORON
We first prove: LEMMA 4.
There exists To > 0 such that if T > To, then Inf F < -1. R(A)
Proof.
From (7) and the convexity of H we deduce that H(u) 2 p uI
Vu E R2"
-C
for some constants p > 0 and C. Therefore He*(v) s C
forI v s p.
Let v =p [-a sin(2J
t)+
(Ja) cos(2j
where a E R2n is arbitrary with Ia = 1. So Kv T2p2
-
F (v) '
t)]
-(T/2 7r)vand
+ TC ' -I
provided T ' To for some large To. In what follows wefix T 2 To. Next we observe (see Lemma 2) that
Fe(v) ? aH|vH|22- C
VvER(A), Vc
where ae> 0 and C are independent of c. Therefore, MinR(A) Fe is achieved at some ve(,E 1/41 X-11)and we have Kv. + (H(*)v(v,) =
XEN(A).
Set Ue = (HI*)v(v,)
569
PERIODIC SOLUTIONS AND HAMILTFONIANSYS I EMS
so that Hu(u,) + CUE= Ve
and Au + Hu(u') + cuc = ? Clearly, II1VEHL2S C; thus IIAuEIIL2 S C and so Next we have, by the convexity of H and (7)
U1EH1]L' S
C.
Vu E R2".
Hu(u) u 2 H(u) 2 pIuI - C Therefore T
IUEI
p
-
CT
The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.
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PERIODIC SOLUTIONS OF NONLINEAR WAVE EQUATIONS AND HAMILTONIAN SYSTEMS By HAIMBREZIsand JEAN-MICHEL CORON
Abstract. We consider the nonlinear vibrating string equation u,,-uxx + h(u) = 0 under Dirichiet boundary conditions on a finite interval. We assume that h is nondecreasing, h(O) = 0 and limjuj_I[h(u)/u] = 0. We prove that for T sufficiently large, there is a nontrivial T-periodic solution. A similar result holds for Hamiltonian systems. 0. Introduction. u,,-u,,,,
Consider the following nonlinear wave equation:
+ h(u)=O
O < x < 7r,tE R.
(1)
under the boundary conditions: u(O, t) = u( 7r,t) = 0,
(2)
where h: R - R is a continuous nondecreasing function such that h(O) = 0. We assume: lim IuI-o
h(u) =0 U
(3)
There exists a constant R such that h(u) ? 0 for Iu ? R. (4) We seek nontrivial solutions of (1), (2) which are T-periodic (in t). By "nontrivial" we mean that h(u(x, t)) ? 0 on a set (x, t) of positive measure; in particular, u(x, t) ? 0 on that set. In Section 1 we prove the following THEOREM 1. There exists To > 0 such that for every T : To, with T/7r rational, Problem (1), (2) admits a nontrivial T-periodic (weak) solution u ELw.
Manuscript received May 15, 1980. AniericaniJournailof Mathemiiatics,Vol. 103, No. 3, pp. 559-570 0002-9327/81/1033-0559 $01.50 Copyright ? 1981 by The Johns Hopkins University Press
559
560
HAIM BREZIS AND JEAN-MICHEL CORON
By a result of [4], weak solutions are in fact smooth if h is smooth and strictly increasing. The existence of nontrivial solutions for (1), (2) has been considered by several authors under assumptions which differ from ours (see [1, 4, 5, 7, 8]). In Section 2 we discuss a comparable result for Hamiltonian systems. Our investigation has been stimulated by the results of [3] (Section 4). Our technique relies on a duality device used in [6] for Hamiltonian systems and subsequently in [5] for the wave equation. We thank P. Rabinowitz for helpful discussions. Proof of Theorem 1. The proof is divided into five steps. Step 1 Step 2 Step 3
Generalities about Au = ut - uxx Determination of To. Existence of a nontrivial solution for Au + h(u) + cu = 0
Step 4 Step 5
(c > 0 small).
Estimates. Passage to the limit as c - 0.
Step 1 Generalities about Au = ut - ux Since T/7r E Q we may write T = 27rb/a where a and b are coprime integers. Let H = L2(Q) with Q (0, 7r) X (0, T). In H we consider the operator Al = U,,acting on functions satisfying (2) and which are T-periodic in t. We summarize some of the main properties of A which we shall use (see e.g. [4] and the references in [4]): i) A* =A ii) N(A) consists of functions of the form N(A) = {p(t + x)-p(t-x),
wherep has period
2ir
T
a
b
-= -and
T/b
)
P= ) Oi
iii) R(A) is closed and R(A) = N(A)'; whenever u EH we shall write U = III + u2with uI E R(A), u2 E N(A).
PERIODIC SOLUTIONS AND HAMILTONIAN SYSTIEMS
iv) The eigenvalues of A arej2 - [(27r/T)k]2, j = 1, 2, 3, ... k = 0, 1, 2, .... The corresponding eigenfunctions are sin jx sin(2J
and sin jx cos2(
kt)
561 and
kt).
We denote by X 1(T) the first negative eigenvalue. Note that X-1(T) - 0 as T - oo. Indeed, let ,u = j2 - [(2 7r/T)k]2 with j = 1 and k = [T/2 7r]+ 1. We have 1- [1 + (2 7r/T)]2 < ,u < 0 and so
X1l(T)l
sH
?y1s4 j
1+ r
v) Givenf ER(A), there exists a unique u ER(A) n C(Q) such that
Au =1. We set i/ =Kf
-(A
')J
We have KfHLOO s CIHIIILIVf E R(A), IKf HI s CIHIfL2 Vf E R(A).
K is a compact self-adjoint operator in R(A). Step 2 Determination of To We set H(u) =
h(s)ds 0
He(u) = H(u) + 2-I
1U2
e > 0
so that He is convex and we denote by H.* its conjugate convex function (He* is C1 and (HE*)' is the inverse function of h(u) + cu). We shall use the same "duality" approach as in [5].
562
HAIM BREZIS AND JEAN-MICHEL CORON
On R(A) we define: ?+
Fe(v) = 2 tKv.v 2 1
*(v).
The following lemma plays a crucial role: LEMMA
1.
There exists To > 0 such that if T > To and T/r is ra-
tional, then Inf Fe
R(A)
Proof of Lemma 1.
-1
v c>
0.
By (4) we may assume that
H(u)2
pIuI -C
Vu
for some constants p > 0 and C. Hence He(u)
2 pIuI
-
C
Vu
and HE*(v) < C
forI v I
p.
As a testing function for evaluating InfR(A) Fe we choose an eigenfunction of A corresponding to the eigenvalue X_1(T). More precisely, let = X I(T). Thus, v = p sinjx sin[(2ir/T)kt] with j2 -[(2ir/T)k]2 2
Fe(v)
IX(T)
V12 1 +
1
7rTp2 -
81 X (T) ?CI?
provided T - To for some large To. In what follows we fix T - T(. Step 3 Existence of a nontrivial solution for Au + h(u) + cu = 0
(c > 0 small)
PERIODIC SOLUTIONS AND HAMILTONIAN SYSTEMS
563
We start with LEMMA
2.
There exists constants
a >
0 and C (independent of
e)
such that Fe(v) 2
CZ V |22 -
Proof of Lemma 2. such that
Vv ER(A), E c l/41X1 I
C
Let 6 =
1/4 IX_ I.
By (3) there is a constant C
Vu.
H(u) s 2 JU12+ C Thus
_ J + C I_lU12
s He(u)~ HIE
Vu e
and He*(v) 21
2v2-C
Vv.
On the other hand, iKvv v -
V
1v 2
VvER(A)
and the conclusion follows. It is now clear that for c s 1/4 X1 X , MinR(A)Fe is achieved at some vE. Indeed if v,1is a minimizing sequence, then by Lemma 2, v,, is bounded in L2 and we may assume that v,, converges weakly to some v in L2. Then lim | Kv,1 v,= Kv v and lim |HE*(v,,) 2 H,*(v) (by the convexity of H.*). Clearly, we have KvE + (He*)' (v,)
-
X E N(A).
Set ue
=
(He*)
(VE)
so that v. = h(u,) + sue and Au, + h(u,) + sue = 0. Note that v, 0 0 since FE(vE) s - 1.
564
HAIM BREZIS AND JEAN-MICHEL CORON
Step 4.
(c
S
Estimates
In what follows we denote by C various constants independent of c 1/4X1 1- ). By Lemma 2 we already know that |VE C. Thus L2 S
IlAuE I|L2
C and so IIU1EHLO I
S
C.
S
We shall now prove
3.
LEMMA
C
S
11UE||L??
Proof of Lemma 3. We follow the same technique as in [2]. We first prove that IIUE I ILI S C. Indeed h(u)u >H(u)>pIul
C
Vu.
Therefore
p
= 0. Finally we prove that u is a nontrivial solution. Indeed we have FE(vE)
KvE.vE+
2
HE*(vE)< -1
and in particular
2
KvE Ve
-1.
566
HAIM BREZIS AND JEAN-MICHEL CORON
On the other hand, v,= Therefore, v 0 0.
h(u,,) +
EUe,,
v and so
-
1/I
Kv*v
?
-1.
2. Nontrivial periodic solutions of Hamiltonian systems. Let - R be a Cl convex function such that H(0) = H`(0) = 0. Consider the Hamiltonian system
H:R2,,
Ji;
Jr
HI,(1) = O
ER
(S)
where
u
(pandq are n-tuples) andJJ
L
We assume lim
H(u)
-
0
(6)
lim H(ii) = oo.
(7)
1tt1-??
We seek nonconstant solutions of (5) which are T-periodic. OUr main result is the following THEOREM 2. There exists To > 0 such that for every T > To, Problem (5) possesses a solution with minimal period T.
Remark. Theorem 2 is closely related to Theorem 4.7 in [3]. In [3] there is no convexity assumption; however, they assume (7) and Hu(u)I > 0, H(u) > O IHu(u)I 5M
VuER2'1\{O} VuER2n.
Theorem 2 is also related to the main result of [6] and our technique has been inspired by the duality device of [6]. Note, however, that we make no assumption about the behavior of H near 0; while the result of [6] requires the additional assumption
PERIODIC SOLUTIONS AND HAMILTONIAN SYSTEMS
567
H(u) >0.
lim
ii7V-o lU 12 Proof of Theorem 2. The proof follows essentially the same pattern as the proof of Theorem 1 and we shall omit some details; it is somewhat simpler since dim N(A) < oo. In H = L2(0, T)2",we consider the operator Au =Ju acting on functions which are T-periodic. We summarize some properties of A: i) A* = A. ii) N(A) consists of constants. iii) R(A) is closed and R(A) = N(A)'; whenever u EH we shall write u = uI + u2 with uI ER(A), u2 EN(A). iv) The eigenvalues of A are (2 7r/T)k, k E Z, and the corresponding eigenfunctions are kt) + Ja cos( 2 kt)
u(t) = a sin(
where a E R2",is arbitrary (a ? 0). Note that X_1 = -2ir/T. v) Given f ER(A) there exists a unique u ER(A) such that Au = We set
f.
u = Kf = (A-If).
K is a compact self-adjoint operator in R(A). Given c > 0 we set
He(u) = H(u) + 22
U 12
and we denote by He*(v) its conjugate convex function. Note that He* is Cl and that (Hf*)v is the inverse mapping of (Hu + JI). On R(A) we define 1 Fe(V = 2
~TT
Kvv
2 oo
H?*(v) +
568
HAIM BREZIS AND JEAN-MICHEL CORON
We first prove: LEMMA 4.
There exists To > 0 such that if T > To, then Inf F < -1. R(A)
Proof.
From (7) and the convexity of H we deduce that H(u) 2 p uI
Vu E R2"
-C
for some constants p > 0 and C. Therefore He*(v) s C
forI v s p.
Let v =p [-a sin(2J
t)+
(Ja) cos(2j
where a E R2n is arbitrary with Ia = 1. So Kv T2p2
-
F (v) '
t)]
-(T/2 7r)vand
+ TC ' -I
provided T ' To for some large To. In what follows wefix T 2 To. Next we observe (see Lemma 2) that
Fe(v) ? aH|vH|22- C
VvER(A), Vc
where ae> 0 and C are independent of c. Therefore, MinR(A) Fe is achieved at some ve(,E 1/41 X-11)and we have Kv. + (H(*)v(v,) =
XEN(A).
Set Ue = (HI*)v(v,)
569
PERIODIC SOLUTIONS AND HAMILTFONIANSYS I EMS
so that Hu(u,) + CUE= Ve
and Au + Hu(u') + cuc = ? Clearly, II1VEHL2S C; thus IIAuEIIL2 S C and so Next we have, by the convexity of H and (7)
U1EH1]L' S
C.
Vu E R2".
Hu(u) u 2 H(u) 2 pIuI - C Therefore T
IUEI
p
-
CT
