VIA EKELAND VARIATIONAL PRINCIPLE
MABEL CUESTA Received 10 January 2003
We prove two minimax principles to find almost critical points ofC1functionals restricted to globally definedC1manifolds of codimension 1. The proof of the theorems relies on Ekeland variational principle.
1. Introduction
LetXbe a Banach space andΦ:X→Rof classC1. We are interested in finding critical points for the restriction ofΦto the manifoldM= {u∈X:G(u)=1}, whereG:X→Ris aC1function having 1 as a regular value. A pointu∈Mis a critical point of the restriction ofΦtoMif and only ifdΦ(u)|TuM=0 (see the definition inSection 2).
Our purpose is to prove two general minimax principles to findalmost critical pointsofΦrestricted toM. A compactness condition of (PS) type will then imply the existence of a critical point.
The applications of minimax principles in the theory of elliptic PDEs are well known and the reader is referred, for instance, to [15] for a thorough introduc- tion to the subject. For applications of minimax principles onC1manifolds, we refer, for instance, to [2,8,11,13,16].
In this paper, we present two general minimax principles, Theorems2.1and 2.6, for functionals Φ restricted toM. The first one,Theorem 2.1, is a theo- rem of “mountain-pass type” and the second one,Theorem 2.6, is a theorem of
“Ljusternik-Schnirelman type.”
A standard approach to prove such results is to first derive a deformation lemma on the manifold M. In the case of Theorem 2.6, one would ask fur- thermore the deformation to be symmetric, that is, equivariant under the ac- tion of the groupZ2. Classically the deformation homotopy is constructed with the help of integral lines of a pseudogradient vector field ofΦonM. Since the construction of the integral lines requires the vector field to be locally Lipschitz
Copyright©2003 Hindawi Publishing Corporation Abstract and Applied Analysis 2003:13 (2003) 757–768 2000 Mathematics Subject Classification: 58E05, 58E30, 35J20 URL:http://dx.doi.org/10.1155/S1085337503303100
758 Minimax theorems onC manifolds
continuous, it seems necessary to assume thatM is at leastC1,1. Deformation lemmas and their equivariant versions forC1,1 manifolds are well-established results and we refer to [15].
However, in some applications the manifold M is merely of class C1 and then one has to construct the deformation more carefully. As a matter of fact, several deformation lemmas onC1 manifolds have already been proved by [3, 4,7,13] and, precisely, the deformation lemma of [7] could be used to prove Theorem 2.1. According to our knowledge, the only symmetric version of the deformation lemma onC1manifolds has been proved by [3,4]. The equivariant deformation lemmas of [13,15] are stated for manifolds of classC1,1 and the symmetric deformation lemma of [7] is stated on Banach spaces. These defor- mation theorems do not seem to apply directly in the proof ofTheorem 2.6or Proposition 2.7.
The main novelty of this paper is that we present a proof that relies mainly on the variational principle of Ekelandwithout any use of a deformation lemma. The only cost of this approach is that we need to assume the spaceXto beuniformly convex. This is not however a restriction for the applications that we have in mind whereX=W01,p(Ω) orLp(Ω) for 1< p <∞andΩis an open set ofRn.
This approach via Ekeland principle to prove a minimax principle similar to Theorem 2.1has already been used by [5,9,14] in the case of no constraint, that is, whenM=X. Our proof follows the general lines of [9]. Our approach also seems to be new in proving the analogue ofTheorem 2.6in the case of no constraint or for the proofs of Theorems2.1and2.6in the case of more regular manifolds.
2. Statement of the theorems
LetXbe a Banach space with norm · X,X∗a dual space, and·,·a dual- ity pairing betweenX∗andX. We will assume throughout this paper thatXis uniformly convex (see [10]).
LetG:X→Rbe given and assume thatG∈C1(X,R) and 1 is a regular value ofG. We consider theC1manifoldMdef= {u∈X:G(u)=1}.ForΦ∈C1(X,R), the norm of the derivative atu∈M of the restriction ˜ΦofΦtoM is defined asΦ˜(u)∗def
= dΦ(u)(TuM)∗, whereTuM= {v∈X:dG(u), v =0}denotes the tangent space toMatuand · (TuM)∗ denotes the norm on the dual space (TuM)∗.
In what followsK is a given compact metric space andK0⊂K is a closed subset.
Theorem 2.1. Let Φ∈C1(X,R)and let h0∈C(K0, M)be fixed. Consider the familyΓ= {h∈C(K, M) :h|K0=h0}and assume thatΓ= ∅. Assume further that the following condition holds:
maxz∈K0Φh0(z)<max
z∈K Φh(z) (2.1)
for allh∈Γ. Definecdef=infh∈Γmaxz∈KΦ(h(z)). Let>0andh∈Γbe such that maxz∈KΦ(h(z))< c+2. Then there existsu∈Msuch that
c≤Φ(u)≤c+2, distu, h(K)≤, Φ˜(u)∗≤. (2.2) The following propositions follow directly fromTheorem 2.1. We recall that Φis said to satisfy the (PS)condition onM at levelc((PS)c,M for short) if any sequenceun∈M, such that limn→∞Φ(un)=cand limn→∞Φ˜(un)∗=0, pos- sesses a convergent subsequence.
Proposition2.2. Let Φ,Γ, andcbe as inTheorem 2.1and assume that (2.1) holds. IfΦsatisfies(PS)c,M, then there existsu∈Msuch thatΦ(u)=candΦ˜(u)
=0.
Proposition2.3. Let Φ,Γ, andcbe as inTheorem 2.1and assume that (2.1) holds. Assume further that there exists a pathh∈Γsuch thatmaxz∈KΦ(h(z))=c.
Then there existsu∈h(K)such thatΦ(u)=candΦ˜(u)=0.
Remark 2.4. Theorem 2.1with the stronger condition maxz∈K0Φh0(z)<inf
h∈Γmax
z∈K Φh(z) (2.3)
instead of condition (2.1) has been proved by [13, Lemma 3.7 and Theorem 3.2]
using a deformation lemma.
Remark 2.5. The result ofProposition 2.3was already observed by [5] in the case of no constraint and it can also be proved using a deformation argument. No- tice that the (PS)c,M condition is not required inProposition 2.3to get a critical point.
Next we state a second minimax principle that will give almost critical points ofΦrestricted toMwhen we minimize along continuous odd maps defined on spheres of finite dimension. To that effect, we assume that the mapGiseven, so in particular,−M=M.
For anyk∈N, we denote bySkthe unit sphere ofRk+1. We also denote Co
Sk, M:=
h∈CSk, M:his odd. (2.4) Theorem2.6. LetΦ∈C1(X,R)be an even function. We define
ddef= inf
h∈Co(Sk,M)max
z∈SkΦh(z) (2.5)
760 Minimax theorems onC manifolds
and assume thatd∈R. Let>0andh∈Co(Sk, M)be such that
maxz∈SkΦh(z)< d+2. (2.6) Then there existsu∈Msuch that
d≤Φ(u)≤d+2, distu, hSk≤, Φ˜(u)∗≤. (2.7) As a consequence of the theorem, we have the following results.
Proposition2.7. LetΦanddbe as inTheorem 2.6. IfΦsatisfies(PS)d,M, then there existsu∈Msuch thatΦ(u)=dandΦ˜(u)=0.
Proposition2.8. LetΦanddbe as inTheorem 2.6and assume that there exists a pathh∈Co(Sk, M)such thatmaxz∈SkΦ(h(z))=d. Then there existsu∈h(Sk) such thatΦ(u)=dandΦ˜(u)=0.
3. Proof ofTheorem 2.1
Before starting the proof ofTheorem 2.1, we will give a result concerning the existence ofC1paths inC(K, M) with a prescribed derivative.
In the sequel we will consider the complete metric spacesC(K, X) andC(K,R) endowed with the supremum norms · X,∞and · ∞, respectively. The space C(K, M) will be inherited with the norm ofC(K, X).
Lemma3.1. Let f ∈C(K, M)and letq∈C(K, X)be such thatq(z)∈Tf(z)Mfor allz∈K. Then there existr0>0andγ∈C1((−r0, r0), C(K, M))such that
γ(0)=f ,
∀r∈
−r0, r0
,∀z∈K, γ(r)(z)= f(z)iffq(z)=0orr=0, γ(0)=q.
(3.1)
Proof. SinceX is uniformly convex, the duality mapJ:X∗→Xdefined byx, J(x) = x2X∗andJ(x) = xX∗is well defined and uniformly continuous on bounded sets (see [10]). For eachu∈M, we define
ᏺ(u)def= JdG(u) dG(u)2X∗
. (3.2)
ThusdG(u),ᏺ(u) =1. We denote
n(z)=ᏺf(z) (3.3)
for eachz∈K. We decompose f(z) as follows: f(z)=v0(z) +t0(z)n(z), where t0(z)=
dGf(z), f(z) , v0(z)= f(z)−
dGf(z), f(z) n(z). (3.4)
Thusv0(·)∈Tf(·)M and it is clear from the definitions thatv0∈C(K, X),n∈ C(K, X), andt0∈C(K,R).
We consider the mapF:C(K, X)×C(K,R)→C(K,R) defined byF(v, t)= G(v+tn). Using thatG∈C1and the uniform continuity on compact sets ofG anddG, one can prove thatFis of classC1. Furthermore,
∂F
∂t(v, t)=
dG(v+tn), n Id, ∂F
∂v(v, t)=dG(v+tn), (3.5) and consequently (∂F/∂t)(v0, t0)=Idis an invertible map. By the implicit func- tion theorem (see, e.g., [1]), there exist two open setsᐂ,ᐁsuch thatv0∈ᐂ⊂ C(K, X),t0∈ᐁ⊂C(K,R) and there exists aC1mapφ:ᐂ→ᐁsuch that
φv0
=t0, Fv, φ(v)=1,
F(v, t)=1, (v, t)∈ᐂ×ᐁ=⇒t=φ(v), (3.6) where1denotes the constant function 1. We now takeqsatisfying the conditions of the lemma and letr0>0 be such that v0+rq∈ᐂfor allr∈(−r0, r0). We define theC1pathγ: (−r0, r0)→C(K, X) as follows:
γ(r)=v0+rq+φv0+rqn. (3.7) Using that1=F(v0+rq, φ(v0+rq))=G(v0+rq+φ(v0+rq)n), it follows that γ(r)(z)∈Mfor allz∈K, that is,γ∈C1((−r0, r0), C(K, M)). It also follows from the definition ofγthat
γ(0)=v0+φv0
n=v0+t0n= f . (3.8) Moreoverγ(r)(z)=f(z) for somez∈Kand somer=0 if and only if
rq(z) +φv0+rq(z)n(z)=t0(z)n(z). (3.9) ApplyingdG(f(z)) to the above identity and using thatdG(f), n =1, we find φ(v0+rq)(z)=t0(z) and thenrq(z)=0.
Finally we differentiate with respect tovthe second identity of (3.6) atv=v0. We finddφ(v0)= −dG(f) and hence
γ(0)=q+dφv0
, q n=q (3.10)
and the proof is complete.
Proof ofTheorem 2.1. We introduce a functionalΘ:C(K,R)→Rdefined byΘ(x)
def=maxz∈Kx(z) and a functionalΨ:Γ→Rdefined byΨ(f)def=Θ(Φ◦f). The familyΓis a complete metric space with the norm inherited fromC(K, X) andΨ is continuous. (To show thatΨis continuous, one uses the uniform continuity ofΦon f(K).) ObviouslyΨis bounded from below and inff∈ΓΨ(f)=c. By the
762 Minimax theorems onC manifolds
Ekeland variational principle [12], there exists f ∈Γsuch that (a)c≤Ψ(f)≤Ψ(h),
(b)f−hX,∞≤,
(c)Ψ(f)<Ψ(g) +f−gX,∞for allg= f,g∈Γ.
Our theorem will be proved if we show the existence of somez∈Ksuch that Φf(z)=Ψ(f), Φ˜f(z)∗≤. (3.11) This will follow from five different claims.
For a given u∈M, we denote byPu the map from X to TuM defined as Pu(v)def=v− dG(u), vn(u).
Claim 3.2. LetΛ0def
= {l∈C(K, X) :l|K0≡0}. Then sup
l∈Λ0
min
µ∈∂Θ(Φ◦f)
µ,dΦ(f), Pf(l) ≤Pf(l)X,∞, (3.12)
where∂Θ(x) stands for the subdifferential ofΘatx(see, e.g., [9]).
Proof ofClaim 3.2. For simplicity we write x=Φ◦f. We fix l∈Λ0. We can assume that Pf(l)=0, otherwise there is nothing to prove. Consider the C1 pathγ: (−r0, r0)→C(K, M) given byLemma 3.1such thatγ(0)= f andγ(0)= Pf(l). SincePf(l)≡0 inK0, thenγ(r)(z)=f(z) for allz∈K0, and consequently γ(r)∈Γfor all r∈(−r0, r0). Moreover, since Pf(l)=0, then γ(r)= f for all r=0. It follows from (c) withg=γ(r), 0< r < r0, that
Θ(x)−ΘΦγ(r)
r ≤
1
rf−γ(r)X,∞. (3.13) We compute the limit asr→0 of both sides of inequality (3.13). The term on the left-hand side can be written as
Θ(x)−Θ(x+r y)
r +Θ(x+r y)−ΘΦγ(r)
r , (3.14)
where for the sake of simplicity we have denotedy= dΦ(f), Pf(l). Using [9, Proposition 5.4], the first term of (3.14) goes to−maxµ∈∂Θ(x)µ, yasr0. The second term of (3.14) goes to 0 becauseΘis Lipschitz continuous and the limit
limr0
Φγ(r)−Φ(f)
r =
dΦ(f), γ(0) =y (3.15) holds uniformly inK.
The limit as r0 of the right-hand side of (3.13) gives γ(0)X,∞= Pf(l)X,∞. Putting all together and passing to the limit in (3.13), we have
− max
µ∈∂Θ(x)µ, y ≤Pf(l)X,∞. (3.16)
The claim follows by replacinglby−land taking the supremum over alll∈
Λ0.
Claim 3.3. LetB= {l∈C(K, X) :Pf(l)X,∞≤1}. Then min
µ∈∂Θ(x) sup
l∈B∩Λ0
µ,dΦ(f), Pf(l) ≤. (3.17)
Proof ofClaim 3.3. It is clear fromClaim 3.2that sup
l∈B∩Λ0
min
µ∈∂Θ(x)
µ,dΦ(f), Pf(l) ≤. (3.18)
We can interchange the sup and the min above because of Ky Fan-von Neumann minimax theorem (see [6]). Indeed, the mapᏲ:ᏹ(K,R)×C(K, X)→Rde- fined byᏲ(µ, l)= µ,dΦ(f), Pf(l)is bilinear and continuous. Hereᏹ(K,R) is the set of Borel measures endowed with theω∗-topology. Moreover the set
∂Θ(x) is a compact convex andB∩Λ0is convex.Claim 3.3is proved.
Claim 3.4. It holds that min
µ∈∂Θ(x)sup
l∈B
µ,dΦ(f), Pf(l) ≤. (3.19)
Proof ofClaim 3.4. We recall (see [9, Proposition 5.6]) that
∂Θ(x)=
µ∈ᏹ(K,R) :µ≥0,µ,1 =1,suppµ⊂K1
, (3.20)
whereK1def
= {z∈K:Φ(f(z))=Θ(x)}. By (2.1)K1andK0are disjoint. Then we can find a continuous mapϕ:K→[0,1] such thatϕ≡1 onK1andϕ≡0 onK0. Given anyl∈Bconsiderl1=ϕl. Thenl1∈Λ0 and by linearityPf(l1)X,∞= ϕPf(l)X,∞≤1. Thusl1∈B∩Λ0. Moreover, since suppµ⊂K1, we have
µ,dΦ(f), Pf(l) =
µ,dΦ(f), Pf
l1
(3.21)
and the claim follows.
Claim 3.5. It holds that
µ∈min∂Θ(x)
µ,sup
l∈B
dΦ(f), Pf(l) ≤. (3.22)
Proof ofClaim 3.5. Letδ >0 andz0∈K. Then there existlz0∈Band an open neighborhoodᐁz0ofz0such that for allz∈ᐁz0,
sup
l∈B
dΦfz0
, Pf(z)l(z) −δ≤
dΦf(z), Pf(z)lz0(z) . (3.23) By compactness we can coverK with a finite subcoveringᐁz1···ᐁzn. Letϕi, i=1, . . . , n, be a continuous partition of unity subordinate toᐁzi, that is,ϕiis
764 Minimax theorems onC manifolds
continuous, 0≤ϕi≤1, with support onᐁzi andni=1ϕi=1 onK. We consider the functionl1=n
i=1ϕilzi. Writing (3.23) forz0=ziand adding fromi=1 to n, we have
sup
l∈B
dΦ(f), Pf(l) −δ1≤
dΦ(f), Pfl1 . (3.24) Composing withµand using thatµ,1 =1 andµ≥0, we get
µ,sup
l∈B
dΦ(f), Pf(l) −δ≤
µ,Φ(f), Pf
l1
. (3.25)
Now observe thatl1∈Band consequently the right-hand side of the above in- equality is less than or equal to supl∈Bµ,dΦ(f), Pf(l). Lettingδ0, we ob- tain
µ,sup
l∈B
dΦ(f), Pf(l) ≤sup
l∈B
µ,dΦ(f), Pf(l) . (3.26)
The result now follows by taking the minimum over allµ∈∂Θ(X) and using
Claim 3.4.
Claim 3.6. We have
µ∈min∂Θ(x)
µ,Φ˜f(·)∗ ≤. (3.27)
Proof ofClaim 3.6. The proof of this claim follows easily fromClaim 3.5and the identity supl∈BdΦ(f), Pf(l) = Φ˜(f)∗. Now letµ∈∂Θ(x) realize the minimum ofClaim 3.6. Sinceµhas mass equal to 1 and is supported inK1, there existsz∈K1such that
Φ˜f(z)∗≤. (3.28)
Then (3.11) hold foru=f(z) and the proof of the theorem is complete.
4. Proof ofTheorem 2.6
The proof ofTheorem 2.6goes along the same lines as the proof ofTheorem 2.1.
We indicate the necessary modifications.
First we give the symmetric version ofLemma 3.1. We will denote byCo(Sk,R) the subspace of odd functions ofC(Sk,R) and byCe(Sk, X) andCe(Sk,R) the sub- spaces of even functions ofC(Sk, X) andC(Sk,R), respectively.
Lemma4.1. Assume thatG is even. Let f ∈Co(Sk, M)and letq∈Co(Sk, X)be such thatq(z)∈Tf(z)Mfor allz∈Sk. Then there existr0>0andγ∈C1((−r0, r0), Co(Sk, M))such that (3.1) is satisfied.
Proof. It is easy to see that the duality mapJ:X∗→X and the mapᏺ:M→ Rare odd. Consequently,t0∈Ce(Sk,R), v0∈Co(Sk, X), and n∈Co(Sk,R) as defined in (3.3) and (3.4).
We consider the mapF:Co(Sk, X)×Ce(Sk,R)→Ce(Sk,R) defined byF(v, t)=G(v+tn). Observe thatFis well defined because
Gv(z) +t(z)n(z)=G−v(−z)−t(−z)n(−z)
=Gv(−z) +t(−z)n(−z) (4.1) for allz∈K,v∈Co(Sk, X), andt∈Ce(Sk,R). By the implicit function theorem applied toF, there exist two open setsᐂ,ᐁsuch thatv0∈ᐂ⊂Co(Sk, X),t0∈ ᐁ⊂Ce(Sk,R) and there exists aC1mapφ:ᐂ→ᐁsuch that (3.6) is satisfied.
It is then clear that the mapγdefined by (3.7) satisfied (3.1) and also thatγ∈
C1((−r0, r0), Co(Sk, M)).
Proof ofTheorem 2.6. We consider the functionalΨ:Co(Sk, M)→Rdefined by Ψ(f)def=Θ(Φ◦f). The functionalΨ is continuous and bounded from below with inff∈Co(Sk,M)Ψ(f)=c. By the Ekeland variational principle, there exists f ∈ Co(Sk, M) satisfying (a), (b), and (c). We proceed now to prove four claims to show that there exists somez∈Sksatisfying (3.11).
Observe that if f ∈Co(Sk, M) andl∈Co(Sk, X), thenPf(l)∈Co(Sk, X).
Claim 4.2. We have sup
l∈Co(Sk,X)
min
µ∈∂Θ(Φ◦f)
µ,dΦ(f), Pf(l) ≤Pf(l)X,∞. (4.2)
Proof ofClaim 4.2. The same proof as in Theorem 2.1 applies since we can choose the path γ of Lemma 4.1 with γ(0)= f and γ(0)=Pf(l). Then γ∈
C1((−r0, r0), Co(Sk, M)).
Claim 4.3. LetB= {l∈C(Sk, X) :Pf(l)X,∞≤1}. Then min
µ∈∂Θ(x) sup
l∈B∩Co(Sk,X)
µ,dΦ(f), Pf(l) ≤. (4.3)
Proof ofClaim 4.3. The proof is the same as its analogue inTheorem 2.1. Notice that in this case,B∩Co(Sk, X) is convex as well.
Claim 4.4. It holds that
µ∈min∂Θ(x)
µ, sup
l∈B∩Co(Sk,X)
dΦ(f), Pf(l)
≤. (4.4)
Proof ofClaim 4.4. We proceed as inClaim 3.5. The only problem is to find a functionl1inB∩Co(Sk, X) satisfying (3.21). For anyδ >0 and anyz0∈Sk, let lz0∈B∩Co(Sk, X) and letᐂz0be an open neighborhood inSksuch that (3.23)
766 Minimax theorems onC manifolds
is satisfied for allz∈ᐂz0. We can assume thatᐂz0= −ᐂ−z0andᐂz0∩ᐂ−z0= ∅ by taking, for instance, ᐂz0= {z∈Sk:z−z0< ρ}withρ small. Let ᐁz0= ᐂz0∪ᐂ−z0. Henceᐁz0= −ᐁz0. For anyz∈ᐁz0, we have that eitherz∈ᐂz0
from which we deduce that sup
l∈B∩Co(Sk,X)
dΦfz0
, Pf(z)l(z) −δ≤
dΦf(z), Pf(z)lz0(z) (4.5) orz∈ᐂ−z0in which case we have
sup
l∈B∩Co(Sk,X)
dΦf−z0
, Pf(z)l(z) −δ≤dΦf(z), Pf(z)l−z0(z) . (4.6) WritingdΦ(f(−z0)), Pf(z)(l(z)) = dΦ(f(z0)), Pf(z)(−l(z)), we see that (4.6) is equivalent to
sup
l∈B∩Co(Sk,X)
dΦfz0
, Pf(z)l(z) −δ≤
dΦf(z), Pf(z)l−z0(z) . (4.7) From (4.5) and (4.7), we find that for allz∈ᐁz0,
sup
l∈B∩Co(Sk,X)
dΦfz0
, Pf(z)
l(z) −δ≤
dΦf(z), Pf(z)
lz0 , (4.8)
wherelz0(z) :=(lz0(z) +l−z0(z))/2 belongs toB∩Co(Sk, X).
We now cover Sk with a finite subcoveringᐁz1···ᐁzn. Let ϕi, i=1, . . . , n, be a continuous partition of unity subordinate toᐁzi and consider the even part ofϕi,ϕei(z)=1/2(ϕi(z) +ϕi(−z)). Clearly, 0≤ϕei ≤1, suppϕei ⊂ᐁzi, and n
i=1ϕei =1. We finally consider the odd functionl1=n
i=1ϕeilzi and observe thatl1∈B. The remaining part of the proof is similar to that ofClaim 3.5.
Claim 4.5. It follows that
µ∈min∂Θ(x)
µ,Φ˜f(·)∗ ≤. (4.9)
Proof ofClaim 4.5. We show that sup
l∈B∩Co(Sk,X)
dΦ(f), Pf(l) =Φ˜(f)∗. (4.10)
The inequality≤is clear becausePf(l(·))∈Tf(·)MandPf(l)X,∞≤1. The in- equality≥comes from the following. Fixz∈Skand letδ >0. Then there exists v∈Tf(z)M,vX≤1, such that
dΦf(z), v ≥Φ˜f(z)∗−δ. (4.11)
Takel∈Co(Sk, X) such thatl(z)=vandl∞≤1 (for instance, ifz=(1,0, . . . ,0), one can takel(x1, x2, . . . , xk+1)=x1v). Hence
Φ˜f(z)∗−δ≤ sup
l∈B∩Co(Sk,X)
dΦ(f), Pf(l) . (4.12)
Lettingδgo to 0, we obtain the desired inequality.Claim 4.5is proved.
The remaining part of the proof of the theorem is identical to its correspond-
ing part inTheorem 2.1.
Acknowledgment
We wish to thank J. Campos, J.-P. Gossez, and E. Lami-Dozo for their useful suggestions.
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Mabel Cuesta: Universit´e du Littoral C ˆote d’Opale (ULCO), 50 rue F. Buisson, BP 699, 62228 Calais Cedex, France
E-mail address:[email protected]
Special Issue on
Time-Dependent Billiards
Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi accelera- tion (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.
This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System athttp://
mts.hindawi.com/according to the following timetable:
Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009
Guest Editors
Edson Denis Leonel,Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil ; [email protected]
Alexander Loskutov,Physics Faculty, Moscow State University, Vorob’evy Gory, Moscow 119992, Russia;
Hindawi Publishing Corporation http://www.hindawi.com
