FOR VARIATIONAL PROBLEMS ON A TORUS

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FOR VARIATIONAL PROBLEMS ON A TORUS

ALEXANDER J. ZASLAVSKI Received 17 July 2001

We study minimal solutions for one-dimensional variational problems on a torus. We show that, for a generic integrand and any rational numberα, there exists a unique (up to translations) periodic minimal solution with rotation num- berα.

1. Introduction

In this paper, we consider functionals of the form I^f(a, b, x)₌

_b

a

ft, x(t), x(t)dt, (1.1)

whereaand bare arbitrary real numbers satisfyinga < b,x∈ W^1,1(a, b) and f belongs to a space of functions described below. By an appropriate choice of representatives,W^1,1(a, b) can be identified with the set of absolutely continuous functionsx: [a, b]→R¹, and henceforth we will assume that this has been done.

Denote byMthe set of integrands f ₌ f(t, x, p) :R³→R¹which satisfy the following assumptions:

(A1) f ∈C³and f(t, x, p) has period 1 int, x;

(A2)δf ≤ fpp(t, x, p)≤δ⁻¹_f for every (t, x, p)∈R³; (A3)|f_{x p}_|+|f_{t p}_{| ≤}c_f(1 +|p_|),|f_xx_|+|f_xt_{| ≤}c_f(1 +p²), with some constantsδ_f _∈(0,1),c_f >0.

Clearly, these assumptions imply that

δ˜fp²−c˜f ≤f(t, x, p)≤δ˜⁻¹_f p²+ ˜cf (1.2) for every (t, x, p)_∈R³for some constants ˜c_f >0 and 0<δ˜_f < δ_f.

In this paper, we analyse extremals of variational problems with integrands f ∈ M. The following optimality criterion was introduced by Aubry and Le

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Daeron [2] in their study of the discrete Frenkel-Kontorova model related to dislocations in one-dimensional crystals.

Let f _∈M. A functionx(_·)∈W_loc^1,1(R¹) is called an (f)-minimal solution if I^f(a, b, y)≥I^f(a, b, x) (1.3) for each pair of numbersa < band eachy_∈W^1,1(a, b) which satisfiesy(a)₌x(a) andy(b)=x(b) (see [2,9,10,12]).

Our work follows Moser [9,10], who studied the existence and structure of minimal solutions in the spirit of Aubry-Mather theory [2,7].

Consider any f ∈M. It was shown in [9,10] that (f)-minimal solutions possess numerous remarkable properties. Thus, for every (f)-minimal solution x(_·), there is a real numberαsatisfying

supx(t)−αt:t∈R¹

<∞ (1.4)

which is called the rotation number ofx(·), and given any realαthere exists an (f)-minimal solution with rotation numberα. Senn [11] established the existence of a strictly convex functionE_f :R¹ →R¹, which is called the minimal average action of f such that, for each realαand each (f)-minimal solutionx with rotation numberα,

T2−T1

₋₁

I^fT1, T2, x−→E_f(α) asT2−T1−→ ∞. (1.5) This result is an analogue of Mather’s theorem about the average energy function for Aubry-Mather sets generated by a diﬀeomorphism of the infinite cylinder [8].

In this paper, we show that for a generic integrand f and any rationalα, there exists a unique (up to translations) (f)-minimal periodic solution with rotation numberα.

Letk_≥3 be an integer. SetMk=M∩C^k(R³). Forf _∈Mkandq₌(q1, q2, q3)∈ {0, . . . , k}³satisfyingq1+q2+q3≤k, we set

|q_|₌q1+q2+q3, D^qf ₌ ∂^|q|f

∂t^q¹∂x^q²∂p^q³. (1.6) ForN,>0 we set

E_k(N,)₌(f , g)_∈Mk×Mk:D^qf(t, x, p)₋D^qg(t, x, p)

≤+maxD^qf(t, x, p),D^qg(t, x, p)

∀q∈ {0,1,2}³satisfying|q| ∈ {0,2},∀(t, x, p)∈R³

∩

(f , g)_∈Mk×Mk:D^qf(t, x, p)₋D^qg(t, x, p)_≤

∀q_{∈ {}0, . . . , k_}³satisfying|q_{| ≤}k,_∀(t, x, p)_∈R³ such that|p| ≤N.

(1.7)

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It is easy to verify that, for the setMk there exists a uniformity which is deter- mined by the baseEk(N,),N,>0, and that the uniform spaceMkis metriz- able and complete [3]. We establish the existence of a setᏲk⊂Mk which is a countable intersection of open everywhere dense subsets ofMk such that, for each f ∈Ᏺkand each rationalα∈R¹, there exists a unique (up to translations) (f)-minimal periodic solultion with rotation numberα.

2. Properties of minimal solutions

Consider any f ∈M. We note that, for each pair of integers jandkthe translations (t, x)→(t+j, x+k) leave the variational problem invariant. Therefore, ifx(_·) is an (f)-minimal solution, so isx(_·+j) +k. Of course, on the torus, this represents the same curve as doesx(·). This motivates the following terminology [9,10].

We say that a functionx(_·)_∈W_loc^1,1(R¹) has no self-intersections if for all pairs of integers j, kthe functiont →x(t+j) +k−x(t) is either always positive, or always negative, or identically zero.

Denote byZthe set of all integers. We have the following result (see [6, Propo- sition 3.2] and [9,10]).

Proposition2.1. (i)Let f ∈ M. Given any realα there exists a nonself-inter- secting(f)-minimal solution with rotation numberα.

(ii) For anyf _∈Mand any(f)-minimal solutionx, there is the rotation num- ber ofx.

For each f ∈M, each rational numberα, and each natural numberqsatisfy- ingqα∈Z, we define

ᏺ(α, q)=

x(·)∈W_loc^1,1R¹

:x(t+q)=x(t) +αq, t∈R¹ ,

ᏹf(α, q)₌x(_·)_∈ᏺ(α, q) :I^f(0, q, x)_≤I^f(0, q, y)_∀y_∈ᏺ(α, q). (2.1) We have the following result [9, Theorems 5.1, 5.2, 5.4, and Corollaries 5.3 and 5.5].

Proposition 2.2. Let f ∈ M, letαbe a rational number, and let p, q ≥1 be integers satisfying pα, qα∈Z. Thenᏹf(α, q)=ᏹf(α, p)=∅, eachx∈ᏹf(α, q) is a nonself-intersecting(f)-minimal solution with rotation numberαand the set ᏹf(α, q)is totally ordered, that is, ifx, y∈ᏹf(α, q), then eitherx(t)< y(t)for all t, orx(t)> y(t)for allt, orx(t)=y(t)identically.

For any f ∈Mand any rational numberαwe setᏹ^per_f (α)=ᏹf(α, q), where qis a natural number satisfyingqα_∈Z.

We have the following result (see [6, Theorem 1.1]).

Proposition2.3. Let f ∈M. Then there exist a strictly convex functionEf :R¹→ R¹ satisfying E_f(α) _{→ ∞}as |α_{| → ∞}and a monotonically increasing function Γf : (0,_∞)_→[0,_∞)such that for each realα, each(f)-minimal solutionxwith

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rotation numberαand each pair of real numbersSandT, I^f(S, S+T, x)₋E_f(α)T_≤Γf

|α_|. (2.2)

ByProposition 2.3for each f _∈Mthere exists a unique numberα(f) such that

E_fα(f)=minE_f(β) :β∈R¹

. (2.3)

Note that assumptions (A1), (A2), and (A3) play an important role in the proofs of Propositions2.1,2.2, and2.3(see [9,10]).

3. The main results

Theorem3.1. Letk _≥3be an integer andαbe a rational number. Then there exists a setᏲkα⊂Mkwhich is a countable intersection of open everywhere dense subsets ofMksuch that for each f _∈Mkthe following assertions hold:

(1)Ifx, y_∈ᏹ^(per)_f (α), then there are integers p,qsuch that y(t)₌x(t+p)₋q for allt∈R¹.

(2)Letx∈ᏹ^(per)_f (α)and>0. Then there exists a neighborhoodᐁof f inMk

such that for eachg∈ᐁand eachy∈ᏹ^(per)g (α)there are integersp,qsuch that

|y(t)−x(t+p) +q| ≤for allt∈R¹.

It is not diﬃcult to see thatTheorem 3.1implies the following result.

Theorem3.2. Letk≥3be an integer. Then there exists a setᏲk⊂Mkwhich is a countable intersection of open everywhere dense subsets ofMksuch that, for each f _∈Mkand each rational numberαthe assertions (1) and (2) ofTheorem 3.1hold.

Note that minimal solutions with irrational rotation numbers were studied in [2,7,9,10,12].

4. An auxiliary result

Letk_≥3 be an integer andβ_∈R¹. For each f _∈Mk, defineᏭf _∈C³(R³) by (Ꮽf)(t, x, u)₌ f(t, x, u)₋βu, (t, x, u)_∈R³. (4.1) ClearlyᏭf _∈Mkfor each f _∈Mk.

Proposition4.1. The mappingᏭ:Mk→Mkis continuous.

Proof. Let f _∈Mkand letN,>0. In order to prove the proposition, it is suﬃ- cient to show that there exists0∈(0,) such that

Ꮽg∈Mk: (f , g)∈E_kN,0

⊂

h∈Mk: (h,Ꮽf)∈E_k(N,). (4.2) Set

∆0=2_|β_|+ 1. (4.3)

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Equation (1.2) implies that there existsc0>0 such that

∆0|u_|−c0≤ f(t, x, u) _∀(t, x, u)_∈R³. (4.4) Choose a number0such that

0<0<min{1,}, 40+ 40

1−0

₋₁

(4 +c0)<. (4.5) It follows from (4.3) and (4.4) that for each (t, x, u)_∈R³,

f(t, x, u)−βu≥f(t, x, u)−|βu| ≥f(t, x, u)−|β|∆⁻¹₀ f(t, x, u) +c0

≥f(t, x, u)1_−|β_|∆⁻¹0

−|β_|∆⁻¹0 c0

≥2⁻¹f(t, x, u)₋2⁻¹c0.

(4.6)

Assume that

g∈Mk, (f , g)∈E_kN,0

. (4.7)

By (1.7) and (4.7) for each (t, x, u)_∈R³,

f(t, x, u)−g(t, x, u)≤0+0maxf(t, x, u),g(t, x, u), maxf(t, x, u),g(t, x, u)₋minf(t, x, u),g(t, x, u)

≤0+0maxf(t, x, u),g(t, x, u), 1−0

maxf(t, x, u),g(t, x, u)≤minf(t, x, u),g(t, x, u)+0, g(t, x, u)_≤1−0−1f(t, x, u)+1−0−10.

(4.8)

We show that (Ꮽf ,Ꮽg)∈Ek(N,). It follows from (1.7), (4.1), (4.5), and (4.7) that, for eachq₌(q1, q2, q3)∈ {0, . . . , k_}³satisfying|q_{| ≤}kand each (t, x, p)_∈R³ satisfying|p| ≤N,

D^q(Ꮽf)(t, x, p)₋D^q(Ꮽg)(t, x, p)₌D^qf(t, x, p)₋D^qg(t, x, p)_≤0<. (4.9) Letq∈ {0,1,2}³,|q| ∈ {0,2}, and (t, x, p)∈R³. Equation (4.1) implies that

D^q(Ꮽf)(t, x, p)−D^q(Ꮽg)(t, x, p)=D^qf(t, x, p)−D^qg(t, x, p). (4.10) If|q_|₌2, then by (1.7), (4.1), (4.5), (4.7), and (4.10),

D^q(Ꮽf)(t, x, p)₋D^q(Ꮽg)(t, x, p)

≤0+0maxD^qf(t, x, p),D^qg(t, x, p)

<+maxD^q(Ꮽf)(t, x, p),D^q(Ꮽg)(t, x, p).

(4.11)

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Assume thatq₌0. By (1.7), (4.1), (4.5), (4.6), (4.7), and (4.8), D^q(Ꮽf)(t, x, p)−D^q(Ꮽg)(t, x, p)

=f(t, x, p)₋g(t, x, p)_≤0+0maxf(t, x, p),g(t, x, p)

≤0+0maxf(t, x, p),1−0−1f(t, x, p)+1−0−10

=0+0

1−0−1f(t, x, p)+²0

1−0−1

≤0+²₀1₋0−1+0

1₋0−1

2f(t, x, p)₋βp+ 2c0

≤0+²0

1−0−1

+ 20

1−0−1c0+ 20

1−0−1f(t, x, p)₋βp

≤20(1−0)⁻¹(Ꮽf)(t, x, p)+≤+(Ꮽf)(t, x, p).

(4.12)

Equations (4.9), (4.11), and (4.12) imply that (Ꮽf ,Ꮽg)_∈E_k(N,).Proposition

4.1is proved.

Let−∞< T1< T2<∞andx∈W^1,1(T1, T2). By (4.1) we have I^Ꮽ^fT1, T2, x=

_T₂

T1

ft, x(t), x(t)−βx(t)dt

=I^fT1, T2, x−βxT2

+βxT1 .

(4.13)

Therefore, eachx∈W_loc^1,1(R¹) is an (Ꮽf)-minimal solution if and only ifx(·) is an (f)-minimal solution.

Let x_∈ W_loc^1,1(R¹) be an (f)-minimal solution with rotation number r. By Proposition 2.1there existsc1>0 such that for alls, t_∈R¹,

x(t+s)₋x(t)₋rs_≤c1. (4.14) Proposition 2.3implies that there exists a constantc2>0 such that for eachs∈ R¹and eacht >0,

I^f(s, s+t, x)−Ef(r)t≤c2, (4.15) I^Ꮽ^f(s, s+t, x)−E_Ꮽf(r)t≤c2. (4.16) It follows from (4.13), (4.14), (4.15), and (4.16) that, for eachs_∈R¹and each t >0,

E_Ꮽ_f(r)t+βtr−E_f(r)t

≤E_Ꮽ_f(r)t₋I^Ꮽ^f(s, s+t, x)+I^Ꮽ^f(s, s+t, x) +βtr₋I^f(s, s+t, x) +I^f(s, s+t, x)₋E_f(r)t

≤c2+βtr−βx(t+s)−x(s)+c2≤2c2+|β|c1.

(4.17)

These inequalities imply that

E_Ꮽ_f(r)₌E_f(r)₋βr _∀r_∈R¹. (4.18)

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5. Proof ofTheorem 3.1 Letg_∈M. We define

µ(g)=inf lim inf

T→∞ T⁻¹I^g(0, T, x) :x(·)∈W_loc^1,1[0,∞). (5.1) In [13, Section 5] we showed that the numberµ(g) is well defined and proved the following result [13, Theorem 5.1].

Proposition5.1. Let f ∈M. Then there exists a constantM0>0such that:

(i) I^f(0, T, x)−µ(f)T≥ −M0for eachx∈W_loc^1,1([0,∞))and eachT >0.

(ii) For eacha∈R¹there existsx∈W_loc^1,1([0,∞))such thatx(0)=aand I^f(0, T, x)−µ(f)T≤4M0 ∀T >0. (5.2) Note that assertion (ii) ofProposition 5.1holds by the periodicity of f inx.

Let f _∈M. A functionx_∈W_loc^1,1([0,_∞)) is called (f)-good (see [5]) if supI^f(0, T, x)₋µ(f)T:T_∈(0,_∞)<_∞. (5.3) By [6, Theorem 4.1],

Ef

α(f)=µ(f) ∀f ∈M. (5.4) For f ∈M,x, y, T1∈R¹, andT2> T1we set

U^fT1, T2, x, y=infI^fT1, T2, v:v∈W^1,1T1, T2 , vT1

=x, vT2

=y. (5.5) It is not diﬃcult to see that for eachx, y, T1∈R¹,T2> T1,

U^fT1, T2, x+ 1, y+ 1₌U^fT1, T2, x, y,

U^fT1+ 1, T2+ 1, x, y₌U^fT1, T2, x, y, _−∞< U^fT1, T2, x, y<_∞, infU^fT1, T2, a, b:a, b_∈R¹

>_−∞. (5.6)

Denote byMperthe set of allf _∈Msuch thatα(f) is rational and denote byMper⁰

the set of allg∈Mperfor which there exist an (g)-minimal solutionw∈C²(R¹), a continuous functionπ :R¹→R¹, and integersm, n such that the following properties hold:

(P1)π(x+ 1)₌π(x),x_∈R¹;

(P2)n≥1 andα(g)=mn⁻¹is an irreducible fraction;

(P3)w(t+n)₌w(t) +mfor allt_∈R¹;

(P4)U^g(0,1, x, y)₋µ(g)₋π(x) +π(y)_≥0 for eachx, y_∈R¹; (P5) for anyu∈W^1,1(0, n), the equality

I^g(0, n, u)=nµ(g) +πu(0)−πu(n) (5.7) holds if and only if there are integersi,jsuch thatu(t)₌w(t+i)₋jfor allt_∈[0, n].

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Consider the manifold (R¹/Z)²and the canonical mappingP:R²→(R¹/Z)². We have the following result [13, Proposition 6.2].

Proposition5.2. LetΩbe a closed subset of(R¹/Z)². Then there exists a bounded nonnegative functionφ_∈C^∞((R¹/Z)²)such that

Ω= x∈

R¹/Z2

:φ(x)=0. (5.8)

Proposition 5.2is proved by using [1, Chapter 2, Section 3, Theorem 1] and the partition of unity (see [4, Appendix 1]).

We also have the following result (see [13, Proposition 6.3]).

Proposition5.3. Suppose that f ∈Mper,α(f)=mn⁻¹is an irreducible fraction (m, nare integers,n≥1) andw ∈W_loc^1,1(R¹)is an(f)-minimal solution satisfy- ingw(t+n)=w(t) +mfor allt∈R¹. Letφ∈C^∞((R¹/Z)²)be as guaranteed in Proposition 5.2with

Ω=

Pt, w(t):t_∈[0, n], (5.9)

and let

g(t, x, p)=f(t, x, p) +φP(t, x), (t, x, p)∈R³. (5.10) Theng∈M⁰perand there is a continuous functionπ:R¹→R¹such that the prop- erties (P1), (P2), (P3), (P4), and (P5) hold withg, w, π, m, nandα(g)=α(f).

In the sequel we need the following two lemmas proved in [13].

Lemma5.4 [13, Lemma 6.6]. Assume thatk _≥3 is an integer,g _∈M⁰per∩Mk, and properties (P1), (P2), (P3), (P4), and (P5) hold with ag-minimal solution w(·)∈C²(R¹), a continuous functionπ:R¹→R¹and integersm, n. Then for each ∈(0,1), there exists a neighborhoodᐁofginMksuch that for eachh_∈ᐁand each(h)-good functionv_∈W_loc^1,1([0,_∞))there are integersp,qsuch that

v(t)₋w(t+p)₋q_≤ for all large enought. (5.11) Lemma5.5 [13, Corollary 6.1]. Assume thatk_≥3is an integer,g _∈M⁰per∩Mk, and properties (P1), (P2), (P3), (P4), and (P5) hold with ag-minimal solution w(·)∈C²(R¹), a continuous functionπ:R¹→R¹and integersm,n. Then there exist a neighborhoodᐁofg inMk and a numberL >0such that for eachh_∈ᐁ and each(h)-good functionv_∈W_loc^1,1([0,_∞)), the following property holds.

There is a numberT0>0such that vt2

−vt1

−α(g)t2−t1≤L (5.12) for eacht1≥T0and eacht2> t1.

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Completion of the proof of Theorem 3.1. Let k _≥3 be an integer and let α ₌ mn⁻¹ be an irreducible fraction (n≥ 1 andm are integers). Let f ∈ Mk. By Proposition 2.2there exists an (f)-minimal solutionwf(·)∈W_loc^1,1(R¹) such that wf(t+n)=wf(t) +m ∀t∈R¹. (5.13) Choose

β∈∂Ef(α). (5.14)

Consider a mapping Ꮽ: Mk →Mk defined by (4.1). ByProposition 4.1 the mappingᏭis continuous. Clearly there exists a continuousᏭ⁻¹:Mk →Mk. Equations (5.14) and (4.18) imply that

0_∈∂E_Ꮽ_f(α), E_Ꮽ_f(α)₌minE_Ꮽf(r) :r_∈R¹

=µ(Ꮽf) (5.15) and thatᏭf _∈Mper. It follows fromProposition 5.2that there exists a bounded nonnegative functionφ∈C^∞((R¹/Z)²) such that

x∈R¹/Z2

:φ(x)=0=

Pt, wf(t):t∈[0, n]. (5.16) Setf^(β)₌Ꮽf and for eachγ_∈(0,1) define

fγ(t, x, u)= f(t, x, u) +γφP(t, x), (t, x, u)∈R³, f_γ^(β)=Ꮽfγ

. (5.17) Proposition 5.3implies that for eachγ∈(0,1),

f_γ^(β)_∈M⁰per∩Mk,

fγ−→ f asγ−→0⁺, f_γ^(β)−→ f^β) asγ−→0⁺inMk. (5.18) Fixγ∈(0,1) and an integern≥1. ByProposition 5.3the properties (P1), (P2), (P3), (P4), and (P5) hold withg=f_γ^(β),α(g)=αandw(·)=wf.

By Lemmas5.4and5.5, there exists an open neighborhoodV(f , γ, n) of fγ^(β)

inMγand a numberL(f , γ, n)>0 such that the following properties hold:

(i) for each h _∈ V(f , γ, n) and each (h)-good functionv _∈ W_loc^1,1([0,_∞)), there are integersp, qsuch that

v(t)₋w_f(t+p)₋q_≤1

n (5.19)

for all large enought;

(ii) for each h ∈ V(f , γ, n) and each (h)-good functionv ∈ W_loc^1,1([0,∞)), there is a numberT0such that

vt2

−vt1

−αf_γ^(β)t2−t1≤L (5.20) for eacht1≥T0and eacht2> t1.

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Leth_∈V(f , γ, n) and letv_∈W_loc^1,1(R¹) be an (h)-minimal solution with rotation numberα(h). Then byProposition 2.3, (2.3), (5.4), and property (ii),v_|_[0,∞) is an (h)-good function and there isT0 such that (5.20) holds for eacht1≥T0and eacht2> t1. Sincev_∈W_loc^1,1(R¹) has rotation numberα(h) it follows fromProposition 2.1that there exists c1>0 such that

v(t+s)−v(t)−α(h)s≤c1 ∀s, t∈R. (5.21) Equations (5.15), (5.17), (5.20), and (5.21) imply that

α(h)₌αf_γ^(β)₌αf^(β)₌α. (5.22) Thus we have shown that

α(h)₌α _∀h_∈V(f , γ, n). (5.23)

Leth_∈V(f , γ, n) and letv_∈W_loc^1,1(R¹) be an (h)-minimal solution with rotation numberα. It follows fromProposition 2.3, (2.3), and (5.4) that v|[0,∞)is an (h)-good function. By property (i) there exist integersp,q such that

v(t)−w_f(t+p)−q≤1

n for all large enought. (5.24) Therefore we proved the following property:

(iii) for eachh_∈V(f , γ, n) and each (h)-minimal solutionv_∈ᏹ^per_h (α), there exist integersp,qsuch that

v(t)₋w_f(t+p)₋q_≤1

n ^∀t_∈R¹. (5.25)

Define

ᐁ(f , γ, n)=Ꮽ⁻¹V(f , γ, n). (5.26) Clearlyᐁ(f , γ, n) is an open neighborhood offγinMk. By property (iii) the following property holds:

(iv) for eachξ_∈ᐁ(f , γ, n) and each (ξ)-minimal solutionv_∈ᏹ^per_ξ (α), there exist integersp,qsuch that (5.25) holds.

Define

Ᏺkα=∩^∞_n=1∪

ᐁ(f , γ, i) :f _∈Mk, γ_∈(0,1), i_≥n. (5.27) It is not diﬃcult to see thatᏲkαis a countable intersection of open everywhere dense subsets ofMk.

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Letg∈Ᏺkα,∈(0,1) andx, y∈ᏹ^(per)g (α). Choose a natural numbern >8⁻¹. By (5.27) there existf _∈Mk,γ_∈(0,1) and an integeri_≥nsuch that

g_∈ᐁ(f , γ, i). (5.28)

It follows from (5.28) and property (iv) that there exist integersp1,q1, p2,q2

such that

x(t)−wf t+p1

−q1≤1

i ^∀t∈R¹, (5.29)

y(t)−wf t+p2

−q2≤1

i ^∀t∈R¹, (5.30)

wherew_f ∈ᏹ^(per)_f (α).

It follows from (5.29) and (5.30) that for allt∈R¹, xt−p1

−wf(t)−q1≤1 i, yt−p2

−wf(t)−q2≤1 i, xt₋p1−q1

−

yt₋p2

−q2≤2 i, xt+p2−p1

−y(t)₋q1+q2≤2 i ^≤

2 n<.

(5.31)

Sinceis any number in (0,1), we conclude that there exist integers p,qsuch that

x(t+p)−q=y(t) ∀t∈R¹. (5.32) Assume thath∈ᐁ(f , γ, i) andz∈ᏹ^(per)_h (α). By the property (iv) there exist integersp3,q3such that

z(t)−wf t+p3

−q3≤1

i ^∀t∈R¹. (5.33)

Combined with (5.29) this inequality implies that zt−p3

−q3−xt−p1

+q1≤2 i ^≤

2

n< (5.34)

for allt∈R¹. This completes the proof ofTheorem 3.1.

References

[1] J.-P. Aubin and I. Ekeland,Applied Nonlinear Analysis, John Wiley & Sons, New York, 1984.

[2] S. Aubry and P. Y. Le Daeron,The discrete Frenkel-Kontorova model and its extensions.

I. Exact results for the ground-states, Phys. D8(1983), no. 3, 381–422.

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[3] J. L. Kelley,General Topology, Van Nostrand, New York, 1955.

[4] S. Kobayashi and K. Nomizu,Foundations of Diﬀerential Geometry. Vol. I, Inter- science Publishers, New York, 1963.

[5] A. Leizarowitz,Infinite horizon autonomous systems with unbounded cost, Appl. Math.

Optim.13(1985), no. 1, 19–43.

[6] A. Leizarowitz and A. J. Zaslavski,Infinite-horizon variational problems with noncon- vex integrands, SIAM J. Control Optim.34(1996), no. 4, 1099–1134.

[7] J. N. Mather,Existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology21(1982), no. 4, 457–467.

[8] ,Minimal measures, Comment. Math. Helv.64(1989), no. 3, 375–394.

[9] J. Moser,Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire3(1986), no. 3, 229–272.

[10] , Recent developments in the theory of Hamiltonian systems, SIAM Rev.28 (1986), no. 4, 459–485.

[11] W. Senn,Strikte Konvexit¨at f¨ur Variationsprobleme auf demn-dimensionalen torus, Manuscripta Math.71(1991), no. 1, 45–65 (German).

[12] A. J. Zaslavski,Ground states in Frenkel-Kontorova model, Math. USSR-Izv.29(1987), 323–354.

[13] , Existence and structure of extremals for one-dimensional nonautonomous variational problems, J. Optim. Theory Appl.97(1998), no. 3, 731–757.

Alexander J. Zaslavski: Department of Mathematics, Technion-Israel Insti- tute of Technology, Haifa32000, Israel

E-mail address:[email protected]

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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 acceleration (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;

[email protected]

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