Closing Aubry sets I

Sophia Antipolis

Description

Niveau: Supérieur, Doctorat, Bac+8
Closing Aubry sets I A. Figalli? L. Rifford† July 7, 2011 Abstract Given a Tonelli Hamiltonian H : T ?M ? R of class Ck, with k ≥ 2, we prove the following results: (1) Assume there exist a recurrent point of the projected Aubry set x¯, and a critical viscosity subsolution u, such that u is a C1 critical solution in an open neighborhood of the positive orbit of x¯. Suppose further that u is “C2 at x¯”. Then there exists a Ck potential V : M ? R, small in C2 topology, for which the Aubry set of the new Hamiltonian H + V is either an equilibrium point or a periodic orbit. (2) If M is two dimensional, (1) holds replacing “C1 critical solution + C2 at x¯” by “C3 critical subsolution”. These results can be considered as a first step through the attempt of proving the Man˜e's conjecture in C2 topology. In a second paper [27], we will generalize (2) to arbitrary dimension. Moreover, such an extension will need the introduction of some new techniques, which will allow us to prove in [27] the Man˜e's density Conjecture in C1 topology. Our proofs are based on a combination of techniques coming from finite dimensional control theory and Hamilton-Jacobi theory, together with some of the ideas which were used to prove C1-closing lemmas for dynamical systems.

• hamiltonian viewpoint


• following man˜e


• nonempty compact


• aubry-mather theory


• c1-closing lemmas


• ?? al


• c1 critical

Sujets

• Antipolis

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 7
• Langue: English

Extrait

1

July 7, 2011

1

Closing Aubry sets I A. Figalli ∗ L. Rifford †

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Introduction 1.1 Aubry-Mather theory from the Lagrangian viewpoint . . . . . . . . . . . . . . . . 1.2 Aubry-Mather theory from the Hamiltonian viewpoint . . . . . . . . . . . . . . . 1.3TheMan˜e´Conjecture.................................

3 Connecting Hamiltonian orbits by potentials 12 3.1 Statement of the result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Controlling the action by potentials 23 4.1 Statement of the result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Proof of Proposition 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 0 resappsuauthorthrpehargoetrotybdoBbocaJ-nooe´htteiAMeKrijoce“mrP0--7AtRN-036BLANmilt1,Ha faible”. AF is also supported by NSF Grant DMS-0969962. ∗ Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin TX 78712, USA (ude.tu.hsaxelialat@migf) † itopil,soShpainAedeNice-iversit´nUce08NiVclaP,ra0,16oresMR,U´enn2166RSCN-.J.obaLodueiD.A Cedex 02, France (.fcerfirdrofinu@)

Contents

Abstract Given a Tonelli HamiltonianH:T ∗ M→Rof classC k , withk≥2, we prove the following results: (1) Assume there exist a recurrent point of the projected Aubry set ¯ d a critical vi ity subsolutionu, such thatuis aC 1 critical solution in an open x scos, an neighborhood of the positive orbit of ¯x. Suppose further thatuis “C 2 atx¯”. Then there exists aC k potentialV:M→R, small inC 2 topology, for which the Aubry set of the new HamiltonianH+V Ifis either an equilibrium point or a periodic orbit. (2)M is two dimensional, (1) holds replacing “C 1 critical solution +C 2 at ¯x” by “C 3 critical subsolution”. These results can be considered as a first step through the attempt of proving the Man˜e´’sconjectureinC 2  a second paper [27], we will generalize (2) to arbitrarytopology. In dimension. Moreover, such an extension will need the introduction of some new techniques, whichwillallowustoprovein[27]theMa˜ne´’sdensityConjectureinC 1 topology. Our proofs are based on a combination of techniques coming from finite dimensional control theory and Hamilton-Jacobi theory, together with some of the ideas which were used to proveC 1 -closing lemmas for dynamical systems.

2 3 5 8

2 Statement of the results

5 Proof of Theorem 2.1 30 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.2 Preliminary steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3 Closing the Aubry set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.4 Control of the action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.5 Construction of a critical subsolution . . . . . . . . . . . . . . . . . . . . . . . . . 42

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Proof of Theorem 2.4 47 6.1 Preliminary step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.2 Modification of the potential and conclusion . . . . . . . . . . . . . . . . . . . . . 48 6.3 Construction of the potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7 Final comments

A Conventions and standing notation

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B Controllability of nonlinear control systems 52 B.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 B.2 Singular controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 B.3 Application to partial controllability I . . . . . . . . . . . . . . . . . . . . . . . . 55 B.4 Application to partial controllability II . . . . . . . . . . . . . . . . . . . . . . . . 56 B.5 Application to partial controllability III . . . . . . . . . . . . . . . . . . . . . . . 57

C Quantitative Inverse Function Theorem

D The Mai Lemma

E

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Proofs of Lemmas 3.3, 4.3 and 6.1 59 E.1 Proof of Lemma 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 E.2 Proof of Lemma 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 E.3 Proof of Lemma 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

References

1 Introduction

63

Let (M gbe a smooth compact Riemannian manifold without boundary of dimension) n≥2. GivenH:T ∗ M→RonhTotmomiHalielsaeconjectureintlnoai,nhtMe˜a´nC k topology (with V∈C k ˜ k≥2) asserts that, for generic potentials (M), the projected Aubry setA(H+V) associated to the HamiltonianH+Vis either an equilibrium point or a periodic orbit. This paper is the first of a series of articles where we plan to make progress toward a proofoftheMan˜e´ConjectureinC 2 topology. The aim of this first paper is to show how to provethedensitypartoftheM˜ne´ConjectureinC 2 topology under the following assumptions a (Theorem 2.1): there exist a recurrent point of the projected Aubry set ¯x, and a critical viscosity subsolutionu, such thatuis aC 1 critical solution in an open neighborhood of the positive orbit ofx¯, anduis “C 2 atx¯”. Then, in two dimensions we show how to replace the above assumption by replacing “C 1 critical solution +C 2 at the point” with “C 3 critical subsolution” (Theorem 2.4). In a second paper we will perform the extension of this last result to arbitrary dimension [27, Theorem 1.1]. Moreover, the proof of this last result will involve the introduction of some newideasandtechniques,whichwillallowustoprovethe(densitypartofthe)Ma˜n´eConjecture inC 1 topology [27, Theorem 1.2].

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Before describing our results in detail, we first introduce the Aubry-Mather theory from both the Lagrangian and the Hamiltonian points of view. Some conventions and standing notation are gathered in Appendix A.

1.1 Aubry-Mather theory from the Lagrangian viewpoint LetL:T M→Rbe aC k Tonelli Lagrangian, that is, a Lagrangian of classC k (withk≥2) satisfying the two following assumptions:

(L1)Superlinear growth:For everyK≥0, there is a finite constantC(K) such that

L(x v)≥Kkvk x +C(K)∀(x v)∈T M (L2)Strict convexity:For every (x v)∈T M, the second derivative along the fibers ∂ 2 v 2 (x v) is positive definite.

Thecritical valueofLis defined as c[L] :=− T in > f 0  T 1 A  γ; [0 T]  |γ∈C 1  [0 T] M   γ(0) =γ(T)  (1.1) whereA  γ; [0 T]  denotes theactionof theC 1 curveγ: [0 T]→Mon the time interval [0 T], that is, A  γ; [0 T]  := Z 0 T L  γ(t)˙γ(t)  dt By the assumptions onL, the critical valuec[L] is necessarily finite, and satisfies

inf (xv)∈T M L(x v)≤ −c[L]≤ x i ∈ nf M L(x0) To each closed curveγ∈C p 1 er  [0 T] M  , we can associate a probability measure γ onT M by f d f  γ(t)˙γ(t)  dt∀f∈C 0 (T MR) Z T M  γ := T 1 Z 0 T FollowingMa˜ne´[35],wecallholonomic probability measureany element in the set H:= n  γ |T >0 γ∈C 1 per  [0 T] M o  where the closure is taken with respect to the weak- ∗  Definetopology on the space of measures. theaction functional

A L :P(T M)−→R∪ {+∞} →−7A L () := R T M Ld

By construction, we have inf  A L ()|∈ H  =−c[L] The setHis a (nonempty) closed convex subset ofP(T M), which is not compact (with respect to the weak- ∗  thanks to (L1), the settopology). However,H 0 :=H ∩ {A L ≤ −c(L) + 1}is a compact convex subset ofH. This implies thatA L attains a minimum onH, that is, c[L] =−m ∈ in H {A L ()} 

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The measures∈ Hsuch thatA L () =−c[L] are calledminimizing measures. It can be shown that they are invariant under the Euler-Lagrange flowφ t L [35], and they minimize the functional A L among all Borel probability measures onT Mwhich are invariant underφ L . t TheMather setofLis the nonempty compact subset ofT Mdefined as ˜ [ Supp() M(L) := A L ()=−c[L] and theprojected Mather setM(L)⊂Mis given by M(L) :=π  M(L)   ˜ In [37], Mather proved the following result: ˜ Mather’s Graph Theorem I.The setM(L)⊂T Mis invariant underφ t L . Moreover the ˜ mapπ |M ˜ (L) :M(L)→Mis injective, and  π |M( ˜ L  −1 :M(L)→ M( ˜ L) )

is Lipschitz.

Following Mather [38], for everyT >0 we define the functionh T :M×M→Ras h T (x y) := inf n A  γ; [0 T]  |γ∈C 1  [0 T] M   γ(0) =x γ(T) =y o  ThePeierls barrierassociated withLis the functionh:M×M→Rdefined by h(x y) := l T i → m + in ∞ f n h T (x y) +c[L]T o  It is immediately seen that the following inequalities hold for allT >0, for everyx y z∈M:

h(x z)≤h(x y) +h T (y z) +c[L]T  h(x z)≤h T (x y) +c[L]T+h(y z) In particular, we deduce that the following “triangle inequality” holds:

h(x z)≤h(x y) +h(y z)∀x y z∈M

By compactness ofMand (1.1), it is not difficult to prove that there is at least one pointx∈M such thath(x x Hence the above triangle inequality shows that) = 0.his finite everywhere on M×M. Theprojected Aubry setA(L) is then defined as the nonempty compact set given by A(L) := n x∈M|h(x x) = 0 o (1.2)

We observe that for everyx∈ A(L) there exist a sequence{T k } k∈N of real numbers tending to +∞, and a sequence{γ k } k∈N ofC 1 curvesγ k : [0 T k ]→M, such thatγ k (0) =γ k (T k ) and k→∞ A  γ k ; [0 T k ]  +c[L]T k = 0 lim ApplyingtheArzela`-AscoliTheorem,itcanbeshownthatthesequence{γ˜ k }of curves  γ k  γ˙ k  : [0 T k ]→T Mis relatively compact, so that for each integerl >0 the sequence of curves ]7→−  ˜γ ˜γ k k ( ( t T ) k +t fi )if t t< ≥0 0 t∈[−l l 4

admits, up to a subsequence, a uniform limit. Then, one can show that such limit curve is uniquely determined [38], and deduce that to eachx∈ A(L) it can be associated in a unique way aC k−1 curveγ x :R→M, withγ x (0) =x, which solves the Euler-Lagrange equation d d t  ∂v ∂L  γ x (t) γ˙ x (t)  = x∂ L∂  γ x (t)˙γ x (t)  ∀t∈R Then, theAubry setofLis the compact subset ofT Mdefined by A( ˜ L) :=  n γ x (t) γ˙ x (t)  |x∈ A(L) t∈R o  ˜ ˜ It can be proved that Aubry setA(L) contains the Mather setM(L). Moreover, in [38] Mather showed the following result:

Mather’s Graph Theorem II.The setA ˜ (L)⊂T Mis invariant underφ L t  the. Moreover ˜ mapπ |A ˜ (L) :A(L)→Mis injective, its image coincides withA(L), and  A(L)  −1 : ˜ π | ˜ A(L)→ A(L)

is Lipschitz.

˜ ˜ In other terms, Mather’s Graph Theorems state thatM(L)⊂ A(L) are contained in the graph of a Lipschitz section ofT M.

1.2 Aubry-Mather theory from the Hamiltonian viewpoint TheTonelli HamiltonianH:T ∗ M→Rassociated toLby Legendre-Fenchel duality is defined as v∈T x M n p(v)−L(x v) o ∀(x p)∈T x ∗ M H(x p) := max Thanks to our assumptions onL, it is well-known thatHis of classC k and satisfies both properties of superlinear growth and strict convexity inT ∗ M: (H1)Superlinear growth:For everyK≥0, there is a finite constantC ∗ (K) such that H(x p)≥Kkpk x +C ∗ (K)∀(x p)∈T ∗ M (H2)Strict convexity:For every (x p)∈T ∗ M, the second derivative along the fibers ∂ 2 p 2 (x p) is positive definite.

Under the above assumptions, the Hamiltonian flowφ t H ofHis of classC k−1 , and is conjugated with the Euler-Lagrange flowφ L t ofL. Thecritical valueora˜Mvlaceulace´nitirofHis defined as

c[H] :=c[L] while the Aubry set “seen inT ∗ M” is defined as ˜  A ˜ (L)   A(H) :=L

(1.3)

whereL:T M→T ∗ M Bydenotes the Legendre transform (see Appendix A). construction A ˜ (H) is a nonempty compact subset ofT ∗ Mwhich is invariant underφ t H  a series of papers. In [16, 17, 18], Fathi established a deep link between the concept of Aubry sets and the concept of viscosity solutions of the Hamilton-Jacobi associated withH, which we now describe.

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A continuous functionu:M→Ris called aviscosity subsolutionof the Hamilton-Jacobi equation H  x du(x)  =c∀x∈M(1.4) if, for everyC 1 functionφ:M→Rsuch thatφ≤uand everyz∈M, the following holds:

φ(z) =u(z) =⇒H(z dφ(z))≤c

This is equivalent to asking that u  γ(b)  −u  γ(a)  ≤  γ(t) γ˙ Z b a L(t)  dt+c(b−a) (1.5) for everyC 1 curveγ: [a b]→M. A continuous functionu:M→Ris called aviscosity solutionof (1.4) if, for everyC 1 functionφ:M→Rsuch thatφ≤uand everyz∈M, the following holds 1 :

φ(z) =u(z) =⇒H(z dφ(z)) =c

As shown by Fathi, a continuous functionu:M→Rsolution of (1.4) if and only ifis a viscosity it is a viscosity subsolution of (1.4) and, for eachx∈M, there is aC k−1 curveγ x : (−∞0]→M such that u  x  −u  γ x (−T)  = Z − 0 T L  γ x (t)˙γ x (t)  dt+cT∀T≥0(1.6) In [16], Fathi proved the following result:

Fathi’s Weak KAM Theorem.The critical Hamilton-Jacobi equation H  x du(x)  =c[H]∀x∈M

admits at least one viscosity solution.

(1.7)

Let us recall that, by the compactness ofM,c[H] is the only value ofcfor which the Hamilton-Jacobi equation (1.4) admits a viscosity solution. Indeed, if a continuous function u:M→Ris a viscosity subsolution of (1.4) for somec∈R, then for everyC 1 curve γ: [0 T]→Mone has uk ∞ ≤u  γ(T)  −u  γ(0)  ≤  γ(t) γ˙ (t)  dt+cT  −2k Z 0 T L 1 We notice that the definitions of viscosity subsolution and viscosity solution given here are equivalent to the usual definitions: usually, a continuous functionu:M→Ris called aviscosity solutionof the first-order partial differential equation F ` x u(x) du(x) ´ = 0∀x∈M if it satisfies the two following properties: (i) (uissupersolution) For everyC 1 functionφ:M→Rsuch thatφ≤uand everyz∈M, it holds φ(z) =u(z) =⇒F(z φ(z) dφ(z))≥c (ii) (uissubsolution) For everyC 1 functionφ:M→Rsuch thatφ≥uand everyz∈M, it holds

φ(z) =u(z) =⇒F(z φ(z) dφ(z))≤c

SinceHis convex in thepsublevel sets, the above definitions are equivalent to the onevariable with bounded given in the paper.

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