Regularity of weak KAM solutions and Man˜e's Conjecture

Sophia Antipolis

Description

Regularity of weak KAM solutions and Man˜e's Conjecture L. Rifford? Abstract We provide a crash course in weak KAM theory and review recent results concerning the existence and uniqueness of weak KAM solutions and their link with the so-called Man˜e conjecture. 1 Introduction In the present paper, (M, g) will be a smooth connected compact Riemannian manifold without boundary of dimension n ≥ 2, and H : T ?M ? R a Ck Tonelli Hamiltonian (with k ≥ 2), that is, a Hamiltonian of class Ck satisfying the two following properties (? · ? denotes the dual norm on T ?M): (H1) Superlinear growth: For every K ≥ 0, there is a finite constant C?(K) such that H(x, p) ≥ K?p?x + C ?(K) ? (x, p) ? T ?M. (H2) Uniform convexity: For every (x, p) ? T ?M , the second derivative along the fibers ∂2H ∂p2 (x, p) is positive definite. The Man˜e critical value of H can be defined as follows. Definition 1.1. We call critical value of H, denoted by c[H], the infimum of the values c ? R for which there exists a function u : M ? R of class C1 satisfying H(x, du(x)) ≤ c ?x ?M.

• ttu


• following properties


• arzela-ascoli theorem


• lax-oleinik semigroup


• inf z?m


• supremum norm


• compact riemannian


• fenchel inequality


• lipschitz curve


• legendre-fenchel duality

Sujets

• Institut universitaire de France
• Antipolis
• Arzelà?Ascoli theorem
• Uniform norm
• Convex conjugate

Informations

• Publié par: pefav
• Nombre de lectures: 8
• Langue: English

Extrait

RegularityofweakKAMsolutionsandMa˜n´e’sConjecture

L. Rifford ∗

Abstract We provide a crash course in weak KAM theory and review recent results concerning theexistenceanduniquenessofweakKAMsolutionsandtheirlinkwiththeso-calledMa˜n´e conjecture.

1 Introduction

In the present paper, (M, gbe a smooth connected compact Riemannian manifold without) will boundary of dimensionn≥2, andH:T ∗ M→RaC k Tonelli Hamiltonian(withk≥2), that is, a Hamiltonian of classC k satisfying the two following properties (k ∙ kdenotes the dual norm onT ∗ 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)Uniform convexity:For every (x, p)∈T ∗ M, the second derivative along the fibers ∂ ∂ 2 pH 2 (x, p) is positive definite.

TheMan˜e´criticalvalueofHcan be defined as follows.

Definition 1.1.We call critical value ofH, denoted byc[H], the infimum of the valuesc∈R for which there exists a functionu:M→Rof classC 1 satisfying

H(x, du(x))≤c∀x∈M.

Remark 1.2.We can check easily thatc[H]satisfies the following inequalities

x m ∈ i M n{H(x,0)} ≤c[H]≤ x m ∈ a M x{H(x,0)}. The study of solutions of the critical Hamilton-Jacobi equation, H  x, du(x)  =c[H]∀x∈M,

(1.1)

is the core of Fathi’s weak KAM theory developed in [9, 10, 11, 12, 13]. The aim of the present paper is to recall briefly the construction of Fathi’s weak KAM solutions and to address uniqueness and regularity issues for the critical Hamilton-Jacobi equation. ∗ inUe´tisrev-SceNidentaAhiop,e´nCRMUeiD.noduo.ab-AJ.olip,LisN8cieso,e6001ParcValrNRS6621, Cedex 02, France & Institut Universitaire de France ([e-mail])

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Critical subsolutions

A priori, the infimum in Definition 1.1 is not necessarily attained. For this reason, we introduce the notion of critical subsolutions. We recall that by Rademacher’s theorem, Lipschitz functions are differentiable almost everywhere. Definition 2.1.A functionu:M→Ris called a critical subsolution forHif it is Lipschitz and satisfies H  x, du(x)  ≤c[H]a.e.x∈M.(2.1) Let us denote by  C 0 (M;R),k ∙ k ∞  the Banach space of continuous functions onMequipped with the supremum norm.

Proposition 2.2.The setSS[H]of critical subsolutions is a nonempty, compact and convex subset ofC 0 (M;R). Proof.Pick a sequence ofC 1 functions{u k }associated with a sequence of real numbers{c k } converging toc[H] such that H  x, du k (x)  ≤c k ∀x∈M,∀k. Thanks to the superlinear growth hypothesis (H1), the sequence{u k }is uniformly Lipschitz. Then by compactness, we may assume that it converges uniformly to some Lipschitz function u:M→R fact that. Theufollows easily from the following lemmais a critical subsolution whose proof is left to the reader. Lemma 2.3.Let{u k }be a sequence ofC 1 functions onMwhich converges uniformly to some Lipschitz functionu:M→R that. Assumeuis differentiable atx∈M there is a. Then sequencex k →xsuch thatdu k (x k )→du(x). Thus we proved thatSS[H] is nonempty. By (H1), any critical subsolution is Lipschitz on M(with universal Lipschitz constant). From the Arzela-Ascoli Theorem, the compactness of SS[H Finally,] follows easily. the convexity ofSS[H] is straightforward from the convexity of Hin thepvariable (H2).

The LagrangianL:T M→Rassociated withHby Legendre-Fenchel duality is defined by L(x, v) := p∈ m T a x ∗ x M n hp, vi −H(x, p) o . Thanks to (H1)-(H2), it can be shown (see [4, 13]) thatLis aC k Tonelli Lagrangian, that is it isC k and satisfies the two following properties (k ∙ kdenotes the norm onT M):

(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)Uniform convexity:For every (x, v)∈T M, ∂ ∂ 2 v2 L (x, v) is positive definite. Note that the Fenchel inequality

hp, vi ≤L(x, v) +H(x, p) (2.2) holds for anyx∈Mandv∈T x M, p∈T x ∗ Mwith equality if and only if (in local coordinates) v= H ∂ ∂ p (x, p)⇔p= L ∂ ∂ v (x, v).(2.3) The Legendre-Fenchel duality allows us to characterize the critical subsolutions in a variational way.

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