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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 dimensionn2, andH:TMRaCkTonelli Hamiltonian(withk2), that is, a Hamiltonian of classCksatisfying the two following properties (k ∙ kdenotes the dual norm onTM):
(H1)Superlinear growth:For everyK0, there is a finite constantC(K) such that H(x, p)Kkpkx+C(K)(x, p)TM.
(H2)Uniform convexity:For every (x, p)TM, the second derivative along the fibers 2pH2(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 valuescR for which there exists a functionu:MRof classC1satisfying
H(x, du(x))cxM.
Remark 1.2.We can check easily thatc[H]satisfies the following inequalities
xmiMn{H(x,0)} ≤c[H]xmaMx{H(x,0)}. The study of solutions of the critical Hamilton-Jacobi equation, Hx, du(x)=c[H]xM,
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 (.srnrfhtc.@damffro.cirdovilu)
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:MRis called a critical subsolution forHif it is Lipschitz and satisfies Hx, du(x)c[H]a.e.xM.(2.1) Let us denote byC0(M;R),k ∙ kthe 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 ofC0(M;R). Proof.Pick a sequence ofC1functions{uk}associated with a sequence of real numbers{ck} converging toc[H] such that Hx, duk(x)ckxM,k. Thanks to the superlinear growth hypothesis (H1), the sequence{uk}is uniformly Lipschitz. Then by compactness, we may assume that it converges uniformly to some Lipschitz function u:MR fact that. Theufollows easily from the following lemmais a critical subsolution whose proof is left to the reader. Lemma 2.3.Let{uk}be a sequence ofC1functions onMwhich converges uniformly to some Lipschitz functionu:MR that. Assumeuis differentiable atxM there is a. Then sequencexkxsuch thatduk(xk)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 MRassociated withHby Legendre-Fenchel duality is defined by L(x, v) :=pmTaxxMnhp, vi −H(x, p)o. Thanks to (H1)-(H2), it can be shown (see [4, 13]) thatLis aCkTonelli Lagrangian, that is it isCkand satisfies the two following properties (k ∙ kdenotes the norm onT M):
(L1)Superlinear growth:For everyK0, there is a finite constantC(K) such that
L(x, v)Kkvkx+C(K)(x, v)T M. (L2)Uniform convexity:For every (x, v)T M,2v2L(x, v) is positive definite. Note that the Fenchel inequality
hp, vi ≤L(x, v) +H(x, p) (2.2) holds for anyxMandvTxM, pTxMwith equality if and only if (in local coordinates) v=Hp(x, p)p=Lv(x, v).(2.3) The Legendre-Fenchel duality allows us to characterize the critical subsolutions in a variational way.
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