On the C 1 and C 2 -convergence to weak K.A.M. solutions
aa r X i v : . [ m a t h . D S ] F e b ON THE C AND C -CONVERGENCE TO WEAK K.A.M.SOLUTIONS MARIE-CLAUDE ARNAUD AND XIFENG SUA bstract . We introduce a notion of upper Green regular solutions to theLax-Oleinik semi-group that is defined on the set of C functions of aclosed manifold via a Tonelli Lagrangian. Then we prove some weak C convergence results to such a solution for a large class of approximatedsolutions as(1) the discounted solution (see [DFIZ16]);(2) the image of a C function by the Lax-Oleinik semi-group;(3) the weak K.A.M. solutions for perturbed cohomology class.This kind of convergence implies the convergence in measure of the sec-ond derivatives.Moreover, we provide an example that is not upper Green regular andto which we have C convergence but not convergence in measure of thesecond derivatives.
1. I ntroduction
This article focuses on some weak solutions of the stationary Hamilton-Jacobi equation H ( · , du ( · )) = c on some closed manifold M ( d ) . Classicalsolutions of this equation are generating functions of Lagrangian subman-ifolds that are invariant by the Hamiltonian flow, but it often happens thatsuch classical solutions don’t exist.The viscosity solutions were then introduced by P.-L. Lions and M.G.Crandall (see [CL83]) and provide generalized solutions under very weakhypotheses for H . In 1997 and in a convex setting, A. Fathi proved hisweak K.A.M. theorem (see [Fat97]) that provides weak K.A.M. solutionsand also proved (see [Fat08]) that these solutions coincide with the viscositysolutions. The weak K.A.M. solutions are fixed points of the so-called Lax-Oleinik semi-group and Fathi proved in [Fat98] the convergence of the Lax-Oleinik semi-group to weak K.A.M. solutions in C -topology.Here we consider various problems of C and C convergence, that cor-respond to a convergence of graphs of discontinuous Lagrangian submani-folds of T ∗ M . Before our results, only results concerning the C -convergencewere known. ‡ member of the InstitutuniversitairedeFrance. We study the problem of the C or C convergence of approximated solu-tions for the Lax-Oleinik semi-group defined on a closed manifold M . Moreprecisely, we will consider the following three problems:(1) the dependence of the weak K.A.M. solution on the cohomologyclass;(2) the convergence of the so-called discounted solution (see [DFIZ16]);(3) the convergence of the Lax-Oleinik semi-group to a weak K.A.M.solution.The problem of C convergence for Point (3) was partially solved in[Arn05]. The Dynamics that we will consider are Hamiltonian or confor-mally Hamiltonian on T ∗ M and are all convex in the fiber, which means thefollowing. Definition 1.
A C function H : ( q , p ) ∈ T ∗ M H ( q , p ) ∈ R is C -convexin the fiber direction if for every x ∈ T ∗ M, the Hessian in the fiber direction ∂ H ∂ p ( x ) , denoted by H p , p ( x ) for short, is positive definite as a quadratic form.The C function H is superlinear in the fiber direction if for any Riemannianmetric on M, for any A > , there exists B such that ∀ ( q , p ) ∈ T ∗ M , H ( q , p ) ≥ A k p k + B . A Tonelli Hamiltonian is a function that is superlinear and C -convex in thefiber direction. We will be interested in conformally Hamiltonian flows associated to aTonelli Hamiltonian, defined by the following equations (see [MS17]) with λ > dqdt = ∂ H ∂ p ( q , p ) and d pdt = − ∂ H ∂ q ( q , p ) − λ p . Observe that the case λ = Definition 2 (Hausdor ff distance) . Let ( X , d ) be a metric space. For anynon-empty compact subsets K , K of X, the Hausdor ff distance between K and K is defined byd H ( K , K ) : = max { ρ ( K , K ) , ρ ( K , K ) } where ρ ( K , K ) = sup x ∈ K d ( x , K ) . N otation . • Choosing a Riemannian metric, we will denote by d H the associatedHausdor ff distance in T ∗ M ; • if Λ : N ⊂ M → T ∗ M is a section of π : T ∗ M → M , its graph isdenoted by G ( Λ ) = { ( q , Λ ( q )) : q ∈ N } ⊂ T ∗ M ; AND C -CONVERGENCE TO WEAK K.A.M. SOLUTIONS 3 • at every x ∈ T ∗ M , the vertical subspace at x is V ( x ) = ker D π ( x ); • if A is a subset of a topological space, we denote its closure by ¯ A . Theorem 1.
Let H : T ∗ M → R be a C Tonelli Hamiltonian. Let ( u λ ) λ ∈ (0 , be the solutions to the associated discounted problem (see [DFIZ16] ) andlet u : M → R be their limit lim λ → + u λ = u. Then lim λ → + d H ( G ( du λ ) , G ( du )) = . Corollary 1.
With the same hypotheses as in Theorem 1, if u is C , then u λ converges to u for the uniform C topology when λ → + . To give a similar statement in the case of varying cohomology classes,we introduce some notations.N otation . For every c in the linear space H ( M , R ), we choose in a con-tinuous way a smooth closed 1-form η c with cohomology class c . When M = T d , we can identify H ( T d , R ) with the set of constant 1-forms. Theorem 2.
Let H : T ∗ M → R be a C Tonelli Hamiltonian. For everyc ∈ H ( M , R ) , we consider the modified Lax-Oleinik semi-group ( T ct ) t ∈ R + that corresponds to the closed -form η c , defined byT ct u ( x ) = inf γ ( u ( γ ( − t )) + Z − t (cid:2) L ( γ ( s ) , ˙ γ ( s )) − h η c , ˙ γ ( s ) i + α ( c ) (cid:3)) where the infimum is taken over all the absolutely continuous curves γ :[ − t , → M such that γ (0) = x . Assume that ( u c ) c ∈ D is a family of fixedpoints of ( T ct ) that uniformly converge to u when c tends to .Then, lim c → d H ( G ( η c + du c ) , G ( η + du )) = . Corollary 2.
With the same hypotheses as in Theorem 2, if u is C , then u c converges to u for the uniform C topology when c → . We will now focus on the case of C topology when M = T d and theconsidered limit solution u satisfies some regularity assumption that we willdetail.For Dynamics that are defined with a Tonelli Hamiltonian, the pieces oforbit with no conjugate points play a special role; for example, in a La-grangian setting, they correspond to locally minimizing orbits. Definition 3.
Let ( ϕ t ) t ∈ R be a flow on T d × R d . α ( c ) is Ma˜n´e critical value for the cohomology class c , see [Fat08]. M.-C. ARNAUD AND X. SU • a piece of orbit ( ϕ t ( x )) t ∈ I with interval I ⊂ R has no conjugate points if ∀ t , s ∈ I , ( D ϕ t − s V ( ϕ s ( x ))) ∩ V ( ϕ t ( x )) = { } ; • for such a piece of orbit, for every s , t ∈ I, we defineG t − s ( ϕ t ( x )) = D ϕ t − s V ( ϕ s ( x )) . For such a piece of orbit with no conjugate points, observe that all theLagrangian subspaces G t − s ( ϕ t ( x )) with t , s are transverse to the vertical V ( ϕ t ( x )) and then are graphs of some symmetric matrix in the usual coordi-nates.N otations . • We denote the set of symmetric matrices with size n by S n . • Let G ⊂ T x ( T d × R d ) be a Lagrangian subspace that is transverse tothe vertical subspace. Its height H ( G ) ∈ S d is the symmetric matrixsuch that G = { ( δθ, H ( G ) δθ ) ; δθ ∈ R d } . In fact, we will identify H ( G ) with a quadratic form.The set of symmetric matrices is endowed with a natural order, the oneof the corresponding quadratic forms. The following proposition is provedin [Arn08] for the Hamiltonian case and we will prove in Section 3 that it isalso true for conformal Hamiltonian flows. Proposition 1. If ( ϕ t ) t ∈ R is a conformal Hamiltonian flow on T d × R d thatis associated to a C -convex in the fiber Hamiltonian and if ( ϕ t ( x )) t ∈ I is apiece of orbit with no conjugate points, then • if t ∈ I, the map s ∈ ( −∞ , t ) ∩ I
7→ H ( G t − s ( ϕ t ( x ))) is increasing andthe map s ∈ ( t , + ∞ ) ∩ I
7→ H ( G t − s ( ϕ t ( x ))) is increasing; • for every s ∈ I ∩ ( −∞ , t ) and s ∈ ( t , + ∞ ) ∩ I, then H ( G t − s ( ϕ t ( x ))) > H ( G t − s ( ϕ t ( x ))) ; • when inf I = −∞ , the limit G + ( ϕ t ( x )) = lim s →−∞ G t − s ( ϕ t ( x )) exists andwhen sup I = + ∞ , the limit G − ( ϕ t ( x )) = lim s → + ∞ G t − s ( ϕ t ( x )) exists; • when I = R , we have H ( G − ) ≤ H ( G + ) .G − and G + are then called Green bundles .As said before, we will consider some special weak K.A.M. solutions u : T d → R of some Hamiltonians. These solutions are always semi-concave ,and then • they are Lipschitz and Lebesgue almost everywhere di ff erentiableby Rademacher Theorem (see [EG15]); See Section 2 for the definition. AND C -CONVERGENCE TO WEAK K.A.M. SOLUTIONS 5 • if θ k ∈ T d converges to θ and if p k ∈ D + u ( θ k ) converges to a vector p ∈ R d , then p ∈ D + u ( θ ) (see [CS04]) ; • by Alexandrov Theorem (see [NP06]), they admit a second deriva-tive D u at Lebesgue almost every θ ∈ T d .It can be proved (see [Fat08]) that at every point θ where the weak K.A.M.solution u is di ff erentiable, the negative orbit ( ϕ t ( θ, du ( θ ))) t ∈ R − has no con-jugate points and thus the Green bundle G + ( θ, du ( θ )) exists. Definition 4.
A weak K.A.M. solution u is upper Green regular if at Lebesguealmost every θ ∈ T d , we have H ( G + ( θ, du ( θ ))) = D u ( θ ) . A weak K.A.M. solution u is lower Green regular if at Lebesgue almost every θ ∈ T d , we have H ( G − ( θ, du ( θ ))) = D u ( θ ) . We will prove in Section 4 that the following examples of restricted Dy-namics to invariant C Lagrangian graphs correspond to a C , upper andlower Green regular weak K.A.M. solution • the restricted Dynamics is Lipschitz conjugated to the one of a rota-tion flow; • the restricted Dynamics is Kupka-Smale; • the degree of freedom is d = C distance between any C and upper(resp. lower) Green regular weak K.A.M. solution and its approximatedsolutions. The quantity that we will estimate is described below.N otation . • We denote by Leb the Lebesgue measure on T d . • If S is a symmetric matrix on R d , its norm is defined by k S k = sup v ∈ R d , k v k = | S ( v , v ) | where k · k is the standard Euclidean norm we take on R d . See Section 2 for the notation. Here D + u ( x ) denotes the set of super-di ff erentials of u at x , see Section 2 for thedefinition. M.-C. ARNAUD AND X. SU • Let u , v : T d → R be two semi-concave functions. Then they admita second derivative Lebesgue almost everywhere and we can define d , ( u , v ) = Z T d k D u ( θ ) − D v ( θ ) k d Leb( θ ) . Theorem 3.
Let H : T d × R d → R be a C Tonelli Hamiltonian. Let ( u λ ) λ ∈ (0 , be the solutions of the associated discounted problem and let u : T d → R be their limit, i.e. lim λ → + u λ = u. Then, if u is C and upper Green regular, uis a weak K.A.M. solution that satisfies lim λ → + d , ( u λ , u ) = . Theorem 4.
Let H : T d × R d → R be a C Tonelli Hamiltonian with as-sociated Lax-Oleinik semi-group ( T t ) t ≥ . Let u ∈ C ( T d , R ) and let us usethe notation u t = T t u and u = lim t → + ∞ u t . Then, if u is C and upper Greenregular, u is a weak K.A.M. solution that satisfies lim t → + ∞ d , ( u t , u ) = . Theorem 5.
Let H : T d × R d → R be a C Tonelli Hamiltonian. Forevery c ∈ R d , we consider the modified Lax-Oleinik semi-group ( T ct ) t ∈ R + that corresponds to the cohomology class c . Assume that ( u c ) c ∈ D is a familyof fixed points of ( T ct ) that uniformly converge to u when c tends to .Then, if u is C and upper Green regular, u is a weak K.A.M. solution thatsatisfies lim c → d , ( u c , u ) = . In [Fat08], the symmetrical Lagrangian e L ( q , v ) = L ( q , − v ) is introducedand the symmetrical Lax-Oleinik semi-group is defined. More precisely, ifwe just here adopt the notation ( T Lt ) t > for the Lax-Oleinik semi-group for L and ( T L ,λ t ) t > for the discounted semi-group for L , we define • the symmetrical Lax-Oleinik semi-group ( ˜ T Lt ) t > is defined by ˜ T Lt u = − T ˜ Lt ( − u ); • the symmetrical discounted semi-group ( ˜ T L ,λ t ) t > is defined by ˜ T L ,λ t u = − T ˜ L ,λ ( − u ).Using its definition, we deduce easily for C and lower Green regularsolutions u to the symmetrical semi-group the d , convergence of • the symmetrical discounted solutions; • the image of an initial condition by the symmetrical Lax-Oleiniksemi-group; The existence of the limit is due to weak K.A.M. theorem, see [Fat08]. AND C -CONVERGENCE TO WEAK K.A.M. SOLUTIONS 7 • the symmetrical solutions depending on the cohomology class.R emarks . • For Theorems 3, 4 and 5, the fact that M = T d is not fundamental.But to give some correct statements on any closed manifold, wewould need to choose a “horizontal” subspace at any point by usinga connection. We preferred to avoid this, but a similar proof (incharts) could be given for any closed manifold. • Observe that this kind of convergence implies the convergence to 0in (Lebesgue) measure of the C -distances to the limit, for instance,in the case of Theorem 5, i.e. ∀ ε > , lim c → Leb (cid:16) { θ ∈ T d ; k D u ( θ ) − D u c ( θ ) k ≥ ε } (cid:17) = . • We will see in Subsection 4.2 by providing some example that wecannot improve this convergence in a uniform one for the C -distance d , .Moreover, we will build in Subsection 4.3 an example on a weak K.A.M.solution that is not upper Green regular nor lower Green regular and we willprove for this example that the conclusion of Theorem 4 is not valid. Notethat for this example, we will not work on a torus T d .We end this introduction by asking some question.Q uestion . Does there exist an example of which a weak K.A.M. solutionis C , upper Green regular but not lower Green regular?1.1. Notations.
As said before, π : T ∗ M → M is the canonical projection and the vertical subspace at x ∈ T ∗ M is V ( x ) = ker D π ( x ).We recall that if q = ( q i ) ≤ i ≤ d are coordinates in M , we define dual coor-dinates ( p i ) ≤ i ≤ d as follows: if η is an element of T ∗ q M , it can be written inthe basis ( dq , . . . , dq d ) as η = d X i = p i dq i , and then the coordinates of η are( q , . . . , q d , p , . . . , p d ).The usual symplectic form ω on T ∗ M is chosen in such a way that all thesecoordinates are symplectic. In other words, we have in dual coordinates ω = d X i = dq i ∧ d p i . M and then T ∗ M are endowed with a Riemannian metric and we denote by B ( x , r ) the open ball with center x and radius r .2. B asic facts about discounted equation Semi-concave functions.
M.-C. ARNAUD AND X. SU
Definition 5.
A function u : U → R defined on an open subset U of R d is semi-concave if there exists some constant K ∈ R such that ∀ x ∈ U , ∃ p ∈ R d , ∀ y ∈ U , u ( y ) ≤ u ( x ) + p ( y − x ) + K k y − x k . We also say that u is K -semi-concave .Then p is a super-di ff erential of u at x and we denote by D + u ( x ) the set ofsuper-di ff erentials of u at x.If M is a closed manifold, we fix a finite atlas A = { ( φ, V ) } . A functionu : M → R is said to be K-semi-concave if every u ◦ φ − is K-semi-concaveand p ∈ D + u ( x ) means that p ◦ D φ ( x ) ∈ D + ( u ◦ φ − ) (x). A good reference for semi-concave functions is [CS04]. We recall that asemi-concave function is always locally the sum of a concave function anda smooth function. We recalled in the introduction the following propertiesof the semi-concave functions. • They are Lipschitz and Lebesgue almost everywhere di ff erentiableby Rademacher Theorem (see [EG15]); • if q k ∈ M converges to q and if p k ∈ D + u ( q k ) converges to some p ∈ T ∗ M , then p ∈ D + u ( q ) (see [CS04]); • by Alexandrov Theorem (see [NP06]), they admit a second deriva-tive D u at Lebesgue almost every q ∈ M .Let H : T ∗ M → R be a C Tonelli Hamiltonian and L : T M → R be itsassociated Lagrangian via the Legendre transformation. α ( H ) is the Ma˜n´ecritical value of H . Definition 6.
A function u ∈ C ( M , R ) is called a weak K.A.M. solution ofnegative type of Hamilton-Jacobi equation (2) H ( x , du ( x )) = α ( H ) if (i) for each continuous piecewise C curve γ : [ t , t ] → M with t < t ,we haveu ( γ ( t )) − u ( γ ( t )) ≤ Z t t (cid:2) L ( γ ( s ) , ˙ γ ( s )) + α ( H ) (cid:3) ds ;(ii) for any x ∈ M, there exists a C curve γ : ( −∞ , → M with γ (0) = x such that for any t ≥ , we haveu ( x ) − u ( γ ( − t )) = Z − t (cid:2) L ( γ ( s ) , ˙ γ ( s )) + α ( H ) (cid:3) ds . A discounted version of (2) is the equation(3) λ u ( x ) + H ( x , du ( x )) = α ( H ) AND C -CONVERGENCE TO WEAK K.A.M. SOLUTIONS 9 where λ >
0. Note that the viscosity solution of (3) is unique and denotedby u λ . We call u λ the discounted solutions of (2) and it can be representedby the following formula u λ ( x ) = inf γ Z −∞ e λ s (cid:2) L ( γ ( s ) , ˙ γ ( s )) + α ( H ) (cid:3) ds , ∀ x ∈ M where the infimum is taken over all absolutely continuous curves γ : ( −∞ , → M with γ (0) = x .2.2. Discounted Dynamics.
We assume that H : T ∗ M → R is a TonelliHamiltonian. Let L : T M → R be the Lagrangian associated to H .We denote by ( ϕ λ t ) the flow that solves Equation (1) that we recall:(1) dqdt = ∂ H ∂ p ( q , p ) and d pdt = − ∂ H ∂ q ( q , p ) − λ p . Recall that the Legendre map L : T ∗ M → T M is a di ff eomorphism that isdefined by L ( q , p ) = ( q , ∂ H ∂ p ( q , p ))and we have L − ( q , v ) = ( q , ∂ L ∂ v ( q , v )) . Then the flow ( f λ t ) = ( L ◦ ϕ λ t ◦ L − ) solves the discounted Euler-Lagrangeequation(4) ddt ∂ L ∂ v ( γ, ˙ γ ) ! − ∂ L ∂ q ( γ, ˙ γ ) + λ ∂ L ∂ v ( γ, ˙ γ ) = . For any λ ∈ R and t >
0, we define the following action on M × M (5) a λ t ( q , q ) = inf γ Z − t e λ s (cid:2) L ( γ ( s ) , ˙ γ ( s )) + α ( H ) (cid:3) ds where the infimum is taken on all the absolutely continuous curves γ :[ − t , → M such that γ ( − t ) = q and γ (0) = q .Then the infimum in Equality (5) is a minimum and every γ where thisminimum is reached corresponds to a solution of the λ -discounted Euler-Lagrange equation, i.e. satisfies(4) ddt ∂ L ∂ v ( γ, ˙ γ ) ! − ∂ L ∂ q ( γ, ˙ γ ) + λ ∂ L ∂ v ( γ, ˙ γ ) = . Then γ is a minimizing curve and the corresponding orbits for the Euler-Lagrange and Hamiltonian flows are said to be minimizing . Proposition 2.
Any minimizing orbit has no conjugate points.
Proof.
Observe that if we define ˜ L ( q , v , t ) = e λ t L ( q , v ), Equation (4) is noth-ing else than the classical Euler-Lagrange equation for the time-dependentLagrangian ˜ L . For such an equation, it is well-known that along any mini-mizing orbit, there are no conjugate points. Using Legendre map, there arealso no conjugate points for the corresponding Hamiltonian orbit. (cid:3) Discounted Lax-Oleinik semi-groups.
Using methods similar to theones used in [Ber08], it can be proved that • every function a λ t is semi-concave; • for every minimizing curve γ in (5), − e − λ t ∂ L ∂ v ( γ ( − t ) , ˙ γ ( − t )) is a super-di ff erential of a λ t ( · , γ (0)) at γ ( − t ) and ∂ L ∂ v ( γ (0) , ˙ γ (0)) is a super-di ff erentialof a λ t ( γ ( − t ) , · ) at γ (0); • at ( q , q ), a λ t admits a derivative with respect to the first variable ifand only if it admits a derivative with respect to the second variableif and only if there is only one minimizing curve γ between ( − t , q )and (0 , q ). Then in this case, we have ∂ a λ t ∂ q ( q , q ) = − e − λ t ∂ L ∂ v ( γ ( − t ) , ˙ γ ( − t )) and ∂ a λ t ∂ q ( q , q ) = ∂ L ∂ v ( γ (0) , ˙ γ (0)) . The discounted Lax-Oleinik semi-group ( T λ t ) t > is defined on the set ofcontinuous functions u : M → R by(6) T λ t u ( q ) = inf γ e − λ t u ( γ ( − t )) + Z − t e λ s (cid:2) L ( γ ( s ) , ˙ γ ( s )) + α ( H ) (cid:3) ds ! , where the infimum is taken on all the absolutely continuous curves γ :[ − t , → M such that γ (0) = q . Then T λ t u is semi-concave for any t > γ : [ − t , → M is min-imizing in Equation (6), then γ is a solution for (4), ∂ L ∂ v ( γ ( − t ) , ˙ γ ( − t )) is asub-di ff erential of u at γ ( − t ) and ∂ L ∂ v ( γ (0) , ˙ γ (0)) is a super-di ff erential of T λ t u at γ (0).Moreover, when u is semi-concave, then u is di ff erentiable at γ ( − t ). Inthis case, we have(7) ϕ λ t ( γ ( − t ) , du ( γ ( − t ))) = ( q , ∂ L ∂ v ( γ (0) , ˙ γ (0))) ∈ G ( dT λ t u ) . As every T λ t u is semi-concave, it is Lipschitz and di ff erentiable on a sub-set D ⊂ M that has full Lebesgue measure. Then if q ∈ D , there is only oneminimizing curve in Equality (6), that is given by γ ( s ) = π ◦ ϕ λ s ( q , dT λ t u ( q ))for any s ∈ [ − t , (cid:0) q s , p s ) s ∈ [ − t , = ( ϕ λ s ( q , dT λ t u ( q )) (cid:1) s ∈ [ − t , is a piece of orbitfor the discounted Hamiltonian flow that joins a point of G ( du ) to a point of G ( dT λ t u ) and then AND C -CONVERGENCE TO WEAK K.A.M. SOLUTIONS 11 • for every s ∈ [ − t , T λ t + s u is di ff erentiable at q s = γ ( s ); • for every s ∈ [ − t , q s , p s ) ∈ G ( dT λ s + t u ) ⊂ ϕ λ t + s ( G ( du )) . Observe that this implies that(8) G ( dT λ t u ) ⊂ ϕ λ t ( G ( du )) . A priori compactness results.Proposition 3 (A priori compactness) . Let L be a Tonelli Lagrangian, λ > and t > . There exist a neighborhood N of ( L , λ ) in the compact-open C topology and a compact set K t ⊂ T M such that if ( L ′ , λ ′ ) ∈ N with L ′ Tonelli and λ ′ > and if γ : [ − t , → M is a minimizing orbit for ( L ′ , λ ′ ) ,then ( γ ( s ) , ˙ γ ( s )) ∈ K t ∀ s ∈ [ − t , . Proof.
We fix ε > g on M and t >
0. Let γ q , q : [ − t , → M be a geodesic for the metric g joining q and q . We have k ˙ γ q , q ( s ) k γ q , q ( s ) = d ( q , q ) t s ∈ [ − t , . Consequently, the compact set K = ( ( q , v ) ∈ T M : k v k ≤ diam( M ) t ) contains all the points ( γ q , q ( s ) , ˙ γ q , q ( s )) for s ∈ [ − t , M L : = max { max K L ( q , v ) , } . a L ,λ t ( q , q ) = inf η Z − t e λ s L ( η ( s ) , ˙ η ( s )) ds ≤ Z − t e λ s L ( γ q , q ( s ) , ˙ γ q , q ( s )) ds ≤ Z − t e λ s M L ds ≤ M L t (9)where the infimum is taken over all the absolutely continuous curves η :[ − t , → M such that η ( − t ) = q and η (0) = q .By the superlinearity of L , there exists R > diam( M ) t such that if k v k ≥ R we have L ( q , v ) ≥ ( M L + ε )(1 + e ( λ + ε ) t ) + ε. We introduce the notation K = { ( q , v ) ∈ T M : k v k ≤ R } . Step 2. We consider any Tonelli Lagrangian L ′ satisfying k L ′ − L k C , K ≤ ε and any λ ′ > | λ ′ − λ | ≤ ε .We deduce from the definition of R and K and the inequality that k L ′ − L k C , K ≤ ε that if k v k = R , we have L ′ ( q , v ) ≥ ( M L + ε )(1 + e ( λ + ε ) t ) . For every ( q , v ) ∈ T M with k v k > R , let w = R k v k v . By the convexity of L ′ ,we have L ′ ( q , w ) ≤ (1 − R k v k ) L ′ ( q , + R k v k L ′ ( q , v ) . So L ′ ( q , v ) ≥ k v k R L ′ ( q , w ) − ( k v k R − L ′ ( q , ≥ k v k R ( M L + ε )(1 + e ( λ + ε ) t ) − ( k v k R − M L + ε ) = k v k R ( M L + ε ) e ( λ + ε ) t + M L + ε > ( M L + ε ) e ( λ + ε ) t . We have then proven that(10) ∀ ( q , v ) < K , L ′ ( q , v ) > ( M L + ε ) e ( λ + ε ) t . Because k L ′ − L k C , K ≤ ε , we have M L ′ ≤ M L + ε and we deduce from (9)that a L ′ ,λ ′ t ( q , q ) ≤ ( M L + ε ) t . That is, if γ is minimizing for ( L ′ , λ ′ ) between − t and 0, we have Z − t e λ ′ s L ′ ( γ ( s ) , ˙ γ ( s )) ds ≤ ( M L + ε ) t . Hence, there exists s ∈ [ − t ,
0] such that L ′ ( γ ( s ) , ˙ γ ( s )) ≤ ( M L + ε ) e − λ ′ s ≤ ( M L + ε ) e ( λ + ε ) t . We deduce from Equation (10) that ( γ ( s ) , ˙ γ ( s )) ∈ K .Hence, if γ is minimizing for ( L ′ , λ ′ ) between − t and 0, we have ∀ s ∈ [ − t , , ( γ ( s ) , ˙ γ ( s )) ∈ φ [ − t , t ] L ′ ,λ ′ ( K ) . To conclude, note that the set [ k L ′ − L k C , K ≤ ε | λ ′ − λ |≤ ε φ [ − t , t ] L ′ ,λ ′ ( K )is relatively compact in T M because of the continuous dependence of thesolutions of a di ff erential equation from the parameters (see e.g. [HW95]). (cid:3) Using Legendre duality, we deduce a similar statement for Tonelli Hamil-tonians. AND C -CONVERGENCE TO WEAK K.A.M. SOLUTIONS 13 Corollary 3.
Let H be a Tonelli Hamiltonian, λ > and t > . There exist aneighborhood N of ( H , λ ) in the compact-open C topology and a compactset K t ⊂ T ∗ M such that if ( H ′ , λ ′ ) ∈ N with H ′ Tonelli and λ ′ > and if ( ϕ Hs ( x )) s ∈ [ − t , is a minimizing piece of orbit for ( H ′ , λ ′ ) , then ϕ Hs ( x ) ∈ K t ∀ s ∈ [ − t , .
3. G reen bundles
Green bundles will be the main ingredient to prove the results of C con-vergence. Here we state some of their properties.3.1. Proof of Proposition 1.
The first goal of this section is to prove Propo-sition 1. The proof is very similar to the one given in [Arn08] for TonelliHamiltonian flows. With the notations of Proposition 1, we use I − = ( −∞ , t ) ∩ I and I + = ( t , + ∞ ) ∩ I .Because there are no conjugate points on I , we have for every s , s ′ in ID ϕ s ′ − s V ( ϕ s ( x )) ∩ V ( ϕ s ′ ( x )) , { } and then by taking their images by D ϕ t − s ′ ( x ), H ( G t − s ( ϕ t ( x ))) − H ( G t − s ′ ( ϕ t ( x )))is always a non-degenerate symmetric matrix. As this continuously dependson s , s ′ , we deduce that its signature is constant on each connected set D − = { ( s , s ′ ) ∈ I − , s < s ′ } ; D + = { ( s , s ′ ) ∈ I + , s < s ′ } ; D = I − × I + . To determine these three signatures, we only consider the case where | s | and | s ′ | are small. We use the notation in usual coordinates for D ϕ s ( y ) M s ( y ) = a s ( y ) b s ( y ) c s ( y ) d s ( y ) ! . Then we have d s ( y ) = + O ( s ) and b s ( y ) = O ( s ) and we deduce fromlinearized discounted equations that ˙ b s ( y ) = H q , p ( y ) b s ( y ) + H p , p ( y ) d s ( y ) = H p , p ( y ) + O ( s ) and then b s ( y ) = sH p , p ( y ) + O ( s ), the O ( s ) being uniform in y . This implies that H ( G s ( ϕ t ( x ))) = d s ( ϕ t − s ( x )) . ( b s ( ϕ t − s ( x ))) − = s ( H p , p ( ϕ t ( x )) − + O ( s )) . We deduce for s > H ( G s ( ϕ t ( x ))) − H ( G s ( ϕ t ( x ))) = − s ( H p , p ( ϕ t ( x )) − + O ( s )) < H ( G s ( ϕ t ( x ))) − H ( G − s ( ϕ t ( x ))) = s ( H p , p ( ϕ t ( x )) − + O ( s )) > H ( G − s ( ϕ t ( x ))) − H ( G − s ( ϕ t ( x ))) = − s ( H p , p ( ϕ t ( x )) − + O ( s )) < . This finishes the proof of Proposition 1.3.2.
Continuity of G c ,λ t and semi-continuity of the two Green bundles. N otation . • We consider a map c ∈ C H c defined on some metric space C thatis continuous for the C open-compact topology and such that every H c : T ∗ M → R is a C -convex in the fiber Hamiltonian. We willdenote by ( ϕ c ,λ t ) the flow associated to the λ -discounted equation for H c . • then we use the notation G c ,λ t ( x ) = D ϕ c ,λ t V ( ϕ c ,λ − t ( x )).Observe that the map g : ( t , x , c , λ ) G c ,λ − t ( x )is continuous.N otation . We then define U as being the set of the ( t , x , c , λ ) ∈ R × T ∗ M × C × R such that there is no conjugate points for H c ,λ on the piece of orbit of x between x and ϕ c ,λ t ( x ).Moreover U − = U ∩ ( R − × T ∗ M × C × R ) and U + = U ∩ ( R + × T ∗ M × C × R ).Because g is continuous, U is open and the map h = H ◦ g : U − ∪ U + →S d is continuous. We deduce from Proposition 1 that h is increasing in thefirst variable on U + (resp. U − ) and that if ( t , x , c , λ ) ∈ U − and ( s , x , c , λ ) ∈U + , we have(11) h ( s , x , c , λ ) < h ( t , x , c , λ ) . N otation . We are interested in infinite time interval, so we introduce U ∞− = { ( x , c , λ ); ∀ t ∈ R − , ( t , x , c , λ ) ∈ U − } and U ∞ + = { ( x , c , λ ); ∀ t ∈ R + , ( t , x , c , λ ) ∈ U + } . We deduce from the continuity of h that U ∞− and U ∞ + are closed. More-over, we can define • for ( x , c , λ ) ∈ U ∞− the Green bundle G c ,λ + ( x ) = lim t → + ∞ G c ,λ t ( x ); • for ( x , c , λ ) ∈ U ∞ + the Green bundle G c ,λ − ( x ) = lim t → + ∞ G c ,λ − t ( x ).Then we have • h − ( x , c , λ ) : = H ( G c ,λ − ( x )) = lim t → + ∞ h ( t , x , c , λ ); • h + ( x , c , λ ) : = H ( G c ,λ + ( x )) = lim t → + ∞ h ( − t , x , c , λ ). AND C -CONVERGENCE TO WEAK K.A.M. SOLUTIONS 15 Observe that because of Equation (11), we have ∀ ( x , c , λ ) ∈ U ∞− ∩ U ∞ + , h − ( x , c , λ ) ≤ h + ( x , c , λ ) . We deduce from the fact that the considered functions are continuous and t -increasing the following proposition about semi-continuity. Proposition 4.
Let us fix ( x , c , λ ) ∈ U ∞ + and ε > . Then there exist aneighborhood N − of ( x , c , λ ) in T ∗ M × C × R and T > such that • for every ( t , y , c , λ ) ∈ U + ∩ ( R + × N − ) with t ≥ T , we haveh ( t , y , c , λ ) ≥ h − ( x , c , λ ) − ε ; • for every ( y , c , λ ) ∈ N − ∩ U ∞ + , we haveh − ( y , c , λ ) ≥ h − ( x , c , λ ) − ε . Proof.
The second point comes from the first point by taking the limit for t → + ∞ . Now we prove the first point.Because lim t → + ∞ h ( t , x , c , λ ) = h − ( x , c , λ ), there exists some T > h ( T , x , c , λ ) > h − ( x , c , λ ) − ε . By continuity of h , there exists a neighborhood N − of ( x , c , λ ) in T ∗ M × C × R such that ∀ ( y , c , λ ) ∈ N − , ( T , y , c , λ ) ∈ U + ⇒ h ( T , y , c , λ ) > h − ( x , c , λ ) − ε . Because h is increasing in t , we have ∀ t ≥ T , ∀ ( y , c , λ ) ∈ N − , ( t , y , c , λ ) ∈ U + ⇒ h ( t , y , c , λ ) > h − ( x , c , λ ) − ε . (cid:3) We have of course in a similar way a statement for the positive times.
Proposition 5.
Let us fix ( x , c , λ ) ∈ U ∞− and ε > . Then there exists aneighborhood N + of ( x , c , λ ) in T ∗ M × C × R and T > such that • for every ( − t , y , c , λ ) ∈ U − ∩ ( R − × N + ) with t ≥ T , we haveh ( − t , y , c , λ ) ≤ h + ( x , c , λ ) + ε ; • for every ( y , c , λ ) ∈ N + ∩ U ∞− , we haveh + ( y , c , λ ) ≤ h + ( x , c , λ ) + ε . Comparison between Green bundles and second derivatives.Proposition 6.
Let u ∈ C ( M , R ) and t > . Then for every point q whereT λ t u is twice di ff erentiable • ( ϕ λ s ( q , dT λ t u ( q ))) s ∈ [ − t , has no conjugate points; • D T λ t u ( q ) ≤ H ( G λ t ( q , dT λ t u ( q ))) .Proof. Let us now consider a point q where T λ t u is twice di ff erentiable.Then the infimum in Equation (6) is attained at a unique solution γ : [ − t , → M for (4) and we have ∂ L ∂ v ( γ (0) , ˙ γ (0)) = dT λ t u ( q ) . As γ is minimizing, ( ϕ λ s ( q , dT λ t u ( q ))) s ∈ [ − t , has no conjugate points.Because of the definition of the semi-group in (6), we have T λ t u ( q ) − e − λ t u ( γ ( − t )) = a λ t ( γ ( − t ) , γ (0)) + α ( H ) t and ∀ q ∈ M , T λ t u ( q ) − e − λ t u ( γ ( − t )) ≤ a λ t ( γ ( − t ) , q ) + α ( H ) t . Subtracting these two equations, we deduce T λ t u ( q ) − T λ t u ( q ) ≤ a λ t ( γ ( − t ) , q ) − a λ t ( γ ( − t ) , q ) . These two functions vanish for q = q and have the same derivative ∂ L ∂ v ( q , ˙ γ (0))at q . If we succeed in proving that(12) ∂ a λ t ∂ q ( γ ( − t ) , q ) = H ( G λ t ( q , dT λ t u ( q ))) , we will deduce that D T λ t u ( q ) ≤ H ( G λ t ( q , dT λ t u ( q ))) . The arguments that we use to prove Equality (12) are similar to the onesgiven in [Arn12].
Lemma 1.
For every t > and every q ∈ M, the function v tq = a λ t ( q , · ) issemi-concave, and satisfies G ( dv tq ) ⊂ ϕ λ t ( T ∗ q M ) . Proof.
Because a λ t is semi-concave, the function v tq is semi-concave andthen Lipschitz. By Rademacher’s theorem v tq is di ff erentiable almost every-where.Moreover, if q is a point where v tq is di ff erentiable, then v tq has exactly onesuper-di ff erential at this point, there is only one minimizing arc η joining( − t , q ) to (0 , q ), and we have: • dv tq ( q ) = ∂ L ∂ v ( η (0) , ˙ η (0)); AND C -CONVERGENCE TO WEAK K.A.M. SOLUTIONS 17 • ( η ( − t ) , ∂ L ∂ v ( η ( − t ) , ˙ η ( − t ))) = ( q , ∂ L ∂ v ( η ( − t ) , ˙ η ( − t ))) ∈ T ∗ q M ; • ϕ λ t (cid:16) q , ∂ L ∂ v ( η ( − t ) , ˙ η ( − t )) (cid:17) = ( η (0) , ∂ L ∂ v ( η (0) , ˙ η (0))) = ( q , dv tq ( q )).Then we have proved that: ϕ λ t ( T ∗ q M ) ⊃ G ( dv tq ). Hence, we have selected apseudograph in the image ϕ λ t ( T ∗ q M ) of the vertical. (cid:3) We come back to the point q where T λ t u is twice di ff erentiable and recallthat ( ϕ λ s ( q , dT λ t u ( q ))) s ∈ [ − t , = ( q s , p s ) s ∈ [ − t , has no conjugate point becauseit is minimizing and that γ = ( π ◦ ϕ λ s ( q , dT λ t u ( q ))) s ∈ [ − t , is the uniqueminimizing arc joining γ ( − t ) to γ (0) = q . Lemma 2.
There exists a neighborhood V of q = γ (0) in M such thatv t γ ( − t ) is as regular as H is (then at least C ) and thenD v t γ ( − t ) ( q ) = H ( G λ t ( q , dT λ t u ( q ))) . Proof.
Lemma 1 proves that G ( dv t γ ( − t ) ) ⊂ ϕ λ t ( T ∗ γ ( − t ) M ). Let us now provethat v t γ ( − t ) is smooth near q .We use now the so-called “a priori compactness lemma” (see Corollary 3)that says to us that there exists a constant K t = K > γ ( s )) s ∈ [0 , t ] of any minimizing arc γ between any points q ∈ M and q ′ ∈ M are bounded by K ; hence if we denote by K the set of the minimizing arcsthat are parametrized by [ − t , K is a compact set for the C topologybecause it is the image by the projection π of a closed set of bounded orbits.Let us denote by K the set of η ∈ K such that η ( − t ) = γ ( − t ); then K iscompact. Let us introduce another notation: K ( q ) = { η ∈ K : η (0) = q } .Then K ( q ) = { γ } and hence, because K is closed, for q close enough to q , all the elements of K ( q ) are C close to γ .Moreover, ϕ λ t ( T ∗ γ ( − t ) M ) is a submanifold of T ∗ M that contains( q , ∂ L ∂ v ( q , ˙ γ (0))) = ( q , p ) . Its tangent space at ( q , p ) is G λ t ( q , p ), which is transverse to the verti-cal because ( q s , p s ) s ∈ [ − t , has no conjugate vectors. Hence, the manifold ϕ λ t ( T ∗ γ ( − t ) M ) is, in a neighborhood U of ( q , p ), the graph of a C sectionof T ∗ M defined on a neighborhood V of q in M . Moreover, because thissubmanifold is Lagrangian (indeed, T ∗ γ ( − t ) M is Lagrangian and ϕ λ t is con-formally symplectic), it is the graph of du where u : V → R is a C function.Now, if q is close enough to q , we know that all the elements η of K ( q )are C close to γ , and then that ( q , ∂ L ∂ v ( η (0) , ˙ η (0))) belongs to the neigh-borhood U of ( q , p ) = ( q , ∂ L ∂ v ( γ (0) , ˙ γ (0))) and to ϕ λ t ( T ∗ γ ( − t ) M ). Because ϕ λ t ( T ∗ γ ( − t ) M ) ∩ U is a graph, this element is unique: K ( q ) has only one element and v t γ ( − t ) is di ff erentiable at q , with dv t γ ( − t ) ( q ) = ∂ L ∂ v ( η (0) , ˙ η (0)) = du ( q ). We deduce that near q , on the set of di ff erentiability of v t γ ( − t ) , dv t γ ( − t ) is equal to du ; because v t γ ( − t ) and u are Lipschitz on V and theirdi ff erentials are equal almost everywhere, we deduce that on V , v t γ ( − t ) − u is constant. Hence, on a neighborhood V of q , v t γ ( − t ) is C and D v t γ ( − t ) ( q ) = H ( G λ t ( q , dT λ t u ( q ))) . (cid:3) (cid:3) On the dynamical criterion in the Hamiltonian case.
We recall heretwo dynamical criteria concerning the Green bundles that are proven in[Arn08].
Proposition 7. (Proposition 3.12 in [Arn08])
Let x ∈ T ∗ M be a point whosenegative orbit under the Tonelli Hamiltonian flow ( ϕ Ht ) has no conjugatepoints and let v ∈ T x ( T ∗ M ) be a tangent vector. Then, if v < G + ( x ) , we have lim t → + ∞ k D ( π ◦ ϕ H − t ) v k = + ∞ . When moreover we pay attention to points that are far from the criticalpoints of H , we can use a symplectic reduction on the level E of H in aneighborhood of such points by using the canonical projection p x : T x E → T x E / R . X H ( x ). The following statement can be deduced from Proposition3.17 in [Arn08]. Proposition 8.
Let x ∈ T ∗ M be a point whose negative orbit under theTonelli Hamiltonian flow ( ϕ Ht ) has no conjugate points and let v ∈ T x ( T ∗ M ) be a tangent vector. We assume that ( t n ) n ∈ N is a sequence of positive realnumbers tending to + ∞ such that the angle of X H ( ϕ H − t n ( x )) with ker D π ( ϕ H − t n ( x )) = V ( ϕ H − t n ( x )) is uniformly bounded from below by some pos-itive constant. Then, if v < G + ( x ) , we have lim t → + ∞ k p ϕ − t ( x ) ◦ D ϕ H − t n v k = + ∞ .
4. E xamples and counter - examples Examples of upper and lower Green regular weak K.A.M. solu-tions.
The following proposition is proven in [Arn14]. It can also be de-duced from the dynamical criterion and Proposition 4.12 of [Arn08].
Proposition 9.
Assume that H : T ∗ T d → R is a Tonelli Hamiltonian thathas a C , weak K.A.M. solution u : T d → R such that there exists t > forwhich ϕ Ht is bi-Lipschitz conjugate to some rotation of T d . Then u is upperand lower Green regular. Observe that it is proved in [Fat08] (Theorem 4.11.5) that every C weak K.A.M.solution is in fact C , . AND C -CONVERGENCE TO WEAK K.A.M. SOLUTIONS 19 The ideas of the proof of the following proposition are contained in[Arn14]. Let us recall that a vector field is Kupka-Smale if all its periodicand fixed points are hyperbolic.
Proposition 10.
Assume that H : T ∗ T d → R is a Tonelli Hamiltonian thathas a C weak K.A.M. solution u : T d → R such that X H |G ( du ) is Kupka-Smale. Then u is upper and lower Green regular.Proof. We denote by O , . . . , O m the periodic (eventually critical) orbits thatare contained in G ( du ) and by W s ( O i , ( ϕ Ht |G ( du ) )) and W u ( O i , ( ϕ Ht |G ( du ) )) theirstable and unstable manifolds.Because the non-wandering set of ( ϕ Ht |G ( du ) ) is O ∪ · · · ∪ O m , then G ( du ) = [ ≤ i , j ≤ n (cid:16) W s ( O j , ( ϕ Ht |G ( du ) )) ∩ W u ( O i , ( ϕ Ht |G ( du ) )) (cid:17) . If O i is not an attractive orbit for ( ϕ Ht |G ( du ) ) then W s ( x i , ( ϕ Ht |G ( du ) ) is an im-mersed manifold whose dimension is less that d and then has zero Lebesguemeasure. We deduce that there is a dense set D in G ( du ) such that for all x ∈ D , ϕ Ht ( x ) tends to a repulsive periodic orbit when t tends to −∞ andtends to an attractive periodic orbit when t tends to + ∞ .Let us consider x ∈ D .We assume that ( ϕ Ht ( x )) tends to a critical attractive fixed point x when t tends to + ∞ . We can choose k ∈ ]0 ,
1[ and a Riemannian metric such thatin a neighborhood V of x : (cid:13)(cid:13)(cid:13) D ϕ H |G ( du ) ( y ) (cid:13)(cid:13)(cid:13) ≤ k , ∀ y ∈ V . If t ≥ T is greatenough, ϕ Ht ( x ) belongs to V and (cid:13)(cid:13)(cid:13) D ϕ H |G ( du ) ( ϕ Ht ( x )) (cid:13)(cid:13)(cid:13) ≤ k . We deduce: ∀ n ∈ N , k D ϕ HT + n ( x ) k ≤ k D ϕ HT ( x ) k n − Y i = k D ϕ H |G ( du ) ( ϕ HT + i ( x )) k ≤ k D ϕ HT |G ( du ) ( ϕ HT ( x )) k k n ;hence the sequence ( D ϕ HT + n ( x )) n ∈ N is bounded.If ( ϕ Ht ( x )) tends to a true attractive periodic orbit O , then O is a nor-mally hyperbolic (attractive) submanifold for ( ϕ Ht |G ( du ) ). Then there exists x ∈ O such that x ∈ W s ( x , ϕ Ht |G ( du ) ) (see for example [HPS77]). Any vec-tor w of T x G ( du ) can be written as the sum of λ X H ( x ) where X H is theHamiltonian vector field and a vector v tangent to W s ( x , ϕ Ht |G ( du ) ). Then D ϕ Ht ( x ) X H ( x ) = X H ( ϕ Ht ( x )) is bounded and D ϕ Ht ( x ) v tends to 0 when t tendsto + ∞ . Finally, the family ( D ϕ Ht |G ( du ) ( x )) t > is bounded. By the dynamicalcriterion, this implies that ∀ x ∈ G ( du ) , T x G ( du ) = G + ( x )and then u is upper Green regular. (cid:3) The following result is more or less proven in [Arn08] (see Proposi-tion 4.18, the statement is di ff erent but the proof is similar). Proposition 11.
Assume that H : T ∗ T → R is a Tonelli Hamiltonian thathas a C weak K.A.M. solution u : T → R such that all the critical points ofH that are contained in the graph of du are hyperbolic for the Hamiltonianflow. Then u is upper Green regular.Proof. As the critical points of H contained in G ( du ) are hyperbolic for theHamiltonian flow, their set S = { s , . . . , s n } is finite.We denote by W u the union of the unstable sets of the critical points of H in G ( du ) W u = n [ i = W u ( s i ) ∩ G ( du ) . Observe that W u and R = G ( du ) \ W u are measurable sets. The strategyis then to show that at Lebesgue almost every q in π ( W u ) and π ( R ), wehave T ( q , du ( q )) G ( du ) = G + ( q , du ( q )). We will conclude that u is upper Greenregular. Case of π ( R ) . For every i ∈ [1 , n ] ∩ Z , we construct a decreasing sequence( D k ( i )) k ∈ N of open discs that are centered at π ( s i ) and denote by ( ˜ D k ( i )) k ∈ N its lift to G ( du ). We denote by f = ϕ H − the time -1 flow. We also use thenotations ˜ D k = S ni = ˜ D k ( i ) and R k = R \ ˜ D k . For every x ∈ R and m ≥
1, weintroduce the notation n m ( x ) = min { n ≥ f n ( x ) ∈ R k } and F km ( x ) = f n m ( x ) ( x ) when n m ( x ) , + ∞ . Then, for every x ∈ R , there exists k ≥ F mk ( x ) are defined for every m ≥ k ≥ k .Hence, if E k is the set of elements of R k for which F mk is defined for every m , we have ∞ [ k = E k = R . We know by [Fat03] that du is Lipschitz and then di ff erentiable Lebesguealmost everywhere by Rademacher Theorem. Then if we use the notations E ′ k = { q ∈ π ( E k ); D u ( q ) exists } and ˜ E ′ k = { ( q , du ( q )); q ∈ E ′ k } , we knowthat [ k ∈ N E ′ k has full Lebesgue measure into π ( R ).We have thenLeb( π ( R k )) ≥ Leb( π ( F mk ( ˜ E ′ k ))) = Z E ′ k d (cid:0) π ◦ F mk ( · , du ( · )) (cid:1) ∗ Leb , and so Leb( π ( R k )) ≥ R E ′ k (cid:12)(cid:12)(cid:12)(cid:12) det (cid:16) D ( π ◦ F mk ( · , du ( · )) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) d Leb = R E ′ k (cid:12)(cid:12)(cid:12)(cid:12) det (cid:16) D ( π ◦ ϕ H − n m )( · , D u ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) d Leb . AND C -CONVERGENCE TO WEAK K.A.M. SOLUTIONS 21 We deduce from Fatou lemma that at Lebesgue almost everywhere point q in E ′ k , we have(13) lim inf m → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det (cid:16) D ( π ◦ ϕ H − n m )( q , du ( q )) (cid:17) T ( q , du ( q )) G ( du ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < + ∞ . Using the definition of R k , let us note that there exists a constant C k suchthat(14) ∀ x , y ∈ R k , C k ≤ k X H ( x ) kk X H ( y ) k ≤ C k . We then use a symplectic reduction on the energy level of x by X H as ex-plained in Subsection 3.4.Let us denote by ℓ a Lipschitz constant for du . Observe that(15) (cid:12)(cid:12)(cid:12)(cid:12) det (cid:16) D ( π ◦ ϕ H − n m )( · , D u ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≥ + ℓ (cid:12)(cid:12)(cid:12)(cid:12) det (cid:16) D ϕ H − n m T G ( du ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) We deduce from equations (13), (14) and (15) that(16) lim inf m → + ∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) p ϕ H − nm ( q , du ( q )) ◦ D ϕ H − n m )( q , du ( q )) (cid:19) T ( q , du ( q )) G ( du ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < + ∞ . Using Proposition 8, we deduce that T ( q , du ( q )) G ( du ) ⊂ G + ( q , du ( q )) and then T ( q , du ( q )) G ( du ) = G + ( q , du ( q )). Case of π ( W u ) . We denote by W u loc the intersection of G ( du ) with theunion of the local unstable submanifolds of the s i . If x ∈ W u , there exists apositive T > x ∈ ϕ HT ( W u loc ). Then we have two cases.1) We say that x is simple if there exists a neighborhood U x of x in G ( du )such that the only points of U x ∩ ϕ HT ( W u loc ) are on the orbit of x . Observethat the set of simple orbits is countable and thus the projection of the set ofsimple points has zero Lebesgue measure.2) Let W ′ be the set of non simple points of W u at which G ( du ) has a tangentsubspace. The projection of this set has full Lebesgue measure in π ( W u ). If x ∈ W ′ , then for some i we have x ∈ ϕ HT ( W u ( s i )) and because x is not simple,we deduce that T x G ( du ) = T x W u ( s i ). As T x W u ( s i ) = G + ( x ), we obtain thewanted result. (cid:3) An example where the convergence is not C -uniform. We willshow that for the pendulum, the dependance on the cohomology class isnot continuous for the C uniform topology. The Hamiltonian is given by H ( q , p ) = p + cos(2 π q ) . We use the notation I + = R p − cos(2 π q )) dq . Then the map c : [1 , + ∞ ) → [ I + , ∞ ) defined by c ( e ) = Z p e − cos(2 π q )) dq . is a homeomorphism and even a di ff eomorphism when restricted to (1 , + ∞ ).For every I ∈ [ I + , + ∞ ), the function u I ( q ) = Z q (cid:16) p c − ( I ) − cos(2 π s )) − I (cid:17) ds is the unique (up to the addition of a constant) weak K.A.M. solution for T I . Observe that every u I is C .Moreover, u ′ I + is smooth on (0 ,
1) and because of the dynamical criterion inProposition 7, we have for every q ∈ (0 , R (1 , u ′′ I + ( q )) = T ( q , I + + u ′ I + ( q )) W u (0 , = G + ( q , I + + u ′ I + ( q )) . Hence u I + is upper Green regular (and also Green lower regular).There exists α ∈ (0 ,
1) such that for every q ∈ (0 , α ), we have u ′′ I + ( q ) = π sin(2 π q ) p − cos(2 π q )) > . For I > I + , u I is smooth and u ′ I ( q ) = p c − ( I ) − cos(2 π q )) − I attains itsminimum at q = u ′′ I (0) =
0. Hence there exists α ( I ) ∈ (0 , α ) suchthat ∀ q ∈ [0 , α ( I )] , u ′′ I ( q ) < . We then deduce ∀ I > I + , k u ′′ I − u ′′ I + k L ∞ ≥ (cid:13)(cid:13)(cid:13)(cid:13) ( u ′′ I − u ′′ I + ) (cid:12)(cid:12)(cid:12) (0 ,α ( I )) (cid:13)(cid:13)(cid:13)(cid:13) ≥ . We don’t have continuous dependence of u I on I for the uniform C dis-tance.4.3. Examples of weak K.A.M. solutions that are not upper Green reg-ular nor lower Green regular and to which the Lax-Oleinik semi-groupdoesn’t d , -converge. Let S be a closed surface with negative curvature.Let us denote by M = T S its unitary tangent bundle and by X the ge-odesic vector field. We then consider the Ma˜n´e Lagrangian (see [Mn92]) L : T M → R that is defined by L ( q , v ) = k v − X ( q ) k . AND C -CONVERGENCE TO WEAK K.A.M. SOLUTIONS 23 The corresponding Hamiltonian is given by H ( q , p ) = k p k + p . X ( q ) . Observe that the critical level is H = { H = } because this level contains anexact Lagrangian graph (see [Fat08]).Then 0 is a weak K.A.M. solution. We denote by Z the zero section in T ∗ M . The set Z is hyperbolic for the restriction of ( ϕ Ht ) to the energy level H = { H = } . We denote by E s , E u the 3-dimensional stable and unstablebundles along Z : they contain the vector field direction and also the strongstable (unstable) bundle. By [Arn12], we have E u ( x ) = G + ( x ) and E s ( x ) = G − ( x ). As ( ϕ Ht | Z ) is Anosov, the intersection of E u ( x ) (resp. E s ( x )) with T x Z is 2-dimensional and then we have(17) ∀ x ∈ Z , G + ( x ) , T x Z . So u is nowhere upper Green regular.Let us now prove that u is the only weak K.A.M. solution (up to theaddition of a constant).As the flow ( ψ t ) of X is transitive, the projected Aubry set for H is the whole M . To prove that, we use the characterization of the projected Aubry set thatis given in [Fat08]. Let q ∈ M be any point. As ( ψ t ) is transitive, for everyneighborhood V of q and any T >
0, there exist q ∈ V and t ≥ T such that q , ψ t ( q ) ∈ V . Let γ : [0 , t + ε ] → M be the closed arc that is made with thethree following pieces.(1) the straight segment that joins q to q with unitary derivative;(2) the arc of orbit ( ψ s q ) s ∈ [0 , t ] ;(3) the straight segment that joins ψ t ( q ) to q with unitary derivative.The Lagrangian action of the first and third parts of this arc are very small,and the second one is zero because we have a piece of orbit. Hence theaction of γ can be very small. Hence q belongs to the Aubry set.This implies that, up to the addition of a constant, there is only one weakK.A.M. solution, and so the only weak K.A.M. solutions are the constantfunctions.We will now build an example of an initial condition u for the Lax-Oleinik semi-group such that the conclusion of Theorem 4 is not satisfied,i.e. such that the family ( d , ( T t u , t ∈ [0 , + ∞ [ doesn’t tend to 0 when t tendsto + ∞ .We choose a large set of points ( q , q n ,
0) in Z , we fix some T > N otation . The Lagrangian action is denoted by a T ( q , q ) = inf R T L ( γ ( t ) , ˙ γ ( t )) dt where the infimum is taken over all the absolutely continuous curves γ :[0 , T ] → M such that γ (0) = q and γ ( T ) = q .The a T is semi-concave and then Lipschitz (see [Ber08]). Define • u Ti ( q ) = a T ( q i , q ); • u T ( q ) = min { u Ti ( q ); 1 ≤ i ≤ n } .All these functions are non-negative and K T -semi-concave. By Lemma 1,we have ∀ i ∈ { , . . . , n } , G ( du Ti ) ⊂ ϕ HT ( T ∗ q i M );and so because of semi-concavity(18) G ( du T ) ⊂ n [ i = ϕ HT ( T ∗ q i M ) . Note that u Ti ( ψ T ( q i )) = i ∈ { , . . . , n } , we have u T ( ψ T ( q i )) =
0. Because u T is K T -semi-concave, non-negative and vanishes at the points ψ T ( q i ), if we choose the q i ’s in such a way that the ψ T ( q i ) are ǫ -dense in M for a small ε , then u T is C close to 0.Let us now prove that u T can be chosen such that the graph of du T isin a small neighborhood of the zero section. We denote by M T the set of T -minimizing orbits for the Euler-Lagrange flow ( f Lt ) t ∈ R . M T = (cid:26) ( f LT ( x )) t ∈ [0 , T ] ; (cid:16) π ◦ f LT ( x ) (cid:17) t ∈ [0 , T ] is minimizing (cid:27) . Observe that M T is compact. We can endow it as well with the C or C topology that are equal. We have • ∀ Γ ∈ M T , A L ( Γ ) = R T L ◦ Γ ( t ) dt ≥ • ∀ Γ ∈ M T , A L ( Γ ) = ⇔ Γ ([0 , T ]) ⊂ G ( X ).We introduce the notation Z T = { Γ ∈ M T ; Γ ([0 , T ]) ⊂ G ( X ) } . Then Z T = { Γ ∈ M T ; A L ( Γ ) = } is compact. We now fix a small neighbor-hood N T of Z T in M T . We introduce ε =
12 inf { A L ( Γ ); Γ ∈ M T \N T } . Then ε >
0. We choose α > ∀ q , q , q ∈ M , d ( q , q ) < α ⇒| a T ( q , q ) − a T ( q , q ) | < ε. AND C -CONVERGENCE TO WEAK K.A.M. SOLUTIONS 25 We choose a finite number of points x , . . . , x n ∈ G ( X ) on the graph of thevector-field X such that M ⊂ n [ i = B ( π ◦ f LT ( x i ) , α ) . and use the notation x i = ( q i , X ( q i )). Then we define the u Ti ’s and u T asbefore. Let us consider q ∈ M . Then q belongs to some ball B ( π ◦ f LT ( x i ) , α )and so we have u Ti ( q ) = a T ( q i , q ) − a T ( q i , π ◦ f LT ( x i )) < ε. We deduce that u T ( q ) ≤ u Ti ( q ) < ε . Let j ∈ { , . . . , n } be such that u Tj ( q ) = u T ( q ). We have u Tj ( q ) = a T ( q j , q ) < ε . Hence for every Γ ∈ M T such that π ◦ Γ (0) = q j , π ◦ Γ ( T ) = q , we have Γ ∈ N T .If now du T ( q ) exists, we have du T ( q ) = du Tj ( q ), the minimizing Γ is uniqueand we denote γ = π ◦ Γ , then ˙ γ is C -close to X ◦ γ because Γ ∈ N T . Byusing Legendre map, this implies that du Tj ◦ γ is C -close to the zero sectionand then du T ( q ) = du Tj ( γ ( T )) is close to zero.So we have proved that we can assume that the graph of du T is containedin a neighborhood N of the zero section that is as small as we want. By[Ber08], observe that(19) ∀ t ≥ , T t u T = u t + T and G ( du T + t ) ⊂ ϕ Ht ( G ( du T )) . By continuity of the flow and compacity of the closure of G ( du T ), thereexists a small τ > ∀ t ∈ [0 , τ ] , G ( du T + t ) ⊂ ϕ Ht ( G ( du T )) ⊂ N . We now use Lemma 7 of [Arn05] and find some β > ∀ u ∈ C ( M , R ) , k u k ∞ < β ⇒ G ( dT τ u ) ⊂ N . We can assume that u T satifies k u T k ∞ < β . Then for every t ≥ τ , we have k T t − τ u T − k ∞ = k T t − τ u T − T t − τ k ∞ ≤ k u T k ∞ < β because of the non-expansiveness of the Lax-Oleinik semi-group (see [Fat08]).We deduce ∀ t ≥ τ, G ( du t + T ) = G ( dT τ ( T t − τ u T )) ⊂ N . We have then proved that(20) ∀ t ≥ , G ( dT t u T ) ⊂ N . Let us recall that the flow ( ψ t ) is Anosov. This implies that the cocyclethat we will now introduce is hyperbolic on Z . The cocycle is defined in a fiber bundle over a neighborhood N of thezero section Z in T ∗ M . At a point x ∈ N , we consider the tangent space T x H x of the energy level H x = { H = H ( x ) } . Then E x is defined as being thereduced linear space T x H x / R . X H ( x ) endowed with the quotient norm k · k and the corresponding projection is denoted by p x : T x H x → E x .As we take the quotient of an Anosov flow by the vector field, the corre-sponding reduced cocycle ( M t ) of ( D ϕ Ht ) restricted to Z is hyperbolic, andhas an invariant splitting E = E s ⊕ E u where the stable and unstable bundlesare 2-dimensional. By [Yoc95], we can translate the hyperbolicity condi-tion by using some cones. This is an open condition and we can extendthese cones to a neighborhood N of Z such that • there exists a continuous splitting E = E ⊕ E on N that coincideswith E = E s ⊕ E u on Z and two norms | · | i on E i such that C x = { v = v + v , v ∈ E x , v ∈ E x , | v | , x ≤ | v | , x } ;the family ( C x ) x ∈N is the associated cone field; the dual cone field isthe family ( C ∗ x ) x ∈N defined by C ∗ x = E x \ int C x . • for some constant c >
0, we have for every x ∈ N , v ∈ E x and v ∈ E x c − k v + v k ≤ max {| v | , x , | v | , x } ≤ c k v + v k x . • there exists an integer m ≥ λ, µ > x ∈ N , M ( C x ) ⊂ e C λ,ϕ H ( x ) where e C λ, x = { v = v + v ∈ E x ; λ | v | , x ≤ | v | , x } ;(2) for x ∈ N , for v ∈ C x , k M m ( v ) k ϕ Hm ( x ) ≥ µ. k v k x ;(3) for x ∈ N , for v ∈ C ∗ x , k M − m ( v ) k ϕ H − m ( x ) ≥ µ. k v k x .Following [Arn12], we now introduce some notations.N otations . • for x ∈ N , we denote by v ( x ) the p -projection of the intersectionof the vertical with the tangent space to the energy level v ( x ) = p x ( T x H x ∩ V ( x )); • when ϕ Hs ( x ) ∈ N for every s between 0 and − t , we denote by g t ( x )the subspace M t ( ϕ H − t x ) v ( ϕ H − t x ). Moreover – if ϕ Hs ( x ) ∈ N for every s ∈ ( −∞ ,
0) and g u ( x ) is transverse to v ( x ) for every u >
0, then g + ( x ) = lim t → + ∞ g t ( x ) exists and isa reduced Green bundle; for x ∈ Z , we have g + ( x ) = E u ( x ) = E ( x ); – if ϕ Hs ( x ) ∈ N for every s ∈ (0 , + ∞ ) and g u ( x ) is transverse to v ( x ) for every u <
0, then g − ( x ) = lim t →−∞ g t ( x ) exists and is AND C -CONVERGENCE TO WEAK K.A.M. SOLUTIONS 27 a reduced Green bundle; for x ∈ Z , we have g − ( x ) = E s ( x ) = E ( x ). – for every x ∈ Z , we also use the notation H x = p x ( T x Z ) anddenote by H the corresponding bundle over Z .On Z , g − = E is well defined and transverse to v ( x ). The hyperbolicityof ( M t ) on Z implies that for every m ≥
1, there exists some n > ∀ x ∈ Z , g n ( x ) = M n ( v ( ϕ H − n x )) ∈ e C λ m + , x , and we can also assume that(22) ∀ x ∈ Z , M n ( e C λ m , x ) ⊂ e C λ m + ,ϕ Hn x . Observe that E is di ff erent from H x (because of Equation (17)). Hence wecan choose m ∈ N large enough such that(23) ∀ x ∈ Z , H x e C λ m , x . We now choose an eventually smaller neighborhood N of Z that satisfiesthe following conditions, where we assume that we choose a metric on E N that allows us to compare tangent vectors of di ff erent fibers.(24) ∀ x ∈ N , ∃ w ∈ H x , k w k = , d ( w , e C λ m , x ) > ε for some ε > ∀ x ∈ N , g n ( x ) = M n ( v ( ϕ H − n x )) ∈ e C λ m , x because of Equation (21);(26) ∀ x ∈ N , M n ( e C λ m , x ) ⊂ e C λ m ,ϕ Hn x because of Equation (22).We now choose u T depending on N as before. We have proved that forevery t ≥ G ( dT t u T ) ⊂ N (see Equation (20)). We also have by Equation(19) that ∀ t ≥ , ∀ s ∈ [0 , t ] , ϕ Hs (cid:16) G ( dT t u T ) (cid:17) ⊂ G ( dT t − s u T ) ⊂ N and by Equation (18)(27) ∀ t > , ∀ x ∈ G ( dT t u T ) , T ϕ t x G ( dT t u T ) = D ϕ t (cid:16) ( T x G ( du T ) (cid:17) = G t ( ϕ Ht x ) . We deduce from Equations (27), (25) and (26) that ∀ k ∈ N , ∀ x ∈ G ( dT nk u T ) , ∃ D T nk u T ( x ) ⇒ p x (cid:16) T x G ( dT nk u T ) (cid:17) ⊂ e C λ m ,ϕ Hnk x and then by Equation (24) ∀ k ∈ N , ∀ x ∈ G ( dT nk u T ) , ∃ D T nk u T ( x ) ⇒ k D T nk u T ( π ( x )) k ≥ ε . and then the quantity d , ( T t u T ,
0) doesn’t tend to 0 when t tends to + ∞ ,hence doesn’t satisfy the conclusion of Theorem 4.
5. P roof of the C convergence Proof of Theorems 1 and 2.
We extend the ideas that were introducedin [Arn05] to a more general setting.We consider a map c ∈ C H c defined on some compact metric space C that is continuous for the C open-compact topology and such that every H c : T ∗ M → R is a Tonelli Hamiltonian. The associated Lagrangian is de-noted by L c . We will denote by ( ϕ c ,λ t ) the flow associated to the λ -discountedequation for H c . Proposition 12.
Let us fix t > and ( u , c , λ ) ∈ C ( M , R ) × C × R + withu fixed point of the semi-group ( T c ,λ τ ) . For every ε > , there exists aneighborhood N of ( u , c , λ ) such that for every ( u , c , λ ) ∈ N , we haved H ( G ( dT c ,λ t u ) , G ( du )) ≤ ε. To finish the proofs of Theorems 1 and 2, we have to apply Proposition12 when • either the space C is only one point, u = u λ is the discounted solu-tion and u the limit weak K.A.M. solution; • or λ = c ∈ H ( M , R ), u = u c is a weak K.A.M.solution for the cohomology class c . Proof.
We recall that because of the a priori compactness Lemma (Corol-lary 3), there exists for every t > K t ⊂ T ∗ M such that,for every c ∈ C and λ ∈ Λ where Λ is any compact subset of R , any mini-mizing orbit ( ϕ c ,λ s ( q , p )) s ∈ [0 , t ] takes all its values in K t . Observe that if T > t ,we can choose K T = K t .Let us fix t >
0. We define the map M t : T ∗ M × C × R + → R by M t ( q , p , c , λ ) = Z − t e λ s L c ( π ◦ ϕ c ,λ s ( q , p ) , ∂∂ s ( π ◦ ϕ c ,λ s ( q , p ))) ds . Observe that this map is continuous with respect to all the variables.We have for every t ≥ T > u : M → R (28) T c ,λ t u ( q ) = min ( q , p ) ∈ K T (cid:16) e − λ t u ( π ◦ ϕ c ,λ − t ( q , p )) + M t ( q , p , c , λ ) (cid:17) . We now define for every u ∈ C ( M , R + ), ( c , λ ) ∈ C × R + and t ≥ T > U t ( u , c , λ ) : K T → R by U t ( u , c , λ )( q , p ) = e − λ t u ( π ◦ ϕ c ,λ − t ( q , p )) + M t ( q , p , c , λ ) . Then every map U t ( u , c , λ ) is continuous and the map U t : C ( M , R ) × C × R + → C ( K T , R ) AND C -CONVERGENCE TO WEAK K.A.M. SOLUTIONS 29 is itself continuous if C ( M , R ) and C ( K T , R ) are endowed with the uniform C distances.This implies that the map ( u , c , λ ) ∈ C ( M , R ) × C × Λ T c ,λ t u ∈ C ( M , R )that is defined by(see Equation (28)) T c ,λ t u ( q ) = min ( q , p ) ∈ K T U t ( u , c , λ )( q , p )is also continuous.Moreover, the corresponding arg min function, that is the function E t : M × C ( M , R ) × C × Λ → K ( K t )that takes its values is the set K ( K t ) of non-empty compact subsets of K t and is defined by E t ( q , u , c , λ ) = { ( q , p ) ∈ T ∗ q M ; T c ,λ t u ( q ) = U t ( u , c , λ )( q , p ) } is an upper-continuous function when K ( K t ) is endowed with the Hausdor ff distance. Hence F t ( u , c , λ ) = [ q ∈ M E t ( q , u , c , λ )is also compact. Observe that G ( dT c ,λ t u ) ⊂ F t ( u , c , λ ) and then(29) G ( dT c ,λ t u ) ⊂ F t ( u , c , λ ) . Let us prove that Equation (29) is an equality for u = u . We recall that u is a fixed point of the semi-group ( T c ,λ τ ). But we know from Equation (7)that for every τ ∈ (0 , t ), we have F t ( u , c , λ ) ⊂ F τ ( u , c , λ ) ⊂ ϕ c ,λ τ ( G ( du )) ⊂ ϕ c ,λ τ (cid:16) G ( du ) (cid:17) and then by taking the limit for τ tending to 0, we deduce that(30) F t ( u , c , λ ) ⊂ G ( du ) . Equations (29 ) and (30 ) give finally(31) G ( du ) = F t ( u , c , λ ) . Let us now fix ε >
0. If ( u , c , λ , q ) ∈ C ( M , R ) × C × [0 , × M , thereexists a neighborhood V of ( u , c , λ ) and a neighborhood V of q such thatfor every ( q , u , c , λ ) ∈ V × V , we have E ( q , u , c , λ ) ⊂ E ( q , u , c , λ ) ε , where we use the following notation for K ⊂ T ∗ M .N otation . K ε = { x ∈ T ∗ M ; d ( x , K ) ≤ ε } . Then we can extract a finite covering of M by ( V i ) ≤ i ≤ n that are built as be-fore, with neighborhoods V i × V i of ( u , c , λ , q i ). Then ( E ( q i , u , c , λ ) ε ) ≤ i ≤ n is a covering of F t ( u , c , λ ) = G ( du ) by equation (31 ) and we have for( u , c , λ ) ∈ V = \ ≤ i ≤ n V i G ( dT c ,λ t u ) ⊂ F t ( u , c , λ ) = [ q ∈ M E t ( q , u , c , λ ) ⊂ n [ i = E t ( q i , u , c , λ ) ε ⊂ ( G ( du )) ε . To obtain the wanted conclusion, we only need to prove that G ( du ) ⊂ ( G ( dT c ,λ t u )) ε for ( u , c , λ ) close to ( u , c , λ ).We denote by D the set of derivability of u and we consider a fi-nite covering of G ( du ) by open balls with radius η = ε and centers at( q i , du ( q i )) ≤ i ≤ n where q i ∈ D . As the map E t is upper semi-continuousand has for value at every ( q i , u , c , λ ) the set { ( q i , du ( q i )) } , there exists α > k u − u k c < α, d ( c , c ) < α, | λ − λ | < α, d ( Q i , q i ) < α ⇒ d (( q i , du ( q i )) , E t ( Q i , u , c , λ )) < ε . Then we choose for every i a Q i where u is di ff erentiable and obtain d (( q i , du ( q i )) , ( Q i , dT c ,λ t u ( Q i )) < ε . Then we have ∀ q ∈ D , ∃ i ∈ [1 , n ] , d (( q , du ( q )) , ( Q i , dT c ,λ t u ( Q i )) ≤ d (( q , du ( q )) , ( q i , du ( q i )) + d (( q i , du ( q i )) , ( Q i , dT c ,λ t u ( Q i )) ≤ ε = ε. (cid:3) Hausdor ff distance in T ∗ M and C convergence. We will prove aproposition that implies that if a family of pseudographs ( G ( η c λ + du ′ λ )) λ ∈ Λ converges to the graph of G ( η c + du ) for the Hausdor ff distance and if themap u : M → R is C , then the derivatives ( du λ ) C - uniformly converge to du . Hence Corollaries 1 and 2 comes easily from Theorems 1 and 2.N otation . If K ⊂ T ∗ M and q ∈ M , we will denote by K q = K ∩ T ∗ q M . Proposition 13.
Let us consider a family ( K λ ) λ ∈ Λ of compact subsets ofT ∗ M and let G = G ( η ) ⊂ T ∗ M be the graph of a continuous map η definedon the whole M. Assume that lim λ → λ d H ( K λ , G ) = . Then lim λ → λ sup q ∈ M sup x ∈ K λ q d ( η ( q ) , x ) = . AND C -CONVERGENCE TO WEAK K.A.M. SOLUTIONS 31 Proof.
Assume that the result is not true. Then there exists a sequence ( λ n )that converges to λ and an ε > ∀ n ∈ N , ∃ q n ∈ M , ∃ x n ∈ K λ n q n , d ( η ( q n ) , x n ) ≥ ε. Extracting a subsequence, we can assume that ( q n ) converges to some q in M . For n ≥ N large enough we havesup x ∈ K λ n d ( x , G ) ≤ . Hence for n ≥ N , we have d ( x n , G ( η )) ≤
1, which means that x n takesits values in a fixed compact set. Extracting a subsequence, we can thenassume that ( x n ) converges to some x ∈ T ∗ M . We deduce from equation(32) and continuity of du that d ( η ( q ) , x ) ≥ ε .This implies that x < G ( η ). Let us recall that the graph of a continuous mapis closed. Hence there exists some β > B ( x , β ) ∩ G ( η ) = ∅ .As ( x n ) converges to x , for n ≥ N ′ large enough, we have x n ∈ B ( x , β ) andthen B ( x n , β ) ∩ G ( η ) = ∅ . Hence we obtain finally ∀ n ≥ N ′ , x n ∈ K λ n and d ( x n , G ( η )) ≥ β ;and then ∀ n ≥ N ′ , d H ( K λ n , G ( η )) ≥ β ;which contradicts the hypothesis. (cid:3)
6. P roof of the C convergence We give a proof that is valid for Theorems 3, 4 and 5.We fix u ∞ the C and upper Green regular weak K.A.M. solution. It isproved in [Fat08] that any C weak K.A.M. solution is C , , so u ∞ is C , and then semi-concave and semi-convex.We recall that we consider a family of semi-concave functions u that • converges to u ∞ in C uniform topology; this comes from Corollar-ies 1 and 2 and also [Arn05] joint with Proposition 13; • satisfies the following lemma. Lemma 3.
For ε > small enough, we haveLeb { θ ∈ T d : D u − D u ∞ ≮ ε } < ε where is the identity matrix of the standard scalar product.Proof. Since u ∞ is semi-concave, D u ∞ : T d → S d is defined (and measur-able) Lebesgue almost everywhere. Due to Lusin’s theorem, there exists acompact K ⊂ T d with Leb( T d \ K ) < ε such that D u ∞ (cid:12)(cid:12)(cid:12) K is continuous. Hence, there exists α >
0, such that for any θ, θ ′ ∈ K with d ( θ, θ ′ ) < α wehave(33) k D u ∞ ( θ ) − D u ∞ ( θ ′ ) k < ε . Let us recall that the approximated solutions u that we consider is semi-concave, C -close to u ∞ and is of one of the three possible kinds that wenow describe. • u = u λ that is a discounted solution for a small λ ; then by Corol-lary1, u is C close to u ∞ ; • u = T t U for some U ∈ C ( T d , R ) and some t > u is C close to u ∞ ; • u = u c that is a weak K.A.M. solution for a cohomology class c close to 0; then by Corollary 2, u is C close to u ∞ .We deduce that for every x = ( θ, du ∞ ( θ )) ∈ G ( du ∞|K ), due to Propositions5 and 6, there exists η x with η x < α such that for every y ∈ B ( θ, η x ) × B ( du ∞ ( θ ) , η x ), we have for some t > D u ( π ( y )) < H ( G t ( y )) < D u ∞ ( θ ) + ε = H ( G + ( x )) + ε . Since G ( du ∞ (cid:12)(cid:12)(cid:12) K ) is compact, there exists n and x i ∈ K with i = , · · · , n such that G ( du ∞ (cid:12)(cid:12)(cid:12) K ) ⊂ n [ i = B ( θ i , η x i ) × B ( du ∞ ( θ i ) , η x i ) : = N Because u converges to u ∞ in C uniform topology, one can choose u suchthat G ( du (cid:12)(cid:12)(cid:12) K ) ⊂ N . For every θ ∈ K , without loss of generality, we assume( θ, du ( θ )) ∈ B ( θ , η x ) × B ( du ( θ ) , η x ). Using (33) and (34), we obtain D u ( θ ) < D u ∞ ( θ ) + ε < D u ∞ ( θ ) + ε . (cid:3) We fix one integer i ∈ { , . . . , d } and we consider one (non-injective) arc γ : t ∈ [0 , → γ ( t ) ∈ T d defined by t = θ i ∈ [0 ,
3] and the other coordi-nates θ j fixed. Because u and u ∞ are semi-concave, they are di ff erentiableLebesgue almost everywhere and admits a second derivative Lebesgue al-most everywhere. By Fubini Theorem, there exists some σ ∈ [0 ,
1) suchthat for Lebesgue almost every choice of ( θ j ) j , i , u is di ff erentiable at γ ( σ )and is twice di ff erentiable Lebesgue almost everywhere along the arc γ andwe can write e ( γ ) = u ( γ (3)) − u ∞ ( γ (3)) − ( u ( γ ( σ )) − u ∞ ( γ ( σ ))) = Z σ ( du ( γ ( t )) − du ∞ ( γ ( t ))) ˙ γ ( t ) dt ; AND C -CONVERGENCE TO WEAK K.A.M. SOLUTIONS 33 and also because u − u ∞ is semi-concave and ¨ γ =
0, we have e ( γ ) ≤ Z σ (cid:18)(cid:0) du ( γ ( σ )) − du ∞ ( γ ( σ )) (cid:1) ˙ γ ( σ ) + Z t σ [( D u ( γ ( s )) − D u ∞ ( γ ( s )))( ˙ γ ( s ) , ˙ γ ( s )) + ( du ( γ ( s )) − du ∞ ( γ ( s ))) ¨ γ ( s )] ds (cid:19) dt i.e. e ( γ ) ≤ (3 − σ ) (cid:0) du − du ∞ )( γ ( σ ) (cid:1) ˙ γ ( σ ) + Z σ (3 − t )( D u ( γ ( t )) − D u ∞ ( γ ( t )))( ˙ γ ( t ) , ˙ γ ( t )) dt , and then because ˙ γ ( t ) = e i and then k ˙ γ k = e ( γ ) − (3 − σ ) " ( du − du ∞ )( γ ( σ )) e i + ε ≤ Z σ (3 − t )( D u ( γ ( t )) − D u ∞ ( γ ( t )) − ε )( e i , e i ) dt . Let E = { t ∈ [0 ,
3] : ( D u − D u ∞ )( γ ( t )) − ε < } and E = [0 , \ E .Because of the uniform semi-concavity of u and because u ∞ is C , , thereexists K > D u − D u ∞ )( θ ) ≤ K for any θ ∈ T d where D u − D u ∞ exists. We deduce e ( γ ) − (3 − σ ) " ( du − du ∞ )( γ ( σ )) e i + ε ≤ Z E ∩ [ σ, (3 − t )( D u ( γ ( t )) − D u ∞ ( γ ( t )) − K )( e i , e i ) dt + K − σ ) Leb ( E ) + Z E ∩ [ σ, (3 − t )( D u ( γ ( t )) − D u ∞ ( γ ( t )) − ε )( e i , e i ) dt Hence as the functions that appears in the integrals are non-positive, wededuce also e ( γ ) − (3 − σ ) " ( du − du ∞ )( γ ( σ )) e i + ε ≤ Z E ∩ [1 , ( D u ( γ ( t )) − D u ∞ ( γ ( t )) − K )( e i , e i ) dt + K − σ ) Leb ( E ) + Z E ∩ [1 , ( D u ( γ ( t )) − D u ∞ ( γ ( t )) − ε )( e i , e i ) dt . We introduce the notation E = { θ ∈ T d : D u − D u ∞ < ε } . ByLemma 3, we have Leb ( T d \E ) < ε . Integrating with respect to ( θ j ) j , i ∈ T d − , we finally obtain − k u − u ∞ k ∞ + k u − u ∞ k C + ε ! ≤ Z T d \E ( D u ( θ ) − D u ∞ ( θ ) − K )( e i , e i ) d θ + K − σ ) Leb ( T d \E ) + Z E ( D u ( θ ) − D u ∞ ( θ ) − ε )( e i , e i ) d θ and then because Leb ( T d \E ) < ε , we have − k u − u ∞ k ∞ + k u − u ∞ k C +
32 ( K + ε ! ≤ Z T d \E ( D u ( θ ) − D u ∞ ( θ ) − K )( e i , e i ) d θ + Z E ( D u ( θ ) − D u ∞ ( θ ) − ε )( e i , e i ) d θ. We deduce that(36) − Z T d \E ( D u ( θ ) − D u ∞ ( θ ) − K )( e i , e i ) d θ ≤ k u − u ∞ k ∞ + k u − u ∞ k C +
32 ( K + ε ! and(37) − Z E ( D u ( θ ) − D u ∞ ( θ ) − ε )( e i , e i ) d θ ≤ k u − u ∞ k ∞ + k u − u ∞ k C +
32 ( K + ε ! . Let v = P v i e i ∈ R d such that k v k =
1. Then we have for every θ ∈ E ≤ − ( D u ( θ ) − D u ∞ ( θ ) − ε )( v , v ) ≤ − d sup ≤ i ≤ d ( D u ( θ ) − D u ∞ ( θ ) − ε )( e i , e i )and for every θ ∈ T d \E ≤ − ( D u ( θ ) − D u ∞ ( θ ) − K )( v , v ) ≤ − d sup ≤ i ≤ d ( D u ( θ ) − D u ∞ ( θ ) − K )( e i , e i ) . We deduce that Z E k ( D u − D u ∞ )( θ ) − ε k d θ ≤ − d X ≤ i ≤ d Z E (( D u − D u ∞ )( θ ) − ε )( e i , e i ) d θ and Z T d \E k ( D u − D u ∞ )( θ ) − K k d θ ≤ − d X ≤ i ≤ d Z T d \E (( D u − D u ∞ )( θ ) − K )( e i , e i ) d θ. AND C -CONVERGENCE TO WEAK K.A.M. SOLUTIONS 35 Observe that d , ( u , u ∞ ) ≤ Z T d \E k ( D u − D u ∞ )( θ ) − ε k d θ + K . Leb ( T d \E ) + Z E k ( D u − D u ∞ )( θ ) − K k d θ + ε. Leb ( E )and then d , ( u , u ∞ ) ≤ Z T d \E k ( D u − D u ∞ )( θ ) − K k d θ + Z E k ( D u − D u ∞ )( θ ) − ε k d θ + ( K + ε so d , ( u , u ∞ ) ≤− d X ≤ i ≤ d Z T d \E (( D u − D u ∞ )( θ ) − K )( e i , e i ) d θ + Z E (( D u − D u ∞ )( θ ) − ε )( e i , e i ) d θ ! + ( K + ε Using Equations (36) and (37), we deduce d , ( u , u ∞ ) ≤ d ( k u − u ∞ k ∞ + k u − u ∞ k C + K + ε ) . This is the wanted result. R eferences [Arn05] M.-C. Arnaud. Convergence of the semi-group of Lax-Oleinik: a geometricpoint of view.
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