WEAK KAM THEORY FOR MULTIDIMENSIONAL MAPS

Transcription

1 WEAK KAM THEORY FOR MULTIDIMENSIONAL MAPS DIOGO A. GOMES* AND CLAUDIA VALLS+ Abstract. In this paper we discuss a weak version of KAM theory for multidimensional maps that arise from the discretization of the minimal action principle. These systems have certain invariant sets, the Mather sets, which are the generalization of KAM tori in the non-differentible case. We generalize viscosity solution methods to study discrete systems. In particular, we show that, under non-resonance conditions, the Mather sets can be approximated uniformly, up to any arbitrary order, by finite perturbative expansions. * Departamento de Matemtica Instituto Superior Tcnico Av. Rovisco Pais Lisboa, Portugal dgomes@math.ist.utl.pt + Departament de Matematica Aplicada i Analisis Universitat de Barcelona Gran Via 08007, Barcelona - España claudia@maia.ub.es * - Supported in part by FCT Portugal through programs POCTI, POCTI/32931/MAT/ Supported in part by CIRIT 2001SGR Introduction In this paper, we discuss a weak version of KAM theory for multidimensional maps that arise from the discretization of the minimal action principle. By discretizing the minimal action principle from classical mechanics, one obtains certain maps { p n+1 p n = hd x Hp n+1, x n x n+1 x n = hd p Hp n+1, x n, 1

2 2 D. Gomes, C. Valls in which h is the time step and H the Hamiltonian. Those maps are discrete analogs of Hamilton s equations { ṗ = D x Hp, x ẋ = D p Hp, x. It is very important to notice that this discretization comes from variational principles, as we discuss in section 2, and has interesting geometric and analytic features as will be pointed out later in the paper. For these maps, it is possible to construct a version of Aubry-Mather theory [Gom02], and one can prove the existence of invariant sets Mather sets supported on Lipschitz graphs. If the Hamiltonian H only depends on p, the dynamics is very simple since p n = p 0, and x n = x 0 nhd p Hp 0. In this case, the system is called integrable. We would like to study Hamiltonians that are close to integrable. That is, when H is of the form Hp, x = H 0 p + Oɛ, for ɛ a small parameter. Our main objective is to understand the dependence on ɛ of the Mather sets. In section 2, we discuss generating functions and integrability methods for these systems. Our results serve to motivate the methods used in the remaining sections to study the dependence on ɛ of the Mather sets. For each P R n, one can construct a Mather set that is supported on a Lipschitz graph. This graph can be determined from a solution u of ux, P uˆx, P + hhp + D x uˆx, P, x hd x uˆx, P D p HP + D x uˆx, P, x = hhp, in which ˆx x = hd p HP +D x uˆx, P, x. In general, since there are no smooth solutions of this equation one has to consider the class of viscosity solutions. The general viscosity solution methods, in particular, definition, existence, regularity, and basic properties, are discussed in section 3. Whereas viscosity solutions may not be smooth, one can develop a formal expansion ũx, P in power series in ɛ and P P 0 of the solution ux, P, in a neighborhood of ɛ = 0 and P = P 0. There is a solvability condition in order to construct those expansions, namely that the rotation vector D P H 0 P 0 is Diophantine. These expansions are constructed in section 4, and some technical estimates are proven in section 5. When considering the Mather set for ɛ > 0 one has two choices. The first one is, given a vector P 0 fixed, trying to compare the formal

3 Weak KAM 3 approximation of the Mather set with the Mather set itself. Unfortunately there are serious problem with this approach as, when ɛ > 0, the Diophantine conditions may be destroyed. The approach we consider is the following: for ɛ > 0 we construct a vector P ɛ which keeps the rotation vector fixed. Then, in section 6, we prove that the viscosity solution and its approximation at P ɛ are close both in the supremum norm and its derivatives. To prove these estimates we use a technique similar to the one in [Gom03], which is based in first proving estimates along trajectories for finite time. Since ergodization times can be controlled in terms of the Diophantine properties of the rotation vector see [BGW98], [DDG96], [Dum99], we extend this estimate for all points using a-priori Lipschitz bounds for the viscosity solution and its approximation. The idea of controlling a viscosity solution during a long time and then extending to nearby points has been used by other authors. For instance in the paper [Bes], a ergodization times techniques are used to study Hamilton-Jacobi equations perturbed by an elliptic operator. This problem is also studied, in a different setting, in [FS86a], and [FS86b]. It has been pointed out to the authors by U. Bessi, that our results are similar to the ones in [BK87], as both imply the stability of Mather sets. However, their techniques are different, and, in the nonressonant case, we obtain an asymptotic expansion, as well as better stability properties. 2. Discrete variational principles and integrability In the variational formulation of classical mechanics, the trajectories xt of a system are minimizers, or at least critical points, of the action T 0 Lx, ẋdt, in which Lx, v : R 2n R, the Lagrangian, is the difference between kinetic and potential energy. We assume that the Lagrangian is smooth, strictly convex in v, and Z n -periodic in x, that is, for all k Z n we have Lx + k, v = Lx, v. This periodicity makes it convenient to look at the Lagrangian as function from T n R n to R T n is the n dimensional flat torus. The minimizing trajectories are solutions to the Euler equation 1 d dt D vlx, ẋ + D x Lx, ẋ = 0.

4 4 D. Gomes, C. Valls In applications, it is important to consider discrete versions of classical mechanics, for instance, for computational purposes. There are two alternatives to make this discretization, one is to discretize the Euler-Lagrange equations 1, the other is to discretize the variational principle. They are not equivalent, and this last approach has several advantages. In fact, in the continuous setting there are certain invariant sets, the Mather sets, which are obtained using a variational principle. Using the discretization of the variational principle, the Mather sets can also be constructed [Gom02]. Furthermore, the map that is obtained this way has better geometrical properties as it preserves the symplectic structure. The discretization of the variational problem can be done by means of the Euler method for the ODE This yields the discrete dynamics ẋ = vt. x n+1 = x n + hv n, in which h is the time-step. The corresponding variational problem consists in minimizing the action N h Lx n, v n, n=0 among all choices of v n, 0 n N. The analog of the Euler-Lagrange equations is 2 D vlx n+1, v n+1 D v Lx n, v n + D x Lx n+1, v n+1 = 0. h In the continuous case, 1 can be written in a Hamiltonian form: { ṗ = D x Hp, x ẋ = D p Hp, x, for p = D v Lx, ẋ, and the Hamiltonian Hp, x = sup [ p v Lx, v]. v Similarly, 2 can be written in the equivalent form [Gom02] { p n+1 p n = hd x Hp n+1, x n 3 x n+1 x n = hd p Hp n+1, x n, with 4 p n+1 = D v Lx n, v n.

5 Weak KAM 5 Note that the dynamics 3 is semi-explicit, that is, implicit in p and explicit in x. This may therefore constraint the size of h for which 3 defines a discrete flow. We consider the case in which the Lagrangian L ɛ x, v : T n R n R, is a perturbation of an integrable one L 0 v. More precisely, 5 L ɛ = L 0 v + ɛl 1 x, v, in which ɛ is a small parameter. When ɛ = 0, the dynamics 3 is very simple since v n is constant. The Hamiltonian corresponding to 5 has an expansion of the form Hp, x = H 0 p + ɛh 1 p, x + ɛ 2 H 2 p, x + As in the case of continuous flows see [Arn89], [AKN97] or [Gol80], one can use generating functions to change coordinates, and, in particular, there is a version of the Hamilton-Jacobi integrability for maps. In the next proposition we prove the main result on generating functions and change of coordinates, which serves to motivate our methods. Theorem 2.1. Let p, x be the original canonical coordinates and P, X be another coordinate system. Suppose there is a smooth function Sx, P such that 6 p = D x Sx, P X = D P Sx, P defines a change of coordinates this function is called a generating function. Additionally, assume that DxP 2 S is non-singular. Furthermore, suppose there is a function HP, X such that 7 Sx, P Sˆx, ˆP + hhd x Sˆx, ˆP, x hd x Sˆx, ˆP D p HD x Sˆx, ˆP, x+ + hd P Sˆx, ˆP D X HP, D P Sˆx, ˆP = = hhp, D P Sˆx, ˆP, in which 8 ˆP P = hd X HP, D P Sˆx, ˆP ˆx x = hd p HD x Sˆx, ˆP, x In the new coordinate system, the equations of motion 3 are { X n+1 X n = hd P HP n, X n+1 9 P n+1 P n = hd X HP n, X n+1

6 6 D. Gomes, C. Valls In particular, if H does not depend on X, these equations simplify to { X n+1 X n = hd P HP n 10 P n+1 P n = 0. Remark. When we perform this change of coordinates we obtain a new dynamics that is semi-explicit as 3, but this time is implicit in X and explicit in P. Remark. In the continuous case, given a Hamiltonian H and the generating function S, the new Hamiltonian is fully determined since Hp, x = HP, X. However, in our case, that is not immediate since 7 is in fact a partial differential equation for H. Proof. For simplicity, we will set h = 1 in the proof. Define Ŝ = Sˆx, ˆP, and we use the convention D x Ŝ = D x Sˆx, ˆP, as well as D P Ŝ = D P Sˆx, ˆP, to simplify the notation. By differentiating 7 with respect to x we obtain, D x S D x ŜD xˆx D P ŜD x ˆP + Dp HD x D x Ŝ + D x H D x D x Ŝ D p H D x ŜD x D p H + D x D P Ŝ = D X HD x D P Ŝ. Then, by canceling some terms we have Observe that and Therefore D X H + D P ŜD x DX H = D x S D x ŜD xˆx D P ŜD x ˆP + Dx H D x ŜD x D p H + D P ŜD x DX H = 0. D xˆx = I D x D p H, D P ŜD x ˆP + D X H = D x S D x Ŝ + D x H = 0. Differentiating 7 with respect to P we get D P S D x ŜD P ˆx D P ŜD P ˆP + Dp HD P D x Ŝ D P D x Ŝ D p H D x ŜD P D p H + D P D P Ŝ D X H + D P ŜD P DX H = = D P H + D X HD P D P Ŝ.

7 Weak KAM 7 By canceling we get D P S D x ŜD P ˆx D P ŜD P ˆP D x ŜD P D p H + D P ŜD P DX H = D P H. Note that therefore D P ˆx = D P D p H, D P S D P ŜD P ˆP + DP ŜD P DX H = D P H. Since we have D P ˆP = I + DP DX H 12 D P S D P Ŝ = D P H. Define ˆp = D x Ŝ, and ˆX = D P Ŝ. Now assume that x n, p n are solutions to the dynamics 3. If we set x = x n and p = p n, by the change of coordinates 6 we have, correspondingly, X = X n and P = P n. Therefore, from 11 we get ˆp = p n+1. Thus, since ˆp = D x Ŝ we have ˆx = x n+1 and ˆP = P n+1. This implies ˆX = X n+1, and so 12 reads and we also have X n X n+1 = D P HP n, X n+1, P n+1 P n = D X HP n, X n+1. The previous theorem suggests that we should look for a generating function Sx, P = P x+ux, P, periodic in x, and a new Hamiltonian H which only depends on P. If such a solution exist, the equations of motion reduce to 10 and we say that such a system is integrable. The function u should then satisfy the equation 13 ux, P uˆx, P + hhp + D x uˆx, P, x hd x uˆxd p HP + D x uˆx, x = hhp.

8 8 D. Gomes, C. Valls 3. Viscosity solutions In general, equation 13 may not admit smooth solutions. However, see [Gom02], one can prove the existence of viscosity solutions, which are the correct notion of weak solution. In this section we review the main facts and prove some preliminary estimates. For our purposes, a convenient definition of viscosity solution is the following: we say that a function u is a viscosity solution of 13 provided that it satisfies the following fixed point identity: N 1 ux, P = h min [Lx j, v j + P v j + HP ] + ux N, P, j=0 in which the minimum is taken over trajectories x n, v n, 0 n N, with initial condition x 0 = x, and x n+1 = x n + hv n. The next proposition shows that a smooth viscosity solution is a solution of 13. Proposition 3.1. Suppose u is smooth and satisfies 14 ux, P = min v h[lx, v + P v + HP ] + ux + hv, P. Then, u solves 13. Proof. and so The optimal velocity v in 14 is given by D v Lx, v = P D x ux + hv, P, Lx, v + P + D x ux + hv, P v = HP + D x ux + hv, P, x, and v = D p HP + D x ux + hv, P, x. Therefore, with ˆx = x + hv = x hd p HP + D x uˆx, P, x, ux, P uˆx, P + hhp + D x uˆx, P, x hd x uˆx, P D p HP + D x uˆx, P, x = hhp. The optimal trajectory x n and the momentum p n, as given by 4, are solutions of 3 for n 0. Let us start by quoting an existence result whose prove is given in [Gom02]: Theorem 3.2. For each P R n there exists a unique number HP and a family of solutions ux, P, periodic in x, that solves 13 in the viscosity sense. Furthermore, HP is convex in P, and ux, P is Lipschitz in x.

9 Weak KAM 9 Motivated by the formal change of variables that was discussed in the previous section, we would like to relate this weak solution with the dynamics 3 - the next theorem makes this connection. Theorem 3.3. Let u be a viscosity solution of 13. Then: - For each P R n, there exists a subset of T n R n which is invariant under the dynamics 3 and is contained in the graph x, p = x, P + D x ux. - There exists a probability measure µx, p on T n R n discrete Mather measure invariant under 3 supported on this invariant set. - The measure minimizes 15 Lx, v + P vdν, with v = D p Hˆp, x, and ˆp p = hd x Hˆp, x, over all probability measures ν on T n R n that satisfy φx + hv φxdν = 0, for all continuous function φ : T n R. Furthermore 16 H = Lx, v + P vdµ. One of the main points in the previous theorem is that one can translate properties of viscosity solutions into properties of Mather sets or measures and vice-versa. In the next proposition we discuss some of the properties of viscosity solutions, and its relations with the dynamics 3. Proposition 3.4. Suppose x, p is a point in the graph G = {x, P + D x ux : u is differentiable at x}. Then, for all n 0, the solution x n, p n of 3 with initial conditions x, p belongs to G. A further result that we need, taken also from [Gom02], is a representation formula for H as a minimax. This is the discrete analog of the minimax formula for flows proven in [CIPP98]. Proposition HP = inf ϕ [ ] ϕx ϕx + hv sup Lx, v P v, x,v h in which the infimum is taken over C 1 periodic functions ϕ.

10 10 D. Gomes, C. Valls Suppose that L ɛ x, v = L 0 v + ɛl 1 x, v. The last estimates in this section show that, even without non-resonance conditions, the viscosity solution of 13 has good bounds for the semiconcavity and semiconvexity constants of u. These estimates should be seen as weak estimate for the second derivatives of u. Roughly speaking we have Dxxu 2 = O ɛ in the Mather set. Proposition 3.6. Suppose L ɛ is as in 5, and let u be a viscosity solution of 13. Then, for any x and y we have ux + y 2ux + ux y C ɛ y 2. Proof. Let x n, v n, with x n+1 = x n + hv n, 0 n N 1 be an optimal trajectory with x 0 = x, such that Then, N 1 ux = ux N + h [L 0 v n + ɛl 1 x n, v n + P v n + HP ]. ux ± y ux N + h Therefore, n=0 N 1 n=0 [L 0 v n +P ux + y 2ux + ux y h N 1 n=0 N 1 + hɛ n=0 [L 0 v n v n Note that, since D 2 L 0 is bounded, L 0 v n Also y y y + ɛl 1 x n ± N n ] + HP. y 2L 0 v n + L 0 v n + [L 1 x n + N n N y, v n y +L 1 x n N n N y, v n + y 2L 0 v n + L 0 v n + L 1 x n + N n N y, v n y + L 1 x n N n N y, v n + y 2L 1 x n, v n + C N y, v n y y ] + 2L 1 x n, v n + ]. y C y 2 N 2 h y 2. N 2 h 2 +

11 Consequently, for h small, ux + y 2ux + ux y C By choosing N = O 1 h ɛ Weak KAM 11 we obtain [ 1 + ɛhn + ɛ ] y 2. hn ux + y 2ux + ux y C ɛ y 2. Proposition 3.7. Suppose L ɛ is as in 5, and let u be a viscosity solution of 13. Then, if x is in the Mather set and y is arbitrary, we have ux + y 2ux + ux y C ɛ y 2. Proof. Let x n, v n, with x n+1 = x n + hv n, 0 n N 1 be an optimal trajectory with x N = x, such that N 1 ux 0 = ux + h [L 0 v n + ɛl 1 x n, v n + P v n + HP ]. n=0 Note that this is possible because x belongs to the Mather set, and therefore there is a minimizing trajectory that passes through x. Then, ux 0 ux ± y + h hence N 1 n=0 [L 0 v n ± +P ux + y 2ux + ux y h N 1 n=0 N 1 hɛ n=0 [L 0 v n + v n ± y y [L 1 x n + n N y, v n + +L 1 x n n N y, v n Note that, since D 2 L 0 is bounded, L 0 v n + y + ɛl 1 x n ± n N y, v n ± ] + HP, y 2L 0 v n + L 0 v n y 2L 1 x n, v n + y 2L 0 v n + L 0 v n ]. y C y 2 h 2 N. 2 y + y ]

12 12 D. Gomes, C. Valls Also Therefore L 1 x n + n N y, v n + y + L 1 x n n N y, v n ux + y 2ux + ux y C By choosing N = O 1 h ɛ we obtain 2L 1 x n, v n + y C [ N 2 h 2 ] y 2. [ 1 + hɛn + ɛ ] y 2. ux + y 2ux + ux y C ɛ y 2. Theorem 3.8. Suppose L ɛ is as in 5, and let u be a viscosity solution of 13. Then, if x is in the Mather set and y is arbitrary then ux uy D x uxy x C ɛ x y 2. Proof. This follows from the proof of a similar theorem in [Gom02] by replacing the semiconcavity and local semiconvexity constants by C ɛ which result from the two previous propositions. 4. Formal Perturbation Theory In this section we discuss formal perturbation methods using an analog of Linstead series. We look for a formal expansion ũ ɛ N x, P, in power series of ɛ and P P 0, that satisfies 13, up to order Oɛ N + P P 0 N, in a neighborhood of ɛ = 0 and P = P 0. More precisely, 18 ũ ɛ Nx, P ũ ɛ N x, P + hhp + D x ũ ɛ N x, P, x hd x ũ ɛ N x, P D p HP + D x ũ ɛ N x, P, x = h H ɛ NP + Oɛ N + P P 0 N, in which 19 x x = hd p HP + D x ũ ɛ N x, P, x. The main difficulty is that an approximate generating function may not yield a Hamiltonian HP, X. However, by setting S = P x + ũ ɛ N x, P, and performing the change of coordinates given by 6 the dynamics can still be written in a simpler form, up to high order terms.

13 Weak KAM 13 Proposition 4.1. Suppose ũ ɛ N x, P is a solution of 18. Let S = P x + ũ ɛ Nx, P, and define new coordinates X, P by 6. Then { X n+1 X n = hd P Hɛ 20 N P n + Oɛ N + P n P 0 N 1 P n+1 P n = Oɛ N + P n P 0 N. Proof. For simplicity, as in the proof of theorem 2.1, we set h = 1. By differentiating 18 with respect to x, and canceling the terms, as in theorem 2.1, we get 21 D x S D x Ŝ + D x H = Oɛ N + P P 0 N, in which Ŝ = Sˆx, P. simplifying, we obtain Differentiating now with respect to P, and 22 D P S D P Ŝ = D P Hɛ N P + Oɛ N + P P 0 N 1. Define ˆp = D x Ŝ and ˆX = D P Ŝ. If we assume that x n, p n are solutions to the dynamics 3, and set x = x n and p = p n, in the new coordinates the corresponding points are X = X n and P = P n. Then, from 21, we get ˆp = p n+1 + Oɛ N + P n P 0 N. Therefore, the inverse function theorem implies P n+1 P n = Oɛ N + P n P 0 N. Then ˆX = X n+1 + Oɛ N + P n P 0 N 1. So, equation 22 reads X n X n+1 = D P Hɛ N P n + Oɛ N + P n P 0 N 1. The Linstead method consists in constructing solutions of 18 by using an iterative procedure that yields an expansion ũ ɛ N of the solution u ɛ as a power series in ɛ and P P 0. Of course, there are some conditions that have to be satisfied in order to construct the approximated solution. These can be expressed in terms of the Diophantine properties of the vector P 0. We say that a vector ω R n is Diophantine if 23 k Z n \{0}, m Z, ω k m C k s, for some C, s > 0. We will assume that the vector ω 0 = D P H 0 P 0 is Diophantine.

14 14 D. Gomes, C. Valls We look for an expansion of the form 24 ũ ɛ Nx, P = ɛv 1 x, P 0 + ɛp P 0 D P v 1 x, P 0 + ɛ 2 v 2 x, P ɛp P 0 2 D 2 P P v 1 x, P 0 + ɛ 2 P P 0 D P v 2 x, P = N 1 j=1 j i=1 1 j i! ɛi P P 0 j i D j i P j i v i x, P 0, with the notation that, for N = 1, ũ ɛ 1x, P = 0. Furthermore, H ɛ NP = H 0 P 0 + ɛ H 1 P 0 + P P 0 D P H0 P ɛ 2 H2 P 0 + ɛp P 0 D P H1 P 0 + P P 0 2 DP 2 P 2 H 0 P 0 + We will try to choose the functions v j in such a way that, formally, u ɛ x, P ũ ɛ Nx, P = Oɛ N + P P 0 N, by matching powers ɛ and P P 0 in both sides of 18. The first term arises from taking ɛ = 0, and P = P 0 in 18. Then, H 0 P 0 = H 0 P 0, and the solution ũ ɛ 0 = 0. The first order terms in ɛ yield v 1 x, P 0 v 1 x hd p H 0 P 0, P 0 + hh 1 P 0, x = h H 1 P 0, and this equation determines v 1 x, P 0 and H 1 P 0. Furthermore D P H0 P 0 = D P H 0 P 0. The function v 2 x, P 0 and H 2 P 0 are determined by solving the equation v 2 x, P 0 v 2 x hd p H 0 P 0, P h 2 D xv 1 x hd p H 0 P 0, P 0 2 D 2 pph 0 P hd p H 1 P 0 D x v 1 x hd p H 0 P 0, P hh 2 P 0, x = h H 2 P 0. To obtain D P v 1 x, P 0 and D P H1 P 0 we consider the equation D P v 1 x, P 0 D P v 1 x hd p H 0 P 0, P hd p H 1 P 0, x = hd P H1 P 0. In general, we will have to solve equations of the form G ω0 u = f + λ,

15 in which the operator G is given by 25 G ω0 u = ux ux ω 0, Weak KAM 15 the function f can be computed in terms of functions that are already known, λ is the unique constant for which 25 has a solution, and ω 0 = D P H 0 P 0. This operator can be analyzed by using Fourier coefficients. Note that, G w0 e 2πikx = e 2πikx 1 e 2πikw 0. Thus, if ηx = k η k e 2πikx and ux = k 26 G ω0 ux = ηx, reduces formally to u k e 2πikx, the equation u k = η ke 2πikx 1 e. 2πikω 0 If ω 0 is non-resonant, the equation in 26 can be solved formally in Fourier coefficients, since no denominator vanishes, except for k = 0. Moreover, in order for 26 to have a solution, we need η 0 = 0. So this implies 27 λ = fxdx. Furthermore, under Diophantine conditions on ω 0, the series for the solution u converges as long as η is smooth enough. From 27 we have H 1 P 0 = H 1 P 0, xdx, D P H1 P 0 = D P H1 P 0, xdx, and 1 H 2 P 0 = 2 D xv 1 x hd p H 0 P 0, P 0 2 DppH 2 0 P 0 dx+ + D p H 1 P 0 D x v 1 x hd p H 0 P 0, P 0 + H 2 P 0, xdx. Therefore, by computing these expansions we obtain, formally, that H ɛ P = H NP ɛ + Oɛ N + P P 0 N. This identity will be made rigorous in the next section.

16 16 D. Gomes, C. Valls 5. Estimates for the Effective Hamiltonian In this section we prove that H N ɛ P is an asymptotic expansion to H ɛ P, therefore proving rigorously some of the results from the previous section. Proposition 5.1. Suppose there is an approximate solution as in 18. Then, 28 H ɛ P H ɛ NP + Oɛ N + P P 0 N. Proof. that We set h = 1 for simplicity. The inf sup formula 17 implies H ɛ P sup [ũ ɛ Nx, P ũ ɛ Nx + v, P Lx, v P v]. x,v So, the optimal point v is given by and so D x ũ ɛ Nx + v, P = L v x, v P, Lx, v = HP + D x ũ ɛ Nx + v, P, x P + D x ũ ɛ Nx + v, P v. Therefore, sup [ũ ɛ Nx, P ũ ɛ Nx + v, P Lx, v P v] = v = ũ ɛ Nx, P ũ ɛ Nˆx, P + HP + D x ũ ɛ Nˆx, P, x D x ũ ɛ Nˆx, P D p HP + D x ũ ɛ Nˆx, P, x = = H ɛ NP + Oɛ N + P P 0 N. Thus by taking the supremum, we obtain the result. Proposition 5.2. If there exists an approximated solution as in Proposition 5.1, then Remark. H ɛ P H N ɛ P + Oε N + P P 0 N. This proposition and the previous one, together, imply 29 H ɛ P = H N ɛ P + Oε N + P P 0 N. Proof. Suppose ũ ɛ N x, P and H N ɛ P solve 18. We can construct a measure µ on T n R n such that for all continuous function φ with compact support 1 φx, vd µ = lim M M M φ x n, ṽ n, n=1

17 in which x n, ṽ n are given by Weak KAM 17 x n+1 x n = hṽ n = hd p HP + D x ũ ɛ N x n+1, P, x n, and the limit is taken through an appropriate subsequence. Note that µ is a probability measure, and for all continuous function φx we have φx + hv φxd µ = 0, and v = D p HP + D x ũ ɛ Nx + hv, P, x, on the support of µ. Note that 30 ũ ɛ Nx, P ũ ɛ Nx + hv, P L 0 v ɛl 1 x, v P v = = H ɛ NP + Oɛ N + P P 0 N, for v = D p HP + D x ũ ɛ N x + hv, P, x, by an argument similar to the one in the previous proposition. Therefore, by integrating 30 with respect to µ, we obtain L 0 v + ɛl 1 x, v + P vd µ = H NP ɛ + Oɛ N + P P 0 N. Then, equation 16 implies H ɛ P H ɛ NP + Oɛ N + P P 0 N. Finally, we prove a very simple result which will be essential in the following section. Lemma 5.3. There exists a point x 0 at which D x u ɛ x 0, P = D x ũ ɛ Nx 0, P. Proof. Since u ɛ x, P ũ ɛ N x, P is a periodic semiconcave function of x, then it has a minimum at some point x 0. Thus, at x 0, the derivative of u ɛ x, P ũ ɛ N x, P with respect to x, exits and is zero. 6. Uniform estimates This section is dedicated to prove two main results. One is that the solution of 18 approximates uniformly the viscosity solution of 13. The other is that the derivatives of the approximate solution are uniformly close to the ones of the viscosity solution. Since the Mather set is supported on the graph x, vx, with v given by vx = D p HP + D x u ɛ x + hvx, P, x, this implies stability of Mather sets.

18 18 D. Gomes, C. Valls These result should be though of as the discrete analogs to the ones in [Gom03] for the continuous problem. The main idea is that the approximated solution, built using the formal expansion, is very close to the viscosity solution along an optimal trajectory. Under non-resonance conditions, these trajectories get close to any point in the torus in finite time. Therefore, since u ɛ and ũ ɛ N are Lipschitz functions, we can extend the estimate to every point. Then we bootstrap these estimates for estimates on the derivatives. Theorem 6.1. Suppose the rotation vector ω 0 = D P H 0 P 0 satisfies the Diophantine property 23. Then, for every M there exists a vector P ɛ = P 0 + Oɛ and N such that D P Hɛ N P ɛ = ω 0, that is, the approximate rotation vector corresponding to P ɛ is the original rotation vector ω 0. Furthermore, sup u ɛ x, P ɛ ũ ɛ Nx, P ɛ = Oɛ M, x in which u ɛ x, P ɛ is a viscosity solution of u ɛ x, P ɛ u ɛ ˆx, P ɛ + hhp ɛ + D x u ɛ ˆx, P ɛ, x hd x u ɛ ˆx, P ɛ D p HP ɛ + D x u ɛ ˆx, P ɛ, x, = hh ɛ P ɛ, with ˆx x = hd p HP ɛ + D x u ɛ ˆx, P ɛ, x, and ũ ɛ N x, P ɛ is the corresponding approximate solution defined by 18. Proof. that is Define P ɛ by solving the equation ω 0 = D P Hɛ N P ɛ, ω 0 = D P H0 P 0 + ɛd P H1 P 0 + P ɛ P 0 D 2 P P H 0 P 0 +, in which the expansion is taken up to order N 1. Under the strict convexity assumption for H 0 = H 0, we have det D 2 P P H 0 P 0 0, and so, the implicit function theorem yields a unique solution of the form P ɛ = P 0 + ɛp 1 +,

19 for ɛ small enough, and P 1 = Weak KAM 19 [ D 2 P P H 0 P 0 ] 1 DP H1 P 0. Define the new coordinates P, X by 6, that is { p = P + D x ũ ɛ N x, P X = x + D P ũ ɛ N x, P. To simplify the notation we denote X = φx, P. Let x 0 be the point given by Lemma 5.3. Let x 0, p 0 = x 0, P ɛ + D x u ɛ x 0, P be the initial conditions for a trajectory x n, p n of 3. In the new coordinates, we have P 0 = P ɛ. Also the dynamics is transformed into { X n+1 X n = hd P Hɛ N P n + Oɛ N + P n P 0 N 1 P n+1 P n = Oɛ N + P n P 0 N. From this equation, it is clear that P n stays close to P ɛ for long times. The next lemma proves a quantitative estimate that reflects this idea. Lemma 6.2. Proof. Note that sup 0 n 1 ɛ N/4 P n P ɛ Oɛ N/8. P n+1 P ɛ 2 P n P ɛ 2 = P n+1 + P n 2P ɛ P n+1 P n δp n+1 + P n 2P ɛ δ P n+1 P n 2 2δ [ P n+1 P ɛ 2 + P n P ɛ 2] + 1 δ P n+1 P n 2. Moreover, P n+1 P n 2 Cɛ 2N + Cɛ 2N P n P ɛ 2, as long as P n P ɛ 2N 2 Cɛ 2N. This last condition is always satisfied for large N, and for the range of values n that will be used in our estimates. By choosing δ = ɛ N and setting a n = P n P ɛ 2, we have a n+1 a n Cɛ N a n+1 + a n + Cɛ N.

20 20 D. Gomes, C. Valls To obtain the final estimate, we need an auxiliary lemma: Lemma 6.3. Suppose a n is a sequence such that a 0 = 0, and Then, for all 0 n 1 3Cɛ N 1 Cɛ N a n Cɛ N a n + Cɛ N. and ɛ small enough, we have a n Cɛ N/2 ne 2n Cɛ N/2. Proof. Set α = Cɛ N. We will proceed by induction over n. For n = 0, the estimate is clear. Therefore, we assume it holds for some n and we will prove it for n + 1. We have a n α 1 α a n + α 1 α 1 + 3αa n + 2α, for α sufficiently small. Then, by the induction hypothesis, a n+1 1+3α αne 2 αn +2α αe 2 αn+1 + αne 2 αn +2 αne 2 αn. Since α e 2 α, we get a n+1 αn + 1e 2 αn+1. We should note that both the proof and the result of the previous lemma are not sharp, however they are sufficient for our purposes. In fact, this previous lemma implies that for all 0 n C. ɛ N/4 P n P ɛ Cɛ N/8, Observe that X = ψx φx, P ɛ is a diffeomorphism, for small ɛ. Let UX = u ɛ ψ 1 X, P ɛ ũ ɛ Nψ 1 X, P ɛ. Define X n = ψx n. Then, we have U X n+1 U X n = = u ɛ x n+1, P ɛ u ɛ x n, P ɛ ũ ɛ Nx n+1, P ɛ + ũ ɛ Nx n, P ɛ = = hh ɛ P ɛ + hhp ɛ + D x u ɛ x n+1, P ɛ, x n hd x u ɛ x n+1, P ɛ D p HP ɛ + D x u ɛ x n+1, P ɛ, x n + + h H ɛ NP ɛ hhp ɛ + D x ũ ɛ N x n+1, P ɛ, x n + + hd x ũ ɛ N x n+1, P ɛ D p HP ɛ + D x ũ ɛ N x n+1, P ɛ, x n + + ũ ɛ N x n+1, P ɛ ũ ɛ Nx n+1, P ɛ + Oɛ N,

21 in which Therefore U X n+1 U X n Weak KAM 21 x n+1 x n = hd p HP ɛ + D x ũ ɛ N x n+1, P ɛ, x n. C D x u ɛ x n+1, P ɛ D x ũ ɛ Nx n+1, P ɛ + C x n+1 x n+1 + Oɛ N. Now, observe that together with yields p n+1 = P n+1 + D x ũ ɛ Nx n+1, P n+1, p n+1 = P ɛ + D x u ɛ x n+1, P ɛ, D x u ɛ x n+1, P ɛ D x ũ ɛ Nx n+1, P ɛ D x u ɛ x n+1, P ɛ D x ũ ɛ Nx n+1, P n D x ũ ɛ Nx n+1, P n+1 D x ũ ɛ Nx n+1, P ɛ C P n+1 P ɛ. Moreover, the equation y x n = hd p HP + D x ũ ɛ Ny, P, x n defines y as smooth function of x n and P. Therefore, x n+1 x n+1 C P n+1 P ɛ. Thus, by using Lemma 6.2 we have U X n+1 U X n Oɛ N/8, for all 0 n C. We may add a constant to u ɛ in such a way that ɛ N/4 U X 0 = 0, and thus sup U X n = Oɛ N/16. 0 n C ɛ N/16 The Diophantine property implies that the map Y n+1 Y n = hd P H 0 P 0 has an ergodization time of order O C δ, for some time depending on r the Diophantine exponent s in 23. That is, given δ and any Y, there exists 0 n O C δ such that Y r Yn δ. Consider the map defined for 0 n C ɛ N/16 X n+1 X n = hd P H 0 P 0 + Oɛ N 1.

22 22 D. Gomes, C. Valls Then given ɛ and any X, there is a 0 n C ɛ rm X n X ɛ M, provided M < N. Furthermore, we have 16r X n = φx n, P n, and So, for 0 n X n = φx n, P ɛ. C, Lemma 6.2 implies ɛ N/16 X n X n Oɛ N/8. such that Consequently, the sequence X n satisfies that, given ɛ and any X, there is 0 n C such that ɛ rm X n X Cɛ M. Since U is a Lipschitz function, by choosing X n as in the previous formula UX UX U X n + U X n Cɛ M. The same estimate carries over to u ɛ x, P ɛ ũ ɛ N x, P ɛ, as ψ is a diffeomorphism. Remark. One should observe that sup u ɛ x, P ɛ ũ ɛ Nx, P ɛ = Oɛ M x implies sup u ɛ x, P ɛ ũ ɛ Mx, P ɛ = Oɛ M, x although this last estimate requires the existence of ũ ɛ N. Theorem 6.4. Let M > 0. Suppose ω 0 = D p H 0 P 0 is Diophantine, and 18 admits an approximated solution ũ ɛ N for N large enough such that the previous theorem holds. Then esssup Duˆx, ɛ P ɛ D x ũ ɛ N x, P ɛ Cɛ M/2, x in which ˆx and x are defined, respectively by 8 and 19. Proof. Subtracting 13 to 18, and using 29, we have Oɛ N =ũ ɛ Nx, P ɛ ũ ɛ N x, P ɛ u ɛ x, P ɛ + u ɛ ˆx, P ɛ + + hhp ɛ + D x ũ ɛ N x, P ɛ, x hhp ɛ + D x u ɛ ˆx, P ɛ, x hd x ũ ɛ N x, P ɛ D p HP ɛ + D x ũ ɛ N x, P ɛ, x+ + hd x u ɛ ˆx, P ɛ D p HP ɛ + D x u ɛ ˆx, P ɛ, x.

23 Weak KAM 23 Moreover, by strict convexity, we have HP ɛ + D x ũ ɛ N x, P ɛ, x HP ɛ + D x u ɛ ˆx, P ɛ, x+ + D p HP ɛ + D x u ɛ ˆx, P ɛ, xd x ũ ɛ N x, P ɛ D x u ɛ ˆx, P ɛ + + γ D x ũ ɛ N x, P ɛ D x u ɛ ˆx, P ɛ 2. Therefore, we get Oɛ N ũ ɛ Nx, P ɛ ũ ɛ N x, P ɛ u ɛ x, P ɛ + u ɛ ˆx, P ɛ + + hd x ũ ɛ N x, P ɛ [D p HP ɛ + D x u ɛ ˆx, P ɛ, x D p HP ɛ + D x ũ ɛ N x, P ɛ, x] + + γ D x ũ ɛ N x, P ɛ D x u ɛ ˆx, P ɛ 2. Since D p HP ɛ + D x u ɛ ˆx, P ɛ, x D p HP ɛ + D x ũ ɛ N x, P ɛ, x = x ˆx, and ũ ɛ N x, P ɛ D x ũ ɛ N x, P ɛ ˆx x ũ ɛ Nˆx, P ɛ Cɛ x ˆx 2, with we obtain, x ˆx 2 C D x ũ ɛ N x, P ɛ D x u ɛ ˆx, P ɛ 2, Oɛ N ũ ɛ Nx, P ɛ ũ ɛ Nˆx, P ɛ u ɛ x, P ɛ + u ɛ ˆx, P ɛ + + γ D x ũ ɛ N x, P ɛ D x u ɛ ˆx, P ɛ 2, for some constant γ, as long as ɛ is small enough. previous theorem yields Therefore, the D x ũ ɛ N x, P ɛ D x u ɛ ˆx, P ɛ 2 Oɛ M. We should point out that this last theorem shows that the Mather sets can be approximated through a perturbative method. In fact, since the Mather set is supported in x, D p HP ɛ + D x u ɛ ˆx, P ɛ, x, the result implies that this graph is approximated uniformly by x, D p HP ɛ + D x ũ ɛ N x, P ɛ, x.

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