Aubry-Mather and Hamilton-Jacobi
Ugo Bessi,*
Abstract We review and extend some results of Jauslin, Kreiss, Moser and Weinan on the smooth approximation of Mather sets.
Introduction We will consider a time-dependent lagrangian L(q, q, ˙ t) satisfying L(q+1, q, ˙ t) = L(q, q, ˙ t) = L(q, q, ˙ t+1); in other words, the phase space is the cylinder S 1 × R and L depends periodically on time; the exact hypotheses on L are listed below. Even in this low-dimensional and simple-looking situation the dynamics of the E-L flow of L is not completely understood. For instance, in general it is impossible to read directly on L whether, given M > 0, there are orbits q satisfying |q(T ˙ ) − q(0)| ˙ > M for some T > 0; this fact being known as Arnold’s diffusion. J. N. Mather has shown in [8] and [9] that much information on the orbits is coded inside the family of functions, depending on a real parameter c, defined by Z n ∞ hc (x, t, ξ) = lim inf min [L(q(s), q(s), ˙ s) − cq(s) ˙ + α(c)]ds : n→∞
t 1
q ∈ A.C.([t, n], S ),
q(t) = x,
q(n) = ξ
where A.C.([t, n], S 1 ) denotes the class of S 1 -valued absolutely continuous functions. Grossly, h∞ c (x, t, ξ) is the minimum action required to go from x to ξ in infinite time; α(c) is the unique real making the lim inf finite and the term −cq˙ has been added to make the functional sensitive to the homotopy class of the orbits. We will not try to give an exposition of Mather theory but we would like to stress the fact that each KAM ∞ torus is the graph of c − ∂x h∞ c (x, t, ξ) for a unique c ∈ R; for other values of c, the graph of c − ∂x hc (x, t, ξ)
gives portions of the stable and unstable manifolds of the “action minimizing” periodic orbits. It is thus natural that this function plays such an important rˆ ole in the theorems regarding the dynamics of L. Let now H denote the Hamiltonian of L; it is a standard calculation (see for instance [7]) that for any ξ ∈ S 1 h∞ c (x, t, ξ) is a viscosity solution of the following Hamilton-Jacobi equation ∞ ∂t h∞ c (x, t, ξ) − H(x, c − ∂x hc (x, t, ξ), t) + α(c) = 0 ∞ ∞ hc (x + 1, t, ξ) = hc (x, t, ξ) = h∞ c (x, t + 1, ξ).
(1)
* Dipartimento di Matematica, Universit`a Roma Tre, Largo S. Leonardo Murialdo, 00146 Roma, Italy. email:
[email protected] 1
Jauslin, Kreiss and Moser proposed in [5] to find h∞ c by the vanishing viscosity method. We recall their results in the appendix; what matters now is that 2 ν 2 ∂xx u + ∂t u − H(x, c − ∂x u, t) + β(ν) = 0 u(x + 1, t) = u(x, t) = u(x, t + 1)
(2)
has, for a suitable choiche of β(ν) ∈ R, a classical solution which is unique up to an additive constant and which we call uν . If νk & 0 then uνk has (always up to an additive constant) a subsequence converging a. e. whose limit solves (1) in the viscosity sense. The problem with this approach is that the viscosity solutions of (1) are not unique and many of them, as shown in [5], do not coincide with h∞ c (x, t, ξ) for any value of ξ. Thus one is lead to the following question: when does the limit of uν as ν → 0 exist and coincide, for a suitable choiche of ξ, with h∞ c (x, t, ξ)? This question has a rather long history, though in a different setting: as we explain in the digression at the end of the paper, the solutions of (2) can be transformed into those of the Schr¨odinger equation. Here ν plays the rˆole of the Plank constant h; identifying the limit of uν as ν → 0 is known in Physics as the problem of the semiclassical approximation. Theorem 5 of [5] concerns the tunnelling effect. Indeed, here Jausslin et al. restrict to the very particular case in which L(q, q, ˙ t) =
1 ˙2 2 |q|
− V (q) and they suppose that the absolute maxima of V are all nondegenerate and of
different second order derivatives. Let c0 be the action functional of the heteroclinic chain connecting all the maxima of V ; if |c| ≥ c0 it is easy to see that the solution of (1) is unique and uν converges to it. If |c| < c0 they show that uν converges, as ν → 0, to h∞ c (x, t, ξ) where ξ is the maximum of V with smallest second order derivative; this is exactly the statement of tunnelling. Our aim is to generalize this result of [5] to a more general class of lagrangians and to periodic orbits of any rational rotation numbers. Assumptions on the lagrangian We suppose that L is of class C 3 , that ∂q˙q˙ L > 0, that lim|q|→∞ ˙
L(q,q,t) ˙ |q| ˙
=
+∞ and that the Euler-Lagrange flow of L is complete. These are exactly the hypotheses under which the Aubry-Mather theory works, save for the fact that L is of class C 3 and not C 2 . Assumptions on the minimal orbits Let n ∈ N, m ∈ Z be relatively prime and let us suppose that the functional I: {q ∈ A.C.([0, n], R) : q(n) = q(0) + m, q(0) ∈ [0, 1)} → R Z n I: q → L(q, q, ˙ t)dt
(3)
0
has only a finite number of minima, q˜1 , . . . , q˜N , all of them hyperbolic. This implies (see for instance [9], section 7) that there is an interval [c1 , c2 ] with c1 < c2 maximal with respect to this property: for c ∈ [c1 , c2 ] and any ξ ∈ S 1 the graph of c − ∂x h∞ c (x, t, ξ) consists of portions of the stable and unstable manifolds of some q˜i . We will always suppose c ∈ [c1 , c2 ]. Moreover, dropping some q˜i we can always suppose that q˜i (· + m) 6= q˜j for i 6= j, ∀m ∈ Z. We define Ai =
Z
n 2 ∂qq L(˜ qi , q˜˙i , t)dt.
0
2
We will prove the following theorem. Theorem 1.
Let L, {˜ q i }N i=1 , c1 < c2 n ∈ N, m ∈ Z and Ai be as above. Let c ∈ [c1 , c2 ] and let there be
only one q˜i with minimal Ai , say q˜j . Then ∂x uν (x, t) → ∂x h∞ ˜j (0)) a. e. as ν → 0. c (x, t, q We will also prove a theorem for a completely opposite situation, the case in which a KAM torus exists. It is a theorem of Birkhoff’s that the KAM tori for this kind of lagrangians are graphs of closed one-forms, T = {(x, c − ∂x u(x, t), t)}(x,t)∈S 1 ×S 1 ; as we explain later, it follows from [10] that ∂x uν → ∂x u a. e.; the authors of [5] ask how fast this convergence is. Theorem 2.
Let there exist a KAM torus T C 1 -conjugate to a rotation of frequency ω; we suppose that
for some τ > 0 ω satisfies |ωq − p| ≥
D qτ
∀p ∈ Z, ∀q ∈ N.
It is well-known that T = {(x, c − ∂x u(x, t), t)}(x,t)∈S 1 ×S 1 for a unique c ∈ R and some u ∈ C 2 (S 1 × S 1 ). Then 1
|∂x uν (x, 0) − ∂x u(x, 0)| ≤ Dν 3τ +2 . The proof is based on the fact that the viscosity solutions u of (1) and uν of (2) have a variational characterization; we recall later that they are given by the minimum value of the action functional on orbits with appropriate boundary conditions, but with a small white noise added for uν . This white noise makes the orbits realizing uν sensitive to what happens in a neighbourhood of q˜i ; that’s the reason for the Ai of theorem 1.
Section 1 We begin recalling some notions of [8] and [9]. We say that q ∈ A.C.(R, S 1 ) is c-minimal if ∀a, b, d, e ∈ Z,
a < b,
we have Z
d < e,
∀˜ q ∈ A.C.([d, e], S 1 ) : q˜(d) = q(a), q˜(e) = q(b)
b
[L(q, q, ˙ s) − cq˙ + α(c)]ds ≤
a
Z
e
[L(˜ q , q˜˙ , s) − cq˜˙ + α(c)]ds
d
where α(c) is the same number defined in the introduction. If we allow for a = d = t we have an analogous definition for orbits defined on the half line [t, ∞); in this case we will speak of orbits c-minimal on [t, ∞). Let us now consider a sequence nk → ∞ and orbits qnk such that qnk (t) = x and Z
nk
qnk (nk ) = ξ
[L(qnk , q˙nk , s) − cq˙nk + α(c)]ds → h∞ c (x, t, ξ).
t
3
Such sequences exist by the definition of h∞ c and it can be shown that, up to a subsequence, they converge on compact sets to an orbit q c-minimal on [t, ∞) and with q(t) = x; we will say that q realizes h∞ c (x, t, ξ). There is one caveat: the orbits realizing h∞ c (x, t, ξ) need not have ξ in their ω-limit; of this the reader may easily convince herself considering the pendulum lagrangian L(q, q, ˙ t) = 12 |q| ˙ 2 + [1 − cos(q)] and the corresponding ∞ h∞ 0 (x, 0, ξ) with x < 0 < ξ: the orbit realizing h0 (x, 0, ξ) is the homoclinic starting at x and accumulating
at the unstable equilibrium 0. We recall from [8] that, given any ω ∈ R, it is possible to choose c ∈ R such that all the recurrent c-minimal orbits have rotation number ω; from [1] or [9] it follows that if ω =
m n
∈Q
then the recurrent c-minimal orbits are all periodic of period n. The relationship between the minimum value of the functional I defined in (3) and α(c) is that, for c ∈ [c1 , c2 ],
Z
n
[L(˜ qi , q˜˙ i , t) − cq˜˙ i + α(c)]dt = 0
∀i.
(4)
0
There are two ways to consider our orbits: as having S 1 or R, the universal cover, as target space; from now on we will stick to the second one. It is a a result of [1] that the order of R induces an order on the recurrent c-minimal orbits; in other words, if q˜i (0) < q˜j (0) then q˜i (t) < q˜j (t)∀t ∈ R. Let now 0 ≤ x1 < x2 < . . . < xnN < 1 0≤l≤n−1 denote the points {˜ qj (l)(mod1)}1≤j≤N ; for 1 ≤ i ≤ nN let us call qi (t) the orbit in {˜ qj (· + l)}j,l such that
qi (0) = xi . Since the recurrent minimal orbits are ordered, qi (t) < qi+1 (t)∀t. The following lemmas are a consequence of the fact that two c-minimal orbits intersect at most once. Lemma 1.
Let c ∈ [c1 , c2 ] and x ∈ (xi , xi+1 ). Let q − (t) and q + (t) be two orbits, c-minimal on [0, ∞)
and satisfying
limt→+∞ |qi (t) − q − (t)| = 0 = limt→+∞ |qi+1 (t) − q + (t)| q + (0) = x = q − (0).
(5)
Then q˙− (0) < q˙+ (0). Proof.
We can rule out that q˙− (0) = q˙+ (0): otherwise q − and q + would coincide contradicting (5).
Let us suppose by contradiction that q˙− (0) > q˙+ (0); we already recalled that qi (t) < qi+1 (t)∀t; by (5) and continuity this implies that there is t0 > 0 such that q − (t0 ) = q + (t0 ). Since both q − and q + are c-minimal on [0, ∞), their action functionals on [0, t0 ] must be the same. Thus also q(t) =
q + (t) 0 ≤ t ≤ t0 q − (t) t ≥ t0
is a c-minimal orbit; this is a contradiction since q does not satisfy the E-L equation at t0 , where it is not smooth. \\\ Let now νk → 0 and let uνk solve (2); as we recall in the appendix, possibly taking a subsequence we can suppose that uνk converges a. e. to a Lipschitz u solving (1) in the viscosity sense. It is proven in [10] 4
that, if x ∈ S 1 is a point of differentiability of u(x, 0) (and almost any point is since u is Lipschitz) then the orbit qx of L with initial conditions (
qx (0) = x (6) q˙x (0) = Hp (x, c − ∂x u(x, 0), 0)
is c-minimal on [0, ∞). Let now c ∈ [c0 , c1 ] and x ∈ [xi , xi+1 ]; from [1] or from lemma 2.15 of [10] it follows that either lim |qx (t) − qi (t)| = 0
t→+∞
Lemma 2.
lim |qx (t) − qi+1 (t)| = 0.
or
t→+∞
There is one and only one point z¯ ∈ [xi , xi+1 ] such that if x ∈ (xi , z¯) then
lim |qx (t) − qi (t)| = 0
t→+∞
if x ∈ (¯ z , xi+1 ) then
Proof.
lim |qx (t) − qi+1 (t)| = 0.
t→+∞
2 We recall in the appendix the result of [5] that ∂xx uν ≤ M as ν → 0; thus ∂x u(x, 0) can have only
downward jumps, i. e. ∀z ∈ (xi , xi+1 )
lim inf ∂x u(x, 0) ≥ lim sup ∂x u(x, 0). x%z
(7)
y&z
Moreover, if ∂x u(z, 0) exists, then it stays between the two limits above. Let us suppose that the thesis does not hold; then there is z0 ∈ (xi , xi+1 ) and two sequences, xn % z0 and yn & z0 such that, for n → ∞, qxn and qyn accumulate on qi+1 and qi respectively. One of these two sequences may coincide with z0 , if ∂x u(z0 , 0) exists. It follows by an easy compactness argument that qxn and qyn converge, up to a subsequence, to two orbits, which we call respectively q + and q − ; they are both c-minimal on [0, ∞) and satisfy q − (0) = q + (0) = z0 . We assert that q + accumulates on qi+1 and q − on qi . Let us postpone the proof of this assertion and let us look at its consequences: by lemma 1 we have that q˙− (0) < q˙+ (0) or equivalently limn→∞ q˙yn (0) < limn→∞ q˙xn (0). But Hp is monotone inreasing in p since we supposed L convex; thus by (6) limn→∞ ∂x u(yn , 0) > limn→∞ ∂x u(xn , 0). Since from [5] we know that ∂x u is B. V., this contradicts (7). We now prove that q − accumulates on qi . We recall that by [1] the c-minimal q − must accumulate either on qi or on qi+1 : let us suppose by contradiction that q − accumulates on qi+1 . We consider k1 ∈ N, depending on n, such that |qyn (k1 ) − qi+1 (k1 )| is minimal. Since qyn converges to q − we have that for n large enough |qyn (k1 ) − qi+1 (k1 )| ≤
1 |qy (0) − qi+1 (0)|. 2 n
(8)
Let us now consider qˆn (t) = qyn (t + k1 ) + k2 ; we will choose k2 ∈ Z such that qi (k1 ) < qˆn (k1 ) < qi+1 (k1 ). Since qˆn is c-minimal on [0, ∞), by [1] qi (t) < qˆn (t) < qi+1 (t) for t ∈ [0, ∞); qˆn accumulates on qi since qyn does. By (8) qˆn (0) > qyn (0) while qˆn (k1 ) ≤ qyn (k1 ) because k1 is minimizing. Thus qˆn (t) and qyn cross at t0 ≤ k1 . In other words, qˆn and qyn cross twice, at t0 and at +∞, which by [1] is absurd. \\\ 5
We recall the terminology of [5]. If the point z¯ defined in the previous lemma belongs to (xi , xi+1 ), we will say that z¯ is a strong jump. Let us consider two abutting intervals (xi−1 , xi ) and (xi , xi+1 ) and let us see what can happen at xi . 1) Let qx be defined as in (6); then if ∀x ∈ (xi − , xi + )
lim |qx (t) − qi (t)| = 0
t→+∞
we call xi a transition point. We note that if xi is a transition point, then by (6) its local stable manifold is given by the graph of c − ∂x u(x, t). 2) If
∀x ∈ (xi − , xi ) ∀x ∈ (xi , xi + )
limt→+∞ |qx (t) − qi−1 (t)| = 0 limt→+∞ |qx (t) − qi+1 (t)| = 0
we call xi a limit jump, since it can be considered as the limit of an interior strong jump. 3) If
∀x ∈ (xi − , xi ) ∀x ∈ (xi , xi + )
limt→+∞ |qx (t) − qi−1 (t)| = 0 limt→+∞ |qx (t) − qi (t)| = 0
∀x ∈ (xi − , xi ) ∀x ∈ (xi , xi + )
limt→+∞ |qx (t) − qi (t)| = 0 limt→+∞ |qx (t) − qi+1 (t)| = 0
or
we call xi a second order discontinuity, retrograde in the first case and advanced in the second. We note that there are no other cases. First, because by [1] if x ∈ (xi , xi+1 ) the c-minimal orbit qx must satisfy limt→+∞ |qx (t) − qj (t)| = 0 for some j ∈ (i, i + 1). Second, because, if is small enough, by lemma 2 all the qx with x ∈ (xi − , xi ) accumulate on the same orbit. Lemma 3.
Let c ∈ (c0 , c1 ); then u has at least one transition point. Moreover, if qi (0) is a transition
point, then all the points qi (l), l ∈ Z are transition. Proof.
We begin to prove that u cannot have only second order discontinuities, all of the same kind; to
fix ideas say advanced. Let us suppose by contradiction that this is the case. Thus, if x ∈ (xj−1 , xj−1 + ) ∪ (xj − , xj ), qx accumulates on qj (t); by lemma 2 qx must accumulate on qj ∀x ∈ (xj−1 , xj ). If we take the limit of qx as x & xj−1 , we obtain a heteroclinic (or homoclinic, if xj−1 and xj are on the same q˜k ) connection between xj−1 and xj ; by the variational characterization of u explained at the end of the lemma its action is u(xj , 0) − u(xj−1 , 0). Since this holds for all j, we have a heteroclinic chain from x1 to x2 to x3 PnN +1 all the way up to x1 + 1; the action of this heteroclinic chain is 2 [u(xj , 0) − u(xj−1 , 0)] that is to say 0 since u(x1 + 1, 0) = u(x1 , 0). By [9] a heteroclinic chain with zero action exists if and only if c = c1 or if c = c2 , a contradiction. To show that u cannot have only second order discontinuities, we note that by lemma 2 no retrograde second order discontinuity can immediately follow an advanced one; thus the second order discontinuities of u are either all retrograde or all advanced and we fall back into the previous case. Let us now suppose by contradiction that u has only second order discontinuities and limit jumps; let xi be a limit jump. Since xi+1 is not a transition point, by lemma 2 it must be an advanced second 6
order discontinuity, as xi+2 , xi+3 , etc., all the way up to xi + 1 which we have supposed a limit jump, a contradiction. Since we have ruled out all other possibilities, u must have a transition point. Let us now come to the last assertion of the lemma. We remark that it is a standard fact (see for instance [10], thm. 2.6) that the graph of c − ∂x u(x, 0) (which by (6) is the graph of q˙x (0) in different coordinates) is carried into itself by the step-1 map of the E-L flow of L; thus, if qi (0) is a transition point also qi (1), qi (2), etc. are such and all the orbit qi is transition. \\\ By the way, we note that the invariance of the graph of ∂x u(x, 0) under the step-1 map of the E-L flow of L is at the basis of a completely different approach to this theory, that of [6]. Because of the previous lemma, from now on we will speak of transition orbits instead of transition points. We now recall the variational characterization of u and uν . If u is any viscosity solution of (H-J), then ∀(x, t) ∈ S 1 × (−∞, 0]
u(x, t) = min
Z
0
[L(q, q, ˙ s) − cq˙ + α(c)]ds + u(q(0), 0) :
t
q ∈ A.C.([0, t], S 1 ),
q(t) = x .
From [4] it follows that the solution uν of (2) satisfies, for t ≤ 0, uν (x, t) = min E Y
Z
0
[L(ξ, η, ˙ s) − cη˙ + β(ν)]ds + uν (ξ(0), 0)
t
where Y varies in the class of smooth, time dependent vector fields; ξ solves for s ≥ t the stochastic differential equation (
dξ(s) = Y (ξ(s), s)ds + νdw(s) ξ(t) = x
and η(s) ˙ = Y (ξ(s), s); w is the standard Brownian motion and E denotes expectation with respect to the Wiener measure. We call Yν the minimal Y (which exists and is unique as shown in [4]) and ξν the solution of the corresponding stochastic differential equation; we set η˙ ν (s) = Yν (ξν (s), s). We recall from [4], formula (4.5), that Yν (x, s) = Hp (x, c − ∂x uν (x, s), s).
(9)
Since uν is periodic, the last formula implies that Yν (x + 1, s) = Yν (x, s) = Yν (x, s + 1). Lemma 4.
Let c ∈ (c0 , c1 ) and let u be the a. e. limit of a sequence uνl , νl & 0. Then u has only one
transition orbit, which is the one with minimal Ai . Proof.
The proof consists in two estimates on α(c) − β(ν), one from above and one from below, which can
agree only if the thesis is true. We begin with the estimate from above. 7
Given any qi , we define for t ≥ 0 ξ(t) = qi (t) + νw(t) where w is the standard Brownian motion. By the variational characterization of uν we have that uν (qi (0), 0) ≤ E
(Z
)
nk
[L(ξ(t), q˙i (t), t) − cq˙i + α(c)]dt + uν (ξ(nk), nk)
+ nk(β(ν) − α(c)).
0
We set 2 gi (t) = ∂qq L(qi (t), q˙i (t), t)
and we get, from the previous formula and a Taylor development, that, for some D1 > 0, uν (qi (0), 0) ≤ E
nZ
nk
[L(qi , q˙i , t) − cq˙i + α(c)]dt + ν
0
Z
kn
0
ν2 gi (t)w(t)2 dt + 2
Z
nk
∂q L(qi , q˙i , t)w(t)dt+
0
Z
kn
D1
0
o ν3 |w(t)|3 dt + uν (ξ(nk), nk) + nk(β(ν) − α(c)). 2
The first integral is zero by (4); we recall (see also the appendix) that in [5] it has been proven that |∂x uν | ≤ D2
∀ν ∈ (0, 1].
By this and Fubini we get that ν2 uν (qi (0), 0) ≤ 2
Z
nk
gi (t)tdt + D3 ν
3
nk
Z
0
3
t 2 dt + uν (qi (nk), nk) + D2 νE(|w(nk)|) + nk(β(ν) − α(c)).
0
Evaluating the first integral and recalling the periodicity of uν and qi we get that ν 2 nk(nk − 1) ν2 uν (qi (0), 0) ≤ Ai + nk 2 2 2
Z 0
1
√ 5 2 gi (t)tdt + D3 ν 3 (nk) 2 + uν (qi (0), 0) + D2 ν nk + nk(β(ν) − α(c)) 5
where Ai has been defined in the introduction. Dividing the last formula by nk we get that ν2 α(c) − β(ν) ≤ min 2 i
(nk − 1) Ai + 2
Z 0
1
3 1 gi (t)tdt + D4 ν 3 (nk) 2 + D2 ν √ nk
(10)
where the minimum comes from the fact that qi is arbitrary. We now prove the estimate from below; let qj be a transition orbit of u. We consider the ξν in the variational characterization of uν (qj (0), 0) and recall from [4] that ξν (t) = ην (t) + νw(t) where with probability one ην is a C 1 curve satisfying ην (0) = qj (0); from the variational characterization of u we have that u(qj (0), 0) ≤ E
Z 0
nk
[L(ην , η˙ ν , t) − cη˙ ν + β(ν)]dt + u(ην (nk), nk) + nk(α(c) − β(ν)) 8
from which we get that uν (qj (0), 0) − u(qj (0), 0) ≥ E
Z
nk
[L(ξν , η˙ ν , t) − L(ην , η˙ ν , t)]dt+
0
uν (ξν (nk), nk) − u(ην (nk), nk) − nk(α(c) − β(ν)). From a Taylor development we get as before that uν (qj (0), 0) − u(qj (0), 0) ≥ −nk(α(c) − β(ν))+ E
(Z
nk
[∂q L(ην , η˙ ν , t)νw(t) +
ν 2 ∂qq L(ην , η˙ ν , t)
2
0
)
2 2
3
3
w(t) − D1 ν |w(t)| ]dt + uν (ξν (nk), nk) − u(ην (nk), nk) . (11)
If we could substitute qj to ην in (11), we would have an estimate similar to (10), but with the sign of the inequality reversed; in the rest of the proof we show that, since qj is a transition orbit, this substitution is possible. The main point is that ∃δ(ν) > 0
:
2
∀t ≥ 0 E
2
sup d(qj (t) − ην (t)) + sup |q˙j (t) − η˙ ν (t)| s∈[0,t]
!
≤ δ 2 (ν)
(12)
s∈[0,t]
where d denotes distances on S 1 and δ(ν) → 0 as ν → 0. We postpone the proof of this inequality; we now show how it implies the thesis. From [5] it follows that |∂x uν (x, t)| is bounded uniformly in ν; together with (9) this implies that Yν is bounded and thus that η˙ ν is bounded uniformly in ν. From a Taylor development we get nk
Z
E
2 ∂qq L(ην , η˙ ν , t)ν 2 w(t)2 dt
!
≥
0
E
ν2
Z
nk 2 ∂qq L(qj , q˙j , t)w(t)2 dt − D5 ν 2
0
ν2
nk
Z
d(qj (t) − ην (t))w(t)2 dt − D5 ν 2
0
Z
nk
gj (t)tdt − D5 ν 2
0
E d(qj (t) − ην (t))w(t)2 dt −
0
|q˙j (t) − η˙ ν (t)|w(t)2 dt
0
nk
Z
nk
Z
nk
Z 0
! E |q˙j (t) − η˙ ν (t)|w(t)2 dt .
From the last formula, the Cauchy-Schwarz inequality and (12) we get that E
Z
nk 2 ∂qq L(ην , η˙ ν , t)ν 2 w(t)2 dt
!
≥ ν2
Z
0
D5 ν
2
Z
nk
gj (t)tdt−
0
nk 2
Ed(qj (t) − ην (t))
0
ν
2
21
Z
4
Ew(t)
12
dt +
nk 2
E |q˙j (t) − η˙ ν (t)|
0
nk
gj (t)tdt − D6 δ(ν)ν
0
ν2
Z
2
Z
nk
tdt =
0
Z
nk
gj (t)tdt − D7 δ(ν)ν 2 (nk)2 .
0
9
12
4
Ew(t)
12
dt
!
≥
!
=
Analogously we get, possibly increasing D7 , E
nk
Z
∂q L(ην , η˙ ν , t)νw(t)dt
!
≥
0
E
nk
Z
∂q L(qj , q˙j , t)νw(t)dt − D5 ν
0
Z
nk
E[d(qj − ην ) · |w(t)|] − D5 ν
Z
0
nk
E[|q˙j − η˙ ν | · |w(t)|] ≥
0 3
−D7 δ(ν)ν(nk) 2 √ E[uν (ξν (nk), nk) − u(ην (nk), nk)] ≥ E[uν (ην (nk), nk) − u(ην (nk), nk)] − D2 ν nk ≥ √ uν (qj (nk), nk) − u(qj (nk), nk) − D2 ν nk − D8 δ(ν). From (11) and the last three formulas we get that uν (qj (0), 0) − u(qj (0), 0) ≥ ν2 2 ( Z E −
nk
Z
3
gj (t)tdt − D7 δ(ν)ν 2 (nk)2 − D7 δ(ν)ν(nk) 2 +
0
)
nk
D1 ν 3 |w(t)|3 dt + uν (ξν (nk), nk) − u(ην (nk), nk)
− nk(α(c) − β(ν)) ≥
0
ν2 2 D9 ν 3
Z
nk
nk
Z
3
gj (t)tdt − D7 δ(ν)ν 2 (nk)2 − D7 δ(ν)ν(nk) 2 −
0
√
3
t 2 dt − D2 ν nk − D8 δ(ν) + uν (qj (nk), nk) − u(qj (nk), nk) − nk(α(c) − β(ν)).
0
The periodicity of uν , u and qj implies as before α(c) − β(ν) ≥ ν 2 nk − 1 ν2 Aj + 2 2 2 If we choose nk '
Z
1
0
1 2 ν
1 3 2 D8 D2 ν gj (t)tdt − D7 δ(ν)ν 2 (nk) − D7 δ(ν)ν(nk) 2 − D9 ν 3 (nk) 2 − δ(ν) − √ . 5 nk nk
(13)
we see that (10) and (13) are in contradiction unless the transition orbit qj has
minimal Aj . We now prove (12). We recall that, since qj is a transition orbit of u, in a neighbourhood of qj the graph of c − ∂x u(x, t) coincides with the stable manifold of qj . Thus in the neighbourhood of qj U = {(q, t) : |q − qi (t)| < δ} we have that c − ∂x u(x, t) is of class C 2 ; because of this greater regularity the method of theorem 4 of [4] implies that, possibly reducing δ, ∂x uν (x, t) → ∂x u(x, t) uniformly in U . We know that ξν satisfies ξν (t) − qi (0) =
Z
t
Yν (ξν (s), s)ds + νw(t).
0
We set Y (x, t) = Hp (x, c − ∂x u(x, t), t). 10
a(t) = ∂x Y (q(t), t). We recall that by (6) q˙j (t) = Y (qj (t), t); by (9), the uniform convergence of ∂x uν and a Taylor development we have that |Yν (x, t) − q˙j (t) − a(t)[x − qj (t)]| ≤ (ν)
∀(x, t) ∈ U
with (ν) ≤ Cδ 2 for ν small enough. Thus we have that, as long as ξν ([0, t]) ⊂ U , ( dξν (s) = [q˙j (s) + a(s)(ξν (s) − qj (s)) + g(s)]ds + νw(s)
(14)
ξν (0) = qj (0) with |g(s)| ≤ (ν)
∀s ∈ [0, t].
(15)
Let q solve (
q(s) ˙ = q˙j (s) + a(s)(q(s) − qj (s)) + g(s) q(0) = qj (0)
and let T be the first time such that ξν (t) 6∈ U ; T is a stopping time. From the explicit solution of (14) we get that Z
E|ξν (t ∧ T ) − q(t ∧ T )|2 ≤ ν 2
t∧T
2
e
0
Rk
where the last inequality comes from the fact that
0
R t∧T s
a(τ )dτ
ds ≤ D11 ν 2
a(τ )dτ < 0 since a(τ ) is the linearized motion on
the stable manifold. The last formula and the Kolmogoroff inequality for martingales (see for instance [3], chapter 7) now yield E sup |ξν (t) − q(t)|2 ≤ D12 ν 2 . 0≤t≤T
If we set A = {ω : sup |ξν (t) − q(t)| ≥ 0≤t≤T
δ } 4
and we denote by 1A its characteristic function by the Tchebitcheff inequality we get E1A ≤
D13 ν 2 . δ2
(16)
On the other side, for t ≤ T we have that |g(t)| ≤ (ν) and from the explicit expression of q we get that, for ν small, |q(t) − qj (t)| ≤
δ 4
t ∈ [0, T ].
Thus from the last inequality and (16) we get that, if ν is small enough and if ω ∈ Ac , then ξν (T ) is in the interior of U , which forces T = +∞. In other words, if ω ∈ Ac , then T = +∞. Let d(x, y) represent distances on S 1 ; since d(x, y) ≤ 1 we have from (16) and the previous formula that, if ν is small, Ed(ξν (s) − qi (s))2 ≤ E(1A ) + E(|ξν (s) − qi (s)|1Ac ) ≤ The estimate on E|ην (s) − q˙i (s)|2 11
D13 2 δ ν + ≤δ δ2 4
∀s.
follows from the previous estimate, the fact that ην (s) = Yν (ξν (s), s) and that Yν → Y uniformly in U . \\\
Proof of Theorem 1 We distinguish two cases: c ∈ (c0 , c1 ) and c ∈ {c0 , c1 }. Let us begin with the first case and let us consider a sequence νk & 0. We know from [5] that, possibly taking a subsequence, ∂x uνk → ∂x u a. e., where u is a viscosity solution of (1). We know from lemma 4 that u has only one transition orbit, qj : the one with minimal Aj . Adding a constant, we can always suppose that u(qj (0), 0) = h∞ c (qj (0), 0, qj (0)) = 0, the second equality being a consequence of (4). Clearly, it suffices to prove that 1 u(x, 0) = h∞ c (x, 0, qj (0)) ∀x ∈ S . Let x ∈ (xi , xi+1 ) be a common differentiability point of u(x, 0) and
h∞ ˇx satisfying c (x, 0, qj (0)); let us consider the orbit q (
qˇx (0) = x qˇ˙x (0) = Hp (x, c − ∂x h∞ c (x, 0, qj (0)), 0).
Clearly, qˇx is c-minimal on [0, ∞) and by [1] it must accumulate on qi or on qi+1 ; to fix ideas, let us suppose it is qi . It is easy to see that h∞ ˇx connecting x c (x, 0, qj (0)) is realized by a c-minimal heteroclinic chain: q with xi , qˆi−1 connecting xi with xi−1 , qˆi−2 connecting xi−1 with xi−2 , etc., until we arrive to qˆs connecting xs+1 with qj (b). In other words, we have that Z
h∞ c (x, 0, qj (0)) =
b
[L(qj , q˙j , t) − cq˙j + α(c)]dt+
0 i−1 X l=s
lim
k→∞
lim
k→∞
nk
Z
Z
[L(ˆ ql , qˆ˙ l , t) − cqˆ˙ l + α(c)]dt+
−nk
nk
[L(ˇ qx , qˇ˙x , t) − cqˇ˙x + α(c)]dt.
0
We assert that also u(x, 0) is realized by a heteroclinic chain connecting x with qj (b0 ). Indeed, let us consider the orbit qx defined by (6) and to fix ideas let us suppose it accumulates on xi+1 . If xi+1 is a transition point (i. e. if xi+1 = qj (b0 ) for some b0 ) we have done; otherwise, xi+1 is neither a transition point nor a limit jump, because qx accumulates on it; thus it must be an advanced second order discontinuity and we can connect it to xi+2 by the heteroclinic q¯i+1 ; we go on in this way until we arrive to q¯s0 connecting xs0 to qj (b0 ). The variational characterization of u now implies u(x, 0) =
b0
Z
[L(qj , q˙j , t) − cq˙j + α(c)]dt+
0 0
s X
l=i+1
lim
k→∞
lim
k→∞
Z
Z
nk
[L(¯ ql , q¯˙ l , t) − cq¯˙ l + α(c)]dt+
−nk
nk
[L(qx , q˙x , t) − cq˙x + α(c)]dt.
0
12
By the variational characterization of u, also this heteroclinic chain is c-minimal and thus must have the ∞ same action as the heteroclinic chain for h∞ c ; by the last two formulas, we have that hc (x, 0, qj (0)) = u(x, 0)
for a. e. x and thus for all x since both functions are continuous. Since every sequence {uνl }νl →0 has a subsequence converging to h∞ c (x, 0, qj (0)), the thesis follows. Let now c ∈ {c0 , c1 }. Adding a constant we can always suppose that u(x0 , 0) = h∞ c (x0 , 0, x0 ) = 0. We want to prove that u(x, 0) = h∞ c (x, 0, x0 )∀x. Let x ∈ (xi , xi+1 ) be a common differentiability point of u and h∞ c ; we consider the orbit qx defined by (6); it tends asymptotically say to xi and we can see as before that it ∞ realizes u(x, 0); since qx is c-minimal it also realizes h∞ c (x, 0, xi ) and thus ∂x u(x, 0) = ∂x hc (x, 0, xi ). Since
c ∈ {c0 , c1 } it follows easily from [9] that h∞ c has no internal jump and no limit jump, only second order ∞ discontinuities all of the same kind; in particular, h∞ c (x, 0, xi ) = hc (x, 0, x0 ) + c(xi ) with c(xi ) a constant
depending only on xi . Thus ∂x u(x, 0) = ∂x h∞ c (x, 0, x0 ) and we have done. \\\
Remark 1. We remark that the definition of Ai is gauge-invariant: in other words, the Lagrangian ˜ q, L(q, ˙ t) = L(q, q, ˙ t) − ∂q S(q, t)q˙ − ∂t S(q, t), which has the same c-minimal orbits as L, has the same Ai . We also note that Ai is not the Lyapoumoff exponent of q˜i , which is given by Z n [Hpq (˜ qi , c − uq (˜ qi , t), t) − Hpp (˜ qi , c − uq (˜ qi , t), t)uqq (˜ qi , t)]dt 0
while a simple calculation yields Ai =
Z
n
{Lq˙q˙ (˜ qi , Hp , t)[Hqp − Hpp uqq ]2 − utqq }dt.
0
Proof of Theorem 2 We now turn to a completely different case, that of the existence of a smooth KAM torus. It is a standard fact that any smooth KAM torus T is the graph of a closed one-form: T = {(q, p, t) = (q, c − ∂q u(q, t), t)}(q,t)∈S 1 ×S 1 . Moreover, u is a smooth solution of (H-J); it follows from theorem 2.19 of [10] that, since the rotation number ω is irrational, then u is, up to an additive constant, the only viscosity solution of (H-J). Since u is unique we have that ∂x uν → ∂x u a. e. as ν & 0. Adding a constant, we can always suppose that uν (x, 0) ≥ u(x, 0) and that uν (¯ x, 0) = u(¯ x, 0) for some x ¯ depending on ν. We will denote by D1 a common bound of |∂x uν | and |∂x u|. Let qk (t) be the c-minimal orbit with qk (k) = x ¯. As in the proof of lemma 4 (and as in [4], theorem 4) by the variational characterization of uν and u we get 0 ≤ uν (qk (0), 0) − u(qk (0), 0) ≤ E
Z
k
[L(qk + νw(t), q˙k , t) − L(qk , q˙k , t)]dt + [β(ν) − α(c)]k + u(¯ x + νw(k), 0) − u(¯ x, 0) ≤
0
13
3
[β(ν) − α(c)]k + (D2 k + D1 )E sup |w(t)| ≤ νD3 k 2
(17)
0≤t≤k
where D2 is a constant depending only on L; in the last inequality D3 does not depend on ν since [β(ν) − α(c)] → 0 as ν → 0. We recall that by theorem D of [2] there is D4 > 0 such that {kω
mod(1) : 0 ≤ k ≤
D4 } τ
is a -net on S 1 ; since the motion on T is C 1 -conjugate to a rotation of frequency ω, also {qk (0)}0≤k≤ Dτ4 is
an -net on S 1 . Using this fact, (17) and the fact that |∂x uν | and |∂x u| ≤ D1 , we get that |uν (x, 0) − u(x, 0)| ≤ D5
ν 3
2τ
+ D1 .
2
If we choose ' ν 3τ +2 we get that 2
|uν (x, 0) − u(x, 0)| ≤ D4 ν 3τ +2
∀x ∈ S 1
(18)
Since uν (¯ x, 0) = u(¯ x, 0) we have that Z x 2 [∂x uν (s, 0) − ∂x u(s, 0)]ds ≤ D4 ν 3τ +2
∀x ∈ [0, 1].
x ¯
Since by [5] (see also the appendix) we know that ∂xx uν (s, 0) − ∂xx u(s, 0) ≤ M as ν & 0, the thesis follows. \\\
Digression Let us consider the Schr¨ odinger equation −
ν2 u ¨ − V (x)u = Eu. 2
We are interested in the eigenvalues E lying in the lowest part of the spectrum. If we were on R and if x2
−V (x) = 12 x2 then we would have that e− 2ν is the eigenfunction corresponding to the lowest-lying eigenvalue. We might expect that in our case the eigenfunctions we are looking for are preturbed gaussians peaked near some minimum of −V . We set u(x) = e−
w(x) ν
and we find
ν 1 2 w ¨ − |w| ˙ − V (x) = E 2 2 which is (2) if L(q, q, ˙ t) = 12 |q| ˙ 2 − V (q) and c = 0. Theorem 5 of [5] says that the solution wν of (2) converges as ν → 0 to h∞ 0 (x, 0, ξ) where ξ is the maximum of V with smallest second order derivative. Recalling that 14
∞ close to ξ the graph of −∂x h∞ 0 (x, 0, ξ) is the local stable manifold of ξ, we have that wν (x) ' h0 (x, 0, ξ) ' p 2 |V 00 (ξ)| |x−ξ| which implies that u is a perturbed gaussian peaked near ξ. More suggestively, if we drop a 2
charge in a minimum of −V different from ξ, this charge will tunnel through the energy barrier to ξ.
Appendix We recall the method used in [5] to show that there exists one solution (and only one, up to an additive constant) of (2). In [5] only lagrangians of the type L(q, q, ˙ t) = 21 |q| ˙ 2 − V (q, t) are considered and this makes some of the details of the proof different. The idea of [5] is to differentiate (2) with respect to x; setting v(x, t) = ∂x u(x, t) we have ν2 2 2 ∂xx v + ∂t v − ∂x H(x, c − v, t) + ∂p H(x, c − v, t)∂x v = 0 R1 v(x, t)dt = 0 0 v(x, t + 1) = v(x, t) = v(x + 1, t).
(19)
In the above equation v has zero mean since u is periodic. To solve (19) one considers the evolution (actually involution, since t ≤ 0) problem 2 2 ν2 ∂xx w + ∂t w − ∂x H(x, c − w, t) + ∂p H(x, c − w, t)∂x w = 0 w(x, 0) = φ(x) w(x, t) = w(x + 1, t)
(20)
where φ is a smooth function of mean 0; it is easy to see that Z
1
w(x, t)dx = 0
∀t ≤ 0.
0
One would like to show that for n ∈ Z, n → −∞, w(x, t + n) converges to a solution of (19). Two things are needed: 1) w(·, t)t≤0 is compact in some topology; 2) the limit of w(x, t + n) as n → −∞ is unique. Point 2) is proven in the same way as in [5] so we won’t speak about it; we show point 1). We begin to show that if ν is small enough then kw(·, t)k∞ is bounded as t → −∞. We recall a lemma of [8]: for any K > 0 there is K 0 (K) > K such that, if q is a c-minimal orbit with
|q(T )−q(0)| |T |
≤ K then for T < t0 < t0 + 1 < 0 we
have |q(t0 + 1) − q(t0 )| ≤ K 0 (K). We supposed that the Euler-Lagrange flow of L is complete; in the remark after proposition 4 in [8] it is proven that this implies that, possibly increasing K 0 (K), |q(t)| ˙ ≤ K 0 (K) for t ∈ [t0 , t0 + 1]. Summing up, we have that
|q(T )−q(0)| |T |
≤ K implies |q(t)| ˙ ≤ K 0 (K) for t ∈ [T, 0]. Let Φ denote
one primitive of φ and let us consider the problem ∂t u − H(x, c − ∂x u, t) + α(c) = 0 u(x, 0) = Φ(x). Let T < 0 and let qx satisfy (
q˙x (T ) = ∂p H(x, c − ∂x u(x, T ), T ) qx (T ) = x. 15
(21)
We define K 00 (K) = sup{|∂p H(q, c − p, t)| + |∂q˙ L(q, q, ˙ t) − c| : |p| ≤ K,
|q| ˙ ≤ K 0 (K) q, t ∈ S 1 }.
By the variational characterization of u we know that qx realizes min
Z
0
[L(q, q, ˙ t) − cq˙ + α(c)]dt + Φ(q(0)) : q(T ) = x .
T d This easily implies that there is K > 0 such that, for any M > 0, if k dx Φk∞ ≤ M , then there is T (M ) < 0 x (0) | < K. In other words, if we wait long enough, the mean speed such that, if T ≤ T (M ), then | qx (T )−q T
d of qx does not depend on k dx Φk∞ . Once K is thus defined we set M = K 00 (K) + 1, T = T (K 00 + 1)
and we choose φ in (20) such that kφk∞ ≤ M . As we already said this choiche of φ and T implies that x (0) | < K and thus by the lemma of [8] that |q˙x (T )| ≤ K 0 (K). Consequently, by (21) and the | qx (T )−q T
definition of K 00 , k∂x u(·, T )k∞ ≤ K 00 . Let w be defined as in (20); its primitive z solves ∂xx z + ∂t z − H(x, c − ∂x z, t) + β(ν) = 0 z(x, 0) = Φ(x) for some suitable β(ν) and after adding a suitable function of time. The technique of theorem 4 of [4] shows that there is ν0 > 0, only depending on L and M , such that, if ν ∈ (0, ν0 ), then k∂x zkL∞ ×[−T,0] ≤ k∂x ukL∞ ×[−T,0] + 1 ≤ K 00 + 1 = M. If we iterate this argument using uν (x, −T ) as a new final condition, we get k∂x zkL∞ ×[−nT,0] ≤ K 00 + 1 for any n ∈ N, which implies that sup{|w(x, t)| : x ∈ S 1 ,
t ≤ 0} ≤ M
(22)
with M independent on ν ∈ (0, ν0 ). We now differentiate (20) with respect to x and we get, setting z = ∂x w, ν2 ∂xx z + ∂t z − ∂xx H(x, c − w, t) + 2∂xp H(x, c − w, t)z− 2 ∂pp H(x, c − w, t)z 2 + ∂p H(x, c − w, t)∂x z = 0. If (x0 , t0 ) is a maximum of u in S 1 ×[−T, 0] we have that ∂x z(x0 , t0 ) = 0, ∂xx z(x0 , t0 ) ≤ 0 and ∂t z(x0 , t0 ) ≤ 0; from the above equation we get 2∂xp H(x, c − w, t)z − ∂pp H(x, c − w, t)z 2 ≥ ∂xx H(x, c − w, t). By (22), the above equation implies that ∂x w(x, t) = z(x, t) ≤ M 0 with M 0 independent on ν ∈ (0, ν0 ); since R1 ∂ w(x, t)dx = 0 we get that the total variation of w(·, t) is bounded by a constant independent on t ≤ 0 0 x and ν ∈ (0, ν0 ). Helly’s theorem now provides the pointwise convergence of w(x, t + n) to a limit which has the desired periodicity properties. The above argument also shows that the solution vν of (19) satisfies |vν (x, t)| ≤ D1 for all x, t ∈ S 1 and ν ∈ (0, ν0 ); thus, if uν is a solution of (2), ∂x uν satisfies the same bound. Moreover, we also have that 16
2 ∂xx uν ≤ M 0 . Since
R1 0
2 ∂xx uν dx = 0, the total variation of ∂x uν is bounded independently on ν ∈ (0, ν0 ] and
thus by Helly’s theorem {∂x uν }ν∈(0,ν0 ] is compact for the a. e. convergence. Thus also {uν } is compact and it is a standard fact that the pointwise limits of uν are viscosity solutions of (2).
References [1] S. Aubry, P. Y. Le Daeron, The Discrete Frenkel-Kontorova model and its extensions, Physica 8D, 381-422, 1983. [2] J. Bourgain, F. Golse, B. Wennberg, On the Distribution of Free Path Lengths for the Periodic Lorentz Gas, Comm. Math. Phys., 190, 491-508, 1998. [3] G. Da Prato, Introduction to Differential Stochastic Equations, Scuola Normale Superiore, Pisa, 1995. [4] W. H. Fleming, The Cauchy Problem for a Nonlinear First Order Partial Differential Equation, J. D. E., 5, 515-530, 1969. [5] H. R. Jauslin, H. O. Kreiss, J. Moser, On the Forced Burgers Equation with Periodic Boundary Conditions, Proceedings of Symposia in Pure Mathematics, 65, 133-153, 1999. [6] Y. Katznelson, D. S. Ornstein, Twist Maps and Aubry-Mather sets, Contemporary Mathematics, 211, 343-357, 1997. [7] C. Mantegazza, A. C. Mennucci, Hamilton-Jacobi Equations and Distance Functions on Riemannian Manifolds, preprint SNS, 1996. [8] J. N. Mather, Action Minimizing Invariant Measures for Positive Definite Lagrangian Systems, Math. Z., 207, 169-207, 1991. [9] J. N. Mather, Variational Construction of Connecting Orbits, Annales de l’Institut Fourier, 43, 1349-1368, 1993. [10] E. Weinan, Aubry-Mather Theory and Periodic Solutions of the Forced Burgers Equation, Comm. on Pure and Appl. Maths., 52, 811-828, 1999.
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