CONVERGENCE TO STEADY STATES OR PERIODIC SOLUTIONS

IN A CLASS OF HAMILTON-JACOBI EQUATIONS

Jean-Michel ROQUEJOFFRE

U.F.R. MIG, U.M.R. CNRS 5640 Université Paul Sabatier 118, route de Narbonne 31062 TOULOUSE CEDEX FRANCE

ABSTRACT. The purpose of this paper is to prove long-time behaviour results for Hamilton-Jacobi equations. For autonomous equations, we give an alternative proof of a convergence theorem obtained by A. Fathi when the equations are posed on a manifold, then extend it to Dirichlet boundary conditions on an open subset. When the equations are time-periodic we prove the convergence in several nontrivial special cases.

INTRODUCTION

Let M be a smooth compact N- dimensional manifold without boundary, and Ω an open bounded subset of IR^{N}. We study in this paper the viscosity solutions of two classes of Hamilton-Jacobi equations, namely:

(1)  u_{t} + H(x, Du) = 0,  x ∈ M resp. x ∈ Ω

and the time-periodic version of (1)

(2)  u_{t} + H(t, x, Du) = 0,  x ∈ M resp. x ∈ Ω

When equation (1) - resp. (2)- is solved on Ω, it will be supplemented by the boundary condition

(3)  u(t, x) = φ(x) on ∂Ω resp. u(t, x) = φ(t, x) on IR × ∂Ω,

where φ(x) - resp. φ(t, x) - is a continuous function from ∂Ω to IR - resp¿ continuous, 1-periodic in t function from IR × ∂Ω to IR.

As is now well-known [9], equation (1) admits travelling front solutions - i.e. solutions of the form λt + ϕ(x), the real number λ being uniquely determined. The function ϕ satisfies

(4)  λ + H(x, Dϕ) = 0,  x ∈ M

On the other hand, when solved in Ω, the steady problem for (1)-(3) has solutions only when certain compatibility conditions are fulfilled; see [8].

In a series of important papers, Fathi [4], [5], [6] proves the convergence to travelling fronts of the solutions of (1), provided that the hamiltonian is smooth, strictly convex and super-linear. He leaves the time-periodic case as an open problem. Two contributions are provided in his proof: first, a local regularity property - that was already lurking around in [8], but not made that explicit - is given; second, the relevance of the Aubry-Mather set in this context is made clear. From the PDE point of view this set is, loosely speaking, the uniqueness set: two steady - or periodic - solutions will coincide on their whole definition set provided that they coincide on the Aubry-Mather set.

Another proof, treating cases not covered by [5], was recently provided by Barles and Souganidis [3] in the autonomous case. In particular they manage to remove, to some extent, the strict convexity assumption. Their techniques are based on considerations quite similar to those allowing to prove instant time-smoothing for strictly convex Hamilton-Jacobi equations.

The purpose of this work is to give a different proof of the convergence in the autonomous case (1), which will enable us to treat the case of Dirichlet conditions and several nontrivial special cases in (2). The idea that we will exploit is a monotonicity property along the extremals of the steady solutions, a prototype of which being noticed and proved in [12]. Thus we never use the fact that the hamiltonian is conserved along the extremals, a fact that is no more true in the time-dependent case. The scheme of the proofs will always be the same: (i) convergence on the Aubry-Mather set, which will usually be the delicate part; (ii) convergence on the rest of the domain; in this second part we will use the Barles-Perthame relaxed semi-limits method [2], which was indicated to us by G. Barles to simplify a dynamic Kruzhkov transform in [10]. This method is indispensable here, and one of the contributions of this paper is to show that this method works in the time-dependent case.

Let us define, in all cases, the Lagrangian

(5)  L(t, x, v) = sup_{p∈T_{x}^{∗}M}(< p, v > −λ − H(t, x, p)),

with trivial changes if H does not depend on t or if the equation is solved on an open subset of IR^{N}. We will always make the two series of assumptions:

(H1). H(t, x, p) is smooth in all its variables, and strictly convex in p; the same is true for L.

(H2). The following growth conditions on L(t, x, v) are satisfied: there exist p > 1 and C > 0 such that such that

(6)  \begin{aligned}C(|v|^{p} − 1) ≤ L(t, x, v) & ≤ C(|v|^{p} + 1) \\ |L_{x}(t, x, v)| + |L_{t}(t, x, v) & ≤ C(1 + |v|^{p}) \\ |L_{v}(t, x, v)| & ≤ C(1 + |v|^{p−1}) \end{aligned}

Assumption (6) may certainly be moderately relaxed. Its main purpose is to ensure a low-cost existence proof for extremals. Under these assumptions, let us first assume that the Hamilton-Jacobi equations that we are considering are defined on M.

Theorem 1. Let u(t, x) be a global viscosity solution of (1). Then there exists a solution ϕ of (4) such that the function t ↦ u(t, x) − λt converges uniformly to ϕ.

The extremals of the Lagrangian associated to (2) define a semiflow for which, when M is a circle, a rotation number ρ may be defined.

Theorem 2. Let u(t, x) be a global viscosity solution of (2). Then there exists a periodic front λt + ϕ of (2), of period q ∈ IN, such that the function t ↦ u(t, x) − λt − ϕ(t, x) converges uniformly to 0 in each of the the following two cases. (A) H(t, x, p) = K(t, x, p) − f(x) with K ≥ 0, K(t, x, 0) = 0, f ≥ 0 with nonempty zero set; moreover q = 1. (B) the manifold M is the unit circle and

- either ρ = 0,

- or ρ is irrational and (2) has at least one smooth enough periodic solution.

In case (A), the assumptions on K may considerably be weakened. This will be made explicit in Section 4. As for the assumptions of Case B of THeorem (6), they are non generic, but non void. Moreover we think that they are almost optimal.

Similar statements are available for Dirichlet conditions, although their formulations are different. Assume now equation (1) to be solved in Ω, with the Dirichlet condition (3).

Theorem 3. Let u(t, x) be a bounded global viscosity solution of (1)-(3). Then there exists a solution ϕ of (4)-(3) such that the function t ↦ u(t, x) converges uniformly to ϕ.

Theorem 4. Let u(t, x) be a bounded global viscosity solution of (2)-(3). Then there exists a 1-periodic solution of (2)-(3), such that the function t ↦ u(t, x)−ϕ(t, x) converges uniformly to 0 in each of the the following two cases. (A) H(t, x, p) = K(t, x, p) − f(x) with K ≥ 0, K(t, x, 0) = 0, f ≥ 0 with nonempty zero set. (B) Ω =]0, 1[.

Once again, in Case (A), the assumptions on K may be relaxed.

The paper is organized as follows. Part 1 is devoted to the statement and consequences of the semi-concavity properties of equations (1) and (2). Section 2 is devoted to gathering the main facts about the steady and periodic solutions to (1)-(2); in Section 3, we will prove the convergence of the solutions of (1); finally the non-autonomous problem (2) will be examined in Section 4.

Acknowledgement. The author wishes to thank Profs. G. Barles and A. Fathi, whose comments and suggestions helped a lot in improving this manuscript.

1. REGULARITY OF THE SOLUTIONS

The purpose of this section is to collect the facts that will be useful to us in establishing the convergence result. The results that are stated here are by no means original; what makes the originality of this part is the brevity of the proofs presented. Given to us are the basic viscosity solutions - existence, uniqueness, a priori estimates, and smoothness as stated, for example, in Lions [8]. The semi-concavity of the solutions will play an important role.

We start from the Cauchy Problem and state the main results - existence and explicit formulae - that will be used as blackboxes throughout the paper. We then give an existence result for extremals - Tonelli’s Theorem -; then we state and prove a regularity result. We end this part with some notations.

Cauchy problem

Let us consider the general equation

(1.1)  u_{t} + H(t, x, Du) = 0  (x ∈ M - resp. x ∈ Ω -)

where M is either a smooth compact manifold without boundary and Ω a bounded open subset of IR^{N}. In this last case, equation (1.1) is supplemented by the Dirichlet data

(1.2)  u(t, x) = φ(t, x)  (x ∈ ∂Ω).

A semi-concave function u defined on IR × Ω is a Lipschitz function for which there exists a constant C > 0 such that

(1.3)  u_{tt} ≤ C, u_{x_{i}x_{i}} ≤ C  in D′(IR_{+} × Ω).

For T > 0, we will say a function defined on a manifold M is locally concave if, for every atlas (ω_{i}, Φ_{i}) on M, Φ_{i} mapping ω_{i} onto the unit ball, every function u_{i}(t, y) = u(t, Φ_{i}^{−1}(y)) is concave the unit ball. The main result that we will use is the following.

Theorem 1.1. ([8]). Assume H to be smooth, locally uniformly strictly convex in t ∈ IR and (x, p) ∈ T^{∗}M - resp. (x, p) ∈ \overline{Ω} × IR^{N}.

(i). Let u(t, x) be a Lipschitz viscosity solution of (1.1) defined on [0, T]. Then u is locally semi-concave on ]0, T [×M - resp. semi-concave in ]0, T] × Ω. It is also the only one coinciding with u(0, .) at t = 0 - and assuming the datum (1.2) when (1.1) is solved on Ω. (ii). (Stability) Let v(t, x) be another Lipschitz viscosity solution of (1.1) defined on [0, T]. Then we have

(1.4)  ∥u(t, .) − v(t, .)∥_{∞} ≤ ∥u(0, .) − v(0, .)∥_{∞}.

(iii). Assume u(0, .) being only continuous and H to have a superlinear growth. Then u is Lipschitz on ]0, T] × M - resp. ]0, T] × Ω. For any ε > 0, the Lipschitz constant of u on ]ε, T] only depends on ε, T, ∥u_{0}∥_{∞} and the Hamiltonian.

When we deal with (1.1) on M, there is essentially no more conditions to be fulfilled for having an existence result to (1.1) than the ones already stated. When it comes to dealing with Ω, compatibility relations between the initial and boundary data are required, as will be seen - but not detailed any further - below.

Remark1.2. We also notice that in Part (i), we have not required H to have a super-linear growth: what really matters here is the strict convexity of H. However, we will need to use Part (iii) to obtain some existence results for periodic solutions - namely, compactness - whereas it will not be needed in the case of steady solutions. Therefore we will always assume that H has a superlinear growth: when it is not needed, one may remember that, because of (1.4), one is allowed to consider only Lipschitz initial data. In that case, we wil only have to change the Hamiltonian outside a ball in p of radius larger than the Lipschitz constant of u.

As a consequence of this remark, Let u_{0} be a Lipschitz initial datum for (1.1), that we now assume to be solved on M. Then we have - [8] again:

(1.5)  u(t, x) = inf_{γ∈C^{0,1}([0,t],M) γ(t)=x}(u_{0}(γ(0)) + ∫_{0}^{t} L((s, γ(s), γ̇(s)) ds).

All the information that we will need on how the above infimum is achieved is contained in the following Theorem 1.3. (Tonelli’s Theorem) Assume the growth assumptions (6) to hold. (i). Let u solve (1.1) on IR_{+} × M. For all t > 0 and x ∈ M, there exists a curve γ_{t,x} such that the infimum in (1.5) is achieved. (ii). Any curve γ_{t,x} achieving the above infimum is a smooth curve. (iii). For every T > 0 and δ > 0, the set Γ_{δ,T} := {γ_{t,x}, x ∈ M, δ ≤ t ≤ T } is relatively compact in C([0, T], M).

Its proof will be postponed to the Appendix. Such a curve γ_{t,x} will be called and extremal, and we will often forget the subscript (t, x). The important part of this section, made explicit in Pierre and Rouy [11] for steady eikonal equations, and [4], [6] for steady general equations, is the following regularity result.

Theorem 1.4. Let u be as above and γ_{t,x} be an extremal. Then u is differentiable at every point (s, γ(s)) with 0 < s < t.

PROOF. Choose s ∈]0, t[ and a small h ∈ IR; because u is locally semi-concave its super-differential at (s, γ(s)), denoted by D^{+}u(s, γ(s)) is non void; choose (t, p_{x}) ∈ D^{+}u(s, γ(s)). Because (1.5) is achieved, we have

\begin{aligned}p_{t} + p_{x}.γ̇(s) & ≥ L(s, γ(s), γ̇(s)) \\ & = max_{p∈T_{γ(s)}^{∗}M}(p.γ̇(s) − H(s, γ(s), p)) \end{aligned}

On the other hand, u is a viscosity solution of (1.1); hence we have p_{t} + H(s, γ(s), p_{x}) ≤ 0. Consequently, we have

p_{x}.γ̇(s) − H(s, γ(s), p_{x}) ≥ max_{p∈T_{γ(s)}^{∗}M}(p.γ̇(s) − H(s, γ(s), p));

this implies

p_{x} = L_{v}(s, γ(s), γ̇(s)),

thus

p_{t} = −H(s, γ(s), p_{x}).

Hence D^{+}u(s, γ(s)) has only one element. Now, a concave function whose super-differential at one point is reduced to one vector is differentiable at that point. •

Assume now that (1.1) is solved in Ω. Let us set, as in [8]:

(1.6)  Q =]0, T [×Ω,  ∂_{0}Q = {0} × Ω̄∪]0, T [×∂Ω.

Then we have [8]:

(1.7)  \begin{matrix}u(t, x) = & inf\{u(s, γ(s)) + ∫_{s}^{t} L(σ, γ(σ), γ̇(σ)) ds, \\ (s, y) ∈ ∂_{0}Q, γ ∈ C^{0,1}([0, t], M), γ(s) = y, γ(t) = x\}. \end{matrix}

Theorem 1.5. Assume the growth assumptions (6) to hold, and let u solve (1.1) on IR_{+} × M. Then Points (i) and (iii) of Theorem 1.3 are true.

The proof is omitted. Notice that, due to the possible reflections of the curve γ on ∂Ω, it is false that γ is smooth, although it is still Lipschitz.

2. STEADY AND PERIODIC SOLUTIONS

Steady solutions Let us first concentrate on the manifold case. Recall that we are studying the

problem

(2.1)  λ + H(x, Dϕ) = 0, x ∈ M.

The following proposition is well-known. Its proof is so short that we may give it without making the length of this paper increase too much. It will be given in Appendix.

Proposition 2.1. (Lions, Papanicolaou, Varadhan [9]) There exists a unique λ ∈ IR such that Problem (2.1) has solutions.

From now on and until the end of the paper, we will assume, without loss of generality: λ = 0. Let ϕ be a solution of (2.1); we have - Lions [8]

(2.2)  ∀t > 0, ϕ(x) = inf_{γ∈C^{1}([−t,0],M), γ(0)=x}(ϕ(γ(−t)) + ∫_{−t}^{0} L(γ, γ̇) ds).

Section 1 tells us three things, that are summarized in the following theorem.

Theorem 2.2. Let ϕ be a solution of (2.1). (i). ϕ is locally semi-concave. (ii). ϕ is differentiable at any point x through which an extremal γ_{t,x} passes. (ii). Let x be a point through which an extremal γ_{t,x} passes. Then every extremal passing through x coincides with γ_{t,x} in a vicinity of x.

Recall that such an extremal, defined on [−T, 0], satisfies

(2.3)  ϕ(γ(0)) − ϕ(γ(−T)) = ∫_{−T}^{0} L(γ, γ̇) ds.

Let γ satisfy (2.3) on [−T, 0]. Due to the strict convexity and superlinearity of L(x, v), the Legendre transform

(2.4)  (x, v) ↦ L(x, v) = (x, L_{v}(x, v)).

is a global diffeomorphism from T M to T^{∗}M. Set (γ(t), p(t)) = L^{−1}(γ(t), γ̇(t)); it is a solution of the hamiltonian system

(2.5)  \{ \begin{matrix}Ẋ & = H_{p}(X, p) \\ ṗ & = −H_{x}(X, p) \end{matrix}

The flow generated by (2.5) will be called T̃_{ϕ}.

A consequence of Theorem 1.4 and equality (2.3) is the following: for all t ∈] − T, 0[ there holds p(t) = Dϕ(γ(t)); furthermore γ(t) is a solution of the reduced system

(2.6)  Ẋ = H_{p}(X, Dϕ(X))

Let E_{ϕ} be the set of all extremals γ defined on ] − ∞, 0] satisfying equality (2.3) for all t ≤ 0. We first claim that E_{ϕ} is nonempty; this is a trivial consequence of Theorem 1.3, Points (ii) and (iii). Let us set

M_{ϕ} = {γ(−1), γ ∈ E_{ϕ}}

Due to Theorem 1.3, M_{ϕ} is a nonempty closed subset of M. Because of Theorem 1.4 and 2.2, equation (2.6) generates a backward semiflow on M_{ϕ}, called T_{ϕ}(−t). For any x ∈ M_{ϕ}, let α_{T_{ϕ}}(x) be the α-limit set of x under the semi-flow T_{ϕ}; it is connected and invariant under T_{ϕ}. On α_{T_{ϕ}}(x) the semiflow T_{ϕ} is in fact a flow; this is a consequence of Theorem 2.2, Point (iii). For y ∈ α_{T_{ϕ}}(x) let ω_{T_{ϕ}}(y) be the ω-limit set of x under T_{ϕ}.

Let us now turn to the case of Equation (1) on Ω; we are therefore interested in

(2.7)  \begin{aligned}H(x, Dϕ) & = 0 (x ∈ Ω) \\ u(x) & = φ(x) (x ∈ ∂Ω) \end{aligned}

The existence theorem is the following.

Theorem 2.3. ([8]) Problem (2.7) has a solution if and only if the compatibility relation

(2.8) ∀(x, y) ∈ (∂Ω)^{2}, φ(x) − φ(y) ≤ inf{∫_{−t}^{0} L(γ, γ̇) ds, γ(0) = x, γ(−t) = y}.

This compatibility condition will not be used as such; therefore the proof of this theorem will be omitted. When it is fulfilled, all the solutions of the Dirichlet problem (2.7) satisfy

(2.9)  \begin{matrix}ϕ(x) = & inf\{ϕ(γ(s)) + ∫_{−s}^{0} L(s, γ(σ), γ̇(σ)) dσ, \\ (s, y) ∈ ∂_{0}Q_{t}, γ ∈ C^{0,1}([−t, 0], M), γ(s) = y, γ(0) = x\}. \end{matrix}

Here we have set ∂_{0}Q_{t} = {−t} × Ω ∪ [−t, 0] × ∂Ω. For every δ > 0, let E_{ϕ}^{δ} be the set of all extremals γ defined on ] − ∞, 0] satisfying equality (2.3) for all t ≤ 0, and which remain in Ω, at distance greater than δ from ∂Ω at all time. The set ∪_{δ>0}E_{ϕ}^{δ} may well be empty, i.e. all extremals reach the boundary in finite time. When nonempty, E_{ϕ}^{δ} generates the same definitions as in the manifold case: the set M_{ϕ}, the backward semi-flow T_{ϕ}, and its α and ω-limit sets; These notions will be used freely in the sequel.

Time-periodic solutions

Although the results will only be useful to us in the one-dimensional setting, there would be no fundamental simplification if we restricted ourselves to the circle. Furthermore, it all amounts to rephrasing the above paragraph in the time-periodic setting.

As before, let us first attack the case when (2) is posed on M. Recall that we are now studying the problem

(2.10)  u_{t} + H(t, x, Du) + λ = 0,  x ∈ M

and the time-periodic solutions of (2.10) are those solutions ϕ(t, x) of (2.10) that additionally satisfy

(2.11)  ∀t ∈ IR,  u(t + 1, x) = u(t, x)

There holds the exact analogue of Proposition 2.1, whose proof is also postponed to the Appendix.

Proposition 2.4. There exists a unique λ ∈ IR such that Problem (2.10)-(2.11) has solutions.

Let ϕ be a solution of (2.10)-(2.11); we have - Lions [8]

(2.12)  \begin{matrix}∀n ∈ IN, ∀t ∈ IR, ϕ(t, x) = inf\{ϕ(t, γ(t − n)) \\ + ∫_{t−n}^{t} L(s, γ, γ̇) ds\}; γ ∈ C^{1}([t − n, t], M), γ(t) = x\}. \end{matrix}

A theorem analogous to Theorem 2.2 holds.

Theorem 2.5. Let ϕ be a solution of (2.10)-(2.11). (i). ϕ is locally semi-concave. (ii). ϕ is differentiable at any point (t, x) through which an extremal γ_{t,x} passes. (ii). Let x be a point through which an extremal γ_{t,x} passes. Then every extremal passing through (t, x) coincides with γ_{t,x} in a vicinity of (t, x).

Let γ ∈ C^{1}([−T, 0], M) such that

(2.14)  ϕ(0, γ(0)) − ϕ(−T, γ(−T)) = ∫_{−T}^{0} L(s, γ, γ̇) ds.

Then from Theorem 1.4, ϕ is differentiable at each point (s, γ(s)). Set (γ(t), p(t)) = L_{t}^{−1}(γ(t), γ̇(t)), where the Legendre transform L_{t} is given by

(2.15)  (x, v) ↦ L_{t}(x, v) = (x, L_{v}(t, x, v));

it is a solution of the hamiltonian system

(2.16)  \{ \begin{matrix}Ẋ & = H_{p}(t, X, p) \\ ṗ & = −H_{x}(t, X, p) \end{matrix}

Similarly to the steady solutions part, there holds p(t) = Dϕ(t, γ(t)); furthermore γ(t) is a solution of the reduced system

(2.17)  Ẋ = H_{p}(t, X, Dϕ(t, X)).

Let E_{ϕ} be the set of all extremals γ defined on ]−∞, 0] satisfying equality (2.14) for all t ≤ 0. The set E_{ϕ} is nonempty; this once again is a trivial consequence of Theorem 1.3. Similarly to the above paragraph, we may first define the closed set M_{ϕ}, on which Problem (2.17) generates, this time, a backwards evolution system that will be denoted by T_{ϕ}(−s, −t). It also generates a discrete backward semi-flow T_{ϕ}, defined as T_{ϕ}x = T_{ϕ}(0, −1)x for all x ∈ M_{ϕ}. The α-limit set of x under the discrete semi-flow will be called α_{T_{ϕ}}(x). Because of Theorem 2.5, Point (ii), For all y ∈ α_{T_{ϕ}}(x), we will denote by ω_{T_{ϕ}}(x) the ω-limit set of x.

To end this section, let us turn to the Dirichlet conditions case. This time we are solving the problem

(2.18)  \begin{aligned}u_{t} + H(x, Dϕ) & = 0 (x ∈ Ω) \\ u(t + 1, x) & = u(t, x) (x ∈ Ω) \\ u(t, x) & = φ(t, x) (x ∈ ∂Ω) \end{aligned}

The existence theorem is the following.

Theorem 2.6. Problem (2.18) has a solution if and only if the compatibility relation

(2.19)  \begin{matrix}∀(x, y) & ∈ (∂Ω)^{2}, ∀s < t, \\ φ(t, x) − φ(s, y) ≤ inf\{∫_{s}^{t} L(σ, γ, γ̇) dσ, γ(t) = x, γ(s) = y\}. \end{matrix}

The proof is the same as Proposition 2.4, and we have a formula completely similar to (2.9). For δ > 0, let E_{ϕ}^{δ} be the set of all extremals γ defined on ] − ∞, 0] satisfying equality (2.14) for all t ≤ 0, and which remain in Ω, at distance greater than δ from ∂Ω. The union of all E_{ϕ}^{δ} may once again be empty, i.e. all extremals reach the boundary in finite time. When nonempty, E_{ϕ}^{δ} generates the set M_{ϕ}, the backwards discrete semi-flow T_{ϕ}, and its α-limit sets.

3. CONVERGENCE IN THE TIME-INDEPENDENT CASE

To prove the convergence we proceed in three steps: first we establish the basic monotonicity identity; it will then be used to give an alternative proof of Fathi’s uniqueness result and to prove the convergence on the Aubry-Mather set. Based on this result, we will employ the relaxed semi-limits method, just as in [10].

When we deal with the manifold case, Problem (1) has a unique solution S(t)u_{0}, which is uniformly Lipschitz on IR_{+} × M. We define the ω-limit set of this trajectory as

ω_{S}(u_{0}) = {ψ ∈ C(M) : ∃(t_{n})_{n}, lim_{n→+∞} t_{n} = +∞, lim_{n→+∞} S(t_{n})u_{0} = ψ}.

Following the considerations of Section 1, this set is non void, connected and relatively compact in C(M) - or C(Ω̄).

When we solve (1) in Ω, we have to assume the existence of a bounded global solution to the Cauchy problem for (1). Given this assumption, Theorem 1.1 tells us that the solution is globally Lipschitz, provided the Hamiltonian is superlinear or the initial datum is Lipschitz. Hence we may define the semigroup S(t) and the ω-limit set ω_{S}(u_{0}).

Monotonicity along the extremals In this paragraph let ϕ be a solution of (4). In order to make the notations lighter,

we will denote by T the semiflow T_{ϕ}.

Let γ(t) be an extremal defined on [0, T] satisfying (2.3).

Lemma 3.1. Let u(t, x) be a solution of (1). Then t ↦ u(t, γ(t)) − ϕ(γ(t)) is a nonincreasing function.

PROOF. For every t ≤ t′ we have

\begin{aligned}u(t′, γ(t′)) − u(t, γ(t)) & ≤ ∫_{t}^{t′} L(γ(s), γ̇(s)) ds \\ ϕ(γ(t′)) − ϕ(γ(t)) & = ∫_{t}^{t′} L(γ(s), γ̇(s)) ds \end{aligned}

Hence we have (u − ϕ)(t′, γ(t′)) ≤ (u − ϕ)(t, γ(t)) •

An important consequence of Lemma 3.1 is the following.

Corollary 3.2. (i). For any steady solution ψ, for any γ ∈ E_{ϕ} and for all y ∈ α_{T} (γ(−1), the function t > 0 ↦ ψ(T (t)x) − ϕ(T (t)x)) is constant. (ii). For a Lipschitz initial datum u_{0}, choose ψ ∈ ω_{S}(u_{0}). For every γ ∈ E_{ϕ} defined on IR and for all x ∈ ω_{T} (γ(0)), the function t > 0 ↦ S(t)ψ(T (t)x) − ϕ(T (t)x)) is constant.

PROOF. Point (i) is trivial by application of Lemma 3.1. Point (ii) is a little more involved: choose t_{0} > 0. Let (s_{n})_{n} be a sequence going to +∞ such that (T (s_{n})γ(0))_{n} converges to x; we may always assume that (S(s_{n})u_{0})_{n} converges to ψ_{1} ∈ ω_{S}(u_{0}). Consider a sequence (t_{n})_{n} going to +∞ such that S(t_{n} + s_{n})u_{0} converges uniformly to ψ.

An application of Lemma 3.1 tells us that, for all σ ∈ IR, the function

s > 0 ↦ S(s + σ)u_{0}(T (s)γ(0)) − ϕ(T (s)γ(0))

is nonincreasing, hence has a finite limit l(σ) as s → +∞; in particular for every σ > 0 the function

s ↦ S(σ + s + t)u_{0}(T (s + t)γ(0)) − ϕ(T (s + t)γ(0))

has the same limit l(σ) for all t ∈ IR; we set l_{n} = l(t_{n}). We may assume, up to the extraction of a subsequence, that the sequence (l_{n})_{n} has a limit.

On the other hand the stability property (1.4) implies, for all t ∈ [−t_{0}, t_{0}]:

∥S(t + t_{n} + s_{n})u_{0} − S(t + t_{n})ψ_{1}∥_{∞} ≤ ∥S(t + s_{n})u_{0} − S(t)ψ_{1}∥_{∞};

hence (S(t+t_{n}+s_{n})u_{0}−S(t+t_{n})ψ_{1})_{n} converges to 0 uniformly on [−t_{0}, t_{0}]×M. Therefore we have, for all n:

S(t_{n} + t)ψ_{1}(T (t)x) − ϕ(T (t)x) = l_{n};

letting n → +∞ implies: S(t)ψ(T (t)x) − ϕ(T (t)x) = l. We conclude by saying that t_{0} is arbitrary. •

The next consequence of Lemma 3.1 is really where we use the strict convexity of H.

Proposition 3.3. Let u_{0}, ψ and γ be as in the above corollary, Point (ii). For every x ∈ ω_{T} (γ(0)), S(t)ψ(x) does not depend on t.

PROOF. Set v(t) = S(t)ψ and γ̄(t) = T (t)x. From Corollary 3.2, we have, for all s < t : v(t, γ̄(t)) − v(s, γ̄(s)) = ϕ(γ̄(t)) − ϕ(γ(s)); hence we have

v(t, γ̄(t)) − v(s, γ̄(s)) = ∫_{s}^{t} L(γ̄, ˙ γ̄) dσ.

This implies, by Theorem 1.1, the differentiability of v at any point (t, γ̄(t)); this in turn implies the following identity, valid at (t, γ̄(t)):

(v − ϕ)_{t} + H(γ(t), Dv) − H(γ(t), Dϕ) = 0.

Expanding around Dϕ we obtain

(3.1)  \begin{matrix}∂_{t}(v − ϕ) + H_{p}(γ̄(t), Dϕ).Dv \\ = & − \frac{1}{2}(∫_{0}^{1}(1 − σ)H_{pp}(γ̄(t), Dϕ + σDv) dσ).D(v − ϕ)^{(2)} \end{matrix}

Because of Corollary 3.2, the left handside of (3.1), which is \frac{d}{dt}((v − ϕ)(t, γ(t))), is exactly 0. This implies |D(v − ϕ)| = 0 at the point (t, γ̄(t)) for all t; hence |D(v − ϕ)(0, x)| = 0. This implies in turn v_{t}(0, x) = 0, which concludes the proof. •

Aubry-Mather set and uniqueness We pause here to construct an Aubry-Mather set to prove Fathi’s uniqueness theorem.

As usual, we treat in detail the manifold setting, and give the necessary modifications when it comes to dealing with open subsets with Dirichlet conditions.

The Aubry-Mather set is defined in [4] as the set πM̃_{0}, where

(3.2) M̃_{0} = \overline{⋃\{suppμ : μ invariant probability measure on T M, ∫_{T M} L dμ = 0\}}

and π is the canonical projection T M → M.

In this paragraph let, once again, ϕ be any solution of (4). We define M_{0}(ϕ) as

(3.3)  M_{1}(ϕ) = ⋃_{x∈M_{ϕ}} α_{T_{ϕ}}(x), M_{0}(ϕ) = ⋃_{x∈M_{1}(ϕ)} ω_{T_{ϕ}}(x)

In other words, it represents the global attractor for the flow T_{ϕ}(t) restricted to the global attractor of the semi-flow T_{ϕ}(−t).

Let H be the set of all solutions of the steady equation (4). We define our Aubry-Mather set as

(3.4)  M_{0} = ⋂_{ϕ∈H} M_{0}(ϕ).

Our set M_{0} coincides with πM_{0}. To see this, one may look at Birkhoff sums of the

form \frac{1}{t} ∫_{−t}^{0} θ(γ(t), γ̇(t)) dt, where θ ∈ C(T M, IR). See [6] for more details. Also, chain

recurrence properties should save us the above two-step construction (3.3), but these considerations are beyond the scope of our paper - and may not even apply to the next section. The next proposition will be of constant use in the sequel.

Proposition 3.4. If ϕ ∈ H, choose x ∈ M_{0}(ϕ) and any y ∈ α_{T_{ϕ}}(x), z ∈ ω_{T_{ϕ}}(y). Then z ∈ M_{0}.

PROOF. Choose x_{0} and x_{1} such that x ∈∈ ω_{T_{ϕ}}(x_{1}), x_{1} ∈ α_{T_{ϕ}}(x_{0}). Finally consider ψ ∈ H; from Corollary 3.2, Point (i) we have ψ(T_{ϕ}(t)x_{1}) = ψ(T_{ϕ}(t)x_{1}) - we always may subtract a constant to ϕ. This means that, if we set γ_{1}(t) = T_{ϕ}(t)x_{1}, then γ_{1} is a global extremal for ψ, and this property is transmitted to γ̄(t) := T_{ϕ}(t)x by compactness. Therefore we have γ̄(t) = T_{ψ}(t)x, and the definition of y trivially implies y ∈ M_{0}(ψ). •

We may now very easily derive Fathi’s uniqueness result.

Theorem 3.5. (Fathi [6]) Two solutions of (4) which coincide on M_{0} coincide on M.

PROOF. Let ϕ and ψ be two such solutions; Choose x ∈ M and let γ(t) be related to ϕ by (2.3), defined on IR_{−}, with γ(0) = x. Choose any x̄ ∈ α_{T} (γ(−1)); then ȳ ∈ ω_{T_{ϕ}}(x̄). Then choose y ∈ α_{T_{ϕ}}(ȳ), z ∈ ω_{T_{ϕ}}(y). From Proposition 3.4 we have On the other hand z ∈ M_{0}. Corollary 3.2, Point (i) implies

ψ(x) − ϕ(x) ≤ ψ(x̄) − ϕ(x̄)

= ψ(ȳ) − ϕ(ȳ) = ψ(y) − ϕ(y) = ψ(z) − ϕ(z) = 0 Hence ψ(x) = ϕ(x) because the roles of ϕ and ψ are symmetric. •

Turn now to the Dirichlet conditions case. For every δ > 0, let us set

(3.5)  M_{1}^{δ}(ϕ) = ⋃_{x∈M_{ϕ}^{δ}} α_{T_{ϕ}}(x), M_{0}^{δ}(ϕ) = ⋃_{x∈M_{1}^{δ}(ϕ)} ω_{T_{ϕ}}(x),

Next let H be the set of all solutions of (4)-(3) and let us set

(3.6)  M_{0}^{δ} = ⋂_{ϕ∈H} M_{0}^{δ}(ϕ), M_{0} = ⋃_{δ>0} M_{0}^{δ}.

Theorem 3.6. Two solutions of (4)-(3) which coincide on M_{0} coincide on M.

PROOF. Let ϕ and ψ be two such solutions; Choose x ∈ Ω and let γ(t) be related to ϕ by (2.3), on ] − T, 0], with γ(0) = x. Assume T = −∞ and x ∈ E_{ϕ}^{δ}; then ψ(x) − ϕ(x) ≥ 0 by the same argument as in Theorem 3.5. If x is in none of the M_{ϕ}^{δ}, then γ hits ∂Ω in finite or infinite time, and Lemma 3.1 and the Dirichlet condition takes care of that case. Assume that T cannot be chosen to be infinite; this means that γ(−t) belongs to ∂Ω for some t ∈]0, T], and the preceding argument applies. Reverting the roles of ϕ and ψ we have ϕ = ψ on Ω. •

Convergence We now come back to the semigroup S(t), and we do not care now whether we are

solving our problem on a manifold or in an open subset. The considerations of the two preceding paragraphs lead to the following

Lemma 3.7. Let u_{0} be a Lipschitz initial datum and ψ ∈ ω_{S}(u_{0}). Then S(t)ψ does not depend on t on \overline{M}_{0}; moreover we have |D(S(t)ψ)| = 0 on M̄_{0}.

PROOF. On the set M_{0}, each flow T_{ϕ} coincide in a flow denoted by T , and M_{0} is invariant under the action of T . Therefore M_{0} can be partitioned into orbits of T , and Proposition 3.3 tells us that S(t)ψ is does not depend on t on such orbits. By continuity, this property extends to \overline{M}_{0}. Proposition 3.3 also tells us that |D(S(t)ψ)| = 0 on those orbits, hence on M_{0}. The property extends to M̄_{0} by application of Theorem 1.3, Point (iii). •

PROOF OF THEOREMS 1 AND 3. Choose ψ ∈ ω_{S}(u_{0})and let us set v(t) = S(t)ψ. The relaxed semi-limits of v(t, .) are given by

\underline{v}(x) = lim sup_{s→+∞, σ→t, y→x} v(s + σ, y), \overline{v}(x) = lim inf_{s→+∞, σ→t, y→x} v(s + σ, y).

We note that these two functions do not depend on t. Clearly, \overline{v} ≤ \underline{v}; on the other hand they are both Lipschitz functions and - see Barles [1] - \underline{v} - resp. \overline{v} is a viscosity sub- - resp. super- - solution to (4). Hence S(t)\underline{v} and S(t)\overline{v} converge to steady solutions.

Let us prove that they S(t)\underline{v} = S(t)\overline{v} = ψ on M_{0}. We have, for all t > 0, s > 0, (x, y) ∈ M^{2}, and for all curve γ such that γ(t) = x and γ(s) = y:

(3.7)  u(s + t, x) ≤ u(s, y) + ∫_{s}^{s+t} L(γ, γ̇) dσ.

Consider a sequence (s_{n})_{n} such that (u(s_{n}, y))_{n} converges to \underline{v}(0, y) and take x ∈ M_{0}; then we have, for all curve γ such that γ(0) = y and γ(t) = x:

ψ(x) = S(t)ψ(x) ≤ \underline{v}(y) + ∫_{0}^{t} L(γ, γ̇) dσ.

Because x ∈ M_{0}, there exists an extremal γ in M_{0} such that γ(t) = x and such that

ψ(x) = S(t)ψ(x) = S(0)ψ(γ(0)) + ∫_{0}^{t} L(γ, γ̇) dσ.

By the invariance of M_{0} under the flow T , we have γ(0) ∈ M_{0}; hence ψ(γ(0)) = \underline{v}(γ(0)). This implies S(t)\underline{v}(x) = ψ(x).

As for the equality S(t)\overline{v}(x) = ψ(x), we deduce from (3.7) that, for all t > 0, s > 0, (x, y) ∈ M^{2}, and for all curve γ such that γ(t) = x and γ(s) = y:

ψ(x) ≤ \overline{v}(y) + ∫_{s}^{t} L(γ, γ̇) ds.

However, we know that ψ ≥ \overline{v}; moreover, the function \overline{v} being a super-solution to (4), we have S(t)\overline{v} ≤ \overline{v}. But then the definition (1.5) applied to S(t)\overline{v} implies that S(t)\overline{v}(x) = ψ(x).

Therefore S(t)\underline{v} and S(t)\overline{v} converge to the same limit ψ_{∞}. But we also have S(t)\underline{v} ≤ S(t)ψ ≤ S(t)\overline{v} by the maximum principle. As a consequence, ψ_{∞} is in ω_{S}(ψ), hence in ω_{S}(u_{0}). The convergence of S(t)u_{0} to ψ_{∞}, follows, due to the L^{∞} stability of the solutions (1.4). •

As a conclusion to this part, we hope that the convergence proof presented here will help in clarifying at least one point: although the dynamics of the extremals may be quite complicated, the behaviour of the solutions of (1) is simple because the Aubry-Mather sets do not vary in time. This will not be true in the next section: in the context of periodic hamiltonians, the Aubry-Mather sets are preserved only under discrete iterations. Hence we will really have to care about the dynamics of the extremals, except when we specifically ask the Aubry-Mather set not to depend on time.

4. CONVERGENCE TO PERIODIC SOLUTIONS

It was necessary to go through the whole extremals theory in order to treat the onedimensional case - Case (B); nevertheless we do not know how to do without it. Case (A) can be treated as in Namah-Roquejoffre [10], this is what we will present first. Case (B) will be examined next.

Preliminary notations and results are in order. When the Cauchy Problem for (2) or (2)-(3) has a global solution, we may define an evolution system S(s, t)u_{0}, which is uniformly Lipschitz on IR_{+} × M or IR_{+} × Ω. We define a discrete semigroup S as Su_{0} = S(0, 1)u_{0}. The ω-limit set of a trajectory is

ω_{S}(u_{0}) = {ψ ∈ C(M) : ∃(k_{n})_{n}, lim_{n→+∞} k_{n} = +∞, lim_{n→+∞} S^{k_{n}}u_{0} = ψ}.

This set is non void and relatively compact in C(M) - or C(Ω̄).

Let us say that a Lipschitz function \overline{u}(x) - resp. \underline{u}(x) - is a super - resp. sub - solution to (2.10) if and only if ū ≥ Sū - resp \underline{u} ≤ S\underline{u}.

Proposition 4.1. Let ū be a super-solution to (2.10). The sequence (S^{n}ū)_{n} decreases uniformly to a fixed point of S. Similarly, let \underline{u} be a sub-solution to (2.10). The sequence (S^{n}ū)_{n} increases uniformly to a fixed point of S.

Now we adapt the relaxed semi-limits methods to our setting. For all t > 0, let Ω′_{t} be an open subset of M or Ω, such that Ω′_{t+1} = Ω′_{t}. Let φ: ⋃{t} × Ω′_{t} → IR be a Lipschitz function, 1-periodic in t. Let H(t, x, p) satisfy the requirements of Theorem 2, and consider the generic problem

(4.1)  \begin{aligned}u_{t} + H(t, x, Du) & = 0  ((t, x) ∈ IR_{+} × Ω′_{t}) \\ u(t, x) & = φ(t, x)  ((t, x) ∈ IR_{+} × ∂Ω′_{t}) \end{aligned}

Let S be the discrete semiflow generated by (4.1) - if it exists.

Proposition 4.2. Assume that, for every Lipschitz initial datum u_{0}, the semi-orbit (S^{n}u_{0})_{n} is bounded, and that (4.1) has at most one fixed point. Then (4.1) has a unique fixed point to which all semi-orbits will converge.

PROOF. Let u(t, x) be a global solution to (4.1) and set u_{n}(t, x) = u(t + n, x), for al (t, x) in, say, [0, 2] × Ω′. Set, as usual

(4.2)

ū(t, x) = lim inf_{n→+∞, s→t,y→x} u_{n}(s, y) = lim inf_{n→+∞} u_{n}(t, x)
\underline{u}(t, x) = lim sup_{n→+∞, s→t,y→x} u_{n}(s, y) = lim sup_{n→+∞} u_{n}(t, x)

The last equality in each line of (4.2) holds because u is globally Lipschitz in t and x. As a consequence, ū and \underline{u} are also Lipschitz in t and x.

Obviously we have

ū(t + 1, x) = lim_{n→+∞} sup{u(t + k + 1, x), k ≥ n}

= lim_{n→+∞} sup{u(t + k, x), k ≥ n + 1}

= ū(t, x)

The same holds for \underline{u}; hence these two functions are 1-periodic in t. On the other hand - Barles [1] - ū - resp. \underline{u} - satisfy the viscosity inequalities

ū_{t} + H(t, x, Dū) ≥ 0, \underline{u}_{t} + H(t, x, D\underline{u}) ≤ 0 ((t, x) ∈ IR_{+} × Ω′_{t})

and ū = \underline{u} = φ on ⋃\{t\} × Ω′_{t} → IR. As a consequence, ū - resp. \underline{u} is a super- - resp. sub- - solution to (4.1), to which Proposition 4.1 may be applied; this in particular yields the existence of a fixed point to S. On the other hand if ψ ∈ ω_{S}(u_{0}), then there holds

S^{n}ū ≤ ψ ≤ S^{n}\underline{u},

and a last application of Proposition 4.1 concludes the proof. •

We now come back to our old notations. Theorems 2 and 4 will be proved as soon as we have managed to find a family of open subsets Ω′_{t} on which the time-periodic Dirichlet problem has at most one solution: this is the role of the Aubry-Mather set. The same argument as in the proof of Theorems 1 and 3 - and that we will not repeat - would show that the Dirichlet problem with suitably chosen data on the boundary of the Aubry-Mather set has solutions.

Convergence in Case (A)

Recall that we now solve

(4.3)  u_{t} + K(t, x, Du) − f(x) + λ = 0 (M)

and

(4.4)  \begin{aligned}u_{t} + K(t, x, Du) − f(x) & = 0 (Ω) \\ u(t, x) & = φ(t, x) (∂Ω) \end{aligned}

with K ≥ 0, K(t, x, 0) = 0, f ≥ 0 with nonempty zero set. Let us denote Z the zero set of f. The only assumption that we shall need is in fact

(H’). Assume K to be convex, nonegative, and to satisfy lim_{|p|→+∞} K(t, x, p) = +∞,

uniformly in t and x.

In particular, no strict convexity nor superlinearity is required. There is first the following

Lemma 4.3. Problem (4.3) has periodic fronts if and only if λ = 0.

PROOF. It is enough to prove that (4.3) has solutions for λ = 0; therefore assume λ = 0 in (4.3). Any constant is a subsolution; moreover set

K̄(x, p) = min_{0≤t≤1} K(t, x, p).

This is a convex hamiltonian, satisfying K̄ ≥ 0, K̄(x, 0) = 0. The equation

K̄(x, Dϕ) − f(x) = 0, x ∈ M

has solutions; see [10]. Let ϕ be such a solution; it is a super-solution to (2); application of Proposition 4.1 to the orbit (S^{n}ϕ)_{n} yields the existence of a fixed point to S. •

PROOF OF THEOREM 2, CASE A. For all x ∈ Z, the function t ↦ S(0, t)u_{0}(x) is nonincreasing; this is intuitively obvious, see [10] for a rigorous proof. Hence there is a function φ(x), constant on any connected component of Z, such that S(0, t)ψ = φ on Z, for every ψ ∈ ω_{S}(u_{0}). On the other hand, the Dirichlet problem for (2.10)-(2.11) on Ω′ = M\Z with fixed values on ∂Z has at most one solution; one may see this by a standard Kruzhkov transform. •

PROOF OF THEOREM 4, CASE A. Similarly to the autonomous case, we have to assume that S^{n}u_{0} is well defined and bounded; once this is known the proof is the same as above; we just have to set, this time Ω′ = Ω\Z. •

Thus we see that, when the Aubry-Mather set - in this context, it is reduced to Z - does not depend on time, we recover convergence for the Hamilton-Jacobi semiflow. When there is a true time-dependence, we are only able to conclude in space dimension one - i.e. when the dynamics of the extremals is known.

Monotonicity and Aubry-Mather sets

First we introduce the Aubry-Mather set. Then we carry out the proof in two steps: first, we prove Theorem 4, then Theorem 2. In any case, we will assume, without loss of generality: λ = 0.

Consider first the manifold case. Choose ϕ(t, x) a periodic solution, and let γ(t) be an extremal defined on [0, T] satisfying (2.14). First we define our Aubry-Mather set: let P be the set of all 1-periodic solutions to (2); for ϕ ∈ P we define M_{0}(ϕ) as

(4.5)  M_{1}(ϕ) = ⋃_{x∈M_{ϕ}} α_{T_{ϕ}}(x), M_{0}(ϕ) = ⋃_{x∈M_{1}(ϕ)} ω_{T_{ϕ}}(x)

and the Aubry-Mather set M_{0} as

(4.6)  M_{0} = ⋂_{ϕ∈P} M_{0}(ϕ).

Just as in the preceding section we have the

Lemma 4.4. Let u(t, x) be a solution of (1). Then the function t ↦ u(t, γ(t)) − ϕ(t, γ(t)) is nonincreasing.

Let ψ belong to ω_{S}(u_{0}) and u(t) = S(0, t)ψ. Unfortunately, not all the information that was available in Part 2 can be retrieved in the periodic setting, for the reason that we do not know how T_{ϕ}(0, t)M_{0} looks like for 0 < t < 1 - in the preceding section it was simply M_{0}; this is why we may not conclude to the convergence as in Section 3. However we may still write the two following propositions.

Proposition 4.5. On the set M_{0}, all the flows T_{ϕ} coincide with a single flow T . The set M_{0} is invariant under T , as well as M̄_{0}.

Proposition4.6. (i). For any x ∈ M_{0}, the function t ↦ S(0, t)ψ(T_{ϕ}(0, t)x)−ϕ(T_{ϕ}(0, t)x) is constant. (ii). The function u is differentiable on M_{0}, and u_{t} = |D(u − ϕ)| = 0 on M_{0}.

The proof of the above results is a copy of the corresponding propositions of Section 3. It is to be noticed here that they do not need the conservation of the Hamiltonian along the extremals; what is used is the invariance of the equation under the transform t ↦ t+1. We next have the uniqueness result.

Theorem 4.7. Two solutions of (4) which coincide on IN × M_{0} coincide on M.

PROOF. See the proof of Theorem 3.5. •

Turn now to the Dirichlet conditions case. For every δ > 0, let us set

(4.7)  M_{1}^{δ}(ϕ) = ⋃_{x∈M_{ϕ}^{δ}} α_{T_{ϕ}}(x), M_{0}^{δ}(ϕ) = ⋃_{x∈M_{1}^{δ}(ϕ)} ω_{T_{ϕ}}(x)

Next we set

(4.8)  M_{0}^{δ} = ⋂_{ϕ∈P} M_{0}^{δ}(ϕ), M_{0} = ⋃_{δ>0} M_{0}^{δ}.

Theorem 4.8. Two solutions of (2)-(3) which coincide on M_{0} coincide on M.

PROOF. Let ϕ and ψ be two such solutions; Choose x ∈ Ω and let γ(t) be related to ϕ by (2.3), on ] − T, 0], with γ(0) = x. Assume T = −∞ and x ∈ M_{ϕ}^{δ}; then just as above we have ψ(−1, x) − ϕ(−1, x) ≤ 0. If x belongs to none of the M_{ϕ}^{δ}, but if γ never hits ∂Ω, this means the existence (i). of a sequence (t_{n})_{n} going to +∞, whose fractional part tends to a limit s ∈ [0, 1[; (ii). of y ∈ ∂Ω such that |T (0, −t_{n})x − y| tends to 0; Lemma 4.4 implies

\begin{matrix}ψ(−1, x) − ϕ(−1, x) ≤ & lim_{n→+∞}(ψ(−t_{n}, T (0, −t_{n})x) − ϕ(−t_{n}, T (0, −t_{n})x)) \\ = & ψ(s, y) − ϕ(s, y) = 0 \end{matrix}

Assume finally that T cannot be chosen to be infinite or γ hits ∂Ω in finite time; this means the existence of t ∈]0, T] such that γ(−t) belongs to ∂Ω. But then Lemma 3.1 implies once again ϕ(−1, x) ≤ ψ(−1, x). Reverting the roles of ϕ and ψ we have ϕ(−1, .) = ψ(−1, .) on Ω. •

Convergence in Case (B) PROOF OF THEOREM 4 (END). We have Ω =]0, 1[. Recall that, on each M_{0}^{δ},

we may construct a flow T , deduced from all the flows T_{ϕ} which are the flows of the extremals which remain δ-bounded away from the boundary points 0 and 1. For x ∈ E_{ϕ}^{δ}, the semigroup property of T implies that (T^{−n}x)_{n} is a monotone sequence which, because it remains bounded, converges to a limit. Hence each M_{0}^{δ} is only made up of fixed points of T , and we conclude from Proposition 3.3, (i) that S^{n}ψ(x) is a constant sequence, hence u(t, .) is 1-periodic on each T (0, t)M_{0}^{δ}, hence on each T (0, t)M_{0}. •

Let us end the proof of Theorem 2 and let us define the homeomorphism of M̄_{0}: h(x) = T x. It is not an homeomorphism of the circle but, because it leaves M̄_{0} invariant and because it is order-preserving, the rotation number ρ(h) for it is well-defined as the

common limit of \frac{h̃^{n}(x)}{n}, where h̃ is the usual lifting of h from IR to IR. See [7] for more details.

PROOF OF THEOREM 2 (END). If ρ(h) = 0, then h has fixed point that can be identified with 0; hence the flow T is bounded on M_{0}. The same argument as in the case of Dirichlet conditions applies.

Assume that ρ(h) is irrational. So far, we have worked with any periodic solution ϕ of (2); we choose the smooth one that is assumed to exist. Then h is a diffeomorphism of the circle, and Denjoy’s Theorem asserts that any orbit T^{n}x is dense in M; therefore we have M̄_{0} = M. We apply Proposition 4.6, (ii) to infer that ψ − ϕ is constant on M, hence S(t)ψ is 1-periodic on M. This ensures the convergence of S(t)ψ to a solution which is 1-periodic in time. •

To conclude, let us check that the assumptions of Case B are non void. It is straightforward to build cases when ρ(h) = 0; see for example Case A. To see that there exist smooth solutions to (2) with irrational rotation numbers. Let us start with a hamiltonian H(x, p) defined on the circle such that there holds

λ > max_{x∈M} min_{P ∈T M} H(x, p);

such a situation is indeed possible and in this case there exists a unique - up to a constant - solution of −λ + H(x, ϕ′) = 0 on M; moreover ϕ is as smooth as the data - see [12]. The rotation number for the discrete flow T is the harmonic mean of x ↦ H_{p}(x, ϕ′(x)). To make it irrational let us observe that we change λ into αλ if we change H into αH, and that ϕ′ does not change. Finally, if we wish to have a true dependence in time, let us notice that λt + ϕ(x + t) is a 1-periodic solution of u_{t} + H(t + x, u_{x}) − u_{x} = 0 and that the rotation number remains unchanged.

APPENDIX

A1. Proof of Theorem 1.3. Set

IL(γ) = u(0, γ(0)) + ∫_{0}^{T} L(s, γ, γ̇) ds.

Let (γ_{n})_{n} be a minimizing sequence; because of (6) it is bounded in W^{1,p}([0, T], M), hence converges uniformly, and weakly in W^{1,p}([0, T], M) - up to a subsequence - to a function γ. Because of (6) again the functions L_{x}(t, γ, γ̇) and L_{t}(t, γ, γ̇) belong to L^{1}([0, T], M); the convexity of L with respect to p and the Lipschitz property of u then imply

IL(γ_{n}) ≥ IL(γ) + ∫_{0}^{T} L_{v}(s, γ, γ̇).(γ̇_{n} − γ̇) ds + o(1), ds

and the second term in the above inequality goes to 0, for L_{v}(s, γ, γ̇) belongs precisely to L^{p^{∗}}([0, T], M). Hence we have IL(γ_{n}) → IL(γ). Finally, an extremal satisfies the Euler-Lagrange equation

(A.1)  \frac{d}{dt}L_{v}(t, γ, γ̇) + L_{x}(t, γ, γ̇) = 0;

because L_{v} is coercive in v we have γ̇ ∈ L^{∞}([0, T], M), and we conclude by noticing that we now have

L_{vv}.\frac{d^{2}γ}{dt^{2}} − L_{xv}.γ̇ − L_{tv} + L_{x} ∈ L^{∞}([0, T], M),

which implies the smoothness of γ by an easy iteration argument. •

A2. Existence proofs for wave solutions

PROOF OF PROPOSITION 2.1. Let us consider the sequence of problems

(A.2)  H(x, Dϕ) + εϕ = 0, x ∈ M.

which have unique Lipschitz solutions ϕ_{ε} in W^{1,∞}(M). Comparing ϕ_{ε} with the test function 0 we have εϕ_{ε} ≥ −∥H(., 0)∥_{∞}; because equation(A.2) holds almost everywhere we have

εϕ_{ε} ≤ − inf_{(x,p)∈T^{∗}M} H(x, p);

thus the sequence (εϕ_{ε})_{ε} is bounded in L^{∞}(M), hence (Dϕ_{ε})_{ε} is bounded in L^{∞}(M). Therefore the sequence (ϕ_{ε}− < ϕ_{ε} >)_{ε} is bounded in W^{1,∞}(M), thus relatively compact in C(M). •

PROOF OF PROPOSITION 2.4. Let us consider the sequence of problems

(A.3)  ϕ_{t} + H(t, x, Dϕ) + εϕ = 0, x ∈ M.

Let ϕ_{0} be a Lipschitz initial datum for (A.3); then there exists - [8], Chapter 9 - a unique solution to the Cauchy Problem, denoted by ϕ(t, x). From the maximum principle, there holds |ϕ(t + 1, x) − ϕ(t, x)| ≤ ∥ϕ_{0}∥_{∞}e^{−εt}; therefore there exists a unique solution ϕ_{ε} to (A.3) which is 1-periodic in time, and to which all solutions of (A.3) converge. The maximum principle also tells us that we have, for all solution ϕ to (2.10): ε|ϕ(t, x)| ≤ ∥H(., ., 0)∥_{L^{∞}IR×M}+ε∥ϕ_{0}∥_{∞}e^{−εt} thus the sequence (εϕ_{ε})_{ε} is uniformly bounded in L^{∞}(IR× M), hence (Dϕ_{ε})_{ε} is bounded in L^{∞}(M).

We end the proof just as in Section 1: the sequence (ϕ_{ε} − ∫_{0}^{1} < ϕ_{ε} > dt)_{ε} is bounded

in W^{1,∞}(IR_{+} × M), thus relatively compact in C([0, 1] × M). •

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