WEAK KAM THEORY FOR NONREGULAR COMMUTING HAMILTONIANS

. In this paper we consider the notion of commutation for a pair of continuous and convex Hamiltonians, given in terms of commutation of their Lax–Oleinik semigroups. This is equivalent to the solvability of an associated multi–time Hamilton–Jacobi equation. We examine the weak KAM theoretic aspects of the commutation property and show that the two Hamiltonians have the same weak KAM solutions and the same Aubry set, thus generalizing a re- sult recently obtained by the second author for Tonelli Hamiltonians. We make a further step by proving that the Hamiltonians admit a common critical sub- solution, strict outside their Aubry set. This subsolution can be taken of class C 1 , 1 in the Tonelli case. To prove our main results in full generality, it is crucial to establish suitable diﬀerentiability properties of the critical subsolutions on the Aubry set. These latter results are new in the purely continuous case and of independent interest.


1.
Introduction. In the last decades, the study of Hamiltonian systems has been impacted by a few new tools and methods. For general Hamiltonians, the framework of symplectic geometry led Gromov to his non-squeezing lemma [33], which gave rise to the key notion of symplectic capacity, now uniformly used in the field.
In the particular case of Tonelli (smooth, strictly convex, superlinear) Hamiltonians, some variational techniques led to significant improvements and results. John Mather led the way in this direction in [42,41]. In the first paper he studied free time minimizers of the Lagrangian action functional, introducing the Aubry set, while in the second one he studied invariant minimizing measures, introducing what is now called the Mather set.
Later Fathi, through his weak KAM Theorem and Theory, showed the link between the variational sets introduced by Mather and the Hamilton-Jacobi equation. This allowed to simplify some proofs of Mather and to establish new PDE results, in connection with the theory of homogenization [39]. This material is presented in [29].
where M stands either for the Euclidean space R N or the N -dimensional flat torus T N , H and G denote two real valued functions on M × R N , and u 0 : M → R is any given Lipschitz continuous initial datum. The first existence and uniqueness results appeared in [40] for Tonelli Hamiltonians independent of x via a representation formula for solutions of the Hamilton-Jacobi equation: the Hopf-Lax formula. Related problems were studied in [36].
A generalization of this result came much later in [8], where dependance on x is introduced (and the convexity hypothesis is kept). As a counterpart, the authors explain the necessity to impose the following commutation property on the Hamiltonians: Note that this condition is automatically satisfied when the Hamiltonians are independent of x. The proof involves an a priori different Hamilton-Jacobi equation with parameters and makes use of fine viscosity solution techniques. In [47], under stronger hypotheses, a more geometrical proof, following the original idea of Lions-Rochet, is given. This equation was then studied under weaker regularity assumptions in [43]. The convexity is dropped in [15] in the framework of symplectic geometry and variational solutions. Finally, let us mention that in [46] the influence of first integrals (not necessarily of Tonelli type) on the dynamics of a Tonelli Hamiltonian is studied.
In [47] the second author has explored relation (2) for a pair of Tonelli Hamiltonians in the framework of weak KAM Theory to discover that the notions of Aubry set, of Peierls barrier and of weak KAM solution are invariants of the commutation property. Similar results were independently obtained in [20,19].
We want to emphasize that the extension of this result to the non-regular setting was far from being straightforward. First, there is a problem of techniques: the crucial point in the proof of [47] is based on a careful study of the flows associated with H and G and exploits properties and tools developed in the framework of symplectic geometry and weak KAM Theory. For instance, a key ingredient is a deep result due to Bernard [10], stating that the Aubry set is a symplectic invariant. The use of all this machinery is possible only when the Hamiltonians are smooth enough.
But there is more: looking at the arguments in [47], one realizes that the commutation hypothesis (2) entails a certain rigidity of the dynamics and of the underlying geometric frame of the equations. Even if in the purely continuous case some analogies can be drawn, all this rich structure is lost. To put it differently, the problem did not seem to us just of technical nature: the role of regularity for the validity of the result had to be clarified.
The proof given here borrows some arguments from [47], but the conclusion is reached via a different and rather simple remark on the time-dependent equations. Incidentally, with this idea the proof in the smooth case can be made considerably simpler. It is also worth noticing that Theorem 1.1 applies, in particular, to a pair of Hamiltonians of class C 1 satisfying (2), that is, to a case not covered by the previous works on the subject [20,19,47]. As a byproduct, our study allows us to obtain a new result also for classical Tonelli Hamiltonians: Theorem 1.2. Let H and G be two commuting Tonelli Hamiltonians. Then they admit a C 1,1 critical subsolution which is strict outside their common Aubry set.
In the end, our research reveals that the invariants observed in the framework of weak KAM Theory are consequence of the commutation of the Lax-Oleinik semigroups only, with no further reference to the Hamiltonian flows, that cannot be even defined in our setting. The only point where a kind of generalized dynamics plays a role is when we establish some differentiability properties of critical subsolutions on the Aubry set, which are crucial to state Theorem 1.1 in its full generality. These results are presented in Section 4, where we will prove a more precise version of the following: Theorem 1. 3. Let H be a continuous, strictly convex and superlinear Hamiltonian on T N × R N . Then there exists a set D ⊂ T N such that any subsolution u of the critical Hamilton-Jacobi equation is differentiable on D. Moreover, its gradient Du is independent of u on D. Last, D is a uniqueness set for the critical equation, that is, if two weak KAM (or critical) solutions coincide on D, then they are in fact equal.
These latter results are new and we believe interesting per se. They generalize, in a weaker form, Theorem 7.8 in [32], and bring the hope of extending to the purely continuous case the results of [32] about the existence of a C 1 critical subsolution, strict outside the Aubry set. Such a generalization, however, seems out of reach without any further idea.
The article is organized as follows. In Section 2.1 we present the main notations and assumptions used throughout the paper, while in Section 2.2 we recall the definitions and the results about Hamilton-Jacobi equations that will be needed in the sequel. Section 3 consists in a brief overview of weak KAM Theory for nonregular Hamiltonians. Some proofs are postponed to Appendix A. In Section 4 we prove the differentiability properties of critical subsolutions above mentioned. In Section 5 we examine the weak KAM theoretic aspects of the commutation property and we establish our main results for continuous and strictly convex Hamiltonians. Some auxiliary lemmas are stated and proved in Appendix B. Appendix C contains an argument showing the equivalence between the notion of commutation considered in this paper and the one given in terms of cancellation of the Poisson bracket when the Hamiltonians are of Tonelli type.

Preliminaries.
2.1. Notations and standing assumptions. With the symbols R + and R − we will refer to the set of nonnegative and nonpositive real numbers, respectively. We say that a property holds almost everywhere (a.e. for short) on R k if it holds up to a negligible subset, i.e. a subset of zero k-dimensional Lebesgue measure.
By modulus we mean a nondecreasing function from R + to R + , vanishing and continuous at 0. A function g : R + → R will be termed superlinear if Given a metric space X, we will write ϕ n ⇒ ϕ on X to mean that the sequence of functions (ϕ n ) n uniformly converges to ϕ on compact subsets of X. Furthermore, we will denote by Lip(X) the family of Lipschitz-continuous real functions defined on X. Throughout the paper, M will refer either to the Euclidean space R N or to the N -dimensional flat torus T N , where N is an integer number. The scalar product in R N will be denoted by · , · , while the symbol | · | stands for the Euclidean norm. Note that the latter induces a norm on T N , still denoted by | · |, defined as We will denote by B R (x 0 ) and B R the closed balls in M of radius R centered at x 0 and 0, respectively. With the term curve, without any further specification, we refer to an absolutely continuous function from some given interval [a, b] to M . The space of all such curves is denoted by W 1,1 ([a, b]; M ), while Lip x,y ([a, b]; M ) stands for the family of Lipschitz-continuous curves γ joining x to y, i.e. such that γ(a) = x and γ(b) = y, for any fixed x, y in M .
With the notation g ∞ we will refer to the usual L ∞ -norm of g, where the latter will be either a measurable real function on M or a vector-valued measurable map defined on some interval.
Let u be a continuous function on M . A subtangent (respectively, supertangent) of u at x 0 is a function φ ∈ C 1 (M ) such that φ(x 0 ) = u(x 0 ) and φ(x) u(x) for every x ∈ M (resp., ). Its gradient Dφ(x 0 ) will be called a subdifferential (resp. superdifferential) of u at x 0 . The set of sub and superdifferentials of u at x 0 will be denoted D − u(x 0 ) and D + u(x 0 ), respectively. We recall that u is differentiable at x 0 if and only if D + u(x 0 ) and D − u(x 0 ) are both nonempty. In this instance, where Du(x 0 ) denotes the differential of u at x 0 . We refer the reader to [13] for the proofs.
When u is locally Lipschitz in M , we will denote by ∂ * u(x 0 ) the set of reachable gradients of u at x 0 , that is the set is upper semi-continuous with respect to set inclusion. When ∂ c u(x 0 ) reduces to a singleton, the function u is said to be strictly differentiable at that point. In this instance, u is differentiable at x 0 and its gradient is continuous at x 0 . We refer the reader to [16] for a detailed treatment of the subject.
A function u will be said to be semiconcave on an open subset U of M if for every x ∈ U there exists a vector p x ∈ R N such that where ω is a modulus. The vectors p x satisfying such inequality are precisely the elements of D + u(x), which is thus always nonempty in U . Moreover, ∂ c u(x) = D + u(x) for every x ∈ U , yielding in particular that Du is continuous on its domain of definition in U , see [13]. This property will be exploited in the proof of Lemma B.3.
Throughout the paper, we will call convex Hamiltonian a function H satisfying the following set of assuptions: (H3) there exist two superlinear functions α, β : We define the Fenchel transform L : The function L is called the Lagrangian associated with the Hamiltonian H; it satisfies the following properties, see Appendix A.2 in [13]: (L3) there exist two superlinear functions α * , β * : Remark 1. The functions α * , β * and α, β in (L3) and (H3), respectively, can be taken continuous (in fact, convex), without any loss of generality.
With the term strictly convex Hamiltonian we will refer to a convex Hamiltonian with (H2) replaced by the following stronger assumption: We point out that, in this event, L enjoys This fact will be exploited in the proof of Proposition 9. Here and in the sequel, D q L(x, q) and D x L(x, q) denote the partial derivative of L at (x, q) with respect to q and x, respectively. An analogous notation will be used for the Hamiltonian. A Tonelli Hamiltonian is a particular kind of Hamiltonian satisfying conditions (H1), (H2) and (H3). It is additionally assumed of class C 2 in M × R N and condition (H2) is strengthen by requiring, for every (x, p) ∈ M × R N , that ∂ 2 H ∂p 2 (x, p) is positive definite as a quadratic form.
The associated Lagrangian has the same regularity as H and enjoys the analogous condition (4). We remark that, under these conditions, the associated Hamiltonian and Lagrangian flows are complete even in the case M = R N .

2.2.
Hamilton-Jacobi equations. Let us consider a family of Hamilton-Jacobi equations of the kind where a ∈ R. In the sequel, with the term subsolution (resp. supersolution) of (5) we will always refer to a continuous function u which is a subsolution (resp. a supersolution) of (5) in the viscosity sense, i.e. for every x ∈ M H(x, p) a for any p ∈ D + u(x) resp. H(x, p) a for any p ∈ D − u(x) .
A function will be called a solution of (5) if it is both a subsolution and a supersolution.

Remark 2.
Since H is coercive, i.e. satisfies the first inequality in (H3), it is well known that any continuous viscosity subsolution v of (5) is Lipschitz, see for instance [4]. In particular, v is an almost everywhere subsolution, i.e. By the convexity assumption (H2), the converse holds as well: any Lipschitz, almost everywhere subsolution solves (5) in the viscosity sense, see [44]. In particular, v is a subsolution of (5) if and only if −v is a subsolution of We define the critical value c as c = min{a ∈ R : equation (5) admits subsolutions }.
Following [32], we carry out the study of properties of subsolutions of (5), for a c, by means of the semidistances S a defined on M × M as follows: where σ a (x, q) is the support function of the a-sublevel Z a (x) of H, namely and Z a (x) := {p ∈ R N : H(x, p) a }. The function σ a (x, q) is convex in q and upper semi-continuous in x (and even continuous at points such that Z a (x) has nonempty interior or reduces to a point), while S a satisfies the following properties: for all x, y, z ∈ M and for some positive constant κ a . The following properties hold, see [32]: (i) A function φ is a viscosity subsolution of (5) if and only if for all x, y ∈ M .
In particular, by maximality, S a (y, ·) is a viscosity solution of (5) in M \ {y}.
It is well known, by classical results of Calculus of Variations, that the infimum in (9) is achieved. The curves that realize the minimum are called Lagrangian minimizers.
The following more precise result will be needed in the sequel, see [3,17,21]: Proposition 2. Let x, y ∈ M , t > 0 and C ∈ R such that h t (x, y) < t C. Then any Lagrangian minimizer γ for h t (x, y) is Lipschitz continuous and satisfies γ ∞ κ, where κ is a constant only depending on C, α * , β * .
We recall some properties of h t , see for instance [21]. Proposition 3. Let t > 0. Then h t is locally Lipschitz continuous in M × M . More precisely, for every r > 0 there exists K = K(r, α * , β * ) such that the map We remark that, for every a c, the following holds: yielding in particular h t (y, x) + a t S a (y, x) for every x, y ∈ M . The next result can be proved by making use of suitable reparametrization techniques, see [22,32].
for every x, y ∈ M , and the infimum is always reached when a > c.
For every t > 0, we define a function on M as follows: for some a, b ∈ R. Any function of this kind will be called admissible initial datum in the sequel. The following properties hold:  (iii) S(t) is monotone and commutes with constants, i.e. for every admissible initial data u, v and any a ∈ R we have In particular, S(t) is weakly contracting, i.e.
The semigroup S(t) t>0 is called Lax-Oleinik semigroup and (11) is termed Lax-Oleinik formula. The relation with Hamilton-Jacobi equations is clarified by the next classical results, see for instance Section 10.3 in [28].
With regard to the stationary equation (5), the following characterization holds: Proposition 5. Let u be a continuous function on M . The following facts hold: This readily implies, by monotonicity of the semigroup, for every t > 0. By taking the infimum in t of the right-hand side term, we obtain u(x) − u(y) S a (y, x) for every x, y ∈ M by Lemma 2.2, i.e. u is a subsolution of (5).
Assertion (ii) easily follows by noticing that u is a solution of (5) if and only if u(x) − at is a solution of (13) with u 0 := u.
We conclude this section by proving a result that we will need later in the paper. Lemma 2.4. Let H be a strictly convex Hamiltonian and u an admissible datum. Then, for each x ∈ M , the function t → S(t)u(x) is locally semiconcave on (0, +∞). Moreover, the modulus of semiconcavity is locally uniform in x.
Proof. For simplicity, we prove the assertion for M = T N . The proof in the general case goes along the same lines, but one has to localize the arguments. More precisely, one needs to use the fact that for any positive real number T and any compact set K ⊂ R N , there exists a compact set K ⊂ R N such that any curve realizing the minimum in (S(t)u)(x), for t T and x ∈ K, is included in K .
Let I be an open interval compactly contained in (0, +∞). By the compactness of T N and condition (L3), it is not hard to see that there exists a constant C such that h t (x, y) C t for every x, y ∈ T N and t ∈ I. Let κ be the constant chosen according to Proposition 2. Fix x ∈ T N , t ∈ I and let γ be a κ-Lipschitz curve verifying γ(t) = x and such that

ANDREA DAVINI AND MAXIME ZAVIDOVIQUE
By definition of the Lax-Oleinik semigroup, we have the following obvious inequality: Therefore, the following holds: We make a Taylor expansion to obtain that where ω is a continuity modulus for D q L in T N × B 2κ . We deduce that where 3. Nonregular weak KAM theory. The purpose of this Section is to present the main results of weak KAM Theory we are going to use in the sequel. This material is not new. It is well known for Tonelli Hamiltonians, see [29], while the extension to the non regular setting is either contained in other papers or can be easily recovered from the results proved therein. Nevertheless, it is less standard and it is not always possible to give precise references. For the reader's convenience, we provide here a brief presentation. Some proofs are postponed to Appendix A. Throughout this Section, M stands either for R N or for T N and conditions (H1), (H2) and (H3) are assumed.
We focus our attention on the critical equation where, we recall, the constant c is the constant defined through (6). A subsolution, supersolution or solution of (14) will be termed critical in the sequel.
To ease notations, we will moreover write S and σ in place of S c and σ c , respectively. Finally, by possibly considering H − c instead of H, we will assume c = 0.
We define the Aubry set A as In the sequel, we will sometimes write S y to denote the function S(y, ·). We will assume that the following holds (A) A is nonempty. This condition is always fulfilled when M is compact, but it may be false in the non compact case.
We define the set E of equilibrium points as This set may be empty, but if not it is a closed subset of the Aubry set A. Next, we define a family of curves, called static. In the next Section we will investigate the behavior of the critical subsolutions on such curves.
We first show that static curves are always contained in the Aubry set.
Proof. The definition of the semidistance S and inequality (10) with a = c readily implies that γ enjoys (15). Let us prove that γ is contained in the Aubry set. If γ is a steady curve, i.e. γ(t) = y for every t ∈ J, then for (a, b) ⊂ J we get Let us then assume that γ is nonsteady. We want to prove that, for every fixed t ∈ J, the point y := γ(t) belongs to A, i.e. that S(y, ·) is a critical solution on M . Of course, we just need to check that S(y, ·) is a supersolution of (14) at y, by Proposition 1. To this purpose, choose a point z ∈ γ(J) with z = y. Since γ is static, we have S(y, z) + S(z, y) = 0. This and the triangular inequality imply that the function w(·) = S(y, z) + S(z, ·) touches S(y, ·) from above at y, hence that is, S y is a supersolution of (14) at y and so a critical solution on M .
The next result states that static curves fully cover the Aubry set. This result is proved in [22] by exploiting some ideas contained in [32]. A more concise and self-contained proof of this fact is proposed in the Appendix A.
We denote by K the family of all static curves defined on R, and by K(y) the subset of K made up by those equaling y at t = 0.
The Peierls barrier is the function h : The following holds: Proof. Take y ∈ M such that h(y, y) = 0 and set u(·) := S(y, ·). We want to prove that u is a critical solution in M ; equivalently, by Proposition 1, that u is a critical supersolution at y. To this purpose, we first note that, since u is a critical subsolution on M , the functions S(t)u are increasing in t, see Proposition 5, and equi-Lipschitz in x for u is Lipschitz continuous, see Proposition 4. Let us set According to what was remarked above, v u. Furthermore, v is Lipschitz continuous provided it is finite everywhere, or, equivalently, at some point. We claim that v(y) = u(y).
Indeed, let (t n ) n∈N be a diverging sequence such that lim n∈N h tn (y, y) = h(y, y) = 0. By definition of S(t) we have as it was claimed. This also implies that v touches u from above at y, yielding D − u(y) ⊆ D − v(y). Furthermore, v is a critical solution since it is a fixed point of the (continuous) semigroup S(t), see Proposition 5, in particular it is a critical supersolution at y. Collecting the information, we conclude that finally showing that u is a critical supersolution at y. Let us prove the opposite inclusion. Take y ∈ A. To prove that h(y, y) = 0, it will be enough, in view of Lemma 2.2, to find a diverging sequence (t n ) n∈N such that lim inf n∈N h tn (y, y) = 0.
To this purpose, let η ∈ K(y). Then for each n ∈ N. By Lemma 2.2 there exist s n > 0 such that S η(n), y h sn η(n), y < S η(n), y + 1 n .

By definition of h t we get
h n+sn (y, y) h n y, η(n) + h sn η(n), y < S y, η(n) + S η(n), y + 1 n = 1 n , and the assertion is proved by taking t n := n + s n .
Here and in the remainder of the paper, byȞ we will denote the Hamiltonian defined asȞ The following holds: The Hamiltonians H andȞ have the same critical value and the same Aubry set.
Proof. The fact that H andȞ have the same critical value immediately follows from the definition in view of Remark 2. Furthermore, the Peierls barrierȟ associated withȞ enjoysȟ(x, y) = h(y, x) for every x, y ∈ M . Hence H andȞ have the same Aubry set in view of Theorem 3.4.
We end this section by proving some important properties of the Peierls barrier.

Proposition 7.
Under assumption (A) the following properties hold: (i) h is finite valued and Lipschitz continuous.
For every x, y, z ∈ M and t > 0 h(y, ·) is a critical solution for every fixed y ∈ M .
Furthermore, when M is compact and condition (H2) is assumed, we have Proof. (i) Let K 1 be the constant given by Proposition 3 with r = 1. It is easily seen that for every bounded open set V ⊂ M × M there exists t V such that the functions {h t : t t V } are K 1 -Lipschitz continuous in V . Moreover we already know, by Theorem 3.4, that h(y, y) = 0 for every y ∈ A. This implies that h is finite valued and Lipschitz-continuous on the whole M × M .
Items (ii) and (iii) follow directly from the definition of h and from assertion (ii) in Proposition 3.
(iv) Let us assume, for definiteness, that y ∈ A. Let (t n ) n∈N be a diverging sequence such that 0 h tn (y, y) < 1/n. Then for every t > 0 and n ∈ N in view of Lemma 2.2.
(v) By Proposition 5-(ii), it suffices to prove that S(t)h y = h y for every fixed t > 0 and y ∈ M , where h y denotes the function h(y, ·). First notice that, by (iii) and Lemma 2.2, that is, h y is a critical subsolution. By Proposition 5-(i), that implies S(t)h y h y .
Let us prove the reverse inequality. For any fixed x ∈ M , pick a diverging sequence (t n ) n∈N with t n > t for every n ∈ N and a family of curves γ n : The functions h tn are equi-Lipschitz, see Proposition 3. This yields, by Proposition 2, that the curves γ n are equi-Lipschitz. Up to extraction of a subsequence, we can then assume that there is a curve γ : . By a classical semi-continuity result of the Calculus of Variations, see Theorem 3.6 in [11], we have Last, let us show that h t uniformly converges to h for t → +∞ when M is compact and condition (H2) is assumed. Let y ∈ M be fixed. Then the convergence of h t (y, ·) to h(y, ·) is actually uniform, in view of the asymptotic convergence results proved in [7,22] and of the equality h t (y, ·) = S(t − 1)u with u = h 1 (y, ·). The assertion follows from the fact that y was arbitrarily chosen in M and the functions {h t : t 1} are equi-Lipschitz in M × M in view of Proposition 3.
4. Differentiability properties of critical subsolutions. The purpose of this Section is to prove some differentiability properties of critical subsolutions on the Aubry set. These results will be exploited in the subsequent section to obtain some information for commuting Hamiltonians.
Let us consider, for any fixed t > 0, the locally Lipschitz function defined on M as where u is an admissible initial datum. If the latter is additionally assumed continuous, then the infimum is actually a minimum, and, as previously noticed, for every fixed y ∈ M there exists a Lipschitz curve γ : [−t, 0] → M with γ(0) = y such that As first step in our analysis, we prove some differentiability properties of S(t)u at y and of u at γ(−t) in terms of γ, thus generalizing to this setting some known results in the regular case, see [29].
We start by dealing with the case when the Hamiltonian is independent of x. We need a lemma first.
with equality holding for every s if γ is of class C 1 .
Remark 3. We remark for later use that, since equality (18)  It is easy to see, by the convexity of L and Jensen's inequality, that while the converse inequality is true since γ is a Lagrangian minimizer. By exploiting the convexity of L again, we get On the other hand, Proof. To ease notations, we set v := γ(0) − γ(−t) t and denote by z the point γ(−t). Let us first prove that D q L (v) ∈ D + S(t)u (y). According to the proof of Lemma 4.1, it is enough to prove the assertion when γ is the segment joining z to y. For every x ∈ M , we define a curve γ x : [−t, 0] → M joining z to x by setting γ x (s) = γ(s) + (s + t)(x − y)/t. Let Then S(t)u (·) ϕ(·) with equality holding at y. It is easy to see, using the local Lipschitz character of L, that ϕ is locally Lipschitz continuous. We want to show that D q L(v) ∈ D + ϕ(y), which clearly implies the assertion as D + ϕ(y) ⊆ D + S(t)u (y).
By the standard result of differentiation under the integral sign, the function ϕ is in fact C 1 and we may compute its differential at y by the following formula: Let us now prove that D q L(v) ∈ D − u(z).
For every x ∈ M , we define a curve η x : [−t, 0] → M joining x to y by setting Then ψ(·) u(·) with equality holding at z. We want to show that D q L(v) ∈ D − ψ(z), which is enough to conclude as D − ψ(z) ⊆ D − u(z). Arguing as above, we actually see that ψ is in fact C 1 and This concludes the proof.
We proceed to show a more general version of the previous result. Proof. Let us choose an R > 1 sufficiently large in such a way that γ ∞ R and γ [−t, 0] ⊆ B R . To ease notations, in the sequel we will call z the point γ(−t).
Let ω : R + → R + be a modulus such that If , and the assertion follows from Proposition 8 when γ is the segment joining z to y, and from Remark 3 when γ is any Lipschitz continuous minimizer. Let us then assume ω(h)/h is unbounded. Without loss of generality, we may require ω to be concave, in particular Let δ : (0, +∞) → (0, +∞) be such that for every h > 0.
Since S(t)u is Lipschitz in M , to prove assertion (i) it is in fact enough to show that the following inequality holds for every ξ ∈ ∂B R : To this purpose, for every h ∈ [0, 1] and for every ξ ∈ ∂B 1 we define a Lipschitz curve γ hξ : [−t, 0] → M joining z to y + hξ by setting By definition of S(t)u , we get For h small enough we have

This yields
To evaluate B, we use the Taylor expansion of L(γ,γ hξ ) to get for a.e. t ∈ [−δ(h), 0], where Θ is a continuity modulus for D q L on B 2R × B 2R . From this we obtain by recalling that t = 0 is a Lebesgue point for D q L γ(·),γ(·) . Relations (22) and (23) together with (21) finally give (20).
To prove (ii), it suffices to show, by the Lipschitz character of u, that for every fixed ξ ∈ ∂B 1 To this purpose, for every h ∈ [0, 1] and for every ξ ∈ ∂B 1 we define a Lipschitz curve η hξ : [−t, 0] → M joining z + hξ to y by setting By definition of S(t)u (y), we get To evaluate B , we argue as above to get B −hω(h). To evaluate A , we use the Taylor expansion of L(γ,η hξ ) to get for a.e. t ∈ [−δ(h), 0]. Arguing as above we finally get and (24) follows.
We now exploit the information gathered to deduce some differentiability properties of critical subsolutions. In what follows, we stress the fact that we have assumed the critical value c to be equal to 0, which is not restrictive up to the addition of a constant to the Hamiltonian.
We start by recalling some results proved in previous works. We underline that the compactness of M , which is assumed in these papers, does not actually play any role for the results we are about to state. The first one has been proved in [32]. Proposition 10. Let H be a convex Hamiltonian. For every y ∈ M \ A the set Z 0 (y) has nonempty interior and D − S y (y) = Z 0 (y).
In particular, S y is not differentiable at y.
Therefore, critical subsolutions are in general not differentiable outside the Aubry set. The situation is quite different on it. A fine result proved in [32] shows that, when H is locally Lipschitz-continuous in x and condition (H2) is assumed, all critical subsolutions are (strictly) differentiable at any point of the Aubry set, and have the same gradient. These results are based upon some semiconcavity estimates which, in turn, depend essentially on the Lipschitz character of the Hamiltonian in x. Something analogous still survives in the case of a purely continuous and convex Hamiltonian by looking at the behavior of the critical subsolutions on static curves, see [22]. Theorem 4.2. Let H be a convex Hamiltonian and γ ∈ K. Then there exists a negligible set Σ ⊂ R such that, for any critical subsolution u, the map u•γ is differentiable on R \ Σ and satisfies d dt Here we want to strengthen Theorem 4.2 by proving that, when condition (H2) is assumed, any critical subsolution is actually differentiable at H 1 -a.e. point of γ(R). We give a definition first. Definition 4.3. Let γ be an absolutely continuous curve defined on R. We will denote by Σ γ the negligible subset of R such that R \ Σ γ is the following set: t ∈ R : t is a differentiability point of γ and a Lebesgue point of D q L γ(·),γ(·) .
Theorem 4.4. Let H be a strictly convex Hamiltonian. Then, for any γ ∈ K, every critical subsolution u is differentiable at γ(t 0 ) for any t 0 ∈ R \ Σ γ , and we have Du Proof. Fix t 0 ∈ R \ Σ. As u is a critical subsolution, it is easily seen that with equality holding at γ(t 0 ) since
Let us denote by SS the set of critical subsolutions for H, i.e. the subsolutions of equation (14). We define the set D := v∈SS {y ∈ M : v and S y are differentiable at y, Dv(y) = DS y (y) } , (27) where S y stands for the function S(y, ·). The following holds: Proposition 11. Let H be a strictly convex Hamiltonian. Then D is a dense subset of A. When M is compact, we have in particular that D is a uniqueness set for the critical equation, i.e. if two critical solutions agree on D, then they agree on the whole M .
Proof. It is clear by Proposition 10 that D is contained in A. Pick y ∈ A and choose a static curve γ ∈ K passing through y. According to Theorem 4.4, there exists a sequence of points y n ∈ γ(R) ∩ D converging to y. This proves that D is dense in A.
The fact that D is a uniqueness set is now a direct consequence of the fact that A is a uniqueness set, see [32]. After deriving a result that will be needed later, we restrict to the case when M is compact and we look into the corresponding critical equations. We discover in the end that the commutation property entails very strong informations.
Throughout this section H and G will denote a pair of Hamiltonians satisfying assumptions (H1), (H2) and (H3). The following notations will be assumed • L H and L G are the Lagrangians associated through the Fenchel transform with H and G, respectively.
• S H and S G denote the Lax-Oleinik semigroups associated with H and G, respectively.
• h t H and h t G will denote, for every t > 0, the functions associated via (9) with H and G, respectively.
• h H and h G are the Peierls barriers associated with H and G, respectively. Definition 5.1. We will say that two convex Hamiltonians H and G commute if for every s, t > 0 and x ∈ M , (28) and for every function admissible initial datum u : M → R ∪ {+∞}.
Remark 5. Note that a Hamiltonian function H always commutes with itself. Also note that, when M is compact, any continuous function is an admissible initial datum.
We emphasize that the notion of commutation given in Definition 5.1 is nothing but a rephrasing of the fact that the the multi-time Hamilton-Jacobi equation (1) admits a solution for every Lipschitz continuous initial datum.
A very natural question is that of finding direct and easy-to-check conditions on the Hamiltonians that ensure the commutation property. As explained in the introduction, the problem has been already considered in literature. Here we recall one of the main results proved in [8], that can be stated in our setting as follows: If either H or G is of class C 1 on M × R N , then (28) holds for any u ∈ Lip(M ). Remark 6. The definition of commutation given above via (28) is actually equivalent to the cancellation of the Poisson bracket, {·, ·}, when the Hamiltonians are additionally assumed of class C 1 . The proof of this fact is sketched in the introduction of [8], and is detailed in Appendix C in the case of Tonelli Hamiltonians. This equivalence will be used to establish Theorem 5.7, see the proof of Lemma 5.8.

Remark 7.
There is another procedure to construct commuting continuous Hamiltonians which already appeared in the literature. In their fundamental work, Cardin and Viterbo [15] proved that if two sequences of smooth Hamiltonians (H n ) and (G n ) verify H n ⇒ H, G n ⇒ G and {H n , G n } ⇒ 0, then H and G admit variational solutions to the multi-time Hamilton-Jacobi equation. Under these hypotheses, H and G are termed C 0 -commuting. In particular, if H and G are smooth, their Poisson bracket vanishes. This is what they call C 0 -rigidity of the Poisson bracket, see [15,27] and references therein for more details.
In the case of convex Hamiltonians, variational solutions coincide with viscosity solutions. This implies that a pair of Hamiltonians satisfying conditions (H1)-(H3) that C 0 -commutes, also commute in the sense of Definition 5.1. The converse, however, is not clear.
It would be interesting to understand if the null Poisson bracket condition can be somehow relaxed to less regular Hamiltonians. For instance, one may wonder if the commutation condition (28) holds for pairs of locally Lipschitz Hamiltonians having Poisson bracket almost everywhere zero. We will describe in Remark 9 how a non-trivial class of locally Lipschitz Hamiltonians enjoying this property can be provided.
We prove a result that will be needed in the sequel. We need three auxiliary results first.

Proposition 13. A pair of continuous convex Hamiltonians H and G commutes if and only if
for every x, y ∈ M and t, s > 0.
Remark 8. Formula (29) holds with minima even when M is non compact. Indeed, and the same is valid for h τ G (z, ζ). This readily implies that the infima in (29) are finite and that every minimizing sequence must stay in a compact subset of M .
Proof. Let u : M → R∪{+∞} be an admissible initial datum. Using the definitions and the commutation of two nested infima we get, for every x ∈ M and t, s > 0 Now, if H and G commute, then (29) follows by plugging in the above equalities as u the function equal to 0 at y and +∞ elsewhere, for every fixed y ∈ M . Conversely, if (29) holds true, then (30) and (31) are equal for any admissible u, so H and G commute.
We set L(x, q) := min{L H1 (x, q), L H2 (x, q)} for all (x, q) ∈ M × R N . To ease notations, in the sequel we will write L i , h t i in place of L Hi , h t Hi . We recall that L * denotes the Fenchel transform of L, defined according to (3).
For every x, y ∈ M and t > 0 Proof. By classical results of Calculus of Variations, see for instance [11], we know that the infimum appearing in (32) agrees with so to conclude we only need to prove that L * * = L G . From the inequalities G H i we derive L G L i for i ∈ {1, 2}, so L G L. By duality Hence G = L * and consequently L G = L * * .
Lemma 5.4. For every x, y ∈ M and t > 0 Proof. The fact that the right-hand side term of the above equality is non smaller than h t G (x, y) is an immediate consequence of the inequalities L i L G for i ∈ {1, 2}. To prove the opposite inequality, in view of Lemma 5.3, it suffices to show that for every ε > 0 and for every curve γ : [0, t] → M of class C 1 joining x to y we have for a suitable choice of n ∈ N, {t i : 0 i n } and σ ∈ {1, 2} n .
To this purpose, choose a sufficiently large positive number R such that γ ∞ < R and γ([0, t]) ⊆ B R . Denote by ω a continuity modulus for L 1 and L 2 in B R × B R . Let r be an arbitrarily chosen positive number and choose n ∈ N large enough in such a way that |γ(s) − γ(τ )| + |γ(s) −γ(τ )| < r for any s, τ ∈ [0, t] with |s − τ | < t n . Let t i := i t/n for 0 i n and define σ ∈ {1, 2} n in such a way that for every 1 i n. Proof of Proposition 12. Let us prove that G commutes with H i , where i has been fixed, say i = 1 for definitiveness. In view of Proposition 13, we need to show that (29) holds with H 1 and G in place of H and G, respectively. Let us fix t, s > 0 and x, y ∈ M . Let us show that In view of Lemma 5.4, it will be enough to prove that, for every n ∈ N, the following inequality holds: The proof will be by induction on n. For n = 1 inequality (34) holds true for and we conclude since H σ(1) and H 1 commute and h t σ (1) h t G , for every σ(1) ∈ {1, 2}.
Let us now assume that (34) holds for n and let us show it holds for n + 1. Let where we used the fact that H 1 and H σ(n+1) commute. Let us denote by z a point realizing the minimum in the last row of the above expression. By making use of the inductive hypothesis we get The opposite inequality in (33) comes in an analogous way. The proof is complete.
Remark 9. By suitably modifying the above arguments, we can prove the following more general version of Proposition 12: let H 1 and H 2 be a pair of commuting continuous Hamiltonians and let f : R 2 → R be a convex and increasing function.
Here increasing means that Then We now restrict our attention to the case M = T N and we investigate on the relation between the associated critical equations. We will denote by c H and c G the corresponding critical values of H and G, respectively. Up to adding a constant to the Hamiltonians, we will assume that c H = c G = 0. Note that this does not affect the commutation property. The symbols S H , S G and A H , A G refer to the critical semidistance and the Aubry set associated with H and G, respectively.
We will also denote by SS H and S H the set of subsolutions and solutions of the critical equation H = 0, respectively, and by SS G and S G the analogous objects for the critical equation G = 0.
We start with two results which exploit the fact that H and G commute. Actually, the first result is a direct consequence of the monotonicity of the semigroups and does not require M to be compact. The second one uses the fact that the Lax-Oleinik semigroups are weakly contracting for the infinity norm and the proof is done applying DeMarr's theorem on existence of common fixed points for commuting weakly contracting maps on Banach spaces [26]. The compactness of M is crucial to assure that such common fixed points exist and are critical solutions for both the Hamiltonians.
The proofs of these results may be found in [47] and will be omitted. We now assume strict convexity of the Hamiltonians and we exploit the differentiability properties of critical subsolutions established in Section 4 to prove the following Proof. It is enough to show that S G ⊆ S H , since the opposite inclusion follows by interchanging the roles of H and G.
Take u ∈ S G . To prove that u ∈ S H , it suffices to show, in view of Proposition 5-(ii), that S H (t)u = u on T N for every t > 0.
Since S H (t)u ∈ S G , according to Proposition 11 it suffices to prove that Let u 0 ∈ SS H ∩ SS G , and pick a point y ∈ D G . By definition of D G , the function u 0 is differentiable at y. Moreover, see Remark 4, for every v ∈ SS G v is differentiable at y and Dv(y) = Du 0 (y).
Then the function w(t, x) := S H (t)u (x) is differentiable at y for every t > 0 and H y, D x w(t, y) = H y, Du 0 (y) = 0 for every t > 0 by Proposition 15. Now we use the fact that w is a solution of the evolutive equation The underlining idea is very simple. To focus this point, we will first establish the result by adding a mild regularity assumption on w. Then we will deal with the general case, where some technicalities arise.
First Case: w is locally semiconcave in (0, +∞) × T N . This condition is always fulfilled if, for instance, H is locally Lipschitz continuous in x, see [13,Theorem 5.3.8]. Since the map t → w(t, y) is Lipschitz continuous, it is differentiable for a.e. t > 0. In view of Lemma B.1 and of (35), we infer ∂ t w(t, y) = ∂ t w(t, y) + H y, D x w(t, y) = 0 for a.e. t > 0, yielding that w(·, y) is constant in R + . Hence S H (t)u (y) = u(y) for every t > 0, as it was to be shown.

The general case.
We only need to prove that w(·, y) is constant, i.e. that ∂ t w(t, y) = 0 for a.e. t > 0.
First we recall that, by convexity of the Hamiltonian, the fact that w is a subsolution of the evolutive equation is equivalent to requiring for every (t, x) ∈ (0, +∞) × T N . Now the functions {w(·, x) : x ∈ T N } are locally equi-semiconcave in (0, +∞), see Lemma 2.4. Moreover w has partial derivatives at (t, y) for a.e. t > 0, so in view of Lemma B.2 we get ∂ t w(t, y) = ∂ t w(t, y) + H y, D x w(t, y) 0 for a.e. t > 0.
Let us prove the opposite inequality, i.e.
∂ t w(t, y) 0 at any differentiability point t > 0 of the function w(·, y).
In fact, if this were not the case, there would exist t 0 > 0 such that w(·, y) is differentiable at t 0 and ∂ t w(t 0 , y) < −ε for some ε > 0.
Since w is locally semiconcave in t, uniformly with respect to x, we infer that there exists r > 0 such that This follows from [13,Theorem 3.3.3], which implies here the continuity of ∂ t w with respect to (t, x) on its domain of definition (via an argument analogous to the one used in the proof of Lemma B.1). By Lemma B.3, we infer that in the viscosity sense. On the other hand, w(t 0 , ·) = S H (t 0 )u ∈ S G , it is hence differentiable at y and H y, D x w(t 0 , y) = 0, yielding a contradiction.
Theorem 5.5 has very strong consequences from the weak KAM theoretic viewpoint. Indeed, we have Theorem 5.6. Let H and G be a pair of commuting, strictly convex Hamiltonians. Then Proof. (i) Let us arbitrarily fix y ∈ T N . By Proposition 7, h H (y, ·) and h G (y, ·) both belong to S H = S G , so for every s, t > 0. Moreover Let us denote by u the function being equal to 0 at y and +∞ elsewhere. For every s > 0 we have We derive and the assertion follows sending s → +∞.
Assertions (ii) and (iii) are direct consequences of (i) in view of Theorem 3.4 and of Proposition 7, respectively.
Next, we show that H and G admit a common strict subsolution. We now invoke the results proved in [32]: by Theorem 6.2, there exists a critical subsolution v for F which is strict and smooth in T N \ A. If H and G are locally Lipschitz, the same holds for F , so v can be additionally chosen of class C 1 on the whole T N in view of Theorem 8.1. The inequalities (38) follow since F H, G.
If now H and G are Tonelli Hamiltonians, the commutation property is equivalent to the fact that the Poisson bracket {H, G} = 0 everywhere, as explained in Appendix C. Starting with a C 1 (or in fact any) common strict subsolution v, it is possible to realize, as in [9], a Lasry-Lions regularization v 0 of v, using alternatively the positive and negative semigroups of H. More precisely, v 0 = S H (t) S + H (s) v for s and t suitably chosen, where the positive Lax-Oleinik semigroup is defined as follows: S + H (s)v = − SȞ (−v) . Note that v 0 is still a subsolution both for G and for H, see Remark 2 and Proposition 14. The fact that it is strict in T N \ A is proved in the next lemma. The fact that v 0 is C 1,1 for t small enough (with respect to s) is proved in [9].
We recall that a Lipschitz subsolution v ∈ SS G is said to be strict in an open set U ⊂ T N if for any x 0 ∈ U there is a neighborhood V of x 0 and a constant ε > 0 such that G x, Dv(x) < −ε almost everywhere in V .
Note that if v is C 1 , it is strict on U if and only if G x, Dv(x) < 0 for any x ∈ U .
Lemma 5.8. Let G and H be two commuting Tonelli Hamiltonians. Assume v is a critical subsolution for G which is strict outside A. Then, for all t > 0, both S G (t) v and S H (t) v are critical subsolutions for G, strict outside A.
Proof. We will only prove the result for S H (t) v. The result for S G (t) v is then a consequence for G = H. We already know by Proposition 14 that S H (t) v is a critical subsolution of G. It is only left to prove the strict part. This is done in two steps: in a first one, we prove a point-wise strictness at differentiability points of S H (t) v. In a second one, we extend this result using Clarke's gradient to any point before concluding. Let x ∈ T N \ A. Consider a curve γ verifying that γ(0) = x and The curve (γ,γ) is then a piece of trajectory of the Euler-Lagrange flow of H. It is also known (see [29] or Proposition 9) that D q L H γ(−t),γ(−t) ∈ D − v γ(−t) and D q L H γ(s),γ(s) ∈ D + S H (t + s) v γ(s) for every s ∈ (−t, 0]. Moreover, the curve γ does not intersect A. Indeed, if this were not the case, the curve (γ,γ) would be included in the lifted Aubry set, which is invariant by the Euler-Lagrange flow of H, see [29], while x = γ(0) / ∈ A. We therefore deduce that γ(−t) / ∈ A and, since v is strict, But this is not sufficient to conclude since the function S H (t) v is Lipschitz continuous in T N , hence differentiable almost everywhere only. We will prove the following: where ∂ * S H (t) v (x) denotes the set of reachable gradients of S H (t) v at x. Note that since S H (t) v is Lipschitz, this set is compact. Let p ∈ ∂ * S H (t) v (x) and consider x n → x a sequence of differentiability points for S H (t) v such that D S H (t) v (x n ) → p. For each n, choose a curve γ n : [−t, 0] → T N (which is in fact unique) such that For each n, the curve (γ n ,γ n ) is the (only) trajectory of the Euler-Lagrange flow with initial condition verifying D q L H x n ,γ n (0) = D S H (t) v (x n ). By continuity of this flow, they uniformly converge, along with their derivatives, to a curve γ. By continuity, we obtain Moreover, by passing to the limit in the equalities By arguing as above and by exploiting the fact that x / ∈ A, we obtain G(x, p) < 0. Since G is convex, we infer . We now exploit the fact that the Clarke differential is upper semi-continuous with respect to the inclusion and point-wise compact, see [16]. Let x 0 / ∈ A and choose ε > 0 in such a way that Then there exists a neighborhood V of x 0 such that G(x, p) < −ε for every p ∈ ∂ c S H (t) v (x) and x ∈ V .
In particular, G x, Dv(x) < −ε for almost every x ∈ V . The proof is complete.
Proof. Let s, t be points of [a, b] with s < t. We want to prove that We set y := γ(b) and observe that equality (39) can be equivalently written as Since S(y, ·) is a critical subsolution, the following hold: Proof. By assumption (H3) there exists a constant κ such that Z 0 (x) ⊆ B κ for every x ∈ M , so σ(x, q) κ|q| for every (x, q) ∈ M × R N . By (L3) and by the superlinear and continuous character of α * , see Remark 1, there exists a constant α 0 > 0 such that (κ + 1)|q| − α 0 α * (|q|) L(x, q) for every (x, q) ∈ M × R N .
The assertion follows by choosing R := α 0 .
Proof of Theorem 3.3. Fix y ∈ A and set u(·) = S(y, ·). The function w(x, t) = u(x) is a solution of the equation hence S(t)u = u for every t > 0. In particular, for each n ∈ N there exists a curve γ n : [−n, 0] → M with γ n (0) = y such that u(y) = u γ n (−n) + 0 −n L(γ n ,γ n ) ds.
Now u(y) = 0 and u γ n (−n) = S y, γ n (−n) , so by Lemma A.1 we derive that γ n is a static curve. Lemma A.2 guarantees that the curves γ n are equi-Lipschitz continuous, in particular there exists a Lipschtiz curve γ : R − → M such that, up to subsequences, γ n ⇒ γ in R − andγ n γ in L 1 loc R − ; R N . By a classical semi-continuity result of the Calculus of Variations [11] we have lim inf n→+∞ b a L(γ n ,γ n ) ds b a L(γ,γ) ds for every a < b 0, yielding in particular that γ is static too.
We now consider the HamiltonianȞ(x, p) = H(x, −p). By Proposition 6, we know that the critical value and the Aubry set ofȞ agree with 0 (i.e. the critical value of H) and A. We can apply the previous argument withŠ andĽ in place of S and L to obtain a curve ξ : R − → M which is static forȞ. We define a curve η : R → M by setting We claim that η is the static curve we were looking for. To prove this, it will be enough, in view of Lemma A.1, to show By upper semi-continuity of Clarke's generalized gradient, the above inequality keeps holding in a neighborhood V of (t 0 , x 0 ), i.e. for every (t, x) ∈ V D x w(t 0 , x 0 ), q < a < p, q for every p ∈ ∂ c w(t, x).
For h = 0, let us consider the ratio By the Nonsmooth Mean Value Theorem, see Theorem 2.3.7 in [16], there exists a point x h on the segment joining x 0 to x 0 + hq and a vector (α h , p x h ) ∈ ∂ c w(t 0 , x h ) such that r(h) = (α h , p x h ), (0, q) = p x h , q . For h small enough, we infer that r(h) > a. On the other hand lim h→0 r(h) = D x w(t 0 , x 0 ), q < a, yielding a contradiction.
We conclude this appendix by proving the following (ii) there exist a constant a ∈ R and two open sets I ⊆ (0, +∞) and U ⊆ T N such that ∂ t w(t, x) < a for a.e. t ∈ I and for a.e. x ∈ U . Then, for every t 0 ∈ I, the function u t0 := w(t 0 , ·) satisfies H(x, Du t0 ) −a in U (43) in the viscosity sense.
Proof. We divide the proof in two steps.
Step 1. Let us additionally assume that, for every t > 0, the function w(t, ·) is locally semiconcave in T N .
Set u t = w(t, ·). By semiconcavity, the inequality (44) means that in the viscosity sense. Here we have used the fact that Du t is continuous on its domain of definition and that the supersolution test is nonempty only at points where u t is differentiable. If now t 0 is any point of I, we choose a sequence of points t n ∈ I converging to t 0 for which (45) holds for every n. Since u tn ⇒ u t0 in T N , by stability of the notion of viscosity supersolution we get (43).
Each w n (t, ·) is semiconcave in T N for every fixed t > 0 and satisfies the following inequality in the viscosity sense ∂ t w n + H(y, D x w n ) −δ n in (0, +∞) × T N , where (δ n ) n∈N is an infinitesimal sequence, see [13]. Let us denote by Y n (t, x) the set of points y ∈ T N which realize the minimum in the definition of w n (t, x). If L is the Lipschitz constant of w in R + × T N , it is well known that |y − x| L n for every y ∈ Y n (t, x) and n ∈ N.
Furthermore ∂ t w n (t , x ) = ∂ t w(t , y) for every y ∈ Y n (t , x ) at any point (t , x ) where w n has partial derivative with respect to t. Indeed, if ϕ(t) is a subtangent to w n (·, x ) at the point t , the function ϕ(t) − n|x − y| 2 is a subtangent to w(·, y) at the point t , so (47) follows by semiconcavity of w n and w with respect to t. In particular, for n big enough, ∂ t w n (t, x) < a for a.e. (t, x) ∈ B r (t 0 ) × B r (x 0 ).

By
Step 1 we infer that the functions u t0 n = w n (t 0 , ·) satisfy H(x, Du t0 n ) −a − δ n in B r (x 0 ) in the viscosity sense. Since u t0 n ⇒ u t0 = w(t 0 , ·) in T N , we conclude by stability that H(x, Du t0 ) −a in B r (x 0 ). The assertion follows since t 0 and x 0 were arbitrarily chosen in I and U , respectively, together with the fact that the notion of viscosity supersolution is local.