On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem

We study the representation formulae for the fundamental solutions and viscosity solutions of the Hamilton-Jacobi equations of contact type. We also obtain a vanishing contact structure result for relevant Cauchy problems which can be regarded as an extension to the vanishing discount problem.


INTRODUCTION
In the previous work ( [21] and [22]), the authors developed an analogy to weak KAM theory for contact systems on compact manifolds. This leads to a representation formula of the viscosity solutions of the Hamilton-Jacobi equation using implicit variational principle, where M is a C 2 connected closed manifold and c is contained in the set of critical values. The power of this celebrating work has been shown to understand certain systems in a much wider and deeper viewpoint. The main purpose of this paper is to understand the limit of the viscosity solutions of (HJ e ) in the case M = R n when H u is uniformly bounded and tends to 0. For the special case when H having the form of a s Hamiltonian H 0 with a discount factor λ > 0, i.e., H = λu + H 0 (x, p), this problem has been widely studies. From calculus of variations and optimal controls point of view, the associated Lagrangian L = −λu + L 0 (x, v), where λ > 0 and L 0 is a Tonelli Lagrangian. A classical problem in ergodic control consists of studying the limit behavior of the optimal value u λ of a discounted cost functional with infinite horizon as the discount factor λ tends to zero. In the literature, this problem has been addressed under various conditions ensuring that the rescaled value function λu λ converges uniformly to a constant limit.
In recent works, for instance, [12], [18] and [19], the behavior of the vanishing discount limit has been widely studied in the compact manifold case, especially applying Aubry-Mather theory and weak KAM theory. Sufficient evidences show that the method we use in this paper is also a way to understand the vanishing contact structure limit by developing the Aubry-Mather theory for contact type systems (HJ s ) under suitable conditions. We will use a Langrangian approach of the solutions of (HJ e ) in the viscosity sense developed in [5] using the generalized variational principle proposed by Gustav Herglotz in 1930 (see [5] and the references therein).
Such a Lagrangian approach also leads to a very clear explanation of the representation formulae of the value function of the associated problem from calculus of variations. That is, due to Proposition 1.1, we conclude that the relevant fundamental solution and u ξ is uniquely determined by (1.2) in classical sense. By solving the ordinary differential equation (1.2), we can have some new representation formulae of the viscosity solution u(t, x) of (HJ e ). ( where u ξ is uniquely determined by (1.2) with u ξ (t) = u. This approach also leads to a result on the vanishing contact structure limit problem. This can be regarded as a generalization of the vanishing discount problem in PDE and control theory.
Main Result I: Suppose that {L λ } λ>0 is a family of Tonelli Lagrangians satisfying conditions (L1), (L2) and (L3') at the beginning of section 2, with {H λ } the family of associated Tonelli Hamiltonians. Let each u λ be the unique viscosity solution of (HJ e ) with respect to H λ and u defined by (2.14) be the unique viscosity solution of (HJ' e ). If φ is Lipschitz and bounded, then Main Result II: Under the same assumptions as above and replacing (L3') by (L3") (at the beginning of section 2), then u λ tends to u uniformly as λ → 0 + on any compact subset of (0, +∞) × R n . This paper is organized as follows. In Section 2.1, we give a representation formula for the equation (HJ e ). In section 2.2, we discuss our vanishing contact structure results for (HJ e ).
Acknowledgments This work is partly supported by Natural Scientific Foundation of China (Grant No. 11631006, No. 11790272 and No.11471238). The authors thank Qinbo Chen and Hitoshi Ishii for for helpful discussion.

REPRESENTATION FORMULA AND VANISHING CONTACT STRUCTURE
We will study (HJ e ) when M = R n . It is not difficult to see that the associated Lagrangian L defined in (1.1) is a function of C 2 class and it satisfies the following conditions: (L2) For each r ∈ R, there exist two superlinear and nondecreasing function θ r , θ r : (L3) There exists K > 0 such that Let {L λ } λ>0 be a family of Tonelli Lagrangians satisfying conditions (L1)-(L3). We denote by H λ the associated Hamiltonians.
For the family {L λ } λ>0 we also need the following conditions: and L λ tends to L 0 uniformly as λ → 0 + on any compact subset of R n × R n .
2.1. Representation formulae for fundamental solutions and viscosity solutions. In this section, we want to give a new representation formula for the viscosity of the Hamilton-Jacobi equation (HJ e ) with H satisfying condition (H1)-(H3). Such a representation formula for Tonelli systems appeared first in [21] and [22] using an implicit variational principle and a fixed point method. In [5], the authors give an alternative approach based on Hergoltz' variational principle. Our following new representation formula for the fundamental solutions is motivated by the multiplier rule (see [10]).
where u ξ is uniquely determined by (1.2).
Therefore, the curve u ξ given by (1.2) does not appear in the representation formula above except for the initial point u ξ (0). Therefore or, equivalently, Since A(t, x, y, u) = inf ξ∈Γ t x,y u ξ (t), then we obtain (2.1). Theorem 2.3. If L satisfies conditions (L1)-(L3), or equivalently, H satisfies conditions (H1)-(H3), and φ is bounded and Lipschitz real-valued function on R n with a Lipschitz constant Lip (φ), then is solution of (HJ e ) in the viscosity sense where A(t, x, y, u) is given by (2.1).
We will postpone the proof of Theorem 2.3. To verify that the infimum in (2.2) is indeed a minimum, we want to show the boundedness of the set For this purpose we need a refinement of Lemma B.1. Notice that when one works on a closed manifold instead of R n , at least the the infimum can be achieved automatically. But a quantitative estimate on the size of ball containing Λ x t has its own interest.
Proof. For any x, y ∈ R n and t > 0. Let ξ y ∈ Γ t y,x be a minimizer of A(t, y, x, φ(y)) and u ξy be the unique solution of (1.2) with initial condition u ξy (0) = φ(y). Based on the estimates of the lower bound of A(t, y, x, φ(y)) and upper bound of A(t, x, x, φ(x)) in Lemma 2.4, we have to deal with estimate of the lower bound of e Kt φ(y) − e −Kt φ(x) when φ(y), φ(x) 0 which is the most difficult case. Indeed, we have that where C 0 = sup x∈R n |φ(x)|. Since there exists C 1 > 0 such that 1 − e −2Kt C 1 t for all t 0. Therefore, for C 2 = C 0 C 1 , we conclude Now, suppose that φ(y) 0 and φ(x) 0, then by Lemma 2.4 and (2.12), we have that Therefore, for any k > 0, we have that − Lip (φ)|y − x| + k|y − x| − (c 0 + C 2 + C + e −2Kt θ * 0 (ke 2Kt ))t Choosing k = Lip (φ) + 1 and taking µ(t) = c 0 + C 2 + C + e −2Kt θ * 0 (e 2Kt (Lip (φ) + 1)), we have that the set Λ x t defined in (2.3) is contained in B(x, µ(t)t). Therefore Λ x t is compact and the infimum in (2.2) is indeed minimum. Moreover, (2.11) is a consequence of (2.3). The other cases can be dealt with in a similar way. Indeed,

This completes the proof.
Remark 2.7. It is not clear whether Lemma 2.6 holds true without the assumption that φ is bounded in general, while it does for the Lagrangian in the form L(x, u, v) = −λu + L 0 (x, v), λ > 0, which is the Lagrangian with respect to the well known discounted Hamiltonian (see, for instance, [12]). Lemma 2.6 not only ensures that the infimum in (2.2) is indeed minimum if φ is a bounded and Lipschitz continuous function on R n , but also plays an essential part for the applications to the study of the global propagation of singularities of the associated Hamilton-Jacobi equations ( [3], [4], [8] and [2]). Lemma 2.8. Let x, y ∈ R n , t > 0 and u ∈ R. For any ξ ∈ Γ t x,y being a minimizer of (1.3), we denote by u ξ (s, u) the unique solution of (1.2) with u ξ (0, u) = u. Then, for any 0 < t ′ < t, the restriction of ξ on [0, t ′ ] is a minimizer for and A(s 1 + s 2 , x, ξ(s 1 + s 2 ), u) = A(s 2 , ξ(s 1 ), ξ(s 1 + s 2 ), u ξ (s 1 )) for any s 1 , s 2 > 0 and s 1 + s 2 t.
Proof. Suppose x, y ∈ R n , t > 0 and u ∈ R. Let ξ ∈ Γ t x,y be a minimizer of (1.3) and u ξ (s) = u ξ (s; u) be the unique solution of (1.
By letting |t 2 − t 1 | → 0, this gives rise to As an application of Fenchel-Legendre dual and since u(t 0 , x 0 ) = u ξ (t 0 ), we obtain which shows that u is a subsolution. Now we turn to the proof that u is a supersolution. Let ϕ be a C 1 test function such that with V be an open neighborhood of (t 0 , x 0 ) in R n . Due to Lemma 2.6 and Lemma 2.8, there exists a C 2 curve ξ : [t,

It follows that
Finally, fix x ∈ R n and let y t,x be any minimizer as in Lemma 2.6, we conclude that lim t→0 + y t,x = x. Thus, |L(ξ(s), u ξ (s),ξ(s))|.
Since Proposition B.1 and Proposition B.2 and φ is bounded, then, for 0 < t 1, we conclude that, there exists C 1 > 0 independent of x, t and u, such that It follows that there exists C 2 > 0 such that max s∈[0,t] |L(ξ(s), u ξ (s),ξ(s))| C 2 , and This leads our conclusion that lim t→0 + u(t, x) = φ(x) and completes the proof.

2.2.
Vanishing contact structure. Let u λ be the viscosity solution of (HJ e ) with respect to H λ defined by Herglotz' variational principle (1.3) under the constrain (1.2). If H 0 is the Fenchel-Legendre dual of L 0 , then u defined by (2.14) u(t, x) = inf y∈R n {φ(y) + A t (y, x)}, x ∈ R n , t > 0, is a viscosity solution of where A t (x, y) is the fundamental solution or the least action with respect to L 0 . In this section, we will begin with an easier problem to show, for Cauchy problem, the vanishing discount problem mentioned in the introduction can be generalized to that of vanishing contact structure. Lemma 2.9. Suppose L satisfies conditions (L1)-(L3). Given x ∈ R n , t, R > 0, u ∈ R and |y − x| R. If ξ ∈ Γ t y,x is a minimizer for A t (y, x) and u ξ is determined by (1.2) with respect to L λ and ξ, then we have that where Proof. Denoting by ξ 0 ∈ Γ t y,x the straight line segment defined by ξ 0 (s) = y + s(y − x)/t for any s ∈ [0, t] and in view of (L2), we have that where κ(r) = θ 0 (r) + 2c 0 . Therefore, Due to Gronwall inequality , we obtain that which leads to our conclusion. Let each u λ be the unique viscosity solution of (HJ e ) with respect to H λ and u defined by (2.14) be the unique viscosity solution of (HJ' e ). If φ is Lipschitz and bounded, then x) ∈ (0, +∞) × R n . Remark 2.11. For the uniqueness of the viscosity solutions of both (HJ e ) and (HJ' e ), see, for instance, [1].
Proof. It is similar to the proof of Theorem 2.10.

APPENDIX A. CARATHÉODORY EQUATIONS
Let Ω ⊂ R n+1 be an open set. A function f : R × R n → R n is said to satisfy Carathéodory condition if -for any x ∈ R n , f (·, x) is measurable; -for any t ∈ R, f (t, ·) is continuous; -for each compact set U of Ω, there is an integrable function m U (t) such that |f (t, x)| m U (t), (t, x) ∈ U.
The classical problem of following Carathéodory equation (A.1)ẋ(t) = f (t, x(t)), a.e., t ∈ I is to find an absolutely continuous function x defined on a real interval I such that (t, x(t)) ∈ Ω for t ∈ I and satisfies (A.1).

Proposition A.1 (Carathéodory).
If Ω is an open set in R n+1 and f satisfies the Carathéodory conditions on Ω, then, for any (t 0 , x 0 ) in Ω, there is a solution of (A.1) through (t 0 , x 0 ). Moreover, if the function f (t, x) is also locally Lipschitzian in x with a measurable Lipschitz function, then the uniqueness property of the solution remains valid.
For the proof of Proposition A.1 and more results related to Carathéodory equation (A.1), the readers can refer to [11] and [14].

APPENDIX B. REGULARITY RESULTS
In this section, we collect some fundamental estimates mainly from [5]. We always suppose that conditions (L1)-(L3) are satisfied.