The study of the regularity properties of the viscosity solutions of the Hamilton-Jacobi equation (1.1) H(x, Du(x

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GLOBAL THEORY

PIERMARCO CANNARSA AND WEI CHENG

ABSTRACT. For autonomous Tonelli systems onRⁿ, we develop an intrinsic proof of the existence of generalized characteristics using sup-convolutions. This approach, together with convexity estimates for the fundamental solution, leads to new results such as the global propagation of singularities along generalized characteristics.

1. INTRODUCTION

LetL(x, v)be a Tonelli Lagrangian onRⁿ(L : Rⁿ×Rⁿ → Ris a function of class C², strictly convex in the fibre, with superlinear growth with respect tov), and letH(x, p) be the associated Hamiltonian given by the Fenchel-Legendre transform. The study of the regularity properties of the viscosity solutions of the Hamilton-Jacobi equation

(1.1) H(x, Du(x)) = 0 (x∈Rⁿ)

is extremely important for several reasons. In the last two or three decades, remarkable progress in the broad area of Hamiltonian dynamical systems was achieved by Mather’s theory for Tonelli systems on compact manifolds in Lagrangian formalism [31, 32], and Fathi’s weak KAM theory in Hamiltonian formalism [25, 27]. Both these theories suc- ceeded in the analysis of some hard dynamical problems such as Arnold diffusion. How- ever, a general variational setting that applies to all the above aspects of Hamiltonian dynamics still has to be developed. As is well known, in both Mather’s and Fathi’s theories various global minimal sets in a variational sense—such as Mather’s set, Aubry’s set and Ma˜n´e’s set—play a crucial role. Similarly, if we want to study the behavior of an orbit after it loses minimality, then we have to face the hard problem of dealing with the cut loci and singular sets of the associated viscosity solutions.

Although significant contributions investigating the singularities of viscosity solutions were already given in [17] and [7], the current approach to this problem goes back to [3], where the propagation of singularities was studied for general semiconcave functions.

Since any viscosity solutionuof (1.1) is locally semiconcave (with linear modulus), we have that singularities propagate along Lipschitz arcs starting from any singular pointxof uat which the superdifferentialD⁺u(x)satisfies the condition

∂D⁺u(x)\D^∗u(x)6=∅,

where∂D⁺u(x)denotes the topological boundary ofD⁺u(x)andD^∗u(x)the set of all reachable gradients ofuatx. A more specific approach to the problem was developed in

Date: January 3, 2017.

2010Mathematics Subject Classification. 35F21, 49L25, 37J50.

Key words and phrases. Hamilton-Jacobi equation, weak KAM theory, viscosity solution, generalized characteristic, singularities.

1

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[4] by solving the generalized characteristic inclusion

˙

x(s)∈coHp x(s), D⁺u(x(s))

, a.e.s∈[0, τ].

More precisely, if the initial pointx0belongs to the singular set ofu, hereafter denoted by Sing(u), and is not a critical point ofurelative toH, i.e.,

06∈coHp(x0, D⁺u(x0)),

then it was proved in [4] that there exists a nonconstant singular arcx from x0 which is a generalized characteristic. The study of the local propagation of singularities along generalized characteristics was later refined in [37] and [18]. For weak KAM solutions, local propagation results were obtained in [19] and the Lasry-Lions regularization procedure was applied in [12] to analyze the critical points of Mather’s barrier functions. An interesting interpretation of the above singular curves as part of the flow of fluid particles has been recently proposed in [28] (see also [35] for related results).

Returning to our dynamical motivations, in this paper we try to give an intrinsic interpretation of generalized characteristics and study the relevant global properties of such curves. For this purpose, we use the Lax-Oleinik semigroupsT_t^± (see, e.g. [25]) defined as follows:

T_t⁺u₀(x) := sup

y∈Rⁿ

{u₀(y)−A_t(x, y)}, T_t⁻u₀(x) := inf

y∈Rⁿ{u₀(y) +A_t(y, x)},

whereu0 : Rⁿ → Ris a continuous function andAt(x, y)is the fundamental solution of (1.1). These operators can be also derived from the Moreau-Yosida approximations in convex analysis ([8]) or the Lasry-Lions regularization technique based on sup- and inf- convolutions ([29], [36]).

By analyzing the maximizersyt, for sufficiently smallt > 0, in the sup-convolution givingT_t⁺u₀(x)we obtain the global propagation of singularities which represents the main result of this paper. For such a result we need the following assumptions.

(L1) Uniform convexity: There exists a nonincreasing functionν : [0,+∞)→ (0,+∞) such that

L_vv(x, v)>ν(|v|)I for all(x, v)∈Rⁿ×Rⁿ.

(L2) Growth conditions: There exist two superlinear functionsθ, θ: [0,+∞)→[0,+∞) and a constantc0>0such that

θ(|v|)>L(x, v)>θ(|v|)−c0 ∀(x, v)∈Rⁿ×Rⁿ.

(L3) Uniform regularity: There exists a nondecreasing functionK: [0,+∞)→[0,+∞) such that, for every multindex|α|= 1,2,

|D^αL(x, v)|6K(|v|) ∀(x, v)∈Rⁿ×Rⁿ,

Recalling that Sing(u)stands for the singular set ofuwe now proceed to state our Propagation result: LetLbe a Lagrangian onRⁿ satisfying conditions (L1)-(L3),

and let ube a globally Lipschitz semiconcave solution of the Hamilton-Jacobi equation (1.1), where H is the Hamiltonian associated withL. Ifxbelongs to Sing(u), then there exists a generalized characteristicx : [0,+∞) → Rⁿ such thatx(0) =xandx(s)∈Sing(u)for alls∈[0,+∞).

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We observe that the study of the global propagation of singularities is much more dif- ficult than the local one. Indeed, to this date the only known results concern geodesic systems, see [2] and [5]. More precisely, [2] studies the global propagation of the so- calledC¹-singular support of solutions which—in this case—coincides with the closure of Sing(u), while [5] investigates the propagation of genuine singularities. For time de- pendent problems, global propagation was addressed in [1] for theC¹-singular support of solutions and, recently, in [15] for singularities of solutions to eikonal equations.

It is worth mentioning that, when considering geodesic systems on Riemannian manifolds, the method of generalized characteristics (or generalized gradient flow) has been successfully applied to reveal topological relations between a compact domainΩand the cut locus enclosed inΩ([5]). Such relations depend on global results for the propagation of singularities along the associated generalized characteristics. We will shortly apply our global propagation results to study singularities on the torus ([13]).

For the proof of the above theorem we need regularity results for the value function of the action functional (also called fundamental solution of (1.1) in [33])

At(x, y) = inf

γ∈Γ^t_x,y

Z t

0

L(γ(s),γ(s))ds˙ (t >0, x, y∈Rⁿ) where

Γ^t_x,y ={γ∈W^1,1([0, t],Rⁿ) :γ(0) =x, γ(t) =y}.

More precisely, we need

Convexity and localC^1,1regularity results: SupposeLis a Tonelli Lagrangian satisfying (L1)-(L3). Then the following properties hold true.

(a) For anyλ > 0, there existst_λ >0such that, for anyx ∈ Rⁿ, the function (t, y)7→A_t(x, y)is semiconvex on the cone

S_λ(x, t_λ) :=

(t, y)∈R×Rⁿ : 0< t < t_λ, |y−x|< λt ,

that is, there existsC_λ>0such that for allx∈Rⁿ, all(t, y)∈S_λ(x, t_λ), all h∈[0, t/2), and allz∈B(0, λt)we have that

At+h(x, y+z) +A_t−h(x, y−z)−2At(x, y)>−Cλ

t (h²+|z|²).

(b) For allt ∈ (0, tλ], At(x,·)is uniformly convex onB(x, λt), that is, there existsC_λ⁰ >0such that for allx∈Rⁿ, ally∈B(x, λt), and allz∈B(0, λt) we have that

At(x, y+z) +At(x, y−z)−2At(x, y)> C_λ⁰ t |z|².

(c) For anyx∈Rⁿthe functions(t, y)7→At(x, y)and(t, y)7→At(y, x)are of classC_loc^1,1on the coneSλ(x, tλ)defined above.

Similar regularity results were obtained in [11] by a different approach, under more restric- tive structural assumptions than those we consider in this paper.

This paper is organized as follows. In section 2, we review basic properties of viscosity solution of Hamilton-Jacobi equations. In section 3, we discuss connections between sup- convolutions and generalized characteristics and we give our global result on the propagation of singularities along generalized characteristics. The paper contains four appendices that contain technical results and background material which is useful for our approach:

in the first one we give a uniform bound for minimizers of the action functional following [6] and [24], in the second one we give detailed proofs of all the required regularity

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results for the fundamental solution, in the third one we adapt the construction of generalized characteristics from [4] to the present context, in the fourth one we provide a global semiconcavity estimate for the weak KAM solution onRⁿconstructed in [26].

AcknowledgmentsThis work was partially supported by the Natural Scientific Foundation of China (Grant No. 11271182 and No. 11471238), the National Basic Research Program of China (Grant No. 2013CB834100), a program PAPD of Jiangsu Province, China, and the National Group for Mathematical Analysis, Probability and Applications (GNAMPA) of the Italian Istituto Nazionale di Alta Matematica “Francesco Severi”. The authors thank Albert Fathi for the discussion during his visit in Nanjing. The second author is grateful to Cui Chen, Liang Jin for helpful discussions, and also Jun Yan for the communications and the hospitality during the visit to Fudan University in 2014-15.

2. PRELIMINARIES

2.1. Semiconcave functions. Let Ω ⊂ Rⁿ be a convex set. We recall that a function u : Ω → Ris said to be semiconcave (with linear modulus) if there exists a constant C >0such that

(2.1) λu(x) + (1−λ)u(y)−u(λx+ (1−λ)y)6 C

2λ(1−λ)|x−y|²

for anyx, y∈Ωandλ∈[0,1]. Any constantCthat satisfies the above inequality is called asemiconcavity constantforuinΩ.

A functionu: Ω→Ris said to besemiconvexif−uis semiconcave.

Whenu: Ω→Ris continuous, it can be proved thatuis semiconcave with constantC if and only if

u(x) +u(y)−2u x+y

2

6C

2|x−y|² for anyx, y∈Ω.

Hereafter, we assume thatΩis a nonempty open subset ofRⁿ.

We recall that a functionu : Ω →Ris said to belocally semiconcave(resp. locally semiconvex) if for eachx ∈ Ωthere exists an open ballB(x, r) ⊂ Ω such thatuis a semiconcave (resp. semiconvex) function onB(x, r).

Letu: Ω ⊂ Rⁿ → Rbe a continuous function. We recall that, for anyx ∈ Ω, the closed convex sets

D⁻u(x) =

p∈Rⁿ: lim inf

y→x

u(y)−u(x)− hp, y−xi

|y−x| >0

, D⁺u(x) =

p∈Rⁿ: lim sup

y→x

u(y)−u(x)− hp, y−xi

|y−x| 60

. are called the(Dini) subdifferentialandsuperdifferentialofuatx, respectively.

Let nowu: Ω →Rbe locally Lipschitz. We recall that a vectorp∈Rⁿis said to be areachable(or limiting)gradientof uatxif there exists a sequence{xn} ⊂ Ω\ {x}, converging tox, such thatuis differentiable atxk for eachk∈Nand

k→∞lim Du(x_k) =p.

The set of all reachable gradients ofuatxis denoted byD^∗u(x).

Now we list some well known properties of the superdifferential of a semiconcave function onΩ⊂Rⁿ(see, e.g., [16] for the proof).

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Proposition 2.1. Letu: Ω⊂Rⁿ →Rbe a semiconcave function and letx∈Ω. Then the following properties hold.

(a) D⁺u(x)is a nonempty compact convex set inRⁿ andD^∗u(x) ⊂ ∂D⁺u(x), where

∂D⁺u(x)denotes the topological boundary ofD⁺u(x).

(b) The set-valued functionx D⁺u(x)is upper semicontinuous.

(c) If D⁺u(x) is a singleton, then uis differentiable atx. Moreover, if D⁺u(x) is a singleton for every point inΩ, thenu∈C¹(Ω).

(d) D⁺u(x) = coD^∗u(x).

Proposition 2.2([16]). Letu: Ω→Rbe a continuous function. If there exists a constant C >0such that, for anyx∈Ω, there existsp∈Rⁿsuch that

(2.2) u(y)6u(x) +hp, y−xi+C

2|y−x|², ∀y∈Ω,

thenuis semiconcave with constantCandp∈D⁺u(x). Conversely, ifuis semiconcave inΩwith constantC, then(2.2)holds for anyx∈Ωandp∈D⁺u(x).

A pointx∈Ωis called asingular pointofuifD⁺u(x)is not a singleton. The set of all singular points ofu, also called thesingular setofu, is denoted by Sing(u).

2.2. Tonelli Lagrangians. In this paper, we concentrate on Lagrangians on Euclidean configuration spaceRⁿ. We say that a functionθ : [0,+∞)→[0,+∞)issuperlinearif θ(r)/r→+∞asr→+∞.

Definition 2.3. A functionF :Rⁿ×Rⁿ→Ris called ageneralized Tonelli functionifF is a function of classC²that satisfies the following conditions:

(T1) Uniform convexity: There exists a nonincreasing functionν : [0,+∞)→ (0,+∞) such that

Fvv(x, v)>ν(|v|)I ∀(x, v)∈Rⁿ×Rⁿ.

(T2) Growth condition: There exist two superlinear functionθ, θ : [0,+∞)→[0,+∞) and a constantc0>0such that

θ(|v|)>F(x, v)>θ(|v|)−c0 ∀(x, v)∈Rⁿ×Rⁿ.

(T3) Uniform regularity: There exists a nondecreasing functionK: [0,+∞)→[0,+∞) such that, for every multindex|α|= 1,2,

|D^αF(x, v)|6K(|v|) ∀(x, v)∈Rⁿ×Rⁿ, The convex conjugate of a superlinear functionθis defined as

(2.3) θ^∗(s) = sup

r>0

{rs−θ(r)}s ∀s>0.

In view of the superlinear growth ofθit is clear thatθ^∗is well defined and satisfies (2.4) θ(r) +θ^∗(s)>rs ∀r, s>0,

which in turn can be used to show thatθ^∗(s)/s→ ∞ass→ ∞.

Definition 2.4. A functionL:Rⁿ×Rⁿ→Ris called aTonelli LagrangianifLis a generalized Tonelli function as in Definition 2.3. IfLis a Tonelli Lagrangian, the associated HamiltonianH is the Fenchel-Legendre dual ofLis defined by

(2.5) H(x, p) = sup

v∈Rⁿ

hp, vi −L(x, v) (x, p)∈Rⁿ×Rⁿ.

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The Hamiltonian H is called aTonelli Hamiltonian ifH is a generalized Tonelli function. We denote by (L1)-(L3) (resp. (H1)-(H3)) the corresponding conditions of a Tonelli LagrangianL(resp. Tonelli HamiltonianH).

Example2.5. A typical example of a Tonelli LagrangianLis the one of mechanical systems which has the form

L(x, v) =f(x)(1 +|v|²)^q² +V(x), (x, v∈Rⁿ),

whereq > 1,f andV are smooth functions onRⁿ with bounded derivatives up to the second order, andinf_Rⁿf >0.

Lemma 2.6. IfLis a Tonelli Lagrangian withHits Fenchel-Legendre dual, thenH is a Tonelli Hamiltonian.

Proof. First, letv_x,p∈Rⁿbe such thatp=L_v(x, v_x,p)or, equivalently,v_x,p =H_p(x, p).

By assumptions (L1)-(L3), we have that

|p||vx,p|>hLv(x, v_x,p), v_x,pi>L(x, v_x,p)−L(x,0)>θ(|vx,p|)−c₀−θ(0)¯

>(|p|+ 1)|vx,p| −c0−θ(0)¯ −θ^∗(|p|+ 1).

This shows that

|Hp(x, p)|=|vx,p|6c0+ ¯θ(0) +θ^∗(|p|+ 1) =C1(|p|),

whereC₁(r) =c₀+ ¯θ(0) +θ^∗(r+ 1). The estimates for the other derivatives in (H3) and (H1) can be proved by using the relations

Hx(x, p) =−Lx(x, Hp(x, p)) Hpp(x, p) =L⁻¹_vv(x, Hp(x, p))

H_xp(x, p) =−Lxv(x, H_p(x, p))H_pp(x, p)

Hxx(x, p) =−Lxx(x, Hp(x, p))−Lxv(x, Hp(x, p))Hpx(x, p) and our conditions (L1)-(L3). This completes the verification of (H1) and (H3).

To check (H2), we have that, for allR>0,

H(x, p)>R|p| −θ(R),¯ ∀(x, p)∈Rⁿ×Rⁿ

by (2.5) and (L2). Setθ1(r) = sup_R>0{Rr−θ(R)},¯ r∈(0,+∞), which is well defined by (L2). Thus,θ1gives the required superlinear function for (H2).

2.3. Hamilton-Jacobi equations. SupposeHis the Hamiltonian associated with a Tonelli LagrangianLand consider the Hamilton-Jacobi equation

(2.6) H(x, Du(x)) = 0 (x∈Rⁿ).

We recall that a continuous functionuis called aviscosity subsolutionof equation (2.6) if, for anyx∈Rⁿ,

H(x, p)60, ∀p∈D⁺u(x). (2.7)

Similarly,uis aviscosity supersolutionof equation (2.6) if, for anyx∈Rⁿ, H(x, p)>0, ∀p∈D⁻u(x).

(2.8)

Finally,uis called aviscosity solutionof equation (2.6), if it is both a viscosity subsolution and a supersolution.

Throughout this paper we will be concerned with solutions of the above equation that are Lipschitz continuous and semiconcave onRⁿ. The existence of such solution is the

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object of the following proposition which is essentially a consequence of the existence theorem of [26] and the semiconcavity results of this paper (see also [16] and [34]).

Proposition 2.7. LetLbe a Tonelli Lagrangian and letHbe the associated Hamiltonian.

Then there exists a constantc(H)∈Rsuch that the Hamilton-Jacobi equation

(2.9) H(x, Du(x)) =c, x∈Rⁿ,

admits a viscosity solutionu:Rⁿ→Rforc=c(H)and does not admit any such solution forc < c(H). Moreover,uis globally Lipschitz continuous and semiconcave onRⁿ.

The proof of Proposition 2.7 is given in Appendix D.

Let us consider the class ofadmissible arcs

At,x={ξ∈W^1,1([0, t];Rⁿ) :ξ(t) =x}

whereW^1,1([a, b];Rⁿ)denotes the space of all absolutely continuousRⁿ-valued functions on[a, b], where−∞< a < b <+∞. The functional

(2.10) J_t(ξ) :=

Z t

0

L(ξ(s),ξ(s))˙ ds+u₀(ξ(0)), ξ∈ A_t,x,

whereu0 ∈C(Rⁿ)is theinitial cost, is usually called theaction functional. A classical problem in the calculus of variations is

(CVt,x) to minimizeJtover all arcsξ∈ At,x. We define the associated value function

(2.11) u(t, x) = min

ξ∈At,x

J_t(ξ).

It is known thatu(t, x)is a viscosity solution of the Cauchy problem (2.12)

(ut(t, x) +H(x,∇xu(t, x)) = 0, t >0, x∈Rⁿ u(0, x) =u0(x), x∈Rⁿ,

where∇xudenotes the spatial gradient ofu. From the uniqueness of viscosity solutions of (2.12) if follows that, if the initial datum u0 is a viscosity solution of (2.6), then the solutionuof (2.12) is constant in time and coincides withu0. In this case, because of the translation invariance of problem (CVt,x), we have that, for allt>0,

(2.13) u₀(x) = min

ξ∈W^1,1([−t,0];Rⁿ)

nZ 0

−t

L(ξ(s),ξ(s))˙ ds+u₀(ξ(−t)) : ξ(0) =xo . Moreover, supposeLsatisfies conditions (L1)-(L3) and letH be the associated Hamilton- ian. Then we have the following result (see [16] or [34]).

Proposition 2.8. Letu: Rⁿ →Rbe a viscosity solution of (2.6)and letx∈Rⁿ. Then p∈D^∗u(x)if and only if there exists a uniqueC²curveγ: (−∞,0]→Rⁿwithγ(0) =x which is a minimizer of the problem in(2.13)for everyt>0andp=Lv(x,γ(0)).˙ 2.4. Generalized characteristics. The study of the structure of the singular set of a viscosity solution is a very important and hard one in many fields such as Riemannian ge- ometry, optimal control, classical mechanics, etc. The dynamics of singularities can be described by using generalized characteristics.

Definition 2.9. A Lipschitz arcx : [0, T] → Rⁿ,(T > 0),is said to be a generalized characteristicof the Hamilton-Jacobi equation (2.6) ifxsatisfies the differential inclusion (2.14) x(s)˙ ∈coH_p x(s), D⁺u(x(s))

, a.e.s∈[0, T].

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A basic criterion for the propagation of singularities along generalized characteristics was given in [4] (see [18, 37] for an improved version and simplified proof of this result).

Proposition 2.10 ([4]). Letu be a viscosity solution of (2.6) and letx0 ∈ Rⁿ. Then there exists a generalized characteristicx : [0, T] → Rⁿ with initial pointx(0) = x0. Moreover, if x0 ∈ Sing(u), thenτ ∈ (0, T)exists such that x(s) ∈ Sing(u) for all s∈[0, τ]. Furthermore, if

(2.15) 06∈coH_p(x₀, D⁺u(x₀)), thenx(s)6=x0for everys∈[0, τ].

Condition (2.15) is the key point to guarantee the propagation of singularities along generalized characteristics. For the cell problem withLin the form

L(x, v) = 1

2hA(x)v, vi −V(x) +E,

a local propagation result can be obtained replacing assumption (2.15) by the energy con- ditionE >maxx∈RⁿV(x)(see [14]).

3. GENERALIZED CHARACTERISTICS ANDLAX-OLEINIK OPERATORS

For anyt >0, givenx, y∈Rⁿ, we set

Γ^t_x,y ={ξ∈W^1,1([0, t];Rⁿ) :ξ(0) =x, ξ(t) =y}

and define

(3.1) At(x, y) = min

ξ∈Γ^t_x,y

Z t

0

L(ξ(s),ξ(s))ds˙ (x, y∈Rⁿ).

The existence of the above minimum is a well-known result in Tonelli’s theory (see, for instance, [25]). Anyξ∈Γ^t_x,yat which the minimum in (3.1) will be called aminimizerfor At(x, y)and such a minimizerξis of classC²by classical results. In the PDE literature, At(x, y)is also called thefundamental solutionof (2.6), see, for instance, [33].

LetLbe a Tonelli Lagrangian satisfying (L1)-(L3) and letH be the associated Hamil- tonian. In this section we study the singularities of a Lipschitz continuous semiconcave solutionuof the Hamilton-Jacobi equation

(3.2) H(x, Du(x)) = 0, x∈Rⁿ.

The existence of such a solution is guaranteed by Proposition 2.7.

3.1. Lax-Oleinik operators. For any Lipschitz continuous functionu:Rⁿ→Rwe set

(3.3) Lip(u) = sup

y6=x

|u(y)−u(x)|

|y−x| . For all(t, x)∈R+×Rⁿlet

(3.4) φ^x_t(y) =u(y)−A_t(x, y) and ψ^x_t(y) =u(y) +A_t(y, x) (y∈Rⁿ) whereAtis the fundamental solution of (3.2). The Lax-Oleinik operatorsT_t⁻andT_t⁺are defined as follows

T_t⁺u(x) = sup

y∈Rⁿ

φ^x_t(y), x∈Rⁿ, (3.5)

T_t⁻u(x) = inf

y∈Rⁿ

ψ_t^x(y), x∈Rⁿ. (3.6)

The functionsφ^x_t andψ^x_t are also calledlocal barrier functions.

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Lemma 3.1. SupposeLis a Tonelli Lagrangian and letube a Lipschitz function onRⁿ. Then the supremum in(3.5)is attained for every(t, x)∈R+×Rⁿ. Moreover, there exists a constantλ0 >0, depending only onLip(u), such that, for any(t, x) ∈R+×Rⁿ and any maximum pointy_t,xofφ^x_t, we have

(3.7) |yt,x−x|6λ0t .

Proof. Letku= Lip(u) + 1. Then, for any(t, x)∈R+×Rⁿandy∈Rⁿ, (2.4) yields At(x, y)> inf

ξ∈Γ^t_x,y

Z t

0

θ1(|ξ(s)|)˙ ds−c0t> inf

ξ∈Γ^t_x,yku

Z t

0

|ξ(s)|˙ ds− θ^∗₁(ku) +c0

t

>k_u|y−x| − θ^∗₁(k_u) +c₀ t,

wherec0is the constant in assumption (L2). Therefore

φ^x_t(y)−φ^x_t(x) =u(y)−u(x)−At(x, y) +At(x, x) 6Lip(u)|y−x| −ku|y−x|+t θ₁^∗(ku) +c0

+tL(x,0) 6− |y−x|+t θ₁^∗(k_u) +c₀+θ₂(0)

whereθ2is given by condition (L2). Now, takingλ0=θ₁^∗(ku) +c0+θ2(0)it follows that (3.8) Λ^x_t :={y:φ^x_t(y)>φ^x_t(x)} ⊂B(x, λ₀t).

ThereforeΛ^x_tis compact and the supremum in (3.5) is indeed a maximum. Moreover, (3.7)

is a consequence of (3.8).

A similar result holds for the inf-convolution defined in (3.6). A more detailed study of the properties of inf/sup-convolutions can be found in [8] with respect to the quadratic HamiltonianH(p) = |p|²/2and, consequently, the kernelA_t(x, y) = _2t¹|x−y|². This type of regularization, also called Moreau-Yosida regularization in convex analysis, was developed into a well-known procedure by Lasry and Lions [29]. In our context, we recover more information from the dynamical systems point of view by replacing quadratic kernels with the fundamental solutions.

3.2. Propagation of singularities. In this section, we will discuss the connection between sup-convolutions, singularities, and generalized characteristics. We begin our analysis with the local propagation of singularities of viscosity solutions along generalized characteristics. For Tonelli systems under rather general conditions, a local propagation result was obtained in [3] by a different method, without relating singular arcs to generalized characteristics. In the following lemma, we construct a singular arc starting from any singular point of the solution. A crucial point of this result is the fact that the interval[0, t0]on which the singular arc is defined turns out to be independent of the starting pointx.

Lemma 3.2. Let Lbe a Tonelli Lagrangian and let H be the associated Hamiltonian.

Supposeu: Rⁿ → Ris a Lipschitz continuous semiconcave viscosity solution of (3.2).

Then there existst0 in(0,1]such that, for all (t, x) ∈ (0, t0]×Rⁿ, there is a unique maximum pointyt,xofφ^x_t and the curve

(3.9) y(t) :=

(x if t= 0 y_t,x if t∈(0, t₀] satisfieslim_t→0y(t) =x.

Moreover, ifx∈Sing(u), theny(t)∈Sing(u)for allt∈(0, t0].

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Proof. LetC1 > 0be a semiconcavity constant for uonRⁿ and letλ0 be the positive constant in Lemma 3.1. By Proposition B.8 with λ = 1 +λ0, we deduce that there existst0 ∈ (0,1]and a constantC2 >0such that for every(t, x)∈(0, t0]×Rⁿ, every y∈B(x, λt), and everyz∈B(0, λt)we have that

A_t(x, y+z) +A_t(x, y−z)−2A_t(x, y)>C₂ t |z|².

Thus,φ^x_t(y) =u(y)−At(x, y)is strictly concave onB(x, λt)for allt∈(0, t0]provided that we further restrictt0in order to have

(3.10) t0<C2

C₁.

Then, for all such numberst, there exists a unique maximum pointyt,xofφ^x_t inB(x, λt).

In fact,y_t,xis an interior point ofB(x, λt)since, by Lemma 3.1, we have that|yt−x|6 λ₀t.

We now prove thaty_t,xis a singular point ofufor everyt∈(0, t₀]. Letξ_t,x∈Γ^t_x,y

t,x

be the unique minimizer forAt(x, yt,x)and let

pt,x(s) :=Lv(ξt,x(s),ξ˙t,x(s)), s∈[0, t0], be the associated dual arc. We claim that

(3.11) p_t,x(t)∈D⁺u(y_t,x)\D^∗u(y_t,x),

which in turn yieldsyt,x ∈ Sing(u). Indeed, ifpt,x(t) ∈ D^∗u(yt,x), then by Proposi- tion 2.8 there would exist aC²curveγt,x: (−∞, t]→Rⁿsolving the minimum problem

min

γ∈W^1,1([τ,t];Rⁿ)

nZ t

τ

L(γ(s),γ(s))˙ ds+u(γ(τ)) : γ(t) =yt,x

o

for allτ 6t. It is easily checked thatγ_t,xandξ_t,xcoincide on[0, t]since both of them are extremal curves forLand satisfy the same endpoint condition aty_t,x, i.e.,

L_v(ξ_t,x(t),ξ˙_t,x(t)) =p_t,x(t) =L_v(γ_t,x(t),γ˙_t,x(t)).

This leads to a contradiction sincex∈Sing(u)whileushould be smooth atγ_t,x(0). Thus,

(3.11) holds true andy_t∈Sing(u).

Next, we proceed to show that the singular arc in Lemma 3.2 is a generalized characteristic.

Lemma 3.3. Lett₀andybe given by Lemma 3.2 for a givenx∈Rⁿ. For anyt∈(0, t₀] letξ_t,x∈Γ^t_x,y(t)be a minimizer forA_t(x,y(t)). Then

(3.12) {ξ˙_t,x(·)}_t∈(0,t₀]is an equi-Lipschitz family.

Proof. Sincey(t)∈B(x, λt)by Lemma 3.1, we have that(ξ_t,x(s), p_t,x(s))∈K^∗_x,λ

0 for alls∈[0, t], where the compact setK^∗_x,λ

0is defined in (B.2). Therefore, being solutions of the Hamiltonian system

(ξ˙_t,x(s) =H_p(ξ_t,x(s), p_t,x(s))

˙

pt,x(s) =−Hx(ξt,x(s), pt,x(s)) s∈[0, t],

both{ξ˙t,x(·)}_t∈(0,t₀_]and{p˙t,x(·)}_t∈(0,t₀_]are uniformly bounded. Consequently, ξ¨t,x(s) =Hpx(ξt,x(s), pt,x(s)) ˙ξt,x(s) +Hpp(ξt,x(s), pt,x(s)) ˙pt,x(s) (s∈[0, t])

is also bounded, uniformly fort∈(0, t₀].

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Proposition 3.4. LetLbe a Tonelli Lagrangian. Lett0 ∈(0,1]be given by Lemma 3.2.

For any fixedx∈Rⁿ, lety: [0, t0]→Rⁿbe the curve constructed in Lemma 3.2. Then (a) yis Lipschitz on[0, t₀].

Moreover, for anyt ∈(0, t0], letξt,x ∈Γ^t_x,y(t)be a minimizer forAt(x,y(t)). Then the following properties hold true:

(b) The right derivativey˙⁺(0)exists and

(3.13) y˙⁺(0) = lim

t→0⁺

ξ˙_t,x(t) =H_p(x, p_x) wherepxis the unique element ofD⁺u(x)such that

H(x, p)>H(x, p_x), ∀p∈D⁺u(x).

(c) The arcp(t) :=Lv(ξt,x(t),ξ˙t,x(t))is continuos on(0, t0]andlim_t→0+p(t) =px. (d) There exist0< ρ6t0and constantsC1, C2>0such that

(3.14) H(y(t),p(t))6H(x, p_x) +C₁t−C₂|p(t)−p_x|², ∀t∈(0, ρ].

Proof. Having fixed x ∈ Rⁿ, we shall abbreviateξt,x = ξt. Let0 < t, s 6 t0 and let ξt ∈ Γ^t_x,y(t),ξs ∈ Γ^s_x,y(s) andη ∈ Γ^t_x,y(s) be minimizers forAt(x,y(t)),As(x,y(s)), andAt(x,y(s))respectively. Settingpt = Lv(ξt(t),ξ˙t(t)),ps = Lv(ξs(s),ξ˙s(s)), and p=L_v(η(t),η(t))˙ we have that

C₂

t |y(t)−y(s)|²

6hpt−p,y(t)−y(s)i=hpt−ps,y(t)−y(s)i+hps−p,y(t)−y(s)i 6C1|y(t)−y(s)|²+hps−p,y(t)−y(s)i,

where we have used the notation of the proof of Lemma 3.2. By Proposition B.9, the function(t, y)7→A_t(x, y)is locallyC^1,1in the set{(t, y)∈R×Rⁿ : 0 < t < t₀,|y− x|< λt}. Moreover, Proposition B.3 together with Proposition B.8 ensures that

|ps−p|6 C3

t |s−t|

for some constantC₃>0. Therefore C₂

t −C₁

|y(t)−y(s)|²6 C₃

t |s−t||y(t)−y(s)|.

Recalling (3.10) we have thatC2/t−C1>0for all0< t6t0. Thus

|y(t)−y(s)|6 C₃ C2−C1t0

|t−s|, and this proves (a).

Now we turn to the proof of (b). Since{ξ˙t(·)}_t∈(0,t₀_]are equi-Lipschitz by Lemma 3.3, for any sequencetk →0⁺such thatvk := (ξt_k(tk)−x)/tkconverges, we obtain

ξ_t_k(t_k)−x tk

−ξ˙_t_k(t_k) 6 1

tk

Z t_k

0

|ξ˙_t_k(s)−ξ˙_t_k(t_k)|ds 6 C

tk

Z t_k

0

(tk−s)ds= C 2tk. (3.15)

This implies that

v₀:= lim

k→∞v_k= lim

k→∞

ξ˙_t_k(t_k).

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By the semiconcavity ofu, for anyp∈D⁺u(x), we have u(x)6u(y(tk)) +hLv(y(tk),ξ˙t_k(tk)), x−y(tk)i+C

2|x−y(tk)|²

6u(x) +hp,y(t_k)−xi+hL_v(y(t_k),ξ˙_t_k(t_k)), x−y(t_k)i+C|x−y(t_k)|². Then, recalling thatξt_k(tk) =y(tk)we have

(3.16) hp−Lv(y(tk),ξ˙t_k(tk)), vki+tkC|vk|²>0, ∀p∈D⁺u(x).

Taking the limit in (3.16) ask→ ∞we obtain

(3.17) hp, v0i>hLv(x, v0), v0i=hpx, v0i, ∀p∈D⁺u(x),

wherep_x:=L_v(x, v₀)∈D⁺u(x)by the upper semicontinuity ofx D⁺u(x). So, (3.18) H(x, p)>hL_v(x, v₀), v₀i −L(x, v₀) =H(x, p_x), ∀p∈D⁺u(x), andpxis the unique minimum point ofH(x,·)onD⁺u(x). The uniqueness ofpximplies the uniqueness ofv0sinceLv(x,·)is injective. This leads to the assertion that

v₀= lim

t→0⁺

ξ_t(t)−x

t = lim

t→0⁺

ξ˙_t(t)

and, together with (3.18), implies (3.13). This completes the proof of (b).

The conclusion (c) is a straight consequence of (a), (b) and the locallyC^1,1regularity property of the function(t, y)7→At(x, y).

Finally, we turn to prove (d). First, using Tailor’s expansion, we have that H(x, p_x)−H(y(s),p(s))

=Hx(y(s),p(s))(x−y(s)) +Hp(y(s),p(s))(px−p(s)) +1

2h(Hxp(y(s),p(s)) +Hpx(y(s),p(s)))(px−p(s)),(x−y(s))i +1

2hHxx(y(s),p(s))(x−y(s)),(x−y(s))i +1

2hHpp(y(s),p(s))(px−p(s)),(px−p(s))i+o(|y(s)−x|²+|p(s)−px|²).

Thus, by (a), (b), (c) and our assumptions onH, there existρ >0such that, fors∈(0, ρ], we have

H(x, px)−H(y(s),p(s))

>−C1s+hξ˙s(s), px−p(s)i −Cεs²−ε|p(s)−px|²+C2|p(s)−px|² Takingε >0small enough, we have

H(x, p_x)−H(y(s),p(s))>−C3s+hξ˙_s(s), p_x−p(s)i+C₄|p(s)−p_x|². In view of (3.15), we have

H(x, px)−H(y(s),p(s))>−C5s+

y(s)−x

s , px−p(s)

+C4|p(s)−px|². Therefore, by the semiconcavity ofu, we obtain

H(x, p_x)−H(y(s),p(s))>−C6s+C₄|p(s)−p_x|²,

which completes the proof of (d).

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Remark3.5. Observe that (3.17), that is,

hp−px, v0i>0, ∀p∈D⁺u(x),

is exactly the key condition for propagation of singularities in [4] and [18].

Theorem 3.6. LetLbe a Tonelli Lagrangian and letH be the associated Hamiltonian.

Supposeu : Rⁿ → Ris a Lipschitz continuous semiconcave viscosity solution of (3.2) and x ∈ Sing(u). Then the singular arc y : [0, t0] → Rⁿ defined in Lemma 3.2 is a generalized characteristic and satisfies

(3.19) y(τ)˙ ∈coHp(y(τ), D⁺u(y(τ))), a.e.τ ∈[0, t0].

Moreover,

(3.20) y˙⁺(0) =H_p(x, p₀),

wherep₀is the unique element of minimal energy:

H(x, p)>H(x, p0), ∀p∈D⁺u(x).

Proof. The conclusion can be derived directly from Lemma 3.2 and Proposition 3.4 except

for (3.19). For the proof of (3.19), see Appendix C.

To study the genuine propagation of singularities along generalized characteristics, we have to check that the singular arcy(t)in Lemma 3.2 does not keep constant locally. As we show below, the following condition can be useful for this purpose:

(3.21) D_yA_t(x, x)6∈D⁺u(x), for allt∈(0, t₀].

Proposition 3.7. Lety: [0, t₀]→ Rⁿ be the singular generalized characteristic in The- orem 3.6, and lett ∈ (0, t0]. Then y(t) = xif and only if DyAt(x, x) ∈ D⁺u(x).

Consequently, if (3.21)holds, theny(t)6=xfor everyt∈(0, t0].

Proof. Letp∈D⁺u(x),p⁰ =DyAt(x, x), and lett∈(0, t0]. Recalling thaty(t)is the unique maximizer ofφ^x_t we have

06φ^x_t(y(t))−φ^x_t(x) =[u(y(t))−u(x)− hp,y(t)−xi −C1

2 |y(t)−x|²]

−[A_t(x,y(t))−A_t(x, x)− hp⁰,y(t)−xi −C₂

2t|y(t)−x|²] +hp−p⁰,y(t)−xi+1

2

C1−C2

t

|y(t)−x|²,

6hp−p⁰,y(t)−xi+1 2

C₁−C₂ t

|y(t)−x|²,

where—like in the proof of Lemma 3.2—C1>0is a semiconcavity constant foruonRⁿ andC2>0a convexity constant forAt(x,·)onB(x,(1 +λ0)t). So,

06|y(t)−x|6 2|p−p⁰| C2/t−C1

.

IfDyAt(x, x) ∈ D⁺u(x), then takingp = p⁰ in the above inequality yieldsy(t) = x.

Conversely, ify(t) =x, then the nonsmooth Fermat rule yields0∈D⁺u(x)−DyAt(x, x)

which completes the proof.

Another condition that ensures the genuine propagation of singularities is related to the notion of critical point.

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Definition 3.8. We say thatx∈Rⁿis acritical pointof a viscosity solutionuof (3.2) if 0∈coHp(x, D⁺u(x)),and astrong critical pointofuif0∈Hp(x, D⁺u(x)).

Remark3.9. For a mechanical Lagrangian of the form

(3.22) L(x, v) = 1

2hA(x)v, vi −V(x),

withhA(x)·,·ithe matrix associated with a Riemannian metric in Rⁿ andV a smooth potential,xis a critical point of a semiconcave solutionuof the corresponding Hamilton- Jacobi equation

1

2hA(x)⁻¹Du, Dui+V(x) = 0

if and only if0∈D⁺u(x), i.e.,xis a critical point ofuin the sense of nonsmooth analysis.

It is already known the condition thatxis not a critical point is a key point to guarantee the genuine propagation of singularities along generalized characteristics (see, for instance, [4]).

Corollary 3.10. Lety : [0, t0] → Rⁿbe the singular generalized characteristic in The- orem 3.6. If xis not a strong critical point ofuthen there exists t ∈ (0, t0]such that y(s)6=xfor alls∈(0, t].

Proof. It suffices to show that0 ∈ Hp(x, D⁺u(x))whenever a sequencetk → 0exists such thaty(tk) =xfor allk∈N. Indeed, denoting byξk ∈Γ^t_x,y(t^k

k)the unique minimizer ofA_t_k(x,y(t_k)), as in the proof of Proposition 3.4 we have that

k→∞lim

ξ˙_k(t_k) = lim

k→∞

ξk(tk)−x t_k = 0

becauseξ_k(t_k) =y(t_k). Therefore the dual arcp_k(s) =L_v(ξ_k(s),ξ˙_k(s))satisfies

k→∞lim Hp(ξk(tk), pk(tk)) = lim

k→∞

ξ˙k(tk) = 0.

Now, sincepk(tk) =DyAt_k(x,y(tk))∈D⁺u(y(tk))by the nonsmooth Fermat rule, the upper semicontinuity ofz Hp(x, D⁺u(z))yields0∈Hp(x, D⁺u(x)).

The above results on the propagation of singularities along generalized characteristics leads to the following global propagation property.

Theorem 3.11. LetLbe a Tonelli Lagrangian and letH be the associated Hamiltonian.

Supposeu:Rⁿ →Ris a Lipschitz continuous semiconcave viscosity solution of (3.2). If x∈Sing(u), then there exists a generalized characteristicx: [0,+∞)→Rⁿsuch that x(0) =xandx(s)∈Sing(u)for alls∈[0,+∞).

For geodesic systems, global propagation results were obtained in [1], [2], and [5] even on Riemannian manifolds. Theorem 3.11 above applies to mechanical systems with a LagrangianLof the form (3.22).

Corollary 3.12. LetL be the Tonelli Lagrangian in(3.22)and let H be the associated Hamiltonian. Suppose thatAandV are bounded together with and all their derivatives up to the second order and letube a viscosity solution of the Hamilton-Jacobi equation(3.2).

Ifx∈Sing(u), then there exists a unique generalized characteristicx: [0,+∞)→Rⁿ such thatx(0) =xandx(s)∈Sing(u)for alls∈[0,+∞).

Proof. Observing that all the conditions (L1)-(L3) are satisfied, the main part of the conclusion is an immediate consequence of Theorem 3.11. The uniqueness of the generalized characteristic is a well-known consequence of the semiconcavity ofu(see, e.g., [16]).

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APPENDIXA. UNIFORMLIPSCHITZ BOUND FOR MINIMIZERS

In this appendix we adapt to the present context a Lipschitz estimate for minimizers of the action functional that was obtained in [24] (see also [6]). We give a detailed proof of this result for the readers’ convenience. We assume that the LagrangianL:Rⁿ×Rⁿ→R is a function of classC²that satisfies the following conditions:

(L1’) Convexity:Lvv(x, v)>0for all(x, v)∈Rⁿ×Rⁿ.

(L2’) Growth condition: There exists a superlinear functionθ: [0,+∞)→[0,+∞)and a constantc0>0such that

L(x, v)>θ(|v|)−c0 ∀(x, v)∈Rⁿ×Rⁿ.

(L3’) Uniform bound: There exists a nondecreasing functionK : [0,+∞) → [0,+∞) such that

L(x, v)6K(|v|) ∀(x, v)∈Rⁿ×Rⁿ. Observe that (L1’)-(L3’) are weaker than assumptions (L1)-(L3).

We define theenergy function

E(x, v) =hv, Lv(x, v)i −L(x, v), (x, v)∈Rⁿ×Rⁿ.

Proposition A.1. Lett, R >0and supposeLsatisfies condition(L1’)-(L3’). Given any x∈Rⁿandy∈B(x, R), letξ∈Γ^t_x,ybe a minimizer forA_t(x, y). Then we have that

sup

s∈[0,t]

|ξ(s)|˙ 6κ(R/t), (A.1)

whereκ: (0,∞)→(0,∞)is nondecreasing. Moreover, ift61, then

(A.2) sup

s∈[0,t]

|ξ(s)−x|6κ(R/t).

Proof. Fixt >0,R > 0,x∈Rⁿ, lety ∈ B(x, R), and letξ∈Γ^t_x,y be a minimizer for At(x, y), i.e.,

At(x, y) = Z t

0

L(ξ(s),ξ(s))˙ ds.

Denoting byσ∈Γ^t_x,ythe straight line segment defined byσ(s) =x+^s_t(y−x),s∈[0, t], in view of (L2’) and (L3’) we have that

Z t

0

θ(|ξ(s)|)˙ ds−c0t6 Z t

0

L(ξ(s),ξ(s))˙ ds6 Z t

0

L(σ(s),σ(s))˙ ds

= Z t

0

L x+s

t(y−x),y−x t

ds6tK(R/t).

Therefore

Z t

0

θ(|ξ(s)|)˙ ds6c₀t+tK(R/t) =tC₁(R/t)

withC₁(r) = K(r) +c₀. Since |ξ(s)|˙ 6 θ(|ξ(s)|) +˙ θ^∗(1), where θ^∗ is the convex conjugate ofθdefined in (2.3), we have that

Z t

0

|ξ(s)|˙ 6tC2(R/t), withC₂(r) =C₁(r) +θ^∗(1). Hence

(A.3) |ξ(s)−x|6

Z s

0

|ξ(s)|˙ ds6tC2(R/t), ∀s∈[0, t],

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and

(A.4) inf

s∈[0,t]|ξ(s)|˙ 61 t

Z t

0

|ξ(s)|˙ ds6C₂(R/t).

Now, definel_ξ(s, λ) =L(ξ(s),ξ(s)/λ)λ˙ for alls∈[0, t]andλ >0. Then we have d

dλlξ(s, λ)|λ=1=L(ξ(s),ξ(s))˙ − hξ(s), L˙ v(ξ(s),ξ(s))i˙ =−E(ξ(s),ξ(s)).˙ Since the energy is constant along a minimizer, there exists a constantcξ such that

d

dλl_ξ(s, λ)|_λ=1=c_ξ, ∀s∈[0, t].

Moreover, a simple computation shows thatl_ξ(s, λ)is convex inλ. So, we have c_ξ>sup

λ<1

l_ξ(s, λ)−l_ξ(s,1)

λ−1 , ∀s∈[0, t].

Let us now take, in the above inequality, λ = 3/4 ands0 ∈ [0, t]such that|ξ(s˙ 0)| = inf_s∈[0,t]|ξ(s)|. Then, by (L2’), (L3’), and (A.4) we conclude that˙

c_ξ>4(l_ξ(s₀,1)−l_ξ(s₀,3/4))>4(−c₀−l_ξ(s₀,3/4))

=−4c0−3L ξ(s0),4

3 ξ(s˙ 0)

>−4c0−3K4

3|ξ(s˙ 0)|

>−4c0−3K4

3C2(R/t)

=−C3(R/t), (A.5)

whereC3(r) = 4c0+ 3K(4C2(r)/3).

By the convexity oflξ(s,·)we also have, for anyε∈(0,1), c_ξ6 lξ(s,2−ε)−lξ(s,1)

1−ε .

In other words,

(1−ε)cξ+εlξ(s,1)6lξ(s,2−ε)−(1−ε)lξ(s,1).

Moreover, again by convexity, we have that lξ(s,2−ε) =lξ

s, ε·1

ε + (1−ε)·1 6εlξ

s,1

ε

+ (1−ε)lξ(s,1).

Therefore

(1−ε)c_ξ+εl_ξ(s,1)6εl_ξ s,1

ε , that is,

(1−ε)cξ+εL(ξ(s),ξ(s))˙ 6L(ξ(s), εξ(s)).˙ Hence, combining (A.5) and condition (L2’), we obtain

−(1−ε)C3(R/t) +ε(θ(|ξ(s)|)˙ −c0)6L(ξ(s), εξ(s)).˙ SetSξ ={s∈[0, t] :|ξ(s)|˙ >2}andε=ε(s) = 1/|ξ(s)|˙ fors∈Sξ. Then

−C3(R/t) + 1

|ξ(s)|˙ C3(R/t) +θ(|ξ(s)|)˙ −c0

|ξ(s)|˙ 6L ξ(s),

ξ(s)˙

|ξ(s)|˙

6K(1), ∀s∈Sξ. Thus

θ(|ξ(s)|)˙ 6(K(1) +C₃(R/t))|ξ(s)|˙ + (c₀−C₃(R/t)), ∀s∈S_ξ.

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Therefore, by the Young-Fenchel inequality we deduce that

|ξ(s)|˙ 6(c0−C3(R/t)) +θ^∗(K(1) +C3(R/t) + 1) :=C4(R/t), ∀s∈Sξ. Consequently,

(A.6) sup

s∈[0,t]

|ξ(s)|˙ 6max{2, C4(R/t)}:=C₅(R/t).

The conclusion follows from (A.6) and (A.3) takingκ(r) = max{C₅(r), C₂(r)}.

Corollary A.2. In Proposition A.1, assume the additional condition:

(L3”) There exists a nondecreasing functionK₁: [0,+∞)→[0,+∞)such that

|Lv(x, v)|6K1(|v|) ∀(x, v)∈Rⁿ×Rⁿ. Then the dual arcp(·)associated withξ(·)satisfies

sup

s∈[0,t]

|p(s)|6κ₁(R/t), (A.7)

whereκ₁: (0,∞)→(0,∞)is nondecreasing.

Proof. By (L3”) together with (A.1) and (A.7) follows from sup

s∈[0,t]

|p(s)|= sup

s∈[0,t]

|L_v(ξ(s),ξ(s))|˙ 6K₁(κ(R/t)) =κ₁(R/t),

whereκ1(r) =K1◦κ(r).

APPENDIXB. CONVEXITY ANDC^1,1ESTIMATE OF FUNDAMENTAL SOLUTIONS

LetLbe a Tonelli Lagrangian (which implies conditions (L1’)-(L3’) and (L3”) in Ap- pendix A). Then we have the following fundamental bounds for the velocity of minimizers.

Fixx∈Rⁿand supposeR >0andLis a Tonelli Lagrangian. For any0< t61and y∈B(x, R), letξ∈Γ^t_x,ybe a minimizer forAt(x, y)and letpbe its dual arc. Then there exists a nondecreasing functionκ: (0,∞)→(0,∞)such that

(B.1) sup

s∈[0,t]

|ξ(s)|˙ 6κ(R/t), sup

s∈[0,t]

|p(s)|6κ(R/t),

by Proposition A.1 and Corollary A.2. Now,x∈Rⁿandλ >0define compact sets Kx,λ :=B(x, κ(4λ))×B(0, κ(4λ))⊂Rⁿ×Rⁿ,

K^∗_x,λ :=B(x, κ(4λ))×B(0, κ(4λ))⊂Rⁿ×(Rⁿ)^∗. (B.2)

The following is one of the key technical points of this paper.

Proposition B.1. SupposeLis a Tonelli Lagrangian. Fixx∈Rⁿ,λ >0,t∈(0,1), and y∈B(x, λt). Letz∈Rⁿandh∈Rbe such that

(B.3) |z|< λt and − t

2 < h <1−t.

Then any minimizerξ∈Γ^t+h_x,y+zforAt+h(x, y+z)and corresponding dual arcpsatisfy the following inclusions

{(ξ(s),ξ(s)) :˙ s∈[0, t+h]} ⊂Kx,λ, {(ξ(s), p(s)) :s∈[0, t+h]} ⊂K^∗_x,λ.

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Proof. Sincet/2< t+h <1andy+z∈B(x,2λt)by (B.3), we can use (A.2) and (B.1) to obtain

sup

s∈[0,t+h]

|ξ(s)−x|6κ 2λt t+h

6κ(4λ) and

sup

s∈[0,t]

|ξ(s)|˙ 6κ 2λt t+h

6κ(4λ).

Since a similar bound holds true forsup_s∈[0,t]|p(s)|, the conclusion follows.

RemarkB.2. For anyx∈Rⁿandy∈B(x, λt), condition (B.3) is satisfied when

(B.4) |h|< t/2 and |z|< λt

provided that0< t <2/3.

B.1. Semiconcavity of the fundamental solution. The role of semiconcavity in optimal control problems has been widely investigated, see [16]. For the minimization problem in (3.1), the local semiconcavity ofAt(x, y)with respect toywas proved in [10]. In this paper, we give a local semiconcavity result of the map(t, y)7→At(x, y).

The following result is essentially known.

Proposition B.3(Semiconcavity of the fundamental solution). SupposeLis a Tonelli La- grangian. Then for anyλ >0there exists a constantC_λ >0such that for anyx∈Rⁿ, t ∈ (0,2/3),y ∈B(x, λt), and(h, z)∈ R×Rⁿsatisfying|h| < t/2and|z|< λtwe have

(B.5) At+h(x, y+z) +At−h(x, y−z)−2At(x, y)6Cλ

t |h|²+|z|² .

Consequently, (t, y) 7→ At(x, y)is locally semiconcave in (0,1)×Rⁿ, uniformly with respect tox.

Remark B.4. For the purposes of this paper, it suffices to assume0 < t < 2/3. In the general caset > 0, a local semiconcavity result holds true forAt(x, y)in the same form as (B.5) withC_λdepending ont.

B.2. Main Regularity Lemma. We begin with the following known properties.

Lemma B.5. SupposeLis a Tonelli Lagrangian onRⁿ. For anyx, y ∈ Rⁿ,t > 0, let ξ∈Γ^t_x,y be a minimizer forAt(x, y). Thenξ∈C²([0, t])is an extremal curve¹, and the dual arcp(s) :=Lv(ξ(s),ξ(s))˙ satisfies the sensitivity relation

(B.6) p(s)∈D_y⁺At(x, ξ(s)), s∈[0, t].

Moreover,At(x,·)is differentiable atyif and only if there is a unique minimizerξ∈Γ^t_x,y. In this case, we have

(B.7) DyAt(x, y) =Lv(ξ(t),ξ(t)).˙

Proof. The sensitivity relation (B.6) is obtained in, for instance, [16, Theorem 6.4.8], for a problem with initial cost. Here the proof is similar. The uniqueness of the minimizer and

regularity are classical results.

1An arcξ(s)is called an extremal curve if it satisfies the associated Euler-Lagrange equation.

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Lemma B.6(Main Regularity Lemma). SupposeLis a Tonelli Lagrangian. Then for any λ >0there existstλ∈(0,1]and constantsCλ, C_λ⁰, C_λ⁰⁰>0such that, for allt∈(0, tλ), x∈Rⁿ,y1, y2∈B(x, λt), and any minimizerξi(i= 1,2)forAt(x, yi), we have

kξ2−ξ1k²_L∞(0,t)6Cλ

t |y2−y1|² (B.8)

Z t

0

|p2−p1|²ds6C_λ⁰

t |y2−y1|² (B.9)

Z t

0

|ξ˙2−ξ˙1|²ds6C_λ⁰⁰

t |y2−y1|², (B.10)

wherepidenotes the dual arc ofξi.

Proof. SinceLis a Tonelli Lagrangian, we have thatξ_i(s) (i= 1,2)of classC²and, by Proposition B.1 withh= 0 =z, it follows that

sup

s∈[0,t]

|ξ˙_i(s)|6κ(4λ), sup

s∈[0,t]

|pi(s)|6κ(4λ),

wherepi(s) =Lv(ξi(s),ξ˙i(s)). Moreover, the pair(ξi(·), pi(·))satisfies the Hamiltonian system

(ξ˙i=Hp(ξi, pi)

˙

p_i=−H_x(ξ_i, p_i) on[0, t]

with

ξi(t) =yi, ξi(0) =x.

Furthermore, owing to Lemma B.5,

pi(s)∈D_y⁺At(x, ξi(s)), ∀s∈[0, t].

Therefore

1 2

d

ds|ξ2−ξ1|²=hHp(ξ2, p2)−Hp(ξ1, p1), ξ2−ξ1i.

Integrating over[s, t], we conclude that

|ξ2(t)−ξ1(t)|²− |ξ2(s)−ξ1(s)|²>−C1

Z t

s

|ξ2−ξ1|²+|p2−p1| · |ξ2−ξ1| dτ

>−C1

Z t

s

|p2−p1|²dτ −2C1

Z t

s

|ξ2−ξ1|²dτ, whereC₁=C₁(λ)>0is an upper bound forD²H(x, p)on{(x, p) : |p|6κ(4λ)}. So, (B.11) kξ2−ξ₁k²_L∞(0,t)6|y2−y₁|²+C₁

Z t

s

|p2−p₁|²dτ+ 2C₁tkξ2−ξ₁k²_L∞(0,t). Now,

d

dshp2−p1, ξ2−ξ1i

=hp2−p1, Hp(ξ2, p2)−Hp(ξ1, p1)i − hHx(ξ2, p2)−Hx(ξ1, p1), ξ2−ξ1i

=hp2−p1,Hbpx(ξ2−ξ1) +Hbpp(p2−p1)i − hHbxp(p2−p0) +Hbxx(ξ2−ξ1), ξ2−ξ1i, where

Hbpx(s) = Z 1

0

Hpx λξ2(s) + (1−λ)ξ1(s), λp2(s) + (1−λ)p1(s) dλ,