Local singular characteristics on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document}R2

The singular set of a viscosity solution to a Hamilton–Jacobi equation is known to propagate, from any noncritical singular point, along singular characteristics which are curves satisfying certain differential inclusions. In the literature, different notions of singular characteristics were introduced. However, a general uniqueness criterion for singular characteristics, not restricted to mechanical systems or problems in one space dimension, is missing at the moment. In this paper, we prove that, for a Tonelli Hamiltonian on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document}R2, two different notions of singular characteristics coincide up to a bi-Lipschitz reparameterization. As a significant consequence, we obtain a uniqueness result for the class of singular characteristics that was introduced by Khanin and Sobolevski in the paper [On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations. Arch. Ration. Mech. Anal., 219(2):861–885, 2016].


Introduction
This paper is devoted to study the local propagation of singularities for viscosity solutions of the Hamilton-Jacobi equations H (x, Du(x)) = 0, x ∈ R n , (HJ s ) where H is a Tonelli Hamiltonian in (HJ s ) and H is of class C 1 and strictly convex in the p-variable in (HJ loc ). In (HJ s ), we assume that 0 on the right-hand side is Mañé's critical value. The existence of global weak KAM solutions of (HJ s ) was obtained in [12]. In (HJ loc ), we suppose ⊂ R n is a bounded domain. Semiconcave functions are nonsmooth functions that play an important role in the study of (HJ s ) and (HJ loc ). For semiconcave viscosity solutions of Hamilton-Jacobi equations, Albano and the first author proved in [1] that singular arcs can be selected as generalized characteristics. Recall that a Lipschitz arc x : [0, τ ] → R n is called a generalized characteristic starting from x for the pair (H , u) if it satisfies the following: a.e. s ∈ [0, τ ], x(0) = x, (1.1) where co stands for the convex hull. If x ∈ Sing (u)-the singular set of u-then [1,Theorem 5] gives a sufficient condition for the existence of a generalized characteristic propagating the singularity of u locally. The local structure of singular (generalized) characteristics was further investigated by the first author and Yu in [11], where singular characteristics were proved more regular near the starting point than the arcs constructed in [1]. Such additional properties will be crucial for the analysis we develop in this paper.
For any weak KAM solution u of (HJ s ), the class of intrinsic singular (generalized) characteristics was constructed in [4] by the authors of this paper, using the positive type Lax-Oleinik semi-group. Such a method allowed to construct global singular characteristics, which we now call intrinsic. Moreover, in [5,6] the "intrisic approach" turned out to be useful for pointing out topological properties of the cut locus of u, including homotopy equivalence to the complement of the Aubry set (see also [7] for applications to Dirichlet boundary value problems). In spite of its success in capturing singular dynamics, it could be argued that the relaxation procedure in the original definition of generalized characteristics-that is, the presence of the convex hull in (1.1)-might cause a loss of information coming from the Hamiltonian dynamics behind. On the other hand, such a relaxation is necessary to ensure convexity of admissible velocities for the differential inclusion in (1.1), since the map x ⇒ H p (x, D + u(x)) fails to be convex-valued, in general.
The most important example where the above relaxation is unnecessary is probably given by mechanical Hamiltonians of the form H (x, p) = 1 2 A(x) p, p + V (x), where A(x) is a symmetric positive definite n × n-matrix smoothly depending on x and V (x) is a smooth function on R n . In this case, (1.1) reduces to the generalized gradient system t > 0 a.e. x(0) = x, (1.2) the solution of which, unique for any initial datum, forms a Lipschitz semi-flow (see, e.g., [1][2][3]8,9]). Unfortunately, the argument that justifies such a uniqueness property cannot be adapted to general Hamiltonians (see [11,15]). Recent significant progress in the attempt to develop a more restrictive notion of singular characteristics is due to Khanin and Sobolevski [13]. In this paper, we will call such curves strict singular characteristic but in the literature they are also refereed to as broken characteristics, see [16,17]. We now proceed to recall their definition: given a semiconcave solution u of (HJ loc ), a Lipschitz singular curve x : [0, T ] → is called a strict singular characteristic from x ∈ Sing (u) if there exists a measurable selection p(t) ∈ D + u(x(t)) such that As already mentioned, the existence of strict singular characteristics (for a time dependent version of (HJ loc )) was proved in [13], where additional regularity properties of such curves were established, including the right-differentiability of x for every t, the right-continuity oḟ x, and the fact that p(·) In Appendix A, we give a proof of the existence and regularity of strict characteristics for solutions to (HJ loc ) for the reader's convenience.
In view of the above considerations, it is quite natural to raise the following questions: (Q1) What is the relation between a strict singular characteristic, x, and a singular characteristic, y, from the same initial point? (Q2) What kind of uniqueness result can be proved for singular characteristics? What about strict singular characteristics?
In this paper, we will answer the above questions in the two-dimensional case under the following additional conditions: (A) n = 2 and y is Lipschitz; (B) the singular initial point x 0 = y(0) of the singular characteristic y is not a critical point with respect the pair (H , u), i.e., 0 / ∈ H p (x 0 , D + u(x 0 )); (C) y is right differentiable at 0 andẏ The meaning of conditions (A) is clear. Condition (B) ensures the fact that singular characteristics are not constant. The right differentiability of singular characteristics at 0 and the essential right continuity ofẏ at 0 are crucial properties to our approach. On the one hand, together with condition (B) they ensure that a singular characteristic is a genuine arc near t = 0. On the other hand, (D) is essential to construct the change of variable on which our uniqueness result is based. Notice that any strict singular characteristic x and the singular characteristic y given in [11] (see also Proposition 2.12) satisfy conditions (A)-(D) provided that the initial point is not critical. The intrinsic singular characteristic z constructed in [4] (see also Proposition 2.13) satisfies just conditions (A)-(C), in general.
The main results of this paper can be described as follows.
• For any pair of singular curves x 1 and x 2 satisfying condition (A)-(D), there exists τ > 0 and a bi-Lipschitz homeomorphism φ : In other words, the singular characteristic staring from a non-critical point x is unique up to a bi-Lipschitz reparametrization (Theorem 3.6). • In particular, if x is a strict singular characteristic and y is a singular characteristic starting from the same noncritical initial point x, then there exists τ > 0 and a bi-Lipschitz homeomorphism φ : [0, τ ] → [0, φ(τ )] such that y(φ(t)) = x(t) for all t ∈ [0, τ ] (Corollary 3.8).
• We have the following uniqueness property for strict singular characteristics: let be strict singular characteristics from the same noncritical initial point x. Then there exists τ ∈ (0, T ] such that x 1 (t) = x 2 (t) for all t ∈ [0, τ ]. (Theorem 3.9) Finally, we remark that the results of this paper cannot be applied to intrinsic singular characteristics because of the mentioned lack of condition (D). Extra techniques will have to be developed to cover such an important class.
The paper is organized as follows. In Sect. 2, we introduce necessary material on Hamilton-Jacobi equations, semiconcavity, and singular characteristics. In Sect. 3, we answer question (Q1)-(Q2) in the two-dimensional case. In the appendix, we give a detailed proof of the existence result for strict singular characteristics.

Hamilton-Jacobi equation and semiconcavity
In this section, we review some basic facts on semiconcave functions and Hamilton-Jacobi equations.

Semiconcave function
Let ⊂ R n be a convex open set. We recall that a function u : → R is semiconcave (with linear modulus) if there exists a constant C > 0 such that for any x, y ∈ and λ ∈ [0, 1]. Let u : ⊂ R n → R be a continuous function. For any x ∈ , the closed convex sets are called the subdifferential and superdifferential of u at x, respectively. The following characterization of semiconcavity (with linear modulus) for a continuous function comes from proximal analysis. Proposition 2.1 Let u : → R be a continuous function. If there exists a constant C > 0 such that, for any x ∈ , there exists p ∈ R n such that

2)
then u is semiconcave with constant C and p ∈ D + u(x). Conversely, if u is semiconcave in with constant C, then (2.2) holds for any x ∈ and p ∈ D + u(x).
Let u : → R be locally Lipschitz. We recall that a vector p ∈ R n is called a reachable (or limiting) gradient of u at x if there exists a sequence {x n } ⊂ \{x} such that u is differentiable at x k for each k ∈ N, and lim k→∞ x k = x and lim k→∞ Du(x k ) = p.
The set of all reachable gradients of u at x is denoted by D * u(x).
The following proposition concerns fundamental properties of semiconcave funtions and their gradients (see [10] for the proof).

Proposition 2.2
Let u : ⊂ R n → R be a semiconcave function and let x ∈ . Then the following properties hold.
is not a singleton. The set of all singular points of u is denoted by Sing (u).
Let us recall a result on the rectifiability of the singular set Sing (u) of a semiconcave function u in dimension two.

Aspects of weak KAM theory
For any x, y ∈ R n and t > 0, we denote by t x,y the set of all absolutely continuous curves ξ defined on [0, t] such that ξ(0) = x and ξ(t) = y. Define We call A t (x, y) the fundamental solution for the Hamilton-Jacobi equation By classical results (Tonelli's theory), the infimum in (2.3) is a minimum. Each curve ξ ∈ t x,y attaining such a minimum is called a minimal curve for A t (x, y).

Definition 2.6
For each u : R n → R, let and T t u andT t u be the Lax-Oleinik evolution of negative and positive type defined, respectively, by The following result is well-known.
Proposition 2.7 [12] There exists a Lipschitz semiconcave viscosity solution of (HJ s ). Moreover, such a solution u is a common fixed point of the semigroup {T t }, i.e., T t u = u for all t ≥ 0.
Clearly, (HJ s ) has no unique solution and we call each solution, given as a fixed point of the semigroup {T t }, a weak KAM solution of (HJ s ).
for all absolutely continuous curves ξ : We say such an absolutely continuous curve ξ is a (u, L)-calibrated curve, or a u-calibrated curve for short, if the equality holds in the inequality above. A curve ξ : In this case, we also say that ξ is a backward calibrated curve (with respect to u).
The following result explains the relation between the set of all reachable gradients and the set of all backward calibrated curves from x (see, e.g., [10] or [14] for the proof).

Proposition 2.9
Let u : R n → R be a weak KAM solution of (HJ s ) and let x ∈ R n . Then which is a backward calibrated curve with respect to u.

Propagation of singularities
In this paper, we will discuss various types of singular arcs describing the propagation of singularities for Lipschitz semiconcave solutions of the Hamilton-Jacobi equations (HJ loc ) and (HJ s ).
Let u be a Lipschitz semiconcave viscosity solution of (HJ loc ) and x ∈ Sing (u).
such that: The following existence of singular characteristic is due to [1,11].

Proposition 2.12
Let u be a Lipschitz semiconcave solution of (HJ loc ) and x ∈ Sing (u).
Then, there exists a singular characteristic y : [0, T ] → with y(0) = x. Now, suppose u is a Lipschitz semiconcave weak KAM solution of (HJ s ). In [4], another singular curve for u is constructed as follows. First, it is shown that there exists λ 0 > 0 such that for any (t, x) ∈ R + × R n and any maximizer y for the function u(·) − A t (x, ·), we have that |y − x| ≤ λ 0 t. Then, taking λ = λ 0 + 1, one shows that there exists We now define the curve (2.5) Proposition 2.13 [4] Let the curve z be defined in (2.5). Then, the following holds: ż + (0) exists andż Definition 2.14 The Lipschitz arc z defined in (2.5) is called the intrinsic characteristic from x ∈ Sing (u).

Singular characteristic on R 2
We now return to questions (Q1) and (Q2) from the Introduction. So far, we have introduced three kinds of singular arcs issuing from a point x 0 ∈ Sing (u), namely • strict singular characteristics, that is, solutions to (1.3), • singular characteristics, introduced in Definition 2.11, and • the intrinsic singular characteristic z given by Proposition 2.13.
In this section, we will compare the first two notions of characteristics when ⊂ R 2 . We begin by introducing the following class of Lipschitz arcs. For any x ∈ Lip 0 (0, T ; ) we set Owing to (3.1), we have that ω x (t) → 0 as t ↓ 0.
Proof Observe that, for any 0 ≤ t 0 ≤ t 1 ≤ T , the identity Sinceẋ Let x ∈ R 2 and let θ ∈ R 2 be a unit vector. For any ρ ∈ (0, 1) let us consider the cone with vertex in x, amplitude ρ, and axis θ . Clearly, C ρ (x, θ) is given by the union of the two cones which intersect each other only at x. Define

Lemma 3.4 Let u be a semiconcave solution of (HJ loc )
and let x ∈ Lip u 0 (0, T ; ) be such thatẋ + (0) = 0. Then there exists T 0 ∈ (0, T ] such that the set Proof The structure of the superdifferential of u along x is described by Proposition 2.5 and Proposition 3.3.15 in [10]. Lemma 3.5 Let u be a semiconcave solution of (HJ loc ) and let x 0 ∈ Sing (u) be such that Then there exist constants r 1 > 0, s 1 ∈ (0, T 0 ], and δ ∈ (0, 1) and such that and, for all s ∈ [0, s 1 ] ∩ S x and r ∈ [0, r 1 ], Proof The existence of backward calibrated curves satisfying (3.14) follows from Proposition 2.9. Moreover, for all r ≥ 0 we have that Now, observe that, since x 0 is not a critical point with respect to (u, H ), by possibly reducing T 0 we have that x(s) is also not a critical point for all s ∈ [0, T 0 ] due to the uppersemicontinuity of the set-valued map s ⇒ H p (x(s), D + u(x(s))). So, for some r 0 > 0, s 0 ∈ (0, T 1 ], and δ 0 ∈ (0, 1), we deduce that for all s ∈ [0, s 0 ] ∩ S x and r ∈ [0, r 0 ]. This proves (3.15). Next, recall that H (x 0 , p i 0 ) = 0 because p i 0 ∈ D * u(x 0 ) (i = 1, 2). So, by the strict convexity of H (x 0 , ·), we deduce that there exists ν > 0 such that Hence, the upper-semicontinuity of the set-valued map s ⇒ H p (x(s), D + u(x(s))) ensures the existence of numbers δ 1 ∈ (0, 1) and s 1 ∈ (0, s 0 ] such that 19) Therefore, combining (3.17) and (3.19), we conclude that, after possibly replacing r 0 by a smaller nummber r 1 > 0, 1 2 for all s ∈ [0, s 1 ] ∩ S x and r ∈ [0, r 1 ]. By (3.18) and the above inequality we have that ξ 1 s (−r ) ∈ C + δ (x(s), θ 2 (s)) with δ = δ 0 δ 1 /2. The analogous statement for ξ 2 s in (3.16) can be proved by a similar argument. We are now ready to state our main result, which ensures that singular curves coincide up to a bi-Lipschitz reparameterization, at least when x is not a critical point.
To complete the proof we observe that x 2 (t s ) = x 1 (s) for all s ∈ [0, σ ], not just on a set of full measure. This fact can be easily justified by an approximation argument.
We are now in a position to prove our main result.
Being continuous, φ is a homeomorphism. It remains to prove that φ is bi-Lipschitz. The continuity of φ at 0 ensures that, after possibly reducing σ , for all s 0 , s 1 ∈ [0, σ ]. Thus, by (3.21) we have that The fact that φ −1 is also Lipschitz follows by a similar argument. Indeed, writing (3.22) for t i = φ(s i ) and appealing to Lemma 3.2 and (3.23) once again we obtain The proof is completed noting that φ is unique due to the injectivity of x 1 and x 2 . For strict singular characteristics, uniqueness holds without reparameterization as we show next. Theorem 3.9 Let u be a semiconcave solution of (HJ loc ) and let x 0 ∈ Sing (u) be such that Proof By Theorem 3.6 there exists a bi-Lipschitz homeomorphism φ : [0, (3.24) Moreover, since x 1 and x 2 are strict characteristics we have that Therefore, where, in addition to (3.24), we have that p 2 (φ(t))) = arg min H (x 2 (φ(t)), p) = arg min Theorems 3.6 and 3.9 establish a connection between the absence of critical points and uniqueness of strict singular characteristics. In this direction, we also have the following global result. Corollary 3.10 Let u be a semiconcave solution of (HJ loc ) and let x 0 ∈ Sing (u). Let x j : Proof On account of Theorem 3.9 we have that is a nonempty set. Let τ 0 = sup T = max T . We claim that τ 0 = T . For if τ 0 < T , applying Theorem 3.9 with initial point x 1 (τ 0 ) we conclude that x 1 (t) = x 2 (t) on some intarval τ 0 ≤ t < τ 0 + δ, contradicting the definition of τ 0 .
Another well-known example where we have uniqueness of the generalized characteristic is the mechanical Hamiltonian with A(x) is positive definite symmetric n×n-matrix C 2 -smooth in x and V a smooth function on . More precisely, if x ∈ Sing (u), then there exists a unique Lipschitz arc y determined byẏ + (t) = A(y(t)) p(t), where y(0) = x and p(t) = arg min p∈D + u(y(t)) A(y(t)) p, p . In this case, uniqueness follows from semiconcavity by an application of Gronwall's lemma (see, e.g., [2,10]) ensuring that, in addition, any generalized characteristic is strict. We now give another justification of such a property from the point of view of this section.

Corollary 3.11
If H is a mechanical Hamiltonian as in (3.25), then the reparameterization φ in Theorem 3.6 is the identity.
Proof We observe that, for almost all t ≥ 0, where λ(t) ∈ [0, 1] and we can assume is also the set of extremal points of the convex set D + u(y(t)). Since x(t) = y(φ(t)), differentiating we obtain thaṫ Therefore, there exists a unique λ t ∈ [0, t] such that It follows that Thus, φ(t) ≡ t and this completes the proof.

Remark 3.12
Observe that our results apply in particular to solutions of (HJ s ).
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Lemma A.3
For everyt ∈ [0, T 0 ) and ε > 0 there exists and integer m ε ≥ 1 and a real number τ ε ∈ (0, T 0 −t) such that where B denotes the closed unit ball of R 2 , centered at the origin.
Proof We begin by showing that for everyt ∈ [0, T 0 ) and ε > 0 there exist m ε ≥ 1 and for all m ≥ m ε . We argue by contradiction: set (t) = H p x(t), Du + (x(t)) and suppose there existt ∈ [0, T 0 ), ε > 0, and sequences m k → ∞ and t k ↓t such that where we have used bound (b) above to justify (ii). We claim thatp ∈ D + u x(t) . Indeed, in view of (c) above we have that, for all k ≥ 1, Hence, in the limit as k → ∞, we get which in turn proves our claim. Thus, we conclude thaṫ in contrast with (i). So, (A.6) is proved.
By appealing to the upper semi-continuity of D + u and assumption (A.1) we conclude that there exists T ∈ (0, T 0 ] such that Now, fix anyt ∈ [0, T ) and letv ∈ R 2 be any vector such that for some sequence τ j 0 ( j → ∞). Observe thatv ∈ co H p x(t), Du + (x(t)) in view of Lemma A.3. So,v = 0 owing to (A.7). Setx = x(t) and definē Notice that Fv(x) is the exposed face of the convex set D + u(x) in the directionv (see, for instance, [10]). The following lemma identifiesp (hencev) uniquely.

Lemma A.4 Supposep ∈ Fv(x). Thenp is the unique element in D + u(x) such that
Proof Sincep ∈ Fv(x), we have that Therefore, by convexity we conclude that Since H is strictly convex in p,p is the unique element in D + u(x) satisfying (A.9).
Notice that the above lemma yields the existence of the right-derivativeẋ + (t) as soon as one shows thatp ∈ Fv(x) for anyv satisfying (A.8).
Next, to show thatp ∈ Fv(x), we proceed by contradiction assuming that Let us define functions α, β : Recall that, since u is semiconcave, (see, for instance, [10]). The following simple lemma is crucial for the proof.
Since M is continuous and D + u(x) is compact, the conclusion follows.
For any ε > 0 set Let 0 < R ≤ R 0 be such that Consider the line segment and fix q ∈ (0, 1). After possible reducing T , we can assume that |γ (t) −x| ≤ q R and |x(t) −x| ≤ q R ∀t ∈ [t, T ].
Consequently, there existsm ∈ N such that for all m ≥m we have Moreover, by cutting T down to size, we can have the following property satisfied: ∈ K δ for all t ∈ (t +3τ j , T ) and m sufficiently large.
Proof Throughout this proof j ∈ N is supposed to be so large that τ j < (T −t)/3. Moreover, in order to simplify the notation, abbreviate τ for τ j and we assumet = 0.