key: cord-0478457-9ahnqro8
authors: Cannarsa, Piermarco; Cheng, Wei
title: Local singular characteristics on $mathbb{R}^2$
date: 2020-08-13
journal: nan
DOI: nan
sha: a480831cfb8b777e6b0e56577a36775a0a089cfd
doc_id: 478457
cord_uid: 9ahnqro8

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 $mathbb{R}^2$, 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].

This paper is devoted to study the local propagation of singularities for viscosity solutions of the Hamilton-Jacobi equations ( , ( )) = 0, ∈ ℝ , (HJ ) ( , ( )) = 0, ∈ Ω, (HJ loc )

where is a Tonelli Hamiltonian in (HJ ) and is of class 1 and strictly convex in the -variable in (HJ loc ). In (HJ ), we assume that 0 on the right-hand side is Mañé's critical value. The existence of global weak KAM solutions of (HJ ) was obtained in [12] . In (HJ loc ), we suppose Ω ⊂ ℝ is a bounded domain.

Semiconcave functions are nonsmooth functions that play an important role in the study of (HJ ) 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 ∶ [0, ] → ℝ is called a generalized characteristic starting from for the pair ( , ) if it satisfies the following: If ∈ Sing ( )-the singular set of -then [1, Theorem 5] gives a sufficient condition for the existence of a generalized characteristic propagating the singularity of 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 of (HJ ), the class of intrinsic singular (generalized) characteristics was introduced in [4] by the authors of this paper, building on properties of the Lax-Oleinik semi-group of positive type. Such a method allowed to construct global singular characteristics, which we now call intrinsic. Moreover, in [5] and [6] such an "intrisic approach" turned out to be useful for pointing out topological properties of the cut locus of , 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 ⇉ ( , + ( )) 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

where ( ) is a symmetric positive definite × -matrix smoothly depending on and ( ) is a smooth function on ℝ . In this case, a much finer theory has been developed, yielding quantitative tools for the analysis of singular characteristics ( [2] , [9] ) and their long time behaviour ( [3] ). This is mainly due to the fact that, for a mechanical Hamiltonian, (1.1) reduces to the generalized gradient system ̇ ( ) ∈ ( ( )) + ( ( )) > 0 a.e. (0) = , (1.2) the solutions of which, unique for any initial datum, form a Lipschitz semi-flow (see, e.g., [1] , [2] , and [8] ). Unfortunately, such a uniqueness property, which for (1.2) is a simple consequence of the quasi-dissipativity of the set-valued map ⇉ ( ) + ( ), breaks down for a general Hamiltonian because ⇉ ( , + ( )) is no longer quasi-dissipative (see [11] and [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 of (HJ loc ), a Lipschitz singular curve ∶ [0, ] → Ω is called a strict singular characteristic from ∈ Sing ( ) if there exists a measurable selection ( ) ∈ + ( ( )) such that

As already mentioned, the existence of strict singular characteristics for time dependent Hamilton-Jacobi equations was proved in [13] , where additional regularity properties of such curves were established including right-differentiability of for every , right-continuity oḟ , and the fact that (⋅) ∶ [0, ] → ℝ satisfies 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, , and a singular characteristic, , 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) = 2 and is Lipschitz; (B) the initial point = (0) of the singular characteristic is not a critical point with respect the pair ( , ), i.e., 0 ∉ ( 0 , + ( )); (C) is right differentiable at 0 anḋ

Notice that any strict singular characteristic and the singular characteristic given in [11] (see also Proposition 2.12) satisfy conditions (A)-(D). The intrinsic singular characteristic 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 1 and 2 satisfying condition (A)-(D), there exists > 0 and a bi-Lipschitz homeomorphism ∶ [0, ] → [0, ( )] such that, 1 ( ( )) = 2 ( ) for all ∈ [0, ]. In other words, the singular characteristic starting from a noncritical point is unique up to a bi-Lipschitz reparameterization (Theorem 3.6). • In particular, if is a strict singular characteristic and is a singular characteristic starting from the same noncritical initial point , then there exists > 0 and a bi-Lipschitz homeomorphism ∶ [0, ] → [0, ( )] such that ( ( )) = ( ) for all ∈ [0, ] (Corollary 3.8). • We have the following uniqueness property for strict singular characteristics: let

be strict singular characteristics from the same noncritical initial point . Then there exists ∈ (0, ] such that 1 ( ) = 2 ( ) for all ∈ [0, ]. (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 of singular arcs.

The paper is organized as follows. In section 2, we introduce necessary material on Hamilton-Jacobi equations, semiconcavity, and singular characteristics. In section 3, we answer question (Q1)-(Q2) in the two-dimensional case. In the appendix, we give a full proof of the existence of strict singular characteristics.

Acknowledgements. Piermarco Cannarsa was supported in part by the National Group for Mathematical Analysis, Probability and Applications (GNAMPA) of the Italian Istituto Nazionale di Alta Matematica "Francesco Severi" and by Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. Wei Cheng is partly supported by National Natural Science Foundation of China (Grant No. 11871267, 11631006 and 11790272). The authors also appreciate the cloud meeting software Zoom for the help to finish this paper in this difficult time of coronavirus.

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

2.1. Semiconcave function. Let Ω ⊂ ℝ be a convex open set. We recall that a function ∶ Ω → ℝ is semiconcave (with linear modulus) if there exists a constant > 0 such that

for any , ∈ Ω and ∈ [0, 1]. Let ∶ Ω ⊂ ℝ → ℝ be a continuous function. For any ∈ Ω, the closed convex sets

are called the subdifferential and superdifferential of at , respectively.

The following characterization of semiconcavity (with linear modulus) for a continuous function comes from proximal analysis. 

then is semiconcave with constant and ∈ + ( ). Conversely, if is semiconcave in Ω with constant , then (2.2) holds for any ∈ Ω and ∈ + ( ).

Let ∶ Ω → ℝ be locally Lipschitz. We recall that a vector ∈ ℝ is called a reachable The set of all reachable gradients of at is denoted by * ( ).

The following proposition concerns fundamental properties of semiconcave funtions and their gradients (see [10] for the proof). Then Sing ( ) is countably (2 − )-rectifiable for = 0, 1, 2. In particular, Sing 2 ( ) is countable.

Aspects of weak KAM theory. For any , ∈ ℝ and > 0, we denote by Γ , the set of all absolutely continuous curves defined on [0, ] such that (0) = and ( ) = . Define

We call ( , ) the fundamental solution for the Hamilton-Jacobi equation

∈ Γ , attaining such a minimum is called a minimal curve for ( , ). The following result is well-known. We say such an absolutely continuous curve is a ( , )-calibrated curve, or a -calibrated curve for short, if the equality holds in the inequality above. A curve ∶ (−∞, 0] → ℝ is called a -calibrated curve if it is -calibrated on each compact sub-interval of (−∞, 0]. In this case, we also say that is a backward calibrated curve (with respect to ).

The following result explains the relation between the set of all reachable gradients and the set of all backward calibrated curves from (see, e.g., [10] or [14] for the proof). Proposition 2.9. Let ∶ ℝ → ℝ be a weak KAM solution of (HJ ) and let ∈ ℝ . Then ∈ * ( ) if and only if there exists a unique 2 curve ∶ (−∞, 0] → ℝ with (0) = and = ( ,̇ (0)), which is a backward calibrated curve with respect to .

2.3. 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 ).

Let be a Lipschitz semiconcave viscosity solution of (HJ loc ) and ∈ Sing ( ).

The following existence of singular characteristic is due to [1, 11] .

Proposition 2.12. Let be a Lipschitz semiconcave solution of (HJ loc ) and ∈ Sing ( ). Then, there exists a singular characteristic

Now, suppose is a Lipschitz semiconcave weak KAM solution of (HJ ). In [4] , another singular curve for is constructed as follows. First, it is shown that there exists 0 > 0 such that for any ( , ) ∈ ℝ + × ℝ and any maximizer for the function (⋅) − ( , ⋅), we have that | − | ⩽ 0 . Then, taking = 0 + 1, one shows that there exists 0 > 0 such that, if ∈ (0, 0 ], then there exists a unique , ∈ ( , ) of (⋅) − ( , ⋅) such that

Moreover, such a 0 is such that − ( , ⋅) is concave with constant 2 ∕ and 1 − 2 ∕ < 0 for 0 < ⩽ 0 . We now define the curve

). Let the curve be defined in (2.5) . Then, the following holds:

̇ + (0) exists anḋ

Hereafter, we will refer to the arc defined in (2.5) as the intrinsic characteristic from .

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 0 ∈ Sing ( ), namely

• strict singular characteristics, that is, solutions to (1.3),

• singular characteristics, introduced in Definition 2.11, and • the intrinsic singular characteristic given by Proposition 2.13.

In this section, we will compare the first two notions of characteristics when Ω ⊂ ℝ 2 .

We begin by introducing the following class of Lipschitz arcs.

Definition 3.1. Given > 0, we denote by Lip 0 (0, ; Ω) the class of all Lipschitz arcs ∶ [0, ] → Ω such that the right derivativė

does exist and satisfies

For any ∈ Lip 0 (0, ; Ω) we set

Owing to (3.1), we have that ( ) → 0 as ↓ 0.

Proof. Observe that, for any 0 ⩽ 0 ⩽ 1 ⩽ , the identity

Let ∈ ℝ 2 and let ∈ ℝ be a unit vector. For any ∈ (0, 1) let us consider the cone

with vertex in , amplitude , and axis . Clearly, ( , ) is given by the union of the two cones

which intersect each other only at . (b) there exists ∈ (0, ] such that for all ∈ (0, ] there exists ( ) ∈ (0, ] such that

Proof. Hereafter, we denote by ( ) ( ∈ ℕ) any (scalar-or vector-valued) function such that

In view of (3.3) we conclude that Now, having fixed ∈ (0, 1) let ∈ (0, 1 ] be such that, for a.e. ∈ [0, ],

and (a) follows. The proof of (b) is similar: sincė + 2 (0) =̇ + 1 (0) by condition (ii), for all ∈ [0, ] and ∈ [0, ] we have that (3.10) 2 ( ) − 1 ( ) = ( − )̇ + 1 (0) + 3 ( ) + 3 ( ). Hence, for all , ∈ (0, ] we deduce that

Next, take the scalar product of each side of (3.10) with 1 ( ) to obtain

Once again, having fixed ∈ (0, 1), we can find ∈ (0, ] satisfying the following: for all ∈ (0, ] there exists ( ) ∈ (0, ] such that

for all ∈ [0, ] and a.e. ∈ [0, ( )]. Then, (3.11) leads directly to (3.6) . Moreover, returning to (3.12) , for all ∈ [0, ] and a.e. ∈ [0, ( )] we conclude that

where we have used (3.11) to deduce the last inequality. Hence, (3.7) follows.

Given a semiconcave solution of (HJ loc ), we hereafter concentrate on singular arcs for , that is, arcs ∈ Lip 0 (0, ; Ω) such that ( ) ∈ Sing ( ) for all ∈ [0, ]. We denote such a subset of Lip 0 (0, ; Ω) by Lip 0 (0, ; Ω). Proof. The structure of the superdifferential of along is described by Proposition 2.5 and Proposition 3.3.15 in [10] .

Lemma 3.5. Let be a semiconcave solution of (HJ loc ) and let 0 ∈ Sing ( ) be such that 0 ∉ ( 0 , + ( 0 )). Let ∈ Lip 0 (0, ; Ω) be such that (0) = 0 anḋ 

Proof. The existence of backward calibrated curves satisfying (3.14) follows from Proposition 2.9. Moreover, for all ⩾ 0 we have that Next, recall that ( 0 , 0 ) = 0 because 0 ∈ * ( 0 ) ( = 1, 2). So, by the strict convexity of ( 0 , ⋅), we deduce that there exists > 0 such that

Hence, the upper-semicontinuity of the set-valued map ⇉ ( ( ), + ( ( ))) ensures the existence of numbers 1 ∈ (0, 1) and 1 ∈ (0, 0 ] such that

Therefore, combining (3.17) and (3.19), we conclude that, after possibly replacing 0 by a smaller nummber 1 > 0,

and ∈ [0, 1 ]. By (3.18) and the above inequality we have that 1 (− ) ∈ + ( ( ), 2 ( )) with = 0 1 ∕2.

The analogous statement for 2 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 is not a critical point. We begin the proof with the following lemma. Indeed, for any ∈ ( 1 ( ), 1 ( )) ∩ ( 1 ( ), 2 ( )) we have that

This yields = 1 ( ) because 2 + 2 > 1. Now, define = min 1 , , ( ) and fix ∈ [0, ] in the set of full measure on which (i) is satisfied together with (ii) and (iii), that is, 2 ( ) ∈ + ( 1 ( ), 1 ( )) and | 2 ( ) − 1 ( )| < 1 where (3.20) has also been taken into account. By possibly reducing , we also have that | 2 ( ) − 1 ( )| < 1 for all ∈ [0, ]. So, the arc 2 , restricted to [0, ], connects the point 2 ( ) of the cone + ( 1 ( ), 1 ( )) with 0 ∈ − ( 1 ( ), 1 ( )), remaining in the open ball of radius 1 centered at 1 ( ). Thus, in view of (3.15) and (3.16), 2 must intersect at least one of the two calibrated curves 1 and 2 . However, this can happen only at 1 (0) = 1 ( ) = 2 (0), because is smooth at all points 2 (− ) with 0 < < ∞, whereas 2 is a singular arc. Finally, such an intersection occurs at a unique time owing to Lemma 3.2.

To complete the proof we observe that 2 ( ) = 1 ( ) for all ∈ [0, ], 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.

Proof of Theorem 3.6. Let ∈ (0, ] be given by Lemma 3.7. Then for each ∈ [0, ] there exists a unique ( ) ∶= ∈ [0, 1 ] with 2 ( ( )) = 1 ( ).

Recalling that, thanks to Lemma 3.2, both 1 (⋅) and 2 (⋅) can be assumed to be injective on [0, ] and [0, ( )], respectively, we proceed to show that is also an injection. Observe that, for any 0 ⩽ 0 , 1 ⩽ ,

Therefore,

2 is given by (3.2) . Thus, returning to 1 = 2 • we derive

(3.21)

Notice that (3.21) leads to

|̇ + 2 (0)| + 2 ( ( 1 ) ∨ ( 0 ))| and this implies that is injective as so is 1 .

Next, we prove that is continuous on [0, ], or the graph of is closed. Let →̄ be any sequence such that ( ) →̄ as → ∞. Then So, 2 ( (̄ )) = 1 (̄ ) = 2 (̄ ). Since 2 (⋅) is injective, it follows that̄ = (̄ ).

Being continuous, is a homeomorphism. It remains to prove that is bi-Lipschitz. The continuity of at 0 ensures that, after possibly reducing , 

The proof is completed noting that is unique due to the injectivity of 1 and 2 . For strict singular characteristics, uniqueness holds without reparameterization as we show next. where, in addition to (3.24), we have that 2 ( ( ))) = arg min ∈ + ( 2 ( ( )) ( 2 ( ( )), ) = arg min ∈ + ( 1 ( ) ( 1 ( ), ) = 1 ( ).

So, ( 1 ( ), 1 ( )) = ′ ( ) ( 1 ( ), 1 ( )) for all ∈ [0, 1 ]. Since 0 ∉ ( 0 , + ( 0 )), we conclude that ′ ( ) = 1, or ( ) = , on some interval 0 ⩽ ⩽ ⩽ . Theorem 3.6 and Theorem 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. Proof. On account of Theorem 3.9 we have that

is a nonempty set. Let 0 = sup  = max  . We claim that 0 = . For if 0 < , applying Theorem 3.9 with initial point 1 ( 0 ) we conclude that 1 ( ) = 2 ( ) on some intarval 0 ⩽ < 0 + , contradicting the definition of 0 .

Another well-known example where we have uniqueness of the generalized characteristic is the mechanical Hamiltonian

with a smooth function on Ω. More precisely, if ∈ Sing ( ), then there exists a unique Lipschitz arc determined bẏ + ( ) = ( ), where (0) = and ( ) = arg min ∈ + ( ( )) | |.

In this case, uniqueness follows from semiconcavity by an application of Gronwall's lemma (see, e.g., [10] ) ensuring that, in addition, any generalised characteristic is strict. We now give another justification of such a property from the point of view of this section. Proof. The proof, which uses ideas from [13] , requires several intermediate steps.

Let 0 > 0 be such that the closed ball ( 0 , 2 0 ) is contained in Ω. Take any sequence of smooth functions

⩽ 2 for some constants 1 , 2 > 0. A sequence with the above properties can be constructed in several ways, for instance by using mollifiers like in [18, 11] . In view of the above uniform bounds, there exists 0 > 0 such that for any ⩾ 1 the Cauchy problem

has a unique solution ∶ [0, 0 ] → ( 0 , 0 ). Moreover, by possibly taking a subsequence, we can assume that converges uniformly on [0, 0 ] to some Lipschitz arc ∶ [0, 0 ] → ( 0 , 0 ). We will show that, after possibly replacing 0 by a smaller > 0, such a limiting curve has the required properties.

in contrast with ( ). So, (A.6) is proved.

Finally, (A.5) can be derived from (A.6) by integration.

By appealing to the upper semi-continuity of + and assumption (A.1) we conclude that there exists ∈ (0, 0 ] such that

Now, fix anȳ ∈ [0, ) and let̄ ∈ ℝ 2 be any vector such that

for some sequence ↘ 0 ( → ∞). Observe that̄ ∈ co (̄ ), + ( (̄ )) in view of Lemma A.3. So,̄ ≠ 0 owing to (A.7). Set̄ = (̄ ) and definē

Notice that ̄ (̄ ) is the exposed face of the convex set + (̄ ) in the direction̄ (see, for instance, [10] ). The following lemma identifies̄ (hencē ) uniquely.

Therefore, by convexity we conclude that

Since is strictly convex in ,̄ is the unique element in + (̄ ) satisfying (A.9).

Notice that the above lemma yields the existence of the right-derivativė + (̄ ) as soon as one shows that̄ ∈ ̄ (̄ ) for anȳ satisfying (A.8).

Next, to show that̄ ∈ ̄ (̄ ), we proceed by contradiction assuming that (A.10)̄ ∉ ̄ (̄ ).

Let us define functions , ∶ + (̄ ) → ℝ by

where we have set ̄ (̄ ) = lim →0 + (̄ + ̄ )− (̄ ) . Recall that, since is semiconcave,

⟨ ,̄ ⟩ (see, for instance, [10] ). The following simple lemma is crucial for the proof. Proof. Observe first that ( , ) ⩾ 0 by convexity and ( ) ⩾ 0 for all ∈ + (̄ ) by (A.11). Since we supposē ∉ ̄ (̄ ), just two cases are possible.

(1) If̄ ∉ + (̄ ), then ≠̄ for all ∈ + (̄ ). So (̄ , ) > 0 by strict convexity.

(2) If̄ ∈ + (̄ ) ⧵ ̄ (̄ ), then ( ) > 0.

In conclusion,

Since is continuous and + (̄ ) is compact, the conclusion follows. Let 0 < ⩽ 0 be such that

Consider the line segment

) and fix ∈ (0, 1). After possible reducing , we can assume that Proof. Throughout this proof ∈ ℕ is supposed to be so large that < ( −̄ )∕3. Moreover, in order to simplify the notation, abbreviate for and we assumē = 0.

For all ∈ (3 , ) we have that where we recall that 1 ⩾ ‖ ‖ ∞ . Next, we fix = with large enough so that To complete the proof it suffices to note that Lemma A.5 and Lemma A.6 ensure that assuming (A.10) leads to a contradiction. Indeed, 

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Lemma A. 3 . For everȳ ∈ [0, 0 ) and > 0 there exists and integer ⩾ 1 and a real number ∈ (0, 0 −̄ ) such thatwhere benotes the closed unit ball of ℝ 2 , centered at the origin.Proof. We begin by showing that for everȳ ∈ [0, 0 ) and > 0 there exist ⩾ 1 andfor all ⩾ . We argue by contradiction: set Φ(̄ ) = (̄ ), + ( (̄ )) and suppose there exist̄ ∈ [0, 0 ), > 0, and sequences → ∞ and ↓̄ such thatwhere we have used bound ( ) above to justify ( ). We claim that̄ ∈ + (̄ ) . Indeed, in view of (c) above we have that, for all ⩾ 1,Hence, in the limit as → ∞, we get ( (̄ ) + ) − ( (̄ )) − ⟨̄ , ⟩ ⩽ 2 | | 2 , ∀| | ⩽ 0 , which in turn proves our claim. Thus, we conclude thaṫ