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Comptes Rendus Mathématique
Volume 355, n° 2
pages 176-180 (février 2017)
Doi : 10.1016/j.crma.2016.12.004
Received : 22 September 2016 ;  accepted : 19 November 2016
On the topology of the set of singularities of a solution to the Hamilton–Jacobi equation
Sur la topologie des singularités d'une solution de l'équation de Hamilton–Jacobi
 

Piermarco Cannarsa a, 1 , Wei Cheng b, 2 , Albert Fathi c, 3
a Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy 
b Department of Mathematics, Nanjing University, Nanjing 210093, China 
c ENS de Lyon & IUF, UMPA, 46, allée d'Italie, 69007 Lyon, France 

Abstract

We address the topology of the set of singularities of a solution to a Hamilton–Jacobi equation. For this, we will apply the idea of the first two authors (Cannarsa and Cheng, Generalized characteristics and Lax–Oleinik operators: global result, preprint, arXiv:1605.07581, 2016) to use the positive Lax–Oleinik semi-group to propagate singularities.

The full text of this article is available in PDF format.
Résumé

Nous étudions l'ensemble des singularités d'une solution de l'équation de Hamilton–Jacobi. Pour cette étude, nous utilisons une idée due aux deux premiers auteurs (Cannarsa and Cheng, Generalized characteristics and Lax–Oleinik operators: global result, preprint, arXiv:1605.07581, 2016) pour propager les singularités en utilisant le semi-groupe positif de Lax–Oleinik.

The full text of this article is available in PDF format.
Version française abrégée

Nous étudions la topologie de l'ensemble des singularités d'une solution de l'équation de Hamilton–Jacobi et leur propagation.

Nous supposons connue la notion de solution de viscosité ainsi que la théorie KAM faible, voir [[6]], qui est bien adapté à nos besoins.

Soit   un hamiltonien de Tonelli sur la variété compacte connexe M . En particulier, le hamiltonien H est au moins C2. On considère une solution de viscosité (ou KAM faible)   de l'équation stationnaire de Hamilton–Jacobi
(1)H(x,dxu)=c[0], où   est la constante critique de Mañé. Nous désignons par   l'ensemble des points   où u n'est pas différentiable, et par   l'ensemble d'Aubry de u . Nous introduisons aussi l'ensemble   des points de coupure de u comme étant l'ensemble des points   où aucune courbe caractéristique en temps négatif de u aboutissant en x ne peut être étendue au-delà de x en une courbe u -calibrée ; de manière équivalente, si   est une courbe u -calibrée avec  , alors  . On a  , et  .

Théorème 0.1

Les inclusions   sont toutes des équivalences d'homotopies.

Il en résulte le corollaire suivant.
Corollaire 0.2

Pour toute composante connexe C de  , les trois intersections  ,   et   sont connexes par arcs.

Théorème 0.3

Les espaces  , and   sont localement contractiles, c'est-à-dire que pour tout   (resp.  ) et tout voisinage V de x dans   (resp.  ), on peut trouver un voisinage W de x dans   (resp.  ), tel que   et que W soit homotope à une constante dans V.

Par conséquent,   et   sont localement connexes par arcs.

Par soucis de brièveté et de clarté, nous avons énoncé nos résultats pour l'équation de Hamilton–Jacobi stationnaire ((1)). Toutefois, ils sont aussi valides pour l'équation de Hamilton–Jacobi sous forme évolution
(2)∂tU+H(x,∂xU)=0, où   est continue et est solution de viscosité de ((2)) sur  .

De plus, sous des conditions appropriées sur l'hamiltonien, les résultats sont encore valables quand M n'est pas compacte. Nous pouvons aussi considérer le cas des problèmes de Dirichlet.

Il est aussi possible d'utiliser nos méthodes pour obtenir les mêmes résultats pour la fonction distance à un fermé dans une variété riemannienne complète.

La version complète de ce travail comportera les détails nécessaires.

Notons que pour propager globalement les singularités, nous utilisons l'idée due aux deux premiers auteurs (Generalized characteristics and Lax–Oleinik operators: global result, preprint, arXiv:1605.07581, 2016) de se servir du semi-groupe positif de Lax–Oleinik. Notons aussi que, bien que les résultats et les démonstrations des deux théorèmes énoncés ci-dessus soient originaux, un cas particulier du premier théorème pour les fonctions distances sur les variétés riemanniennes était déjà connu, voir [[1]].

Introduction

We address the problem of propagation of singularities and the topology of the set of singularities of a viscosity solution to a Hamilton–Jacobi equation.

We assume familiarity with the notion of viscosity solution and weak KAM theory, see [[6]], which is relevant to our manifold framework. Let   be a Tonelli Hamiltonian on the compact connected manifold M ; in particular, the Hamiltonian H is at least C2. We consider   a viscosity (or weak KAM) solution to the Hamilton–Jacobi equation
(3)H(x,dxu)=c[0], where   is Mañé's critical value. We denote by   the set of points  , where u is not differentiable, and by   the Aubry set of u . We also introduce the set   of cut points of u , as the set of points   where no backward characteristic for u ending at x can be extended to a u -calibrating curve beyond x ; equivalently, if   is a u -calibrating curve with   then  . We have  , and  .

Theorem 1.1

The inclusion   are all homotopy equivalences.

This theorem obviously implies the following corollary (see, for instance, [[5]]).

Corollary 1.2

For every connected component C of   the three intersections  ,  , and   are path-connected.

Theorem 1.3

The spaces  , and   are locally contractible, i.e. for every   (resp.  ) and every neighborhood V of x in   (resp.  ), we can find a neighborhood W of x in   (resp.  ), such that  , and W in null-homotopic in V.

Therefore   and   are locally path connected.

For the sake of brevity and clarity, we stated our result for the stationary Hamilton–Jacobi equation ((3)), but they are also valid for the Hamilton–Jacobi equation in its evolution form
(4)∂tU+H(x,∂xU)=0, where   is continuous, and a viscosity solution to ((4)) on  .

Moreover, with the appropriate condition on the Hamiltonian, the results are also valid when M is not compact. We can also deal with Dirichlet-type problems.

It is also possible, using our method, to obtain the same results for the distance function to a closed set in a complete Riemannian manifold.

The details will appear in the complete version of this work.

We would like to note here that to obtain the global propagation of singularities, see Lemma 2.1 below, we apply the idea of the first two authors [[4]] to use the positive Lax–Oleinik group to propagate singularities. We also note that although the results and proofs of Theorem 1.1, Theorem 1.3 are new, a particular version of Theorem 1.1 for distance functions in Riemannian manifolds is already known, see [[1]].

Proof of the theorems

The proof of both theorems uses the following lemma.

Lemma 2.1

There exists some  , and a (continuous) homotopy  , with the following properties:

(a)
for all  , we have  ;
(b)
if  , for some  , and  , then the curve   is u-calibrating on  ;
(c)
if there exists a u-calibrating curve  , with  , then  , for every  .

The proof of the lemma will be sketched in the next section.

We extend F to a homotopy  , using induction on  , by
F(x,s)=F(F(x,nt),s−nt), for s∈[nt,(n+1)t]. It is not difficult to check that this extended F has the same properties (a), (b), and (c) stated above.

These properties imply:

(1)
 ;
(2)
if   never enters  , then  , and   is the forward calibrating curve through x ;
(3)
if  , then  , for every  .
Before proceeding further, we note that this extended homotopy F shows that we have propagation of singularities in infinite time. It is convenient to introduce the cut time function   for u , where   is the supremum of the   such that there exists a u -calibrating curve  , with  . The properties of τ are:
(i)
  if and only if  ;
(ii)
  if and only if  ;
(iii)
the function τ is upper semi-continuous.
By (b) of Lemma 2.1, we have  , for all  .

Proof of Theorem 1.1

The function τ is upper semi-continuous and finite on  ; therefore (by Proposition 7.20 in [[3]]) we can find a continuous function  , with   on  . We now define   by
G(x,s)=F(x,sα(x)). The map G is a homotopy of   into itself starting with the identity, such that  , and  . It is not difficult to check that the time one map of G gives a homotopy inverse for each one of the inclusions  . □

Proof of Theorem 1.3

We first construct for every open subset  , an open subset  , such that   and  , together with a homotopy  , which satisfies:

(i)
  for every  ;
(ii)
 ;
(iii)
 .
To define  , we introduce the function   defined byηO(x)=sup⁡{t∈[0,+∞[:F(x,s)∈O, for all s∈[0,t]}. Since O is open and F is continuous, the function   is lower semi-continuous and everywhere >0 on O . Using that   is lower semi-continuous and the cut time function τ is upper semi-continuous, we conclude that the subsetO˜={x∈O:τ(x)<ηO(x)}⊂O is indeed open. Furthermore, since τ is 0 on  , we get   and  .

It remains to construct the homotopy  . For this, we observe that   on  , with τ upper semi-continuous, and   lower semi-continuous. Hence, Baire's interpolation theorem (Proposition 7.21 in [[3]] or Section VIII.4.3 in [[5]]) guarantees the existence of a c ontinuous function   such that   everywhere on  . It is not difficult to check that the map  , defined by
GO(x,s)=F(x,sαO(x)), satisfies the required conditions (i), (ii), and (iii).

From the properties of  , we obtain that the identity on   (resp.  ) is homotopic to the restriction   (resp.  ) of the time one map of   as maps with values in   (resp.  ). Let B be an open subset included in  , which is homeomorphic to an Euclidean ball. Since B is contractible, the restriction of   to B is homotopic to a constant as maps from B to  . Therefore the restriction of   to   (resp. to  ) is homotopic to a constant as maps with values in   (resp. in  ). Since   (resp.  ), it follows that the inclusion   (resp.  ) is homotopic to a constant as maps with values in   (resp. in  ). □

Proof of Lemma 2.1

By [[2]] we know that there exists   such that   is   for all  , where   is given by
(5)Ts+u(x):=supy∈M⁡{u(y)−hs(x,y)} and   is the minimal action of a curve joining x to y in time s .

The supremum defining   in ((5)) is attained at some point  . We first show that   is unique. Indeed, if   is a minimizer with   and  , we know that
(6)∂L∂v(x,γ˙(0))=dxTs+u(x)and∂L∂v(ys,γ˙(s))∈D+u(ys). The first equality implies that   is unique. Therefore, we can define the map   by  . Continuity of F follows from compactness and the fact that the set{(x,y,s)∈M×M×[0,+∞[:Ts+u(x)=u(y)−hs(x,y)} is closed.

Moreover, if  , the second part of ((6)) implies
dyu(ys)=∂L∂v(ys,γ˙(s)). If we denote by   a u -calibrating curve such that  , we also know thatdyu(ys)=∂L∂v(ys,γ˜˙(0)). Since both γ and   are solutions to the Euler–Lagrange equation, we get  , for all  . Therefore, the minimizer   is a u -calibrating curve with  .

To finish the proof, it remains to show that if   is a u -calibrating curve, then
(7)Tσ+(γ(0))=u(γ(σ))−hσ(γ(0),γ(σ)), for all σ∈[0,s]. Indeed, since u is a weak KAM solution, and γ is u -calibrating, we get  , with equality at  . Therefore, we conclude that  , with equality at  . This clearly proves ((7)).


Acknowledgements

This work was started during the “Workshop on Hamilton–Jacobi Equations”, which was held on 24–30 July 2016, at Fudan University. The authors would like to particularly thank Prof. Jun Yan for his hospitality.

References

Albano P., Cannarsa P., Nguyen K.T., Sinestrari C. Singular gradient flow of the distance function and homotopy equivalence Math. Ann. 2013 ;  356 : 23-43 [cross-ref]
Bernard P. Existence of   critical sub-solutions of the Hamilton–Jacobi equation on compact manifolds Ann. Sci. Éc. Norm. Supér. (4) 2007 ;  40 (3) : 445-452 [cross-ref]
Brown A., Pearcy C. An Introduction to Analysis  New York: Springer-Verlag (1995). 
Cannarsa P., Cheng W. Generalized characteristics and Lax–Oleinik operators: global resultpreprint. arXiv:1605.075812016
Dugundji J. Topology  Boston, MA, USA: Allyn and Bacon, Inc. (1966). 
Fathi A. Weak KAM from a PDE point of view: viscosity solutions of the Hamilton–Jacobi equation and Aubry set Proc. Roy. Soc. Edinburgh Sect. A 2012 ;  120 : 193-1236

1  Work supported by the University of Rome Tor Vergata: Consolidate the Foundation 2014 Project “Irreversibility in Dynamic Optimization” and Istituto Nazionale di Alta Matematica: GNAMPA 2016 Project “Controllo, regolarità e viabilità per alcuni tipi di equazioni diffusive” (INdAM).
2  Work supported by the Natural Scientific Foundation of China (Grants No. 11271182 and No. 11471238) and the National Basic Research Program of China (Grant No. 2013CB834100).
3  Work supported by ANR-12-BS01-0020 WKBHJ.


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