Comptes Rendus
no. 2
Partial differential equations/Calculus of variations
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]

Présenté par : Cédric Villani

Piermarco Cannarsa1 ; Wei Cheng2 ; Albert Fathi3
1 Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy
2 Department of Mathematics, Nanjing University, Nanjing 210093, China
3 ENS de Lyon & IUF, UMPA, 46, allée d'Italie, 69007 Lyon, France
Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 176-180.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.12.004
Piermarco Cannarsa 1 ; Wei Cheng 2 ; Albert Fathi 3

1 Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy
2 Department of Mathematics, Nanjing University, Nanjing 210093, China
3 ENS de Lyon & IUF, UMPA, 46, allée d'Italie, 69007 Lyon, France 
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Piermarco Cannarsa; Wei Cheng; Albert Fathi. On the topology of the set of singularities of a solution to the Hamilton–Jacobi equation. Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 176-180. doi : 10.1016/j.crma.2016.12.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.12.004/

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[2] P. Bernard Existence of C1,1 critical sub-solutions of the Hamilton–Jacobi equation on compact manifolds, Ann. Sci. Éc. Norm. Supér. (4), Volume 40 (2007) no. 3, pp. 445-452

[3] A. Brown; C. Pearcy An Introduction to Analysis, Graduate Texts in Mathematics, vol. 154, Springer-Verlag, New York, 1995

[4] P. Cannarsa; W. Cheng Generalized characteristics and Lax–Oleinik operators: global result, 2016 (preprint) | arXiv

[5] J. Dugundji Topology, Allyn and Bacon, Inc., Boston, MA, USA, 1966

[6] A. Fathi Weak KAM from a PDE point of view: viscosity solutions of the Hamilton–Jacobi equation and Aubry set, Proc. Roy. Soc. Edinburgh Sect. A, Volume 120 (2012), pp. 193-1236

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