no. 9
Partial differential equations
A generalization of the Hopf–Cole transformation for stationary Mean-Field Games systems
[Une généralisation de la transformation de Hopf–Cole pour des systèmes de jeux à champ moyen stationnaires]

Présenté par : Olivier Pironneau

Cirant, Marco1
1 Dipartimento di Matematica “F. Enriques”, Università di Milano, Via Cesare Saldini, 50, 20133 Milano, Italy
Comptes Rendus. Mathématique, Tome 353 (2015) no. 9, pp. 807-811.

On propose dans cette Note une transformation qui découple les systèmes de jeux à champ moyen stationnaires pour des hamiltoniens superlinéaires de la forme |p|r′,r′>1, et qui transforme l'équation de Hamilton–Jacobi–Bellman en une équation quasi linéaire introduisant le r-laplacien. Une telle transformaton nécessite une hypothèse sur la solution : cette hypothèse est satisfaite, par exemple, dans le cas unidimensionnel ou dans le cas où la solution est radiale.

In this note we propose a transformation that decouples stationary Mean-Field Games systems with superlinear Hamiltonians of the form |p|r′, r′>1, and turns the Hamilton–Jacobi–Bellman equation into a quasi-linear equation involving the r-Laplace operator. Such a transformation requires an assumption on solutions to the system, which is satisfied for example in space dimension one or if solutions are radial.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.06.016
Cirant, Marco 1

1 Dipartimento di Matematica “F. Enriques”, Università di Milano, Via Cesare Saldini, 50, 20133 Milano, Italy 
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Cirant, Marco. A generalization of the Hopf–Cole transformation for stationary Mean-Field Games systems. Comptes Rendus. Mathématique, Tome 353 (2015) no. 9, pp. 807-811. doi : 10.1016/j.crma.2015.06.016. http://www.numdam.org/articles/10.1016/j.crma.2015.06.016/

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