Comptes Rendus
no. 9
Partial differential equations
A generalization of the Hopf–Cole transformation for stationary Mean-Field Games systems

Presented by: Olivier Pironneau

Marco Cirant1
1 Dipartimento di Matematica “F. Enriques”, Università di Milano, Via Cesare Saldini, 50, 20133 Milano, Italy
Comptes Rendus. Mathématique, Volume 353 (2015) no. 9, pp. 807-811.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.06.016

Marco Cirant 1

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

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