A-priori estimates for stationary mean-field games

Diogo A. Gomes; Gabriel E. Pires; Héctor Sánchez-Morgado; Diogo A. Gomes; Gabriel E. Pires; Héctor Sánchez-Morgado

doi:10.3934/nhm.2012.7.303

Networks and Heterogeneous Media

2012, Volume 7, Issue 2: 303-314. doi: 10.3934/nhm.2012.7.303

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A-priori estimates for stationary mean-field games

1.
Departamento de Matemática and CAMGSD, IST Avenida Rovisco Pais, Lisboa
2.
Instituto de Matem, Universidad Nacional Aut, M

Received: 01 November 2011 Revised: 01 March 2012
35J47, 49L25, 49N70.

In this paper we establish a new class of a-priori estimates for stationary mean-field games which have a quasi-variational structure. In particular we prove $W^{1,2}$ estimates for the value function $u$ and that the players distribution $m$ satisfies $\sqrt{m}\in W^{1,2}$. We discuss further results for power-like nonlinearities and prove higher regularity if the space dimension is 2. In particular we also obtain in this last case $W^{2,p}$ estimates for $u$.
- Mean field games,
- a-priori estimates,
- variational and quasi-variational methods
Citation: Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games[J]. Networks and Heterogeneous Media, 2012, 7(2): 303-314. doi: 10.3934/nhm.2012.7.303

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Abstract

In this paper we establish a new class of a-priori estimates for stationary mean-field games which have a quasi-variational structure. In particular we prove $W^{1,2}$ estimates for the value function $u$ and that the players distribution $m$ satisfies $\sqrt{m}\in W^{1,2}$. We discuss further results for power-like nonlinearities and prove higher regularity if the space dimension is 2. In particular we also obtain in this last case $W^{2,p}$ estimates for $u$.

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