# American Institute of Mathematical Sciences

June  2012, 7(2): 303-314. doi: 10.3934/nhm.2012.7.303

## A-priori estimates for stationary mean-field games

 1 Departamento de Matemática and CAMGSD, IST Avenida Rovisco Pais, Lisboa, Portugal, Portugal 2 Instituto de Matem, Universidad Nacional Aut, M, Mexico

Received  November 2011 Revised  March 2012 Published  June 2012

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 $u$.
Citation: Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks & Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303
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