A-priori estimates for stationary mean-field games

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

doi:10.3934/nhm.2012.7.303

Article Contents

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

This issue Previous Article Long time average of mean field games Next Article New numerical methods for mean field games with quadratic costs

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: October 31, 2011

Revised: February 29, 2012

Published: May 2012

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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 $u$.

Keywords:

Mean field games,
a-priori estimates,
variational and quasi-variational methods.

Mathematics Subject Classification: 35J47, 49L25, 49N70.

Citation:

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References

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