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On the variational formulation of some stationary second-order mean field games systems.

Mészáros, Alpár Richárd and Silva, Francisco J. (2018) 'On the variational formulation of some stationary second-order mean field games systems.', SIAM journal on mathematical analysis., 50 (1). pp. 1255-1277.

Abstract

We consider the variational approach to prove the existence of solutions of second-order stationary Mean Field Games systems on a bounded domain $\Omega\subseteq {\mathbb R}^{d}$ with Neumann boundary conditions and with and without density constraints. We consider Hamiltonians which grow as $|\cdot|^{q'}$, where $q'=q/(q-1)$ and $q>d$. Despite this restriction, our approach allows us to prove the existence of solutions in the case of rather general coupling terms. When density constraints are taken into account, our results improve those in [A. R. Mészáros and F. J. Silva, J. Math. Pures Appl., 104 (2015), pp. 1135--1159]. Furthermore, our approach can be used to obtain solutions of systems with multiple populations.

Item Type:Article
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1137/17M1125960
Publisher statement:© 2018, Society for Industrial and Applied Mathematics.
Date accepted:30 October 2017
Date deposited:28 February 2020
Date of first online publication:15 February 2018
Date first made open access:28 February 2020

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