Existence of weak solutions to stationary mean-field games through variational inequalities

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22 Scopus citations


Here, we consider stationary monotone mean-field games (MFGs) and study the existence of weak solutions induced by monotonicity. First, we introduce a regularized problem that preserves the monotonicity. Next, using variational inequality techniques, we prove the existence of solutions to the regularized problem. Then, using Minty's method, we establish the existence of weak solutions for the original MFG. Finally, we examine the properties of these weak solutions in several examples. Our methods provide a general framework to construct weak solutions to stationary MFGs with local, nonlocal, or congestion terms.

Original languageEnglish (US)
Pages (from-to)5969-6006
Number of pages38
JournalSIAM Journal on Mathematical Analysis
Issue number6
StatePublished - 2018

Bibliographical note

Publisher Copyright:
© 2018 Society for Industrial and Applied Mathematics.


  • Mean-field games
  • Stationary problems
  • Variational inequalities

ASJC Scopus subject areas

  • Computational Mathematics
  • Analysis
  • Applied Mathematics


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