Local Mean-Field Games: Existence

Diogo A. Gomes*, Edgard A. Pimentel, Vardan Voskanyan

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

5 Scopus citations


In this last chapter, we address the existence problem for local mean-field games. First, we illustrate the bootstrapping technique and put together the previous estimates. Thanks to this technique, we show that solutions of stationary MFGs are bounded a priori in all Sobolev spaces. This is an essential step for the two existence methods developed next. The first method is a regularization procedure in which we perturb the original local MFG into a non-local problem. By the results of the preceding chapter, this non-local problem admits a solution. Then, a limiting procedure gives the existence of a solution. The second method we consider is the continuation method. The key idea is to deform the original MFG into a problem that can be solved explicitly. Then, a topological argument shows that it is possible to deform the solution of the latter MFG into the former. This argument uses both the earlier bounds and an infinite dimensional version of the implicit function theorem.

Original languageEnglish (US)
Title of host publicationSpringerBriefs in Mathematics
PublisherSpringer Science and Business Media B.V.
Number of pages14
StatePublished - 2016

Publication series

NameSpringerBriefs in Mathematics
ISSN (Print)2191-8198
ISSN (Electronic)2191-8201

Bibliographical note

Publisher Copyright:
© 2016, Springer International Publishing Switzerland.


  • Continuation Method
  • Implicit Function Theorem
  • Regularization Method
  • Sobolev Space
  • Weak Solution

ASJC Scopus subject areas

  • General Mathematics


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