Time-Dependent Mean-Field Games with Logarithmic Nonlinearities

Diogo A. Gomes, Edgard Pimentel

Research output: Contribution to journalArticlepeer-review

26 Scopus citations

Abstract

In this paper, we prove the existence of classical solutions for time-dependent mean-field games with a logarithmic nonlinearity and subquadratic Hamiltonians. Because the logarithm is unbounded from below, this nonlinearity poses substantial mathematical challenges that have not been addressed in the literature. Our result is proven by recurring to a delicate argument which combines Lipschitz regularity for the Hamilton-Jacobi equation with estimates for the nonlinearity in suitable Lebesgue spaces. Lipschitz estimates follow from an application of the nonlinear adjoint method. These are then combined with a priori bounds for solutions of the Fokker-Planck equation and a concavity argument for the nonlinearity.
Original languageEnglish (US)
Pages (from-to)3798-3812
Number of pages15
JournalSIAM Journal on Mathematical Analysis
Volume47
Issue number5
DOIs
StatePublished - Oct 6 2015

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