Some estimates for the planning problem with potential

Tigran Bakaryan, Rita Ferreira, Diogo A. Gomes

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


In this paper, we study a priori estimates for a first-order mean-field planning problem with a potential. In the theory of mean-field games (MFGs), a priori estimates play a crucial role to prove the existence of classical solutions. In particular, uniform bounds for the density of players’ distribution and its inverse are of utmost importance. Here, we investigate a priori bounds for those quantities for a planning problem with a non-vanishing potential. The presence of a potential raises non-trivial difficulties, which we overcome by exploring a displacement-convexity property for the mean-field planning problem with a potential together with Moser’s iteration method. We show that if the potential satisfies a certain smallness condition, then a displacement-convexity property holds. This property enables Lq bounds for the density. In the one-dimensional case, the displacement-convexity property also gives Lq bounds for the inverse of the density. Finally, using these Lq estimates and Moser’s iteration method, we obtain L∞ estimates for the density of the distribution of the players and its inverse. We conclude with an application of our estimates to prove existence and uniqueness of solutions for a particular first-order mean-field planning problem with a potential.
Original languageEnglish (US)
JournalNonlinear Differential Equations and Applications
Issue number2
StatePublished - Mar 11 2021

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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