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    Uniform estimates for the planning problem with potential

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    Type
    Preprint
    Authors
    Bakaryan, Tigran cc
    Ferreira, Rita cc
    Gomes, Diogo A. cc
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Applied Mathematics and Computational Science Program
    Date
    2020-03-05
    Permanent link to this record
    http://hdl.handle.net/10754/662281
    
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    Abstract
    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 $L^q$ bounds for the density. In the one-dimensional case, the displacement-convexity property also gives $L^q$ bounds for the inverse of the density. Finally, using these $L^q$ estimates and Moser's iteration method, we obtain $L^\infty$ estimates for the density of the distribution of the players and its inverse.
    Publisher
    arXiv
    arXiv
    2003.02591
    Additional Links
    https://arxiv.org/pdf/2003.02591
    Collections
    Preprints; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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