The planning problem in mean field games as regularized mass transport

Graber, P. Jameson; Mészáros, Alpár R.; Silva, Francisco J.; Tonon, Daniela

Abstract

In this paper, using variational approaches, we investigate the first order planning problem arising in the theory of mean field games. We show the existence and uniqueness of weak solutions of the problem in the case of a large class of Hamiltonians with arbitrary superlinear order of growth at infinity and local coupling functions. We require the initial and final measures to be merely summable. At the same time [relying on the techniques developed recently in Graber and Mészáros (Ann Inst H Poincaré Anal Non Linéaire 35(6):1557–1576, 2018)], under stronger monotonicity and convexity conditions on the data, we obtain Sobolev estimates on the solutions of the planning problem both for space and time derivatives.

Citation

Graber, P. J., Mészáros, A. R., Silva, F. J., & Tonon, D. (2019). The planning problem in mean field games as regularized mass transport. Calculus of Variations and Partial Differential Equations, 58(3), Article 115. https://doi.org/10.1007/s00526-019-1561-9

• Journal Article Type: Article
• Acceptance Date: Apr 30, 2019
• Online Publication Date: Jun 10, 2019
• Publication Date: Jun 30, 2019
• Deposit Date: Oct 1, 2019
• Publicly Available Date: Jun 10, 2020
• Journal: Calculus of Variations and Partial Differential Equations
• Print ISSN: 0944-2669
• Electronic ISSN: 1432-0835
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 58
• Issue: 3
• Article Number: 115
• DOI: https://doi.org/10.1007/s00526-019-1561-9
• Public URL: https://durham-repository.worktribe.com/output/1284847
• Related Public URLs: https://arxiv.org/abs/1811.02706