TIME-DEPENDENT MEAN-FIELD GAMES WITH LOGARITHMIC NONLINEARITIES

Autor(es)

Gomes, Diogo A. ^{[1, 2]} ; Pimentel, Edgard ^[3]

Afiliação do(s) autor(es)

^[1] King Abdullah Univ Sci & Technol, Ctr Uncertainty Quantificat Computat Sci & Engn, CEMSE Div, Thuwal 239556900 - Saudi Arabia

^[2] King Abdullah Univ Sci & Technol, Ctr Uncertainty Quantificat Computat Sci & Engn, KAUST SRI, Thuwal 239556900 - Saudi Arabia

^[3] Univ Fed Sao Carlos, Dept Math, BR-13560 Sao Carlo, SP - Brazil

• Tipo de documento: Artigo Científico
• Fonte: SIAM JOURNAL ON MATHEMATICAL ANALYSIS; v. 47, n. 5, p. 3798-3812, 2015.
• Citações Web of Science: 19

Resumo

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. (AU)

• Processo FAPESP: 15/13011-6 - Equações diferenciais parciais não-lineares: boa colocação e teoria de regularidade
• Beneficiário: Edgard Almeida Pimentel
• Modalidade de apoio: Auxílio à Pesquisa - Regular