We investigate time-dependent mean-field games with superquadratic Hamiltonians and a power dependence on the measure. Such problems pose substantial mathematical challenges as key techniques used in the subquadratic case, which was studied in a previous publication of the authors, do not extend to the superquadratic setting. The main objective of the present paper is to address these difficulties. Because of the superquadratic structure of the Hamiltonian, Lipschitz estimates for the solutions of the Hamilton−Jacobi equation are obtained here through a novel set of techniques. These explore the parabolic nature of the problem through the nonlinear adjoint method. Well-posedness is proven by combining Lipschitz regularity for the Hamilton−Jacobi equation with polynomial estimates for solutions of the Fokker−Planck equation. Existence of classical solutions is then established under conditions depending only on the growth of the Hamiltonian and the dimension. Our results also add to current understanding of superquadratic Hamilton−Jacobi equations.

DOI: 10.1051/cocv/2015029

Keywords: Mean-field games, initial terminal value problem, superquadratic Hamiltonians, nonlinear adjoint method

^{1}; Pimentel, Edgard

^{2}; Sánchez-Morgado, Héctor

^{3}

@article{COCV_2016__22_2_562_0, author = {Gomes, Diogo A. and Pimentel, Edgard and S\'anchez-Morgado, H\'ector}, title = {Time-dependent mean-field games in the superquadratic case}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {562--580}, publisher = {EDP-Sciences}, volume = {22}, number = {2}, year = {2016}, doi = {10.1051/cocv/2015029}, mrnumber = {3491784}, zbl = {1339.35090}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2015029/} }

TY - JOUR AU - Gomes, Diogo A. AU - Pimentel, Edgard AU - Sánchez-Morgado, Héctor TI - Time-dependent mean-field games in the superquadratic case JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 562 EP - 580 VL - 22 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2015029/ DO - 10.1051/cocv/2015029 LA - en ID - COCV_2016__22_2_562_0 ER -

%0 Journal Article %A Gomes, Diogo A. %A Pimentel, Edgard %A Sánchez-Morgado, Héctor %T Time-dependent mean-field games in the superquadratic case %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 562-580 %V 22 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2015029/ %R 10.1051/cocv/2015029 %G en %F COCV_2016__22_2_562_0

Gomes, Diogo A.; Pimentel, Edgard; Sánchez-Morgado, Héctor. Time-dependent mean-field games in the superquadratic case. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 2, pp. 562-580. doi : 10.1051/cocv/2015029. http://archive.numdam.org/articles/10.1051/cocv/2015029/

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