We introduce several PDE tools which are useful in the study of mean field game systems with local couplings. Due to the lack of regularity of solutions, refined compactness and renormalization arguments are needed for a general approach leading to existence and uniqueness results. If time is enough, congestion models will be treated by similar techniques.
Alessio Porretta is a full Professor of Mathematical Analysis at the University of Rome Tor Vergata (Italy). He got his Ph.D. in 2000 at the University of Rome La Sapienza. His research activity is mainly focused on convection-diffusion equations, Hamilton-Jacobi, control theory and mean field games. In Italy, his research was awarded with the “Miranda Prize” (2002) and “Fichera Prize” (2018).He has given talks and courses in more than 20 countries in the world and he was appointed as visiting professor in several universities in France including the Universities of Paris VI, Paris VII, Paris Dauphine, and Toulouse. In recent years he has been invited to give courses on mean field games in Paris, Chicago, ETH Zurich and KAUST (Saudi Arabia). He authored over 80 research papers, with more than 1300 citations in math journals.