Marco Cirant - Università di Padova The study of gradient regularity of solutions to Hamilton-Jacobi equations is a crucial step in the analysis of Mean-Field Games systems, and it is in particular the lynchpin of smoothness in the viscous case. We address here the so-called problem of "maximal regularity" for stationary equations, conjectured some years ago by P.-L. Lions, and discuss a proof based on a refined version of the classical Bernstein method. Joint work with A. Goffi (Padova).