Quantitative maximal $L^2$-regularity for viscous Hamilton-Jacobi PDEs in 2D and Mean Field Games

We discuss quantitative Calder\'on-Zygmund estimates in $W^{2,2}$ for 2D viscous Hamil\-ton-Jacobi equations with natural growth in the gradient. We apply the result to obtain the existence of classical solutions for stationary second order Mean Field Games systems in 2D with (defocusing) coupling behaving like $m^\alpha$ for any $\alpha>0$. We also survey on the known results for the regularity of viscous Hamilton-Jacobi equations and second order Mean Field Games providing a list of several open problems.