The talk will be devoted to the study of regularity properties of solutions to time-dependent viscous Hamilton-Jacobi equations with data in Lebesgue spaces and coercive Hamiltonians in the gradient variable, obtained in collaboration with M. Cirant (Padova). More precisely, I will discuss the so-called maximal L^p-regularity problem for evolutive Hamilton-Jacobi equations, which is at the core of the smoothness of MFG systems with general power-like coupling. This is accomplished by coupling the adjoint method and Gagliardo-Nirenberg interpolation inequalities. In the subquadratic and quadratic regime the results answer positively to a parabolic version of a conjecture by P.-L. Lions recently addressed by Cirant-Goffi in the elliptic case.
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