Hamilton-Jacobi equations on infinite dimensional spaces

Daniela Tonon (Università di Padova)

Abstract:  In this talk, we present a comparison principle for the Hamilton Jacobi (HJ) equation corresponding to linearly controlled gradient flows of an energy functional defined on a metric space. The main difficulties are given by the fact that the geometrical assumptions we require on the energy functional do not give any control on the growth of its gradient flow nor on its regularity. Therefore this framework is not covered by previous results on HJ equations on infinite dimensional spaces (whose study has been initiated in a series of papers by Crandall and Lions). Our proof of the comparison principle combines some rather classical ingredients, such as Ekeland’s perturbed optimization principle, with the use of the Tataru distance and of the regularizing properties of gradient flows in evolutional variational inequality formulation, that we exploit for constructing rigorous upper and lower bounds for the formal Hamiltonian.   Our abstract results apply to a large class of examples, including gradient flows on Hilbert spaces and  Wasserstein spaces equipped with a displacement convex energy functional satisfying McCann’s condition. However,  with respect to the existing literature about the Master equation in Mean Field Games our assumptions have a different nature. Nevertheless, some ideas could be of use for further studies.