Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

Understand

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces (X,d,m).

• Our main results are: - A general study of the relations between the Hopf-Lax semigroup and Hamilton-Jacobi equation in metric spaces (X,d).
• - The equivalence of the heat flow in L^2(X,m) generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional in the space of probability measures P(X).
• - The proof of density in energy of Lipschitz functions in the Sobolev space W^{1,2}(X,d,m).