A Crandall-Liggett Approach to Gradient Flows in Metric Spaces
The notion of convex functions and of a gradient system in general metric spaces has been introduced and investigated in great detail by L. Ambrosio, N. Gigli, and G. Savar´e. In particular, existence and uniqueness of solutions to gradient systems have been proved. Here we give an alternative proof of existence, showing that the Crandall-Liggett approach to nonlinear semigroups can be carried over to this situation. It turns out that the crucial ingredient is a variational inequality satisfied by the minimizer of the Moreau-Yosida functional associated with the problem.