Weak KAM theory was originally developed initially for Lagrangian systems to connect viscosity solutions of the Hamilton-Jacobi
equation and the Aubry-Mather aspects of these Lagragian systems.
In the years 2000, with Antonio Siconolfy, we showed how to adapt these methods to deal with time functions in Lorentzian manifolds.
We have lately generalized the weak kAM theory to semi-metric (or costs) on metric spaces.
We will explain this generalization and explore the possible applications in Non-Regular Spactime Geometry.