Optimal transportation of currents and Maxwell equations.

Our purpose is to extend to the framework of Electromagnetism the Monge-Kantorovich theory which originally comes from Continuum Mechanics and has become very popular in the last ten years in the field of nonlinear PDEs, especially because of its connection with the Monge-Ampere equation.

We construct, in the framework of Electromagnetism, an Action (related to the Nambu-Goto Action for extremal surfaces) which is analogous to the Monge-Kantorovich (or Wasserstein) distance. From this Action, we derive a system of PDEs that look like non-linear Maxwell equations. Then, through suitable approximations, we recover not only the linear Maxwell equation, as expected, but, more surprisingly, the complete (pressureless) Euler-Maxwell system.