Viscosity solutions of the evolutive Hamilton-Jacobi equation by limiting variational methods. With a look to the stationary case.

We consider two notions of weak solutions for the evolutive Hamilton-Jacobi equation: the viscosity and the variational solutions. For globally compactly supported Hamiltonians, we introduce iterative min-max procedures -for time intervals tending to zero- in order to recover viscosity solutions. Finally, we discuss an application to the stationary Hamilton-Jacobi equation.