Perturbed minimizing movements of families of functionals

We consider the well-known minimizing-movement approach to the definition of a solution of gradient-flow type equations by means of an implicit Euler scheme depending on an energy and a dissipation term. We perturb the energy by considering a (Gamma-converging) sequence and the dissipation by varying multiplicative terms. The scheme depends on two small parameters epsilon and tau, governing energy and time scales, respectively. We characterize the extreme cases when epsilon/tau and tau/epsilon converges to 0 sufficiently fast, and exhibit a sufficient condition that guarantees that the limit is indeed independent of epsilon and tau. We give examples showing that this in general is not the case, and apply this approach to study some discrete approximations, the homogenization of wiggly energies and geometric crystalline flows obtained as limits of ferromagnetic energies.