Minimizing movements for oscillating energies: The critical regime

Minimizing movements are investigated for an energy which is the superposition of a convex functional and fast small oscillations. Thus a minimizing movement scheme involves a temporal parameter τ and a spatial parameter with τ describing the time step and the frequency of the oscillations being proportional to 1/. The extreme cases of fast time scales τ â and slow time scales â τ have been investigated in [4]. In this paper, the intermediate (critical) case of finite ratio /τ > 0 is studied. It is shown that a pinning threshold exists, with initial data below the threshold being a fixed point of the dynamics. A characterization of the pinning threshold is given. For initial data above the pinning threshold, the equation and velocity describing the homogenized motion are determined.