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Minimising movements for oscillating energies: the critical regime

Minimising movements are investigated for an energy which is the superposition of a convex functional and fast small oscillations. Thus a minimising movement scheme involves a temporal parameter $τ$ and a spatial parameter $ε$, with $τ$ describing the time step and the frequency of the oscillations being proportional to $\frac 1 ε$. The extreme cases of fast time scales $τ<< ε$ and slow time scales $ε<< τ$ have been investigated in Braides, Springer Lecture Notes 2094 (2014). In this article, the intermediate (critical) case of finite ratio $ε/τ>0$ is studied. It is shown that a pinning threshold exists, with initial data below the threshold being a fixed point of the dynamics. A characterisation of the pinning threshold is given. For initial data above the pinning threshold, the equation and velocity describing the homogenised motion are determined.