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 $\tau$ and a spatial parameter $\epsilon$, with $\tau$ describing the time step and the frequency of the oscillations being proportional to $\frac 1 \epsilon$. The extreme cases of fast time scales $\tau << \epsilon$ and slow time scales $\epsilon << \tau$ have been investigated in Braides, Springer Lecture Notes 2094 (2014). In this article, the intermediate (critical) case of finite ratio $\epsilon/\tau>0$ 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. 
Nadia Ansini and Andrea Braides and Johannes Zimmer
math.AP math.DS