Minimising movements for oscillating energies: the critical regime
Abstract
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.
 Publication:

arXiv eprints
 Pub Date:
 May 2016
 arXiv:
 arXiv:1605.01885
 Bibcode:
 2016arXiv160501885A
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Dynamical Systems;
 35B27 (49K40;
 49J10 49J45)