Minimising movements for oscillating energies: the critical regime

TL;DR

This paper studies the behavior of minimising movements for energies combining convex functionals with oscillations, focusing on the critical regime where the oscillation frequency and time step are proportionally related, revealing a pinning threshold and homogenized motion.

Contribution

It characterizes the critical regime of minimising movements with oscillating energies, identifying a pinning threshold and deriving the homogenized motion equations.

Findings

Existence of a pinning threshold for initial data below it.

Initial data above the threshold leads to homogenized motion.

Characterization of the pinning threshold and velocity in 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.