Calculus of Variations and Geometric Measure Theory

N. Ansini - A. Braides - J. Zimmer

Minimising movements for oscillating energies: the critical regime

created by ansini on 18 Mar 2016
modified by braidesa on 25 Jul 2019


Published Paper

Inserted: 18 mar 2016
Last Updated: 25 jul 2019

Journal: Proc. Roy. Soc. Edinburgh A
Volume: 149
Pages: 719-737
Year: 2019
Doi: 10.1017/prm.2018.46


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 $\varepsilon$, with $\tau$ describing the time step and the frequency of the oscillations being proportional to $1/ \varepsilon$. The extreme cases of fast time scales $\tau\ll\varepsilon$ and slow time scales $\varepsilon\ll\tau$ have been investigated in the book by Braides (2014). In this article, the intermediate (critical) case of finite ratio $\varepsilon/\tau>0$ 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.

Keywords: Gamma-convergence, Gradient Flow, minimizing movements, wiggly energy