Calculus of Variations and Geometric Measure Theory
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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 09 Mar 2017

[BibTeX]

Accepted Paper

Inserted: 18 mar 2016
Last Updated: 9 mar 2017

Journal: Proc. Roy. Soc. Edinburgh A
Year: 2017

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 $\varepsilon$, 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.


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