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Minimizing movements for oscillating energies: the critical regime

Part of: Existence theories Optimality conditions Partial differential equations

Published online by Cambridge University Press:  27 December 2018

Nadia Ansini ,
Andrea Braides  and
Johannes Zimmer
Nadia Ansini
Affiliation:
Dept. of Mathematics, Sapienza University of Rome, P.le Aldo Moro 2, 00185 Rome, Italy
Andrea Braides
Affiliation:
Dept. of Mathematics, University of Rome ‘Tor Vergata’, Via della Ricerca Scientifica, 00133 Rome, Italy
Johannes Zimmer
Affiliation:
Dept. of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK (zimmer@maths.bath.ac.uk)
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Abstract

Minimizing movements are investigated for an energy which is the superposition of a convex functional and fast small oscillations. Thus a minimizing movement scheme involves a temporal parameter τ and a spatial parameter ε, with τ describing the time step and the frequency of the oscillations being proportional to 1/ε. The extreme cases of fast time scales τ ≪ ε and slow time scales ε ≪ τ have been investigated in [4]. In this paper, the intermediate (critical) case of finite ratio ε/τ > 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 characterization of the pinning threshold is given. For initial data above the pinning threshold, the equation and velocity describing the homogenized motion are determined.

Keywords

Gradient flowwiggly energyΓ-convergenceminimizing movements

MSC classification

Primary: 35B27: Homogenization; equations in media with periodic structure
Secondary: 49K40: Sensitivity, stability, well-posedness 49J10: Free problems in two or more independent variables 49J45: Methods involving semicontinuity and convergence; relaxation
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 3 , June 2019 , pp. 719 - 737
Copyright
Copyright © Royal Society of Edinburgh 2018 

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