Minimizing movements for oscillating energies: The critical regime

Nadia Ansini, Andrea Braides, Johannes Zimmer

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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 with τ describing the time step and the frequency of the oscillations being proportional to 1/. The extreme cases of fast time scales τ â and slow time scales â τ have been investigated in [4]. In this paper, the intermediate (critical) case of finite ratio /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.

Original languageEnglish
Pages (from-to)719-737
Number of pages19
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
Issue number3
Early online date27 Dec 2018
Publication statusPublished - 1 Jun 2019


  • Gradient flow
  • minimizing movements
  • wiggly energy
  • Γ-convergence

ASJC Scopus subject areas

  • Mathematics(all)


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