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Minimizing Movements Along a Sequence of Functionals

  • Andrea Braides
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2094)

Abstract

In this chapter we give a notion of minimizing movement along a sequence \(F_{\varepsilon }\) (with time step τ), which will depend in general on the interaction by the time scale τ and the parameter \(\varepsilon\) in the energies. In general, the final evolution depends on the \(\varepsilon\)-τ regime. The extreme case are the minimizing movements for the Γ-limit F (for “fast-converging \(\varepsilon\)”) and the limits of minimizing movements for \(F_{\varepsilon }\) as \(\varepsilon \rightarrow 0\) (for “fast-converging τ”). Heuristically, minimizing movements for all other regimes are “trapped” between these two extreme cases. We show that for scaled Lennard-Jones interactions all minimizing movements coincide with the one of the Mumford–Shah functional, while for oscillating energies we have a critical \(\varepsilon\)-τ regime, in which we show phenomena of pinning, non-uniqueness and homogenization of the velocity. In this example, the case \(\varepsilon =\tau\) gives an effective motion, from which the ones in all regimes can be deduced, including the extreme ones.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Andrea Braides
    • 1
  1. 1.Dipartimento di MatematicaUniversità di Roma Tor VergataRomaItaly

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