Minimizing Movements Along a Sequence of Functionals

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.

Appendix

The definition of minimizing movement along a sequence of functionals formalizes a natural extension to the notion of minimizing movement, and follows the definition given in the paper by Braides et al. [2].

The energies in Examples 8.2 and 8.6 have been taken as a prototype to model plastic phenomena by Puglisi and Truskinovsky [7]. More recently, that example has been recast in the framework of quasistatic motion in the papers by Mielke and Truskinovsky [4, 6].

The example of the minimizing movement for Lennard-Jones interactions is part of results of Braides et al. [1]. It is close in spirit to a semi-discrete approach (i.e., the study of the limit of the gradient flows for the discrete energies) by Gobbino [3].

For the notion of BV-solution we refer to Mielke et al. [5].