Different Time Scales

Abstract

In this section we examine the effect of scaling the energies in the resulting minimizing movements. One application is the possibility of defining and study long-time behaviour of variational motions, such as the ones connected to Mumford–Shah or Perona–Malik energies, Lennard-Jones discrete systems, or the gradient theory of phase transitions. These are obtained by suitably choosing a diverging (but positive) scaling of the energies. Negative scalings may be used to define a backward motion of some energy F, after properly choosing a family of functional Γ-converging to F.

Appendix

The literature on long-time behaviour and backward equations, even though not by the approach by minimizing movements, is huge. The long-time motion of interfaces in one space dimension by energy methods has been studied in a paper by Bronsard and Kohn [1].

Example 10.4 has been part of the course exam of C. Sorgentone and S. Tozza at Sapienza University in Rome, who kindly provided the pictures for the numerical simulations.

Example 10.7 is contained in a paper by Braides and Scilla [2]. It is a pleasure to acknowledge the suggestion of J.W. Cahn to use finite-dimensional approximations to define backward motion of sets.