Skip to main content

Different Time Scales

  • 1735 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 2094)

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.

Keywords

  • Minimizing Movement Scheme
  • Mumford Shah
  • Backward Movement
  • Crystalline Motion
  • Time Scale Arguments

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   49.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions
Fig. 10.1
Fig. 10.2
Fig. 10.3
Fig. 10.4
Fig. 10.5
Fig. 10.6

References

  1. Bronsard, L., Kohn, R.V.: On the slowness of phase boundary motion in one space dimension. Comm. Pure Appl. Math. 43, 983–997 (1990)

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Braides, A., Scilla, G.: Nucleation and backward motion of discrete interfaces. Preprint Scuola Normale Superiore, Pisa (2013). http://cvgmt.sns.it/paper/2239/

Download references

Author information

Authors and Affiliations

Authors

Appendix

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.

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Braides, A. (2014). Different Time Scales. In: Local Minimization, Variational Evolution and Γ-Convergence. Lecture Notes in Mathematics, vol 2094. Springer, Cham. https://doi.org/10.1007/978-3-319-01982-6_10

Download citation