Skip to main content

Minimizing Movements

  • 1847 Accesses

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

Abstract

In this chapter we give a brief account of the variational motion defined by the limit of Euler schemes at vanishing time step. This notion is linked to the study of local minimizers, which provide stationary solutions for such motions, and is a way of defining a gradient flow for smooth energies, but is defined for a wide class of (non-smooth) energies. For the sake of simplicity of exposition, we limit our analysis to a Hilbert setting, even though many results can be proven in general metric spaces. As an example we define a one-dimensional motion for Griffith Fracture energy, that we may compare with the ones obtained as energetic solutions in the quasistatic setting, and as delta-stable evolutions.

Keywords

  • Boundary Displacement
  • Euler Scheme
  • Variational Motion
  • Give Boundary Condition
  • Vary Boundary Condition

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. 7.1
Fig. 7.2

References

  1. Ambrosio, L.: Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19, 191–246(1995)

    MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L., Braides, A.: Energies in SBV and variational models in fracture mechanics. In: Cioranescu, D., Damlamian, A., Donato, P. (eds.) Homogenization and Applications to Material Sciences, pp. 1–22. GAKUTO, Gakkōtosho, Tokio (1997)

    Google Scholar 

  3. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH, Zürich. Birkhhäuser, Basel (2008)

    Google Scholar 

  4. Chambolle, A., Doveri, F.: Minimizing movements of the Mumford and Shah energy. Discr. Cont. Dynam. Syst. 3, 153–174 (1997)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. De Giorgi, E.: New problems on minimizing movements. In: De Giorgi, E. (ed.) Selected Papers, pp. 699–713. Springer, Berlin (2006)

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Appendix

Appendix

The terminology ‘(generalized) minimizing movement’ has been introduced by De Giorgi in a series of papers devoted to mathematical conjectures (see [5]). We also refer to the original treatment by Ambrosio [1].

A theory of gradient flows in metric spaces using minimizing movements is described in the book by Ambrosio et al. [3].

Minimizing movements for the Mumford–Shah functional in more that one space dimension (and hence also for the Griffith fracture energy) with the condition of increasing fracture have been defined by Ambrosio and Braides [2], and partly analyzed in a two-dimensional setting by Chambolle and Doveri [4].

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Braides, A. (2014). Minimizing Movements. 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_7

Download citation