Minimizing Movements

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.

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].