Small-Scale Stability

Abstract

While it is possible to deduce the existence and convergence of local minimizers of $F_{\varepsilon }$ from the existence of an isolated local minimizer of their Γ-limit F, the knowledge of the existence of local minimizers of $F_{\varepsilon }$ is not sufficient to deduce the existence of local minimizers for F. In this chapter we examine a notion of stability such that, loosely speaking, a point is stable if it is not possible to reach a lower-energy state from that point without crossing an energy barrier of a specified height. This notion is a quantification of the notion of local minimizer, which instead is “scale-independent” and will allow us to state a convergence theorem for sequences of stable points. Even though a general result is not available, we will show how Γ-converging sequences often give rise to stable convergence.

Appendix

The notion of stable points has been introduced by Larsen in [3], where also stable fracture evolution has been studied; in particular there it is shown that the scheme in Sect. 6.4 can be applied to Griffith fracture energies.

The notions of stability for sequences of functionals have been analyzed by Braides and Larsen in [1], and are further investigated by Focardi in [2].