Local Minimization as a Selection Criterion

Abstract

The Γ-limit F of a sequence $F_{\varepsilon }$ is often interpreted as a simplified description of the energies $F_{\varepsilon }$, where unimportant details have been averaged out still keeping the relevant information about minimum problems. As far as global minimization problems are concerned this is ensured by the fundamental theorem of Γ-convergence, but this is in general false for local minimization problems. Nevertheless, if some information on the local minima is known, we may use the fidelity of the description of local minimizers as a way to “correct” Γ-limits. In order to do that, we first introduce some notions of equivalence by Γ -convergence, and then show how to construct simpler equivalent theories as perturbations of the Γ-limit F in some relevant examples. We will exhibit a sharp-interface theory equivalent to the scalar Ginzburg–Landau theory, and derive Barenblatt theory of fracture as a perturbation of Griffith theory maintaining the pattern of minimizers of Lennard-Jones systems.

Appendix

The notion of equivalence by Γ-convergence is introduced and analyzed in the paper by Braides and Truskinovsky [5].

Local minimizers for Lennard-Jones type potentials (also with external forces) are studied in the paper by Braides et al. [3]

More details on the derivation of fracture energies from interatomic potentials and the explanation of the $\sqrt{\varepsilon }$-scaling can be found in the paper by Braides et al. [4] (see also the quoted paper by Braides and Truskinovsky for an explanation in terms of uniform Γ-equivalence).

For general reference on sets of finite perimeter and BV functions we refer to [1, 2, 6].