Evolution of Planar Lattices

Abstract

AbstractIn this chapter we describe some evolutions in the plane using the method of minimizing movements along families of ferromagnetic energies as those studied in the previous chapter. As shown in Theorem 3.5 (and also Theorem 3.4) the discrete-to-continuum Γ-limits of such energies are crystalline perimeters. In Sect. 4.1 we first describe motion by square crystalline curvature, which is obtained as a minimizing movement for the square perimeter (flat flow) using a dissipation D introduced by Almgren and Taylor. This evolution justifies the definition of discrete dissipations D ε introduced in Sect. 2.2.1 and provides a natural environment for a general evolution of lattices whose energies are asymptotically described by a square perimeter in the extreme regime described in Theorem 2.1(a). In the final and most important section of this chapter we will examine a number of energies which have the square perimeter as a Γ-limit but whose minimizing movements are influenced in different ways by the local arrangements of interactions, showing how local minimization may influence the evolution through homogenization and pinning effects.