Motion and pinning of discrete interfaces
The authors consider energy-driven motions based on a wide class of lattice energies depending on a discrete variable \(u=\{u_i\}\) indexed by the nodes \(i\) of a lattice (also considered as interfacial energies) in the framework of the analysis of lattice systems by a continuous variational approximation. The analysis of these motions can be interesting in view of their connections to physical phenomena linked to phase separation and motion of dislocations. The mathematical models for those phenomena often exhibit a transition in dependence of the time scales, between a regime where no motion is present and an `averaged motion' is achieved. In the paper we have the study of the fine behavior at the transition threshold. The paper consists of three major sections. The first one is an introduction to the analyzed problem. In the second one we have the formulation of the problem and the definitions of all energies the are considered, both as functional of discrete variables and as continuous energies of space of piecewise-constant functions. Section 3 contains the proof of the convergence of that scheme for increasingly general initial data. Section 3.1 considers the case of a rectangular initial set: Theorem 1 (quantization of the limit speed) stands for the general phenomenon of `quantization' of the speed of the motion and Theorem 2 (unique limit motions) stands for the cases when the limit motion is unique. Section 3.2 gives Theorem 3 (characterization of the motion of polyrectangles) while Sect. 3.3 gives the description of evolution of more general sets with Theorem 4 that treats the general case of a crystalline-convex initial datum, showing how the characterization encountered in the case study can be generalized to a general class. In the paper we have also a set of examples that are very interesting illustrations of considered problems.
scientific article; zbMATH DE number 5675032
scientific article