From Discrete Systems to Continuous Variational Problem

Mini-course on Nonconvex Variational Problems and Applications

From Discrete Systems to Continuous Variational Problem

The aim of the course is the description of the overall properties of variational discrete systems when the number of elements of the system gets larger. I will describe some results and methods that have been developed in the last years for large lattice systems (e.g. atomistic system with pair potentials, variational models of vision, resistor networks, etc.). Upon scaling, some of the questions may be equivalently stated in terms of the limit of variational problems for lattice systems as the lattice spacing tends to zero, that are described by a continuous energy. This limit must be understood in a variational sense (Gamma-limit) and is in general (quite) different from the pointwise limit.

Some issue that have been addressed are:

determination of the scaling properties of the discrete energies that correspond to continuous energies of specific type ('elastic', 'brittle fracture', 'softening' type, etc.)
multi-scale analysis describing 'phase-transition' and 'boundary-layer' type effects
homogenization formulas for the effective energies, and interpretation of such formulas in terms of the Cauchy-Born hypothesis
optimal bounds for discrete composites and comparison with continuous bounds percolation analysis of random discrete problems

Tentative plan of the lectures:

A short Introduction to Gamma-convergence
Continuous descriptions of limits of discrete systems
Effective properties of linear networks
Phase transitions for non-convex lattice energies
Deduction of energies in Fracture Mechanics from interatomic potentials

Department of Mathematics VIGRE University of Utah