L1

For atomistic (and related) material models, global minimization

gives the wrong qualitative behaviour; a theory of equilibrium

solutions needs to be defined in different terms. In this talk, a

process based on gradient flow evolutions is presented, to describe

local minimization for simple atomistic models based on the Lennard-

Jones potential. As an application, it is shown that an atomistic

gradient flow evolution converges to a gradient flow of a continuum

energy, as the spacing between the atoms tends to zero. In addition,

the convergence of the resulting equilibria is investigated, in the

case of both elastic deformation and fracture.

In this talk we will discuss a theory of divergence-measure fields and related

geometric measures, developed recently, and its applications to some fundamental

issues in mathematical continuum physics and nonlinear conservation laws whose

solutions have very weak regularity, including hyperbolic conservation laws,

degenerate parabolic equations, degenerate elliptic equations, among others.