Maria Stella Gelli

Continuum Limits of Discrete Systems without Convexity Hypotheses

We describe the variational limit of one-dimensional nearest-neighbour systems of interactions, under no structure hypotheses on the discrete energy densities. We show that the continuum limit is characterized by a bulk and a interfacial energy density, which can be explicitly computed from the discrete energies through operations of limit, scaling and regularization that highlight possible bulk oscillations and multiple cracking.

Local and Nonlocal Continuum Limits of Ising-Type Energies for Spin Systems

We study, through a

Fracture mechanics in perforated domains: A variational model for brittle porous media

\Gamma

Asymptotic analysis of Mumford-Shah type energies in periodically-perforated domains

-convergence procedure, the discrete to continuum limit of Ising-type energies of the form

Asymptotic analysis of microscopic impenetrability constraints for atomistic systems

F_\varepsilon(u)=-\sum_{i,j}c_{i,j}^\varepsilon u_i u_j,

Limits of Discrete Systems With Long-Range Interactions

where

Minimal measures, one-dimensional currents and the Monge-Kantorovich problem

u

Finite-difference approximation of energies in fracture mechanics

is a spin variable defined on a portion of a cubic lattice

Motion and Pinning of Discrete Interfaces

\varepsilon{\Bbb Z}^d\cap\Omega

Continuity of an optimal transport in Monge problem

,