Andrea Braides

Approximation of free-discontinuity problems

Functions of bounded variation.- Special functions of bounded variation.- Examples of approximation.- A general approach to approximation.- Non-local approximation.

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.

SURFACE ENERGIES IN NONCONVEX DISCRETE SYSTEMS

We analyze the variational limit of one-dimensional next-to-nearest neighbours (NNN) discrete systems as the lattice size tends to zero when the energy densities are of multiwell or Lennard–Jones type. Properly scaling the energies, we study several phenomena as the formation of boundary layers and phase transitions. We also study the presence of local patterns and of anti-phase transitions in the asymptotic behaviour of the ground states of NNN model subject to Dirichlet boundary conditions. We use this information to prove a localization of fracture result in the case of Lennard–Jones type potentials.

REITERATED HOMOGENIZATION OF INTEGRAL FUNCTIONALS

We consider the homogenization of sequences of integral functionals defined on media with several length-scales. Our general results connected to the corresponding homogenized functional are used to analyze new types of structures and to illustrate the wide range of effective properties achievable through reiteration. In particular, we consider a two-phase structure which is optimal when the integrand is a quadratic form and point out examples where the macroscopic behavior of this structure underlines an effective energy density which is lower than that of the best possible multirank laminate. We also present some results connected to a reiterated structure where the effective property is extremely sensitive of the growth of the integrand.

Phase and anti-phase boundaries in binary discrete systems: a variational viewpoint

A variational description of nearest-neighbours and next-to-nearest neighbours binary lattice systems is provided. By studying the -limit of proper scaling of the energies of the systems, phase and anti-phase boundary phenomena are highlighted and it is shown how they depend on the geometry of the lattice. [ DOI : 10.1685 / CSC06007] About DOI

Asymptotic analysis of periodically-perforated nonlinear media

We give a direct proof of the nonlinear vector-valued variational version of the Cioranescu Murat result on the asymptotic behaviour of Dirichlet problems in perforated domains giving rise to extra terms. Our method is based on a lemma which allows to modify sequences of functions in the vicinity of the perforation, in the spirit of a method proposed by De Giorgi to match boundary conditions. We describe the extra term by a capacitary formula involving a quasiconvexification process. Nonexistence and nonpositive homogeneity phenomena are discussed.

Homogenization of oscillating boundaries and applications to thin films

The study carried on in this paper draws its motivation from the problem of the asymptotic description of nonlinearly elastic thin films with a fast-oscillating profile. The behaviour of such films is governed by an elastic energy, where two parameters intervene: a first parameter e represents the thickness of the thin film and a second one δ the scale of the oscillations. The analytic description of the elastic energy is given by a functional of the form

A derivation of linear elastic energies from pair-interaction atomistic systems

Pair-interaction atomistic energies may give rise, in the framework of the passage from discrete systems to continuous variational problems, to nonlinear energies with genuinely quasiconvex integrands. This phenomenon takes place even for simple harmonic interactions as shown by an example by Friesecke and Theil [19]. On the other hand, a rigorous derivation of linearly elastic energies from energies with quasiconvex integrands can be obtained by

Homogenization of almost periodic monotone operators

\Gamma

NON-LOCAL VARIATIONAL LIMITS OF DISCRETE SYSTEMS

-convergence following the method by Dal Maso, Negri and Percivale [14]. We show that the derivation of linear theories by