# Mini-Workshop

##
Andrea Braides
(Dip. Mat. Univ. Roma "Tor Vergata")
,
Adriana Garroni
(Dip. Mat. Univ. Roma ``La Sapienza'')
,
Piero Villaggio

created by magnani on 24 Apr 2006

modified by paolini on 13 May 2017

27 apr 2006

**Abstract.**

Dear All,
I am glad to announce that

. on Thursday, 27 April
. in ``Aula Magna'' of the Mathematics Department

there will be a miniworkshop with three seminars.

At 15:00. Adriana Garroni (Rome University, I )
will present

``A variational model for dislocations''
At 16:00. Andrea Braides (Rome University, II)
will present

``The use of Gamma-convergence in the study of asymptotic problems ''

At 17:00. Piero Villaggio (Pisa University)
will present

``Problemi variazionali della locomozione''

The abstracts of the first two seminars follow, respectively:

ABSTRACT I.

Dislocations are line defects which are present on slip planes
of crystals and are consider responsible for many interesting phenomena,
like plasticity and hardening. Those defects can be described by a
multi-phase field variational model recently introduced by Koslowski and
Ortiz. This is a 2d vector phase-transition functional, with a non local
singular perturbation and a non linear potential which vanishes on a
lattice. We describe, by means of $\Gamma$-convergence, the asymptotic
behaviour of these functionals as the lattice parameter goes to zero and
we obtain, in the limit, an anisotropic line tension energy.
The anisotropic line tension energy density can be completely
described in the scalar case
(where the scalar phase describe the activation of one slip system in
the slip plane) and exhibits
a one dimensional character (i.e. the optimal profile is one
dimensional), while in the vector case we can show by means of
explicit construction that the optimal transition may produce
oscillations.

ABSTRACT II.

Gamma-convergence is a commonly recognized tool for the description of
complex variational problems involving small parameters (homogenization,
phase transitions, atomistic systems, etc.). Anyhow, its application is
sometimes criticized (besides not giving correct information for local
minimizers) either for not giving an accurate description with respect to
varying parameters (e.g., boundary conditions, applied forces, etc.) or
simply not providing the same approximate theories that are used by
practitioners. I will present some proposal on how to use
Gamma-convergence to fit (some of) those requirements.