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Symmetry in materials science models under the divergence constraint

Presented by: 
Radu Ignat Université Paul Sabatier Toulouse III
Date: 
Thursday 17th January 2019 - 11:00 to 11:45
Venue: 
INI Seminar Room 1
Abstract: 
The symmetry of the order parameter is one of the most important features in materials science.
In this talk we will focus on the one-dimensional symmetry of transition layers u in some variational
models (such as smectic liquid crystals, thin film blisters, micromagnetics...)
where the divergence div(u) vanishes.
Namely, we determine a class of nonlinear potentials such that the minimal transition layers are
one-dimensional symmetric. In particular, this class includes in dimension N=2 the nonlinearities w^2
with w being an harmonic function or a solution to the wave equation,
while in dimensions N>2, this class contains a perturbation of the standard Ginzburg-Landau potential
as well as potentials having N+1 wells with prescribed transition cost between the wells.
For that, we develop a theory of calibrations for divergence-free maps in R^N (similar to the theory of entropies
for the Aviles-Giga model when N=2).
This is a joint work with Antonin Monteil (Louvain, Belgium).
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons