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).