**Liquid crystals and harmonic maps**

A liquid crystal (lc) is, roughly, a fluid equipped with a field
of directions (think of the local direction of an oblong molecule), that is:
a mapping into the unit sphere *S*. Associated with this mapping the fluid
exhibits an energy density, which measures the local nonuniformness of the
"director field". If no flow is present, an equilibrium situation is
typically a stationary point of the corresponding energy integral over the
region *R*, occupied by the lc.
This establishes a connection with the theory
of harmonic maps with the unit sphere as target. We discuss the situation
where *R* is an infinitely long cylinder and where the situation is
cylindrically symmetric. So, *R* may be replaced
by the unit disk *D*. In this
situation, we are confronted with the following key question:

Given a
sufficiently smooth map *v*: *D* --> *S*,
is there a harmonic map *u*, homotopic
with *v* ?

Eells and Sampson proposed to study this question through the
associated "heat flow", which is the parabolic problem corresponding to the
(elliptic) harmonic map equation. A central theme of the lecture will be the
blow-up behavior of this harmonic map heat flow. Regretfully, this heat flow
problem is too simple to describe the dynamical behavior of an lc. We shall
describe the character of the additional problems.
Most of this is joint work with Michiel Bertsch, Roberta Dal-Passo and
Elisabetta Vilucchi (Roma II).

*Reinier van der Hout*

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