Mathematics Colloquium
Rein van der Hout
(VU Amsterdam)
"Energy concentration in harmonic map heat flows on the disk"
| Date: |
Mon, February 22, 2010 |
| Time: |
17:15 |
| Place: |
Research II Lecture Hall |
Abstract: A harmonic map from the unit disk into the unit sphere
is a stationary point of a certain energy functional. The
problem: Under Dirichlet boudary data, does there exist a harmonic map,
homotopic to a given map with the same boundary data? has a negative answer in general. This is reflected in the behavior of the corresponding gradient
flow ("heat flow"): given appropriate data, it exhibits blow-up (generically in
finite time, and only a finite number of times). As long as the flow is smooth,
the energy is nonincreasing and it is known that "nonincreasing energy and
2D-domain" implies a uniqueness class (Freire). In the radially symmetric
case, blow-up has here been described in various ways:
- energy-concentration in the origin (=only possible location);
- blow-up of a derivative;
- the image of the origin jumps from north pole to south pole (or vice
versa).
These descriptions are equivalent (not trivial!). Energy concentration makes
the energy jump downwards, so that we remain in the uniqueness class.
(Actually, there are infinitely many solutions outside this class; and some of
them may have a physical interpretation.) In this lecture, we shall answer
the questions:
- How much energy precisely concentrates?
- Is it possible to have simultaneous multiple jumps (say: north
pole to south pole to north pole)?
This is joint work with M. Bertsch (Roma II) and J. Hulshof (VU Amsterdam).
