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:

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: This is joint work with M. Bertsch (Roma II) and J. Hulshof (VU Amsterdam).