2009 DG-MP-PDE Seminar - 04

  • Date: 04/07/2009
Yves van Gennip (Simon Fraser University)

University of British Columbia


Curved is the norm


The H^{-1}-norm appears naturally in many areas of science. For example
it is used to express the electrostatic energy of a charge
distribution. We study this norm on tubular neighbourhoods of curves,
mainly motivated by its appearance in energy functionals that describe
the formation of partially localised patterns. These patterns are
structures that are localised in some directions and extended in
others. In two dimensions they resemble fattened curves. We present two
results of a slightly different nature. On the tubular neighbourhood of
a single fixed smooth curve we give a rigorous asymptotic expansion of
the H^{-1}-norm, where the small parameter epsilon is the thickness of
the neighbourhood. Restricting ourselves to closed curves, we also
prove a Gamma-convergence result. Both these results show the influence
of a variety of geometric properties of the curve (length, curvature,
open or closed) entering the H^{-1}-norm on different levels of epsilon.


3:30pm, WMAX 110