## 2009 DG-MP-PDE Seminar - 04

- Date: 04/07/2009

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