2009 DG-MP-PDE Seminar - 04

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.