The H^–1-norm of tubular neighbourhoods of curves

¹ Department of Mathematics Simon Fraser University, 8888 University Drive, Burnaby, British Columbia V5A 1S6, Canada.
² Dept. of Mathematics and Computer Science and Institute for Complex Molecular Systems, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.

Received: 21 March 2009

Abstract

We study the H^–1-norm of the function 1 on tubular neighbourhoods of curves in . We take the limit of small thickness ε, and we prove two different asymptotic results. The first is an asymptotic development for a fixed curve in the limit ε → 0, containing contributions from the length of the curve (at order ε³), the ends (ε⁴), and the curvature (ε⁵). The second result is a Γ-convergence result, in which the central curve may vary along the sequence ε → 0. We prove that a rescaled version of the H^–1-norm, which focuses on the ε⁵ curvature term, Γ-converges to the L²-norm of curvature. In addition, sequences along which the rescaled norm is bounded are compact in the W^1,2-topology. Our main tools are the maximum principle for elliptic equations and the use of appropriate trial functions in the variational characterisation of the H^–1-norm. For the Γ-convergence result we use the theory of systems of curves without transverse crossings to handle potential intersections in the limit.