The H^{-1}-norm of tubular neighbourhoods of curves

TL;DR

This paper analyzes the asymptotic behavior of the H^{-1}-norm of functions on tubular neighborhoods of curves in R^2 as the neighborhood thickness approaches zero, revealing contributions from length, ends, and curvature.

Contribution

It provides new asymptotic expansions and a Gamma-convergence result linking the H^{-1}-norm to the curvature's L^2-norm, considering variable curves.

Findings

Asymptotic expansion includes length, ends, and curvature contributions.

Rescaled H^{-1}-norm Gamma-converges to the L^2-norm of curvature.

Sequences with bounded rescaled norm are compact in W^{1,2} topology.

Abstract

We study the H^{-1}-norm of the function 1 on tubular neighbourhoods of curves in R^2. We take the limit of small thickness epsilon, and we prove two different asymptotic results. The first is an asymptotic development for a fixed curve in the limit epsilon to 0, containing contributions from the length of the curve (at order epsilon^3), the ends (epsilon^4), and the curvature (epsilon^5). The second result is a Gamma-convergence result, in which the central curve may vary along the sequence epsilon to 0. We prove that a rescaled version of the H^{-1}-norm, which focuses on the epsilon^5 curvature term, Gamma-converges to the L^2-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 Gamma-convergence result we use the theory of systems of curves without transverse crossings to handle potential intersections in the limit.