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We study the H^{-1}-norm of the function 1 on tubular neighbourhoods ofcurves in R^2. We take the limit of small thickness epsilon, and we prove twodifferent asymptotic results. The first is an asymptotic development for afixed curve in the limit epsilon to 0, containing contributions from the lengthof 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 curvemay vary along the sequence epsilon to 0. We prove that a rescaled version ofthe H^{-1}-norm, which focuses on the epsilon^5 curvature term, Gamma-convergesto the L^2-norm of curvature. In addition, sequences along which the rescalednorm is bounded are compact in the W^{1,2} -topology.  Our main tools are the maximum principle for elliptic equations and the useof appropriate trial functions in the variational characterisation of theH^{-1}-norm. For the Gamma-convergence result we use the theory of systems ofcurves without transverse crossings to handle potential intersections in thelimit.

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