The ${H}^{-1}$-norm of tubular neighbourhoods of curves
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 1, pp. 131-154.

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 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 W1,2-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.

DOI: 10.1051/cocv/2009044
Classification: 49Q99
Keywords: gamma-convergence, elastica functional, negative Sobolev norm, curves, asymptotic expansion
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title = {The $H^{-1}$-norm of tubular neighbourhoods of curves},
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van Gennip, Yves; Peletier, Mark A. The $H^{-1}$-norm of tubular neighbourhoods of curves. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 1, pp. 131-154. doi : 10.1051/cocv/2009044. http://www.numdam.org/articles/10.1051/cocv/2009044/

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