1 results
The H –1-norm of tubular neighbourhoods of curves
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- Journal:
- ESAIM: Control, Optimisation and Calculus of Variations / Volume 17 / Issue 1 / January 2011
- Published online by Cambridge University Press:
- 04 December 2009, pp. 131-154
- Print publication:
- January 2011
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- Article
- Export citation
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We study the H –1-norm of the function 1 on tubular neighbourhoods of curves in ${\mathbb R}^{2}$
. 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 ε 3), the ends (ε 4), and the curvature (ε 5). 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 ε 5 curvature term, Γ-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 Γ-convergence result we use the theory of systems of curves without transverse crossings to handle potential intersections in the limit.
