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.

Keywords: gamma-convergence, elastica functional, negative Sobolev norm, curves, asymptotic expansion

@article{COCV_2011__17_1_131_0, author = {van Gennip, Yves and Peletier, Mark A.}, title = {The $H^{-1}$-norm of tubular neighbourhoods of curves}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {131--154}, publisher = {EDP-Sciences}, volume = {17}, number = {1}, year = {2011}, doi = {10.1051/cocv/2009044}, zbl = {1213.49052}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2009044/} }

TY - JOUR AU - van Gennip, Yves AU - Peletier, Mark A. TI - The $H^{-1}$-norm of tubular neighbourhoods of curves JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2011 DA - 2011/// SP - 131 EP - 154 VL - 17 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2009044/ UR - https://zbmath.org/?q=an%3A1213.49052 UR - https://doi.org/10.1051/cocv/2009044 DO - 10.1051/cocv/2009044 LA - en ID - COCV_2011__17_1_131_0 ER -

%0 Journal Article %A van Gennip, Yves %A Peletier, Mark A. %T The $H^{-1}$-norm of tubular neighbourhoods of curves %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 131-154 %V 17 %N 1 %I EDP-Sciences %U https://doi.org/10.1051/cocv/2009044 %R 10.1051/cocv/2009044 %G en %F COCV_2011__17_1_131_0

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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