The H–1-norm of tubular neighbourhoods of curves

Yves van Gennip; Mark A. Peletier

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 1, page 131-154
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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van Gennip, Yves, and Peletier, Mark A.. "The H–1-norm of tubular neighbourhoods of curves." ESAIM: Control, Optimisation and Calculus of Variations 17.1 (2011): 131-154. <http://eudml.org/doc/197294>.

@article{vanGennip2011,
abstract = { 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 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. },
author = {van Gennip, Yves, Peletier, Mark A.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Gamma-convergence; elastica functional; negative Sobolev norm; curves; asymptotic expansion; -convergence},
language = {eng},
month = {2},
number = {1},
pages = {131-154},
publisher = {EDP Sciences},
title = {The H–1-norm of tubular neighbourhoods of curves},
url = {http://eudml.org/doc/197294},
volume = {17},
year = {2011},
}

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
DA - 2011/2//
PB - EDP Sciences
VL - 17
IS - 1
SP - 131
EP - 154
AB - 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 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.
LA - eng
KW - Gamma-convergence; elastica functional; negative Sobolev norm; curves; asymptotic expansion; -convergence
UR - http://eudml.org/doc/197294
ER -

References

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