# 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

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topvan 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 -

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