# 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

## Access Full Article

top## Abstract

top## How to cite

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/272951>.

@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 thicknessε, 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},

number = {1},

pages = {131-154},

publisher = {EDP-Sciences},

title = {The H–1-norm of tubular neighbourhoods of curves},

url = {http://eudml.org/doc/272951},

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

PY - 2011

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 thicknessε, 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/272951

ER -

## References

top- [1] W. Allard, On the first variation of a varifold. Ann. Math.95 (1972) 417–491. Zbl0252.49028MR307015
- [2] G. Bellettini and L. Mugnai, Characterization and representation of the lower semicontinuous envelope of the elastica functional. Ann. Inst. H. Poincaré Anal. Non Linéaire21 (2004) 839–880. Zbl1110.49014MR2097034
- [3] G. Bellettini and L. Mugnai, A varifolds representation of the relaxed elastica functional. Journal of Convex Analysis14 (2007) 543–564. Zbl1127.49032MR2341303
- [4] T. D'Aprile, Behaviour of symmetric solutions of a nonlinear elliptic field equation in the semi-classical limit: Concentration around a circle. Electronic J. Differ. Equ.2000 (2000) 1–40. Zbl0954.35057
- [5] A. Doelman and H. van der Ploeg, Homoclinic stripe patterns. SIAM J. Appl. Dyn. Syst.1 (2002) 65–104. Zbl1004.35063MR1972974
- [6] I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications. Oxford University Press Inc., New York, USA (1995). Zbl0852.47030MR1373430
- [7] O. Gonzalez and J. Maddocks, Global curvature, thickness, and the ideal shape of knots. Proc. Natl. Acad. Sci. USA96 (1999) 4769–4773. Zbl1057.57500MR1692638
- [8] J. Hutchinson, Second fundamental form for varifolds and the existence of surfaces minimising curvature. Indiana Univ. Math. J.35 (1986) 45–71. Zbl0561.53008MR825628
- [9] M. Kac, Can one hear the shape of a drum? Amer. Math. Monthly73 (1966) 1–23. Zbl0139.05603MR201237
- [10] M.A. Peletier and M. Röger, Partial localization, lipid bilayers, and the elastica functional. Arch. Rational Mech. Anal.193 (2008) 475–537. Zbl1170.74034MR2525110
- [11] N. Sidorova and O. Wittich, Construction of surface measures for Brownian motion, in Trends in stochastic analysis: a Festschrift in honour of Heinrich von Weizsäcker, LMS Lecture Notes 353, Cambridge UP (2009) 123–158. Zbl1176.58020MR2562153
- [12] Y. van Gennip and M.A. Peletier, Copolymer-homopolymer blends: global energy minimisation and global energy bounds. Calc. Var. Part. Differ. Equ.33 (2008) 75–111. Zbl1191.49006MR2413102
- [13] Y. van Gennip and M.A. Peletier, Stability of monolayers and bilayers in a copolymer-homopolymer blend model. Interfaces Free Bound.11 (2009) 331–373. Zbl1179.93106MR2546603

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.