# 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 -

## References

top- W. Allard, On the first variation of a varifold. Ann. Math.95 (1972) 417–491.
- 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.
- G. Bellettini and L. Mugnai, A varifolds representation of the relaxed elastica functional. Journal of Convex Analysis14 (2007) 543–564.
- 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.
- A. Doelman and H. van der Ploeg, Homoclinic stripe patterns. SIAM J. Appl. Dyn. Syst.1 (2002) 65–104.
- I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications. Oxford University Press Inc., New York, USA (1995).
- O. Gonzalez and J. Maddocks, Global curvature, thickness, and the ideal shape of knots. Proc. Natl. Acad. Sci. USA96 (1999) 4769–4773.
- J. Hutchinson, Second fundamental form for varifolds and the existence of surfaces minimising curvature. Indiana Univ. Math. J.35 (1986) 45–71.
- M. Kac, Can one hear the shape of a drum? Amer. Math. Monthly73 (1966) 1–23.
- M.A. Peletier and M. Röger, Partial localization, lipid bilayers, and the elastica functional. Arch. Rational Mech. Anal.193 (2008) 475–537.
- 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 Notes353, Cambridge UP (2009) 123–158.
- 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.
- 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.

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