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 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.

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/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

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  1. [1] W. Allard, On the first variation of a varifold. Ann. Math.95 (1972) 417–491. Zbl0252.49028MR307015
  2. [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. [3] G. Bellettini and L. Mugnai, A varifolds representation of the relaxed elastica functional. Journal of Convex Analysis14 (2007) 543–564. Zbl1127.49032MR2341303
  4. [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. [5] A. Doelman and H. van der Ploeg, Homoclinic stripe patterns. SIAM J. Appl. Dyn. Syst.1 (2002) 65–104. Zbl1004.35063MR1972974
  6. [6] I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications. Oxford University Press Inc., New York, USA (1995). Zbl0852.47030MR1373430
  7. [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. [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. [9] M. Kac, Can one hear the shape of a drum? Amer. Math. Monthly73 (1966) 1–23. Zbl0139.05603MR201237
  10. [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. [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. [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. [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

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