The H -1 -norm of tubular neighbourhoods of curves
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 131-154.

We study the H-1-norm of the function 1 on tubular neighbourhoods of curves in 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.

DOI : 10.1051/cocv/2009044
Classification : 49Q99
Mots clés : gamma-convergence, elastica functional, negative Sobolev norm, curves, asymptotic expansion
@article{COCV_2011__17_1_131_0,
     author = {van Gennip, Yves and Peletier, Mark A.},
     title = {The $H^{-1}$-norm of tubular neighbourhoods of curves},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {131--154},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {1},
     year = {2011},
     doi = {10.1051/cocv/2009044},
     zbl = {1213.49052},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2009044/}
}
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
SP  - 131
EP  - 154
VL  - 17
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2009044/
DO  - 10.1051/cocv/2009044
LA  - en
ID  - COCV_2011__17_1_131_0
ER  - 
%0 Journal Article
%A van Gennip, Yves
%A Peletier, Mark A.
%T The $H^{-1}$-norm of tubular neighbourhoods of curves
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2011
%P 131-154
%V 17
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2009044/
%R 10.1051/cocv/2009044
%G en
%F COCV_2011__17_1_131_0
van Gennip, Yves; Peletier, Mark A. The $H^{-1}$-norm of tubular neighbourhoods of curves. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 131-154. doi : 10.1051/cocv/2009044. http://www.numdam.org/articles/10.1051/cocv/2009044/

[1] W. Allard, On the first variation of a varifold. Ann. Math. 95 (1972) 417-491. | MR | Zbl

[2] G. Bellettini and L. Mugnai, Characterization and representation of the lower semicontinuous envelope of the elastica functional. Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004) 839-880. | Numdam | MR | Zbl

[3] G. Bellettini and L. Mugnai, A varifolds representation of the relaxed elastica functional. Journal of Convex Analysis 14 (2007) 543-564. | MR | Zbl

[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. | Zbl

[5] A. Doelman and H. Van Der Ploeg, Homoclinic stripe patterns. SIAM J. Appl. Dyn. Syst. 1 (2002) 65-104. | MR | Zbl

[6] I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications. Oxford University Press Inc., New York, USA (1995). | MR | Zbl

[7] O. Gonzalez and J. Maddocks, Global curvature, thickness, and the ideal shape of knots. Proc. Natl. Acad. Sci. USA 96 (1999) 4769-4773. | MR | Zbl

[8] J. Hutchinson, Second fundamental form for varifolds and the existence of surfaces minimising curvature. Indiana Univ. Math. J. 35 (1986) 45-71. | MR | Zbl

[9] M. Kac, Can one hear the shape of a drum? Amer. Math. Monthly 73 (1966) 1-23. | MR | Zbl

[10] M.A. Peletier and M. Röger, Partial localization, lipid bilayers, and the elastica functional. Arch. Rational Mech. Anal. 193 (2008) 475-537. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

Cité par Sources :