A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity

Andrew Lorent

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 2, page 383-400
  • ISSN: 1292-8119

Abstract

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The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain Ω ⊂ ℝ2 the functional is I ϵ ( u ) = 1 2 Ω ϵ -1 1 Du 2 2 + ϵ D 2 u 2 d z where u belongs to the subset of functions in W 0 2 , 2 ( Ω ) whose gradient (in the sense of trace) satisfies Du(x)·ηx = 1 where ηx is the inward pointing unit normal to ∂Ω at x. In [Ann. Sc. Norm. Super. Pisa Cl. Sci. 1 (2002) 187–202] Jabin et al. characterized a class of functions which includes all limits of sequences u n W 0 2 , 2 ( Ω ) with Iϵn(un) → 0 as ϵn → 0. A corollary to their work is that if there exists such a sequence (un) for a bounded domain Ω, then Ω must be a ball and (up to change of sign) u: = limn → ∞un = dist(·,∂Ω). Recently [Lorent, Ann. Sc. Norm. Super. Pisa Cl. Sci. (submitted), http://arxiv.org/abs/0902.0154v1] we provided a quantitative generalization of this corollary over the space of convex domains using ‘compensated compactness’ inspired calculations of DeSimone et al. [Proc. Soc. Edinb. Sect. A 131 (2001) 833–844]. In this note we use methods of regularity theory and ODE to provide a sharper estimate and a much simpler proof for the case where Ω = B1(0) without the requiring the trace condition on Du.

How to cite

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Lorent, Andrew. "A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity ." ESAIM: Control, Optimisation and Calculus of Variations 18.2 (2012): 383-400. <http://eudml.org/doc/221934>.

@article{Lorent2012,
abstract = {The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain Ω ⊂ ℝ2 the functional is \hbox\{$I_\{\ep\}(u)=\frac\{1\}\{2\}\int_\{\Omega\} \ep^\{-1\}\lt|1-\lt|Du\rt|^2\rt|^2+\ep\lt|D^2 u\rt|^2 \{\rm d\}z$\} where u belongs to the subset of functions in \hbox\{$W^\{2,2\}_\{0\}(\Omega)$\} whose gradient (in the sense of trace) satisfies Du(x)·ηx = 1 where ηx is the inward pointing unit normal to ∂Ω at x. In [Ann. Sc. Norm. Super. Pisa Cl. Sci. 1 (2002) 187–202] Jabin et al. characterized a class of functions which includes all limits of sequences \hbox\{$u_n\in W^\{2,2\}_0(\Omega)$\} with Iϵn(un) → 0 as ϵn → 0. A corollary to their work is that if there exists such a sequence (un) for a bounded domain Ω, then Ω must be a ball and (up to change of sign) u: = limn → ∞un = dist(·,∂Ω). Recently [Lorent, Ann. Sc. Norm. Super. Pisa Cl. Sci. (submitted), http://arxiv.org/abs/0902.0154v1] we provided a quantitative generalization of this corollary over the space of convex domains using ‘compensated compactness’ inspired calculations of DeSimone et al. [Proc. Soc. Edinb. Sect. A 131 (2001) 833–844]. In this note we use methods of regularity theory and ODE to provide a sharper estimate and a much simpler proof for the case where Ω = B1(0) without the requiring the trace condition on Du. },
author = {Lorent, Andrew},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Aviles Giga functional; liquid crystals; compensated compactness},
language = {eng},
month = {7},
number = {2},
pages = {383-400},
publisher = {EDP Sciences},
title = {A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity },
url = {http://eudml.org/doc/221934},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Lorent, Andrew
TI - A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/7//
PB - EDP Sciences
VL - 18
IS - 2
SP - 383
EP - 400
AB - The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain Ω ⊂ ℝ2 the functional is \hbox{$I_{\ep}(u)=\frac{1}{2}\int_{\Omega} \ep^{-1}\lt|1-\lt|Du\rt|^2\rt|^2+\ep\lt|D^2 u\rt|^2 {\rm d}z$} where u belongs to the subset of functions in \hbox{$W^{2,2}_{0}(\Omega)$} whose gradient (in the sense of trace) satisfies Du(x)·ηx = 1 where ηx is the inward pointing unit normal to ∂Ω at x. In [Ann. Sc. Norm. Super. Pisa Cl. Sci. 1 (2002) 187–202] Jabin et al. characterized a class of functions which includes all limits of sequences \hbox{$u_n\in W^{2,2}_0(\Omega)$} with Iϵn(un) → 0 as ϵn → 0. A corollary to their work is that if there exists such a sequence (un) for a bounded domain Ω, then Ω must be a ball and (up to change of sign) u: = limn → ∞un = dist(·,∂Ω). Recently [Lorent, Ann. Sc. Norm. Super. Pisa Cl. Sci. (submitted), http://arxiv.org/abs/0902.0154v1] we provided a quantitative generalization of this corollary over the space of convex domains using ‘compensated compactness’ inspired calculations of DeSimone et al. [Proc. Soc. Edinb. Sect. A 131 (2001) 833–844]. In this note we use methods of regularity theory and ODE to provide a sharper estimate and a much simpler proof for the case where Ω = B1(0) without the requiring the trace condition on Du.
LA - eng
KW - Aviles Giga functional; liquid crystals; compensated compactness
UR - http://eudml.org/doc/221934
ER -

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