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

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/272816>.

@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 ⊂ ℝ2the functional is $I_\{\}(u)=\frac\{1\}\{2\}\int _\{\Omega \} ^\{-1\}|1-|Du|^2|^2+|D^2 u|^2 \{\rm d\}z$ I ϵ ( u ) = 1 2 ∫ Ω ϵ -1 1 − Du 2 2 + ϵ D 2 u 2 d z whereubelongs to the subset of functions in $W^\{2,2\}_\{0\}(\Omega )$W02,2(Ω) whose gradient (in the sense of trace) satisfiesDu(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 $u_n\in W^\{2,2\}_0(\Omega )$ u n ∈ W 0 2 , 2 ( Ω ) withIϵn(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; Aviles Giga functional; liquid crystals; compensated compactness},
language = {eng},
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/272816},
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
PY - 2012
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 ⊂ ℝ2the functional is $I_{}(u)=\frac{1}{2}\int _{\Omega } ^{-1}|1-|Du|^2|^2+|D^2 u|^2 {\rm d}z$ I ϵ ( u ) = 1 2 ∫ Ω ϵ -1 1 − Du 2 2 + ϵ D 2 u 2 d z whereubelongs to the subset of functions in $W^{2,2}_{0}(\Omega )$W02,2(Ω) whose gradient (in the sense of trace) satisfiesDu(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 $u_n\in W^{2,2}_0(\Omega )$ u n ∈ W 0 2 , 2 ( Ω ) withIϵn(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; Aviles Giga functional; liquid crystals; compensated compactness
UR - http://eudml.org/doc/272816
ER -