# 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 (2012)

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

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topLorent, 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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