Issue |
ESAIM: COCV
Volume 18, Number 2, April-June 2012
|
|
---|---|---|
Page(s) | 383 - 400 | |
DOI | https://doi.org/10.1051/cocv/2010102 | |
Published online | 13 April 2011 |
A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity
Mathematics Department, University of Cincinnati, 2600
Clifton Ave., Cincinnati, Ohio
45221,
USA
lorentaw@uc.edu
Received:
13
April
2010
Revised:
6
September
2010
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
where
u belongs to the subset of functions in
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
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.
Mathematics Subject Classification: 49N99 / 35J30
Key words: Aviles Giga functional
© EDP Sciences, SMAI, 2011
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