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

Lorent, Andrew

doi:10.1051/cocv/2010102

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

Lorent, Andrew

ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 2, pp. 383-400.

Résumé

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 $Ω \subset ℝ^{2}$ the functional is $I_{} (u) = \frac{1}{2} \int_{Ω} ϵ^{- 1} {| 1 - | D u |}^{2} |^{2} + ϵ {| D^{2} u |}^{2} d z$ where $u$ belongs to the subset of functions in $W_{0}^{2, 2} (Ω)$ whose gradient (in the sense of trace) satisfies $D u (x) \cdot η_{x} = 1$ where $η_{x}$ is the inward pointing unit normal to $\partial Ω$ 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 $u_{n} \in W_{0}^{2, 2} (Ω)$ with $I_{ϵ_{n}} (u_{n}) \to 0$ as $ϵ_{n} \to 0$ . A corollary to their work is that if there exists such a sequence $(u_{n})$ for a bounded domain $Ω$ , then $Ω$ must be a ball and (up to change of sign) $u : = {lim}_{n \to \infty} u_{n} = dist (\cdot, \partial Ω)$ . 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 $Ω = B_{1} (0)$ without the requiring the trace condition on $D u$ .

Zbl | 1 citation dans Numdam

DOI : 10.1051/cocv/2010102

Classification : 49N99, 35J30
Mots-clés : aviles giga functional

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     title = {A simple proof of the characterization of functions of low {Aviles} {Giga} energy on a ball \protect\emph{via }regularity},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {383--400},
     publisher = {EDP-Sciences},
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     year = {2012},
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     zbl = {1259.49077},
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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, Tome 18 (2012) no. 2, pp. 383-400. doi : 10.1051/cocv/2010102. https://www.numdam.org/articles/10.1051/cocv/2010102/

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