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.

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 $\Omega \subset {ℝ}^2$ the functional is $I_{}\left(u\right)=\frac{1}{2}{\int }_{\Omega }{ϵ}^{-1}{|1-|Du|}^2{|}^2+ϵ{\left|D^{2}u\right|}^2\mathrm{d}z$ where $u$ belongs to the subset of functions in $W_0^{2,2}\left(\Omega \right)$ whose gradient (in the sense of trace) satisfies $Du\left(x\right)·{\eta }_x=1$ where ${\eta }_x$ is the inward pointing unit normal to $\partial \Omega$ 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 $u_n\in W_0^{2,2}\left(\Omega \right)$ with $I_{{ϵ}_n}\left(u_n\right)\to 0$ as ${ϵ}_n\to 0$. A corollary to their work is that if there exists such a sequence $\left(u_n\right)$for a bounded domain $\Omega$, then $\Omega$ must be a ball and (up to change of sign) $u:={lim}_{n\to \infty }{u}_n=\mathrm{dist}(·,\partial \Omega )$. 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 $\Omega =B_1\left(0\right)$ without the requiring the trace condition on $Du$.

[1] F. Alouges, T. Riviere and S. Serfaty, Neel and cross-tie wall energies for planar micromagnetic configurations. ESAIM : COCV 8 (2002) 31-68. | Numdam | MR | Zbl

[2] L. Ambrosio, C. Delellis and C. Mantegazza, Line energies for gradient vector fields in the plane. Calc. Var. Partial Differential Equations 9 (1999) 327-355. | MR | Zbl

[3] L. Ambrosio, M. Lecumberry and T. Riviere, Viscosity property of minimizing micromagnetic configurations. Commun. Pure Appl. Math. 56 (2003) 681-688. | MR | Zbl

[4] P. Aviles and Y. Giga, A mathematical problem related to the physical theory of liquid crystal configurations, in Miniconference on geometry and partial differential equations 2, Canberra (1986) 1-16, Proc. Centre Math. Anal. Austral. Nat. Univ. 12, Austral. Nat. Univ., Canberra (1987). | MR

[5] P. Aviles and Y. Giga, The distance function and defect energy. Proc. Soc. Edinb. Sect. A 126 (1996) 923-938. | MR | Zbl

[6] P. Aviles and Y. Giga, On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg-Landau type energy for gradient fields. Proc. Soc. Edinb. Sect. A 129 (1999) 1-17. | MR | Zbl

[7] G. Carbou, Regularity for critical points of a nonlocal energy. Calc. Var. 5 (1997) 409-433. | MR | Zbl

[8] S. Conti, A. Desimone, S. Müller, R. Kohn and F. Otto, Multiscale modeling of materials - the role of analysis, in Trends in nonlinear analysis, Springer, Berlin (2003) 375-408. | Zbl

[9] A. Desimone, S. Müller, R. Kohn and F. Otto, A compactness result in the gradient theory of phase transitions. Proc. Soc. Edinb. Sect. A 131 (2001) 833-844. | Zbl

[10] A. Desimone, S. Müller, R. Kohn and F. Otto, A reduced theory for thin-film micromagnetics. Commun. Pure Appl. Math. 55 (2002) 1408-1460. | MR | Zbl

[11] L.C. Evans, Partial differential equations, Graduate Studies in Mathematics 19. American Mathematical Society (1998). | Zbl

[12] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics, CRC Press (1992). | MR | Zbl

[13] G. Gioia and M. Ortiz, The morphology and folding patterns of buckling-driven thin-film blisters. J. Mech. Phys. Solids 42 (1994) 531-559. | MR | Zbl

[14] R. Hardt and D. Kinderlehrer, Some regularity results in ferromagnetism. Commun. Partial Differ. Equ. 25 (2000) 1235-1258. | MR | Zbl

[15] R. Ignat and F. Otto, A compactness result in thin-film micromagnetics and the optimality of the Néel wall. J. Eur. Math. Soc. (JEMS) 10 (2008) 909-956. | MR | Zbl

[16] P. Jabin, F. Otto and B. Perthame, Line-energy Ginzburg-Landau models : zero-energy states. Ann. Sc. Norm. Super. Pisa Cl. Sci. 1 (2002) 187-202. | Numdam | MR | Zbl

[17] W. Jin and R.V. Kohn, Singular perturbation and the energy of folds. J. Nonlinear Sci. 10 (2000) 355-390. | MR | Zbl

[18] A. Lorent, A quantitative characterisation of functions with low Aviles Giga energy on convex domains. Ann. Sc. Norm. Super. Pisa Cl. Sci. (submitted). Available at http://arxiv.org/abs/0902.0154v1. | Zbl

[19] T. Riviere and S. Serfaty, Limiting domain wall energy for a problem related to micromagnetics. Commun. Pure Appl. Math. 54 (2001) 294-338. | MR | Zbl

[20] E.M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series 30. Princeton University Press (1970). | MR | Zbl