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.

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 2 the functional is I ( u ) = 1 2 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 ) · η x = 1 where η x 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 W 0 2,2 (Ω) with I ϵ n ( u n ) 0 as ϵ n 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 u n = 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 = B 1 ( 0 ) without the requiring the trace condition on D u .

DOI : 10.1051/cocv/2010102
Classification : 49N99, 35J30
Mots clés : aviles giga functional
@article{COCV_2012__18_2_383_0,
     author = {Lorent, Andrew},
     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},
     volume = {18},
     number = {2},
     year = {2012},
     doi = {10.1051/cocv/2010102},
     zbl = {1259.49077},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2010102/}
}
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
SP  - 383
EP  - 400
VL  - 18
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2010102/
DO  - 10.1051/cocv/2010102
LA  - en
ID  - COCV_2012__18_2_383_0
ER  - 
%0 Journal Article
%A Lorent, Andrew
%T A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2012
%P 383-400
%V 18
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2010102/
%R 10.1051/cocv/2010102
%G en
%F COCV_2012__18_2_383_0
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. http://archive.numdam.org/articles/10.1051/cocv/2010102/

[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

Cité par Sources :