# 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 -

## References

top- F. Alouges, T. Riviere and S. Serfaty, Neel and cross-tie wall energies for planar micromagnetic configurations. ESAIM : COCV8 (2002) 31–68.
- L. Ambrosio, C. Delellis and C. Mantegazza, Line energies for gradient vector fields in the plane. Calc. Var. Partial Differential Equations9 (1999) 327–355.
- L. Ambrosio, M. Lecumberry and T. Riviere, Viscosity property of minimizing micromagnetic configurations. Commun. Pure Appl. Math.56 (2003) 681–688.
- P. Aviles and Y. Giga, A mathematical problem related to the physical theory of liquid crystal configurations, in Miniconference on geometry and partial differential equations2, Canberra (1986) 1–16, Proc. Centre Math. Anal. Austral. Nat. Univ.12, Austral. Nat. Univ., Canberra (1987).
- P. Aviles and Y. Giga, The distance function and defect energy. Proc. Soc. Edinb. Sect. A126 (1996) 923–938.
- 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. A129 (1999) 1–17.
- G. Carbou, Regularity for critical points of a nonlocal energy. Calc. Var.5 (1997) 409–433.
- 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.
- A. DeSimone, S. Müller, R. Kohn and F. Otto, A compactness result in the gradient theory of phase transitions. Proc. Soc. Edinb. Sect. A131 (2001) 833–844.
- A. DeSimone, S. Müller, R. Kohn and F. Otto, A reduced theory for thin-film micromagnetics. Commun. Pure Appl. Math.55 (2002) 1408–1460.
- L.C. Evans, Partial differential equations, Graduate Studies in Mathematics19. American Mathematical Society (1998).
- L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics, CRC Press (1992).
- G. Gioia and M. Ortiz, The morphology and folding patterns of buckling-driven thin-film blisters. J. Mech. Phys. Solids42 (1994) 531–559.
- R. Hardt and D. Kinderlehrer, Some regularity results in ferromagnetism. Commun. Partial Differ. Equ.25 (2000) 1235–1258.
- 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.
- 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.
- W. Jin and R.V. Kohn, Singular perturbation and the energy of folds. J. Nonlinear Sci.10 (2000) 355–390.
- 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 . URIhttp://arxiv.org/abs/0902.0154v1
- T. Riviere and S. Serfaty, Limiting domain wall energy for a problem related to micromagnetics. Commun. Pure Appl. Math.54 (2001) 294–338.
- E.M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series30. Princeton University Press (1970).

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