Asymptotics for the minimization of a Ginzburg-Landau energy in n dimensions

Paweł Strzelecki

Colloquium Mathematicae (1996)

  • Volume: 70, Issue: 2, page 271-289
  • ISSN: 0010-1354

Abstract

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We prove that minimizers u ∈ W 1 , n of the functional E ( u ) = 1 / n ∫ | ∇ u | n d x + 1 / ( 4 n ) ∫ ( 1 - | u | 2 ) 2 d x , ⊂ ℝ n , n ≥ 3, which satisfy the Dirichlet boundary condition u = g on for g: → S n - 1 with zero topological degree, converge in W 1 , n and C l o c α for any α<1 - upon passing to a subsequence k → 0 - to some minimizing n-harmonic map. This is a generalization of an earlier result obtained for n=2 by Bethuel, Brezis, and Hélein. An example of nonunique asymptotic behaviour (which cannot occur in two dimensions if deg g = 0) is presented.

How to cite

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Strzelecki, Paweł. "Asymptotics for the minimization of a Ginzburg-Landau energy in n dimensions." Colloquium Mathematicae 70.2 (1996): 271-289. <http://eudml.org/doc/210412>.

@article{Strzelecki1996,
abstract = {We prove that minimizers $u ∈ W^\{1,n\}$ of the functional $E_\{\}(u) = 1/n ∫_ |∇u|^\{n\} dx + 1/(4^\{n\}) ∫_ (1-|u|^\{2\})^\{2\} dx$, ⊂ $ℝ^\{n\}$, n ≥ 3, which satisfy the Dirichlet boundary condition $u_\{\} = g$ on for g: → $S^\{n-1\}$ with zero topological degree, converge in $W^\{1,n\}$ and $C^α_\{loc\}$ for any α<1 - upon passing to a subsequence $_\{k\} → 0$ - to some minimizing n-harmonic map. This is a generalization of an earlier result obtained for n=2 by Bethuel, Brezis, and Hélein. An example of nonunique asymptotic behaviour (which cannot occur in two dimensions if deg g = 0) is presented.},
author = {Strzelecki, Paweł},
journal = {Colloquium Mathematicae},
keywords = {-harmonic map; nonunique asymptotic behaviour},
language = {eng},
number = {2},
pages = {271-289},
title = {Asymptotics for the minimization of a Ginzburg-Landau energy in n dimensions},
url = {http://eudml.org/doc/210412},
volume = {70},
year = {1996},
}

TY - JOUR
AU - Strzelecki, Paweł
TI - Asymptotics for the minimization of a Ginzburg-Landau energy in n dimensions
JO - Colloquium Mathematicae
PY - 1996
VL - 70
IS - 2
SP - 271
EP - 289
AB - We prove that minimizers $u ∈ W^{1,n}$ of the functional $E_{}(u) = 1/n ∫_ |∇u|^{n} dx + 1/(4^{n}) ∫_ (1-|u|^{2})^{2} dx$, ⊂ $ℝ^{n}$, n ≥ 3, which satisfy the Dirichlet boundary condition $u_{} = g$ on for g: → $S^{n-1}$ with zero topological degree, converge in $W^{1,n}$ and $C^α_{loc}$ for any α<1 - upon passing to a subsequence $_{k} → 0$ - to some minimizing n-harmonic map. This is a generalization of an earlier result obtained for n=2 by Bethuel, Brezis, and Hélein. An example of nonunique asymptotic behaviour (which cannot occur in two dimensions if deg g = 0) is presented.
LA - eng
KW - -harmonic map; nonunique asymptotic behaviour
UR - http://eudml.org/doc/210412
ER -

References

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  1. [1] F. Bethuel, H. Brezis et F. Hélein, Limite singulière pour la minimisation de fonctionnelles du type Ginzburg-Landau, C. R. Acad. Sci. Paris 314 (1992) 891-895. Zbl0773.49003
  2. [2] F. Bethuel, H. Brezis et F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calculus of Variations and PDE 1 (1993), 123-148. Zbl0834.35014
  3. [3] F. Bethuel, H. Brezis et F. Hélein, Tourbillons de Ginzburg-Landau et energie renormalisée, C. R. Acad. Sci. Paris 317 (1993), 165-171. 
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  8. [8] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, 1983. 
  9. [9] Z. Han and Y. Li, Degenerate elliptic systems and applications to Ginzburg-Landau type equations, I, preprint, Rutgers University, 1995. 
  10. [10] R. Hardt and D. Kinderlehrer, Mathematical questions of liquid crystals theory, in: Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Vol. IX (Paris, 1985-1986), Pitman Res. Notes Math. Ser. 181, Longman Sci. Tech., 1988, 276-289. 
  11. [11] R. Hardt, D. Kinderlehrer and F. H. Lin, The variety of configurations of static liquid crystals, in: H. Berestycki, J.-M. Coron, and I. Ekeland (eds.), Variational Methods, Birkhäuser, 1990, 115-131. Zbl0723.58018
  12. [12] M. C. Hong, Asymptotic behavior for minimizers of a Ginzburg-Landau functional in higher dimensions associated with n-harmonic maps, preprint, 1995. 
  13. [13] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), 219-240. Zbl0372.35030

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