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

Colloquium Mathematicae (1996)

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

## Access Full Article

top## Abstract

top## How to cite

topStrzelecki, 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

top- [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] 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] F. Bethuel, H. Brezis et F. Hélein, Tourbillons de Ginzburg-Landau et energie renormalisée, C. R. Acad. Sci. Paris 317 (1993), 165-171.
- [4] F. Bethuel, H. Brezis et F. Hélein, Ginzburg-Landau Vortices, Progr. Nonlinear Differential Equations Appl. 13, Birkhäuser, Boston, 1994.
- [5] F. Bethuel and X. Zheng, Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal. 80 (1988), 60-75. Zbl0657.46027
- [6] B. Bojarski and T. Iwaniec, p-harmonic equation and quasiregular mappings, in: Banach Center Publ. 19, PWN, Warszawa, 1987, 25-38.
- [7] E. DiBenedetto and A. Friedman, Regularity of solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math. 349 (1984), 83-128. Zbl0527.35038
- [8] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, 1983.
- [9] Z. Han and Y. Li, Degenerate elliptic systems and applications to Ginzburg-Landau type equations, I, preprint, Rutgers University, 1995.
- [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] 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] M. C. Hong, Asymptotic behavior for minimizers of a Ginzburg-Landau functional in higher dimensions associated with n-harmonic maps, preprint, 1995.
- [13] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), 219-240. Zbl0372.35030

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.