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1996 | 70 | 2 | 271-289
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We prove that minimizers $u ∈ W^{1,n}$ of the functional $E_{𝜀}(u) = 1/n ∫_𝛺 |∇u|^{n} dx + 1/(4𝜀^{n}) ∫_𝛺 (1-|u|^{2})^{2} dx$, 𝛺 ⊂ $ℝ^{n}$, n ≥ 3, which satisfy the Dirichlet boundary condition $u_{𝜀} = g$ on 𝜕𝛺 for g:𝜕𝛺 → $S^{n-1}$ with zero topological degree, converge in $W^{1,n}$ and $C^α_{loc}$ for any α<1 - upon passing to a subsequence $𝜀_{k} → 0$ - 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.
Słowa kluczowe
Rocznik
Tom
70
Numer
2
Strony
271-289
Opis fizyczny
Daty
wydano
1996
otrzymano
1995-10-02
Twórcy
  • Institute of Mathematics Warsaw University Banacha 2 02-097 Warszawa, Poland
Bibliografia
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-cmv70i2p271bwm
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