Advances in Differential Equations

Asymptotic behavior for minimizers of a Ginzburg-Landau-type functional in higher dimensions associated with $n$-harmonic maps

Min-Chun Hong

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Abstract

We describe the behavior as $\varepsilon\to 0$ of minimizers for a Ginzburg-Landau functional $$ E_{\varepsilon}(u;\Omega )=\int_{\Omega}\bigl [ \frac {|\nabla u|^n}n +\frac 1{4\varepsilon^n}(1-|u|^2)^2\bigr ]\,dx $$ in the space $H^{1,n}_g(\Omega ;\Bbb R^n)$, where $\Omega\subset\Bbb R^n$ and the boundary data $g:\partial \Omega\to S^{n-1}$ has a nonzero topological degree. Some recent results of Bethuel, Brezis and H\'elein, and of Struwe on the two-dimensional problem, are extended to higher-dimensional cases. New proofs for their results are also presented in this paper.

Article information

Source
Adv. Differential Equations Volume 1, Number 4 (1996), 611-634.

Dates
First available in Project Euclid: 25 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1366896030

Mathematical Reviews number (MathSciNet)
MR1401406

Zentralblatt MATH identifier
0857.35120

Subjects
Primary: 58E20: Harmonic maps [See also 53C43], etc.
Secondary: 35B40: Asymptotic behavior of solutions 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 49J45: Methods involving semicontinuity and convergence; relaxation

Citation

Hong, Min-Chun. Asymptotic behavior for minimizers of a Ginzburg-Landau-type functional in higher dimensions associated with $n$-harmonic maps. Adv. Differential Equations 1 (1996), no. 4, 611--634. https://projecteuclid.org/euclid.ade/1366896030.


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