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2000 On the asymptotic behavior of the solutions of the Landau-Lifshitz equation
Takeshi Isobe
Adv. Differential Equations 5(7-9): 1033-1090 (2000). DOI: 10.57262/ade/1356651294


Let $\Omega\subset\Bbb{R}^2$ be a bounded domain and $H\in\Bbb{R}^3$. The Landau-Lifshitz equation with external field $H$ and boundary data $\gamma\in C^{\infty}(\partial\Omega;\Bbb{S}^2)$ is the following: $$\Delta u+|\nabla u|^2u-(H,u)u+H=0\quad\text{in $\Omega$},\quad u=\gamma\quad\text{on $\partial\Omega$}.$$ Here $u\in C^{\infty}(\Omega;\Bbb{S}^2)$. We study the asymptotic behavior of the solutions of this equation as $H\to 0$. We show that the "large solutions" obtained by Hong and Lemaire blow up only when $\gamma\equiv\text{const.}$ and in such a case blow-up occurs only at a single point in $\Omega$. We characterize the blow-up point as a critical point of a certain function defined in $\Omega$. We also give the asymptotic value estimate of $\|\nabla u\|_{{L^{\infty}(\Omega)}}$ as $H\to 0$.


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Takeshi Isobe. "On the asymptotic behavior of the solutions of the Landau-Lifshitz equation." Adv. Differential Equations 5 (7-9) 1033 - 1090, 2000.


Published: 2000
First available in Project Euclid: 27 December 2012

zbMATH: 1223.35155
MathSciNet: MR1776348
Digital Object Identifier: 10.57262/ade/1356651294

Primary: 35J60
Secondary: 35B40 , 35Q55 , 35Q60 , 58E20

Rights: Copyright © 2000 Khayyam Publishing, Inc.


Vol.5 • No. 7-9 • 2000
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