Advances in Differential Equations

On the asymptotic behavior of the solutions of the Landau-Lifshitz equation

Takeshi Isobe

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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 $H\to 0$.

Article information

Adv. Differential Equations, Volume 5, Number 7-9 (2000), 1033-1090.

First available in Project Euclid: 27 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B40: Asymptotic behavior of solutions 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 35Q60: PDEs in connection with optics and electromagnetic theory 58E20: Harmonic maps [See also 53C43], etc.


Isobe, Takeshi. On the asymptotic behavior of the solutions of the Landau-Lifshitz equation. Adv. Differential Equations 5 (2000), no. 7-9, 1033--1090.

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