Differential and Integral Equations

On the construction of solutions of the Landau-Lifshitz equation

Takeshi Isobe

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Let $\Omega\subset\mathbb {R}^2$ be a smooth bounded domain and $H\in\mathbb {R}^3$. The static Landau-Lifshitz equation with external magnetic field $H$ and boundary data $\gamma\in C^{\infty}( \Omega;\mathbb {S}^2)$ 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 $u\in C^{\infty}(\Omega;{\mathbb {S}^2})$. We establish some connection between solutions of the Landau-Lifshitz equation and critical points of the function $\Psi$ introduced in [9] (see $\S1$ Theorem B for the definition). In particular, we show that local nondegenerate minimums of $\Psi$ have their associated solutions of the Landau-Lifshitz equation for small $|H|$ and $\gamma\equiv\,{\rm const}$.

Article information

Differential Integral Equations, Volume 13, Number 1-3 (2000), 159-188.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J50: Variational methods for elliptic systems
Secondary: 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations 35J65: Nonlinear boundary value problems for linear elliptic equations 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)


Isobe, Takeshi. On the construction of solutions of the Landau-Lifshitz equation. Differential Integral Equations 13 (2000), no. 1-3, 159--188. https://projecteuclid.org/euclid.die/1356124295

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