## Differential and Integral Equations

### On the construction of solutions of the Landau-Lifshitz equation

Takeshi Isobe

#### Abstract

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

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

Dates
First available in Project Euclid: 21 December 2012