On the construction of solutions of the Landau-Lifshitz equation

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 $\gamma\equiv\,{\rm const}$.