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

Abstract

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 $H\to 0$.