## Differential and Integral Equations

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

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

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356124295

**Mathematical Reviews number (MathSciNet)**

MR1811954

**Zentralblatt MATH identifier**

0974.35031

**Subjects**

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.)

#### Citation

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.