Multiple solutions of the static Landau-Lifshitz equation from \(B^ 2\) into \(S^ 2\)
type: Item
instance of: scholarly article
title: Multiple solutions of the static Landau-Lifshitz equation from \(B^ 2\) into \(S^ 2\)
zbMATH Open document ID: 0843.35029
DOI: 10.1007/BF02572616
author: Min-Chun Hong
author: Luc Lemaire
published in: Mathematische Zeitschrift
publication date: 1996-08-22T00:00:00Z
full work available at URL: https://eudml.org/doc/174811
review text: We consider maps \(u: B^2\to S^2\) from the disc in Euclidean space to the Euclidean sphere, which have constant value \(\gamma\) on the boundary of \(B^2\), and we study the question of existence of extremals of the functional \[ E_H(u)= \int_{B^2} (|\nabla u|^2- 2u\cdot H)dx, \] where \(H\) is a constant vector in \(\mathbb{R}^3\) and \(u\cdot H\) is the Euclidean scalar product. The Euler-Lagrange equation is \(\Delta u+ |\nabla u|^2 u- (H\cdot u) u+ H= 0\), an equation introduced in physics by Landau and Lifshitz. When \(H= 0\), it reduces to the equation of harmonic maps, and it is known that the only solution of the problem is the constant map with value \(\gamma\). We show that if \(\gamma\) is not proportional to \(H\), then the problem has at least two solutions as soon as \(H\neq 0\). When \(\gamma= H/|H|\), it has only one solution, and when \(\gamma= - H/|H|\), the number of solutions increases with \(|H|\), and in particular if \(|H|\) is larger than the first eigenvalue of \(\Delta\) on \(B^2\), there are at least three solutions. The proofs use on one hand the methods introduced by [\textit{H. Brezis} and \textit{J. M. Coron}, Commun. Math. Phys. 92, 203-215 (1993; Zbl 0532.58006)] and [\textit{J. Jost}, J. Differ Geom. 19, 393-401 (1984; Zbl 0551.58012)], and on the other a reduction to an ordinary differential equation when \(H\) and \(\gamma\) are proportional.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 856566
zbMATH Keywords: equation of harmonic maps
zbMATH Keywords: two solutions
zbMATH Keywords: three solutions
MaRDI profile type: Publication
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OpenAlex ID: W2040309644
description: scientific article; zbMATH DE number 856566