The Landau-Lifshitz equation with the external field -- a new extension for harmonic maps with values in \(S^ 2\)
type: Item
instance of: scholarly article
title: The Landau-Lifshitz equation with the external field -- a new extension for harmonic maps with values in \(S^ 2\)
zbMATH Open document ID: 0843.35028
DOI: 10.1007/BF02572608
author: Min-Chun Hong
published in: Mathematische Zeitschrift
publication date: 1996-03-17T00:00:00Z
full work available at URL: https://eudml.org/doc/174803
review text: We consider critical points of the functional \[ E_H(u; \Omega):= \int_\Omega \textstyle{{1\over 2}} [|\nabla u|^2- H\cdot u] dx \] in the class \(H^{1,2}_\gamma(\Omega, S^2)\), where \(H\) is a constant vector in \(\mathbb{R}^3\) and \(\Omega\) is a domain in \(\mathbb{R}^3\). The Euler-Lagrange equation of \(E_H\), \(\Delta u+ |Du|^2 u- (H\cdot u) u+ H= 0\) in \(\Omega\), is equivalent to a static Landau-Lifshitz equation of physics. We prove non-existence and existence results for nontrivial smooth solutions of the static Landau-Lifshitz equation depending on the boundary conditions. We also show a gap phenomenon for the functional \(E_H\).
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 856558
zbMATH Keywords: non-existence
zbMATH Keywords: existence
zbMATH Keywords: gap phenomenon
MaRDI profile type: Publication
cites work: On global weak solutions for Landau-Lifshitz equations: Existence and nonuniqueness
cites work: The Dirichlet problem for harmonic maps from the disk into the Euclidean n-sphere
cites work: The weak solutions to the evolution problems of harmonic maps
cites work: Heat flow of \(p\)-harmonic maps with values into spheres
cites work: Existence and partial regularity results for the heat flow for harmonic maps
cites work: Another Report on Harmonic Maps
cites work: Harmonic Mappings of Riemannian Manifolds
cites work: The gap phenomenon for variational integrals in Sobolev spaces
cites work: The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps
cites work: A remark on \(H^ 1\) mappings
cites work: Multiple solutions of the static Landau-Lifshitz equation from \(B^ 2\) into \(S^ 2\)
cites work: Q5759616
cites work: Applications harmoniques de surfaces riemanniennes
cites work: Q3344873
cites work: Boundary regularity and the Dirichlet problem for harmonic maps
cites work: On the evolution of harmonic mappings of Riemannian surfaces
cites work: On the evolution of harmonic maps in higher dimensions
cites work: Q4011223
OpenAlex ID: W1977172502
description: scientific article; zbMATH DE number 856558