Harmonic Maps with Potential from $\mathbb{R}^2$ into $S^2$

TL;DR

This paper investigates the existence of harmonic maps with potential from the plane into the sphere, providing conditions for solutions and extending results related to the Landau-Lifshitz equation.

Contribution

It establishes necessary and sufficient conditions for equivariant solutions of harmonic maps with potential from  into S^2, generalizing previous Landau-Lifshitz results.

Findings

Derived existence criteria for equivariant solutions

Extended Landau-Lifshitz equation results

Improved understanding of harmonic maps with potential

Abstract

We study the existence problem of harmonic maps with potential from $\mathbb{R}^2$ into $S^2$. For a specific class of potential functions on $S^2$, we give the sufficient and necessary conditions for the existence of equivariant solutions of this problem. As an application, we generalize and improve the results on the Landau-Lifshitz equation from $\mathbb{R}^2$ into $S^2$ in \cite{G_S} due to Gustafson and Shatah.