Local well-posedness of the Schr\"odinger flow into $\mathbb{S}^2$ with natural boundary conditions

TL;DR

This paper establishes local well-posedness for the Schr"odinger flow into the 2-sphere with natural boundary conditions using a novel approximation scheme.

Contribution

It introduces a new approximation method to prove local well-posedness for the Landau-Lifshitz equation with boundary conditions.

Findings

Proved local existence and uniqueness of solutions.

Developed a new approximation scheme for the problem.

Addressed boundary condition challenges in Schr"odinger flows.

Abstract

In this paper, we develop a new approximation scheme to resolve the local well-posedness problem for the Landau-Lifshitz equation (i.e., the Schr\"odinger flow into the standard unit 2-sphere $\mathbb{S}^2\subset \mathbb{R}^3$) with natural boundary conditions.