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

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