Initial–boundary value problems for anisotropic Landau–Lifshitz equations in three or four space dimensions

Abstract

In this paper, we will show the existence of partially regular solutions to the initial–boundary value problem for Landau–Lifshitz equations with nonpositive anisotropy constants in three or four space dimensions. The partial regularity is proved up to the boundary both for the Dirichlet problem and for the Neumann problem. In addition, for the Neumann case, a generalized stability condition which ensures the partial regularity is given. For equations with positive or negative anisotropy coefficients, we will give two results of existence and uniqueness for the solutions corresponding to ground states.