Periodic solutions for the Landau–Lifshitz–Gilbert equation

Ferromagnetic materials tend to develop very complex magnetization patterns whose time evolution is modeled by the so-called Landau–Lifshitz–Gilbert equation (LLG). In this paper, we construct time-periodic solutions for LLG in the regime of soft and small ferromagnetic particles which satisfy a certain shape condition. Roughly speaking, it is assumed that the length of the particle is greater than its hight and its width. The approach is based on a perturbation argument and the spectral analysis of the corresponding linearized problem as well as the theory of sectorial operators.