Modelization of a split in a ferromagnetic body by an equivalent boundary condition: Part II

We continue the study of equivalent boundary conditions in ferromagnetic domains crossed by a thin split. In this second part, we add nonhomogenous boundary conditions arising from interactions such as surface anisotropy and superexchange. We expand the problem up to the first order and establish equivalent boundary conditions in presence of surface anisotropy and super-exchange. In particular, the well-posedness of the expansion problem with equivalent boundary condition and the convergence in some meaning of the expansion are proven. Introduction Following part I [8], we study the behavior of a ferromagnetic domain crossed by a thin split. In order to efficiently compute the evolution of the magnetization on such a geometry, we expanded the magnetization m = m + εm on Ωε and derived an equivalent boundary condition on the contact surface: ∂m ∂ν = ∂m ∂ν2 . In this second part of the article, we extend our results when boundary interactions such as super-exchange and surface anisotropy are present [4]. The mathematical effect of these interactions is to modify the Neumann boundary condition in a nonlinear way. The new terms will be described in section 1. In the same paper, the existence, but not the uniqueness, of infinite time weak solutions is also proved. We denote bym the dimensionless magnetization. In the expansion,m represents the term of order 0 and m the term of order 1. Formally m = m + εm, where ε is the half-thickness of the split. The considered geometry is presented in Figure 1. We use the same notations as in part I. Let • ε the half thickness of the split, always verifying ε≪ min(L, L). • B a bounded convex open set of R, with a smooth boundary. • L, L be two nonzero positive numbers. • Ωε = B × (ε, L ) and Ωε = B × (−L ,−ε) for all ε < min(L, L)/2 are the domains filled with the ferromagnetic material. • Ω = B × (0, L) and Ω = B × (−L, 0). 2 K. Santugini-Repiquet / Modelization of a split in a ferromagnetic body. Part II