A generalization of the fundamental theorem of Brown for fine ferromagnetic particles

In this paper we extend the Brown's fundamental theorem on fine ferromagnetic particles to the case of a general ellipsoid. By means of Poincaré inequality for the Sobolev space H1(Ω,R3)H1(Ω,R3), and some properties of the induced magnetic field operator, it is rigorously proven that for an ellipsoidal particle, with diameter d, there exists a critical size (diameter) dc such that for d