In this paper we extend the Browns fundamental theorem on fine ferromagnetic particles to the case of a general ellipsoid. By means of Poincaré inequality for the Sobolev space ~(H1)(Ω, R~3), 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) d _c such that for d< ~(dc) the uniform magnetization states are the only global minimizers of the GibbsLandau free energy functional ~(GL). A lower bound for d _c is then given in terms of the demagnetizing factors.
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