This thesis is motivated by studying the properties of ferromagnetic materials usingthe Landau-Lifshitz theory of micromagnetics. In this theory the state of a ferromagneticmaterial is described by the magnetization vector m in terms of a total micromagnetic energythat consists of several competing sub-energies: exchange energy, anisotropy energy, externalinteraction energy and magnetostatic energy. For large ferromagnetic materials and undersome limiting regimes of the model, the exchange energy can be negligible and the totalenergy becomes a reduced model. Our investigations focus on the study of such a reducedmodel of Landau-Lifshitz theory.The primary focus of the thesis includes two parts: the minimization (static) study andthe evolution (dynamic) study. We investigate a new method for the existence of minimizersof the reduced micromagnetic energy based on a duality method. In this method, the reducedmicromagnetic energy is closely related to a convex functional (the dual functional) on thecurl-free vector functions. Our minimization and dynamics studies are based on the studyof the minimization and gradient ow of this dual functional. Much of the thesis is focusedon the minimization problem of two special cases: soft case and uniaxial case on the annulusdomain; in particular, in the soft case, for some range of the parameter, the energy minimizersof the original micromagnetic energy are constructed through the Euler-Lagrange equationof the dual functional using the characteristics method for a reduced Eikonal type equation.The second direction of our study of this thesis is an attempt to obtain certain reasonabledynamic process for the evolution of m, where the asymptotic behavior of the gradient owof the reduced energy functional is investigated.
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