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CPAA

In this paper, a class of minimization problems, associated with the micromagnetics of thin films,
is dealt with. Each minimization problem is distinguished by the thickness of the thin film,
denoted by $ 0 < h < 1 $, and it is considered under spatial indefinite and degenerative
setting of the material coefficients.
On the basis of the fundamental studies of the governing energy functionals,
the existence of minimizers, for every $ 0 < h < 1 $, and the 3D-2D asymptotic analysis
for the observing minimization problems, as $ h \to 0 $, will be demonstrated in
the main theorem of this paper.

PROC

In this paper, a class of minimization problems, labeled by an
index 0 < $h$ < 1, is considered. Each minimization problem is for a

*free-energy*, motivated by the magnetics in 3D-ferromagnetic thin film, and in the context, the index $h$ denotes the thickness of the observing film. The Main Theorem consists of two themes, which are concerned with the study of the solvability (existence of minimizers) and the 3D-2D asymptotic analysis for our minimization problems. These themes will be discussed under degenerate setting of the material coecients, and such degenerate situation makes the energy-domain be variable with respect to $h$. In conclusion, assuming some restrictive conditions for the domain-variation, a denite association between our 3D-minimization problems, for very thin $h$, and a 2D-limiting problem, as $h \searrow$ 0, will be demonstrated with help from the theory of $\Gamma$-convergence.## Year of publication

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