3d-2d Asymptotic Observation for Minimization Problems Associated with Degenerative Energy-coefficients

In this paper, a class of minimization problems, labeled by an index 0 < h < 1, is considered. Each minimization problem is for a freeenergy, 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 degenerative setting of the material coefficients, and such degenerative situation makes the energy-domain be variable with respect to h. In conclusion, assuming some restrictive conditions for the domain-variation, a definite association between our 3D-minimization problems, for very thin h, and a 2D-limiting problem, as h ց 0, will be demonstrated with helps from the theory of Γ-convergence.