# American Institute of Mathematical Sciences

2011, 2011(Special): 624-633. doi: 10.3934/proc.2011.2011.624

## 3D-2D asymptotic observation for minimization problems associated with degenerate energy-coefficients

 1 Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées, CNRS UMR 8050, UFR des Sciences et Technologie, 61, Avenue du Général de Gaulle, P3, 4e étage, 94010 Créteil Cedex 2 Department of Mathematics, Faculty of Education, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan

Received  July 2010 Revised  February 2011 Published  October 2011

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 coecients, 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 de nite 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.
Citation: Rejeb Hadiji, Ken Shirakawa. 3D-2D asymptotic observation for minimization problems associated with degenerate energy-coefficients. Conference Publications, 2011, 2011 (Special) : 624-633. doi: 10.3934/proc.2011.2011.624
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