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

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

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

Rejeb Hadiji ^1, and Ken Shirakawa ^2,

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

Abstract
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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.

Keywords: 3D-2D asymptotic analysis, degenerate free-energy in micromagnetics, $\Gamma$-convergence.

Mathematics Subject Classification: Primary: 74G65, 35J70; Secondary: 74K35, 82D40.

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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