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 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 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 tends to 0, will be demonstrated with helps from the theory of Γ-convergence

In this paper, let us imagine the situation that a ferromagnetic thin film is applied on a thin region Ω (h) := S × (0, h) with a (small) thickness 0 < h < 1. As a possible free-energy for the magnetic study in such situation, the following functional, denoted by E  (2) |m| = m s , L 3 -a.e. in Ω; (3) was proposed by Brown [7] (1963), where Ψ α is the lower semi-continuous envelopment of a functional: onto the space L 2 (Ω (h) ; R 3 ).
In (1), the value of E α are supposed to represent the most probable profile of the magnetization distribution applied on Ω (h) . Here, the given function α is the so-called material coefficient, and this coefficient is supposed to be degenerative somewhere in Ω. ϕ : R 3 −→ [0, ∞) is a given continuous function, which is involved in the magnetization anisotropy.
The function ζ : R 3 −→ R as in (1)-(2) denotes the magnetic field potential, and hence, it is prescribed as the solution of the simplified Maxwell equation (2). Here, the notation " 0 " denotes the zero-extension of functions. In addition to the above, let us note that the free-energy E In this paper, we set: L 2 (S) = 1 (and hence L 3 (Ω) = 1), and m s = 1; for simplicity. On that basis, let us denote by T (h) the scale transform, defined as: to consider a rescaled minimization problem, denoted by (MP) (h) .
of two variables, which minimizes the following functional: where Φ • α is a convex function on L 2 (S; R 3 ), defined as: by using a class of approximating sequences:

REJEB HADIJI AND KEN SHIRAKAWA
Now, the main theme of this paper is to verify whether analogous observation is available even under h-variable situation of A (h) 0 (or energy domains), or not. To this end, we here impose the following two conditions for the material coefficient α: Consequently, a certain positive answer for our theme will be demonstrated in the main theorem, stated as follows.
Main Theorem. (I) Let us assume the condition (a1). Then, for any 0 < h < 1, the problem (MP) (h) admits at least one solution (minimizer) m (h) . (II) Under the conditions (a1)-(a2), there exist a sequence {h i | i = 1, 2, 3, · · · } ⊂ (0, 1) and a function m • ∈ L 2 (S; R 3 ) of two variables, such that: The content of this paper is as follows. In the next Section 2, some key-properties for the minimization problems (MP) (h) , 0 < h < 1, and (MP) • are briefly mentioned as preliminaries. In subsequent Section 3, the continuous dependence between the energy sequence {F (h) α | 0 < h < 1} and the energy F • α , as h ց 0, will be shown by means of the notion of Γ-convergence (cf. [9]). On that basis, the final Section 4 will be devoted to the proof of Main Theorem.
Notation. Throughout this paper, the Lebesgue measure is denoted by L n , for any observing dimension n ∈ N.
For any abstract Banach space, the norm of X is denoted by | · | X . However, when X is an Euclidean space, the norm is simply denoted by | · |. Besides, for any functional F : X −→ (−∞, ∞], we denote by Dom(F ) the domain of F , and for any r > 0, we denote by L(r; F ) the sublevel set of F , more precisely: For any abstract Hilbert space H, the inner product of H is denoted by (·, ·) H . However, when H is an Euclidean space, the inner product between two vectors ξ, η ∈ H is simply denoted by ξ · η.

2.
Preliminaries. Let us start with summarizing the known-facts, concerned with the coupled Maxwell equation (5).  (5) is prescribed in the scope of a Hilbert space: endowed with a h-dependent inner product: where B Ω is an (fixed) open ball containing Ω. Then, the solution ζ (h) ∈ V (h) is supposed to fulfill a weak formulation by the following variational identity: Moreover, taking more one functionm ∈ L 2 (Ω; R 3 ), arbitrarily, and taking another solutionζ (h) of (10) when m =m, it follows that: Hence, the variational problem (10) by using the solution ζ (h) of the variational identity (10). On that basis, let us assume that . Then: α is a maximal functional in the class of l.s.c. functionals on L 2 (Ω; R 3 ), supporting the functional: Moreover: α is a maximal functional in the class of l.s.c. functionals on L 2 (S; R 3 ), supporting the functional: Moreover:
Taking into account of Lemma 2.1, Remark 1 and [16, Corollary 2], we can derive the following corollary.

Corollary 1. (Compactness) Let us assume the condition (a1), as in introduction.
Then, for any 0 < h < 1 and any r > 0, the sublevel sets L(r; Φ Proof. Let us assume (a1), and let us fix any 0 < h < 1 and any r > 0. Here, taking the solution ζ (h) as the test function of (10), we have: Subsequently, we see from (4), (13) and (i-1) of Lemma 2.1 that: Since the compactness of L(r; Φ α ) are also so. Just as in the above, the assertion for the sublevel sets L(r; Φ • α ) and L(r; F • α ) (resp. for the unions U Φ (r) and U F (r)) can be concluded, with the helps from the condition (a2) and [ 3. Continuous dependence of energies. The objective in this section is summarized in the following theorem, concerned with continuous dependence (Γ-convergence) of lower semi-continuous envelopments, as h ց 0.
The above Theorem 3.1 actually implies the Γ-convergence of free-energies, stated as follows.  Proof of Corollary 2. This corollary is immediately concluded, by taking into account of Theorem 3.1 and (Fact 2) in Section 2.
In response to the above, we infer from the lower semi-continuity of F (h) α that the limit m * is one of minimizers of (MP) (h) .