Junction of ferromagnetic thin films | SpringerLink
This service is more advanced with JavaScript available, learn more at http://activatejavascript.org
Junction of ferromagnetic thin filmsAntonio GaudielloRejeb HadijiArticle
In this paper, starting from the classical 3D micromagnetic energy, we determine, via an asymptotic analysis, the free energy of two joined ferromagnetic thin films. We distinguish different regimes depending on the limit of the ratio between the small thicknesses of the two films.78A25 49S05 78M35 This is a preview of subscription content,
to check accessUnable to display preview.&1.Alicandro R., Leone C.: 3D-2D asymptotic analysis for micromagnetic thin films. ESAIM Control Optim. Calc. Var. 6, 489–498 (2001)2.Ammari H., Halpern L., Hamdache K.: Asymptotic behavior of thin ferromagnetic films. Asymptot. Anal. 24, 277–294 (2000)3.Baía M., Zappale E.: A note on the 3D-2D dimensional reduction of a micromagnetic thin film with nonhomogeneous profile. Appl. Anal. 86(5), 555–575 (2007)4.Bethuel F., Zheng X.: Density of smooth functions between two manifolds in Sobolev spaces. J. Funct. Anal. 80(2), 60–75 (1988)5.Brown W.F.: Micromagnetics. Wiley, New York (1963)6.Carbou G.: Thin layers in micromagnetism, M3AS. Math. Models Methods Appl. Sci. 11(9),
(2001)7.Ciarlet P.G., Destuynder P.: A justification of the two-dimensional linear plate model. J. Mècanique 18(2), 315–344 (1979)8.De Giorgi, E., Franzoni, T.: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58 (8), n.6, 842–850 (1975)9.Desimone A., Kohn R.V., Muller S., Otto F.: A reduced theory for thin-film micromagnetics. Commun. Pure Appl. Math. 55(11),
(2002)10.Gaudiello A., Hadiji R.: Asymptotic analysis, in a thin multidomain, of minimizing maps with values in S
2. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26(1), 59–80 (2009)11.Gaudiello, A., Hadiji, R.: Junction of one-dimensional minimization problems involving S
2 valued maps. Adv. Differ. Equ. 13, 9–10, 935–958 (2008)12.Gioia G., James R.D.: Micromagnetism of very thin films. Proc. R. Lond. A 453, 213–223 (1997)13.Hardt R., Kinderlehrer D., Lin F.H.: Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105, 547–570 (1986)14.Hubert A., Schafer R.: Magnetic Domains: The Analysis of Magnetic Microstructures. Springer-Verlag, Berlin (1998)15.Irudayaraj, S.S., Emadi, A.: Micromachines: principles of operation, dynamics, and control, electric machines and drives. In: 2005 IEEE International Conference, pp.
(2005)16.James R.D., Kinderlehrer D.: Frustation in ferromagnetic materials. Contin. Mech. Thermodyn. 2, 215–239 (1990)17.Kohn R.V., Slastikov V.V.: Another thin-film limit of micromagnetics. Arch. Rational Mech. Anal. 178, 227–245 (2005)18.Landau, L.D., Lifshitz, E.M.: On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phy. Z. Sowjetunion 8: 153 (1935). In: ter Haar, D. (eds.) Reproduced in Collected Papers of L. D. Landau, pp.101–114. Pergamon Press, New York (1965)19.Le Dret, H.: Problèmes variationnels dans le multi-domaines: modélisation des jonctions et applications. Research in Applied Mathematics, vol. 19 (1991)20.Visintin A.: On Landau-Lifschitz’ equations for ferromagnetism. Jpn. J. Appl. Math. 2, 69–84 (1985)Antonio Gaudiello1Rejeb Hadiji21.DAEIMIUniversità degli Studi di CassinoCassino (FR)Italy2.Laboratoire d’Analyse et de Mathématiques Appliquées, UFR des Sciences et TechnologieUniversité Paris-Est, CNRS UMR 8050Créteil CedexFrance
We use cookies to improve your experience with our site.3D-2D Asymptotic observation for minimization problems associated with degenerative energy-coefficients (PDF Download Available)
See all >4 CitationsSee all >19 References
19.14Université Paris-Est Créteil Val de Marne - Université Paris 12AbstractIn 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 coefficients, 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 definite association between our 3D-minimization problems, for very thin h, and a 2D-limiting problem, as h 0, will be demonstrated with help from the theory of Γ-convergence.Discover the world's research14+ million members100+ million publications700k+ research projects
Manuscript submitted to Website: http://AIMsciences.orgAIMS’ JournalsVolume X, Number 0X,XX201X pp. X–XX3D-2D ASYMPTOTIC OBSERVATION FOR MINIMIZATIONPROBLEMS ASSOCIATED WITHDEGENERATIVE ENERGY-COEFFICIENTSRejeb HadijiUniversit?e Paris-Est, Laboratoire d’Analyse et de Math?ematiques Appliqu?eesCNRS UMR 8050, UFR des Sciences et Technologie61, Avenue du G?en?eral de Gaulle, P3, 4e ?etage, 94010 Cr?eteil Cedex, FranceKen ShirakawaDepartment of Applied MathematicsGraduate School of System Informatics, Kobe University1-1 Rokkodai, Nada, Kobe, 657-8501, JapanAbstract. In this paper, a class of minimization problems, labeled by anindex 0 &h&1, is considered. Each minimization problem is for a free-energy, motivated by the magnetics in 3D-ferromagnetic thin film, and in thecontext, the index hdenotes the thickness of the observing film. The MainTheorem consists of two themes, which are concerned with the study of thesolvability (existence of minimizers) and the 3D-2D asymptotic analysis for ourminimization problems. These themes will be discussed under degenerativesetting of the material coefficients, and such degenerative situation makes theenergy-domain be variable with respect to h. In conclusion, assuming some restrictive conditions for the domain-variation, a definite association betweenour 3D-minimization problems, for very thin h, and a 2D-limiting problem, as h&0, will be demonstrated with helps from the theory of Γ-convergence.1. Introduction. Let SΩR2be a two-dimensional bounded domain with asmooth boundary, and let OhmΩR3be a three-dimensional cylindrical domain, givenby Ohm:= S?(0,1). Let AE:Ohm°! [0,1) be a given nonnegative and continuousfunction.In this paper, let us imagine the situation that a ferromagnetic thin film isapplied on a thin region Ohm(h):= S?(0,h) with a (small) thickness 0 &h&1.As a possible free-energy for the magnetic study in such situation, the followingfunctional, denoted by E(h)AE:E(h)AE(m):=Ψ(h)AE(m)+°ZOhm(h)u&(m)+12r≥·m?dL3,for any m=(m1,m2,m3)2L2(Ohm(h);R3);(1)2000 Mathematics Subject Classification. Primary: 74G65,35J70; Secondary: 74K35,82D40.Key words and phrases. 3D-2D asymptotic analysis, degenerative free-energy, Γ-convergence.The second author is supported by Grant-in-Aid for Encouragement of Young Scientists (B)(No. ) JSPS.1
2 REJEB HADIJI AND KEN SHIRAKAWAsubject to:div (°r≥+0m)=0,in R3,(2)|m|=ms,L3-a.e. in Ohm;(3)was proposed by Brown [7] (1963), where ΨAEis the lower semi-continuous envelop-ment of a functional:√2W1,2(Ohm(h);R3)\L2(Ohm(h);S2)7! °ZOhm(h)AE|r√|2dL3;onto the space L2(Ohm(h);R3).In (1), the value of E(h)AEdenotes an energy quantity, per unit volume in Ohm(h),and the variable m=(m1,m2,m3) denotes the magnetization in the region Ohm(h)ofmagnetic thin film. In this light, the minimizers of E(h)AEare supposed to represent themost probable profile of the magnetization distribution applied on Ohm(h). Here, thegiven function AEis the so-called material coefficient, and this coefficient is supposedto be degenerative somewhere in Ohm.&:R3°! [0,1) is a given continuous function,which is involved in the magnetization anisotropy.The function ≥:R3°! Ras 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 tothe above, let us note that the free-energy E(h)AEis considered under the constrainedcondition (3), by a positive constant msof the magnetization saturation.In this paper, we set:L2(S) = 1 (and hence L3(Ohm) = 1), and ms= 1;for simplicity. On that basis, let us denote by T(h)the scale transform, defined as:T(h):x=(x1,x2,x3)2R37! (x1,x2,hx3)2R3;to consider a rescaled minimization problem, denoted by (MP)(h).(MP)(h)Find a vectorial function m(h)=(m(h)1,m(h)2,m(h)3)2L2(Ohm;R3) of threevariables, which minimizes the following functional on L2(Ohm;R3):F(h)AE(m):=Φ(h)AE(m)+ZOhm&(m)dL3+12ZOhmurP≥·mP+1h@3≥m3?dL3,for any m=(m1,m2,m3)2L2(Ohm;R3);(4)subject to:rP·(°rP≥+0mP)+ 1h@3u°1h@3≥+0m3?=0,in R3; (5)where the subscript “ P” denotes the restriction of the situation onto thetwo-dimensional plane R2, e.g.:yP:= (y1,y2),for y=(y1,y2,y3)2R3,uP:= (u1,u2)2L2(Ohm;R2),for u=(u1,u2,u3)2L2(Ohm;R3),and the distributional gradientrPu:= 0@@1u1@2u1@1u2@2u2@1u3@2u31A,for u=(u1,u2,u3)2L2(Ohm;R3);
3D-2D ASYMPTOTIC OBSERVATION FOR MINIMIZATION PROBLEMS 3and Φ(h)AEis the rescaled version of the lower semi-continuous envelopment Ψ(h)AEby T(h), and it is rigorously defined as:Φ(h)AE(m):=8&&&&&:inf{√(i)}2Q≠(m)lim infi!1 ZOhmAE(h)u|rP√(i)|2+1h2|@3√(i)|2?dL3,if |m|= 1, L3-a.e. in Ohm,1, otherwise, for any m2L2(Ohm;R3);(6)by using a composition:AE(h)(x):=(AE±T(h))(x)=AE(x1,x2,hx3),for all x=(x1,x2,x3)2Ohm;and a class of approximating functions:QOhm(m):=Ω{√(i)}√(i)2W1,2(Ohm;R3)\L2(Ohm;S2), i=1,2,3,···,and √(i)!min L2(Ohm;R3) as i!1 ae; (7)for any m2L2(Ohm;S2).As is easily seen, the inverse transform (T(h))°1provides a bijective corre-spondence between the minimizers m(h)of (MP)(h)and the minimizers m(h)org :=m(h)±(T(h))°1of the original free-energy E(h)AE. Besides, let us note that the do-mains Dom(F(h)AE) of free-energies are not uniform, but variable with respect to0&h&1, and the variation is directly governed by the degenerating part of thecoefficient:A(h)0:= (AE(h))°1(0) ΩOhm,for 0 &h&1.Under very thin situation of the thickness h, it is naturally expected that theminimization problem (MP)(h)can be reduced to a simpler problem, considered intwo-dimensional domain S. Such reduction will be realized through the limitingobservation for (MP)(h)as h&0, and then, the binary function:AE±(x1,x2):=AE(x1,x2,0) for any (x1,x2)2S,with the degenerating part A±0:= (AE±)°1(0);will be the material coefficient in the limiting problem. Actually, in the h-inde-pendent case of A(h)0, a number of like-minded study results, such as [1,2,3,4,5,6,8,10,11,12,13,14,15,16,17,18], were reported, from various viewpoints,and some of them concluded a definite association between the limiting profile of(MP)(h), as h&0, and the following minimization problem, denoted by (MP)±.(MP)±Find a vectorial function m±=(m±1,m±2,m±3)2L2(S;R3) of two variables,which minimizes the following functional:F±AE(m):=Φ±AE(m)+ZS&(m)dL2+12ZS|m3|2dL2,for any m=(m1,m2,m3)2L2(S;R3);(8)where Φ±AEis a convex function on L2(S;R3), defined as:Φ±AE(m):=8&&&&&:inf{√(i)}2QS(m)lim infi!1 ZSAE±|r√(i)|2dL2,if |m|= 1, L2-a.e. in S,1, otherwise, for any m2L2(S;R3);(9)by using a class of approximating sequences:QS(m):=Ω{√(i)}√(i)2W1,2(S;R3)\L2(S;S2), i=1,2,3,···,and √(i)!min L2(S;R3) as i!1 ae.
4 REJEB HADIJI AND KEN SHIRAKAWANow, the main theme of this paper is to verify whether analogous observation isavailable even under h-variable situation of A(h)0(or energy domains), or not. Tothis end, we here impose the following two conditions for the material coefficient AE:(a1) L3(A(h)0) = 0, for 0 &h&1;(a2) L2(A±0) = 0, and AE±(xP)∑AE(x), for all x=(x1,x2,x3)2Ohm.Consequently, a certain positive answer for our theme will be demonstrated in themain 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 {hi|i=1,2,3,···} Ω(0,1) and a function m±2L2(S;R3)of two variables, such that:(i) hi&0,m(hi)!m±in L2(Ohm;R3),F(hi)AE(m(hi))!F±AE(m±),asi!1;(ii) the limit m±solves the problem (MP)±;where {m(h)|0&h&1}is the sequence of minimizers m(h),0&h&1,obtained in (I).The content of this paper is as follows. In the next Section 2, some key-propertiesfor the minimization problems (MP)(h),0&h&1, and (MP)±are briefly mentionedas preliminaries. In subsequent Section 3, the continuous dependence between theenergy sequence {F(h)AE|0&h&1}and the energy F±AE, as h&0, will be shown bymeans of the notion of Γ-convergence (cf. [9]). On that basis, the final Section 4will be devoted to the proof of Main Theorem.Notation. Throughout this paper, the Lebesgue measure is denoted by Ln, forany observing dimension n2N.For any abstract Banach space, the norm of Xis denoted by |·|X. However,when Xis an Euclidean space, the norm is simply denoted by |·|. Besides, forany functional F:X°! (°1,1], we denote by Dom(F) the domain of F, andfor any r&0, we denote by L(r;F) the sublevel set of F, more precisely:Dom(F):=(C)?2X F (?)&1(TM)and L(r;F):=(C)?2X F (?)∑r(TM).For any abstract Hilbert space H, the inner product of His denoted by (·,·)H.However, when His an Euclidean space, the inner product between two vectors?,?2His simply denoted by ?·?.2. Preliminaries. Let us start with summarizing the known-facts, concerned withthe coupled Maxwell equation (5).(Fact 1) (Summary of [18, Lemma 3.1]) Let us fix any 0 &h&1. Then, for anyfunction m=(m1,m2,m3)2L2(Ohm;R3), the solution ≥(h)of the Maxwellequation (5) is prescribed in the scope of a Hilbert space:V(h):= Ωv2H1loc(R3)rv2L2(R3;R3) and ZB≠vdL3=0 ae;endowed with a h-dependent inner product:(u, v)V(h):= (rPu, rPv)L2(R3;R3?2)+1h2(@3u, @3v)L2(R3;R3),for u, v 2V(h);
3D-2D ASYMPTOTIC OBSERVATION FOR MINIMIZATION PROBLEMS 5where BOhmis an (fixed) open ball containing Ohm. Then, the solution ≥(h)2V(h)is supposed to fulfill a weak formulation by the following variational identity:ZOhmu(rP≥(h)°mP)·rPv+1hu1h@3≥(h)°m3?@3v?dL3=0,for any v2V(h);(10)Moreover, taking more one function ~m2L2(Ohm;R3), arbitrarily, and takinganother solution ~≥(h)of (10)whenm=~m, it follows that:|≥(h)°~≥(h)|V(h)∑|m°~m|L2(Ohm;R3).(11)Hence, the variational problem (10) is well-posed.(Fact 2) (Summary of [15, Proposition 4.1]) Let us set:8&&&&&:F(h)mag(m):= 12ZOhmurP≥(h)·mP+1h@3≥(h)m3?dL3,F±mag(m):= 12ZOhm|m3|2dL3,for any m=(m1,m2,m3)2L2(Ohm;R3);(12)by using the solution ≥(h)of the variational identity (10). On that basis, letus assume that {?m(h)|0&h&1}ΩL2(Ohm;R3), and ?m(h)!?min L2(Ohm;R3)as h&0, for some ?m=(?m1,?m2,?m3)2L2(Ohm;R3). Then:F(h)mag(?m(h))!F±mag(?m),as h&0.Next, let us look toward the key-properties of the lower semi-continuous envel-opments Φ(h)AE,0&h&1, and Φ±AE.Lemma 2.1. (I) For an y 0&h&1, the functional Φ(h)AEis a maximal functionalin the class of l.s.c. functionals on L2(Ohm;R3), supporting the functional:√2W1,2(Ohm;R3)\L2(Ohm;S2)7! ZOhmAE(h)u|rP√|2+1h2|@3√|2?dL3.Moreover:(i-1) Φ(h)0,AE(m):=Z≠\A(h)0AE(h)u|rPm|2+1h2|@3m|2?dL3∑Φ(h)AE(m),for any m2W1,2loc (Ohm\A(h)0;R3)\L2(Ohm;S2);(i-2) W1,2(Ohm;R3)\L2(Ohm;S2)ΩDom(Φ(h)AE)ΩDom(Φ(h)0,AE)ΩW1,2loc (Ohm\A(h)0;R3),and Φ(h)0,AE=Φ(h)AEon W1,2(Ohm;R3)\L2(Ohm;S2);(i-3) for any m2Dom(Φ(h)AE), there exists a sequence {u(i)|i=1,2,3,···}ΩW1,2(Ohm;R3)\L2(Ohm;S2)such thatu(i)!min L2(Ohm;R3)and Φ(h)0,AE(u(i))!Φ(h)AE(m),asi!1.(II) The functional Φ±AEis a maximal functional in the class of l.s.c. functionalson L2(S;R3), supporting the functional:√2W1,2(S;R3)\L2(S;S2)7! ZSAE±|r√|2dL2.Moreover:
6 REJEB HADIJI AND KEN SHIRAKAWA(ii-1) Φ±0,AE(m):=ZS\A±0AE±|rm|2dL2∑Φ(h)AE(m),foranym2W1,2loc (S\A±0;R3)\L2(S;S2);(ii-2) W1,2(S;R3)\L2(S;S2)ΩDom(Φ±AE)ΩDom(Φ±0,AE)ΩW1,2loc (S\A±0;R3),andΦ±0,AE=Φ±AEon W1,2(S;R3)(ii-3) for any m2Dom(Φ(h)AE), there exists a sequence {u(i)|i=1,2,3,···}ΩW1,2(S;R3)\L2(S;S2)such thatu(i)!min L2(S;R3)and Φ±0,AE(u(i))!Φ±AE(m),asi!1.Proof. This lemma follows, directly, from the definition formulas (6) and (9).Remark 1. (Key-properties for free-energies) By virtue of (4), (8) and Lemma2.1, the functional F(h)AE(resp. F±AE) turns out to be l.s.c. in L2(Ohm;R3)(resp. inL2(S;R3)), and Dom(F(h)AE) = Dom(Φ(h)AE), for 0 &h&1 (resp. Dom(F±AE)=Dom(Φ±AE)). Furthermore, under the conditions (a1)-(a2), as in introduction, thevariation of energy-domains Dom(F(h)AE), with respect to h,willberestrictiveinthesense that Dom(F(h)AE)willbeincludedinW1,2loc (Ohm\(A±0?(0,1)); R3), uniformly, forall 0 &h&1.Taking into account of Lemma 2.1, Remark 1and [16, Corollary 2], we can derivethe following corollary.Corollary 1. (Compactness) Let us assume the condition (a1), as in introduction.Then, for any 0&h&1and any r&0, the sublevel sets L(r;Φ(h)AE)and L(r;F(h)AE)are compact in L2(Ohm;R3).Additionally,ifweassumetheconditions(a1)-(a2),asin introduction, then for any r&0, the sublevel sets L(r;Φ±AE)and L(r;F±AE)arecompact in L2(S;R3),andtheunionsUΦ(r):= [0&h&1L(r;Φ(h)AE)and UF(r):= [0&h&1L(r;F(h)AE);are relatively compact in L2(Ohm;R3).Proof. Let us assume (a1), and let us fix any 0 &h&1 and any r&0. Here, takingthe solution ≥(h)as the test function of (10), we have:F(h)mag(m)= 12|≥(h)|2V(h)∏0,for any m2L2(Ohm;R3). (13)Subsequently, we see from (4), (13) and (i-1) of Lemma 2.1 that:L(r;F(h)AE)ΩL(r;Φ(h)AE)ΩL(r;Φ(h)0,AE).Since the compactness of L(r;Φ(h)0,AE)inL2(Ohm;R3) is already concluded in [16, Corol-lary 2], we can say that its closed subsets L(r;Φ(h)AE) and L(r;F(h)AE) are also so.Just as in the above, the assertion for the sublevel sets L(r;Φ±AE) and L(r;F±AE)(resp. for the unions UΦ(r) and UF(r)) can be concluded, with the helps from thecondition (a2) and [16, Corollary 2] (resp. [16, Theorem 3.4]).
3D-2D ASYMPTOTIC OBSERVATION FOR MINIMIZATION PROBLEMS 73. Continuous dependence of energies. The objective in this section is summa-rized in the following theorem, concerned with continuous dependence (Γ-convergence)of lower semi-continuous envelopments, as h&0.Theorem 3.1. (Γ-convergence from Φ(h)AEto Φ±AEas h&0) Let us assume theconditi ons (a1)-(a2), as in introduction. Then, the sequence {Φ(h)AE|0&h&1}of the lower semi-continuous envelopments converges to Φ±AE, in the sense of Γ-convergence, as h&0. More precisely, referring to [9](or [1, Lemma 2.3]), theabove assertion is equivalent to:(∞1) lim infh&0Φ(h)AE(ˇm(h))∏Φ±AE(ˇm±),if{ˇm(h)|0&h&1}ΩL2(Ohm;R3),ˇm±2L2(Ohm;R3),and ˇm(h)!ˇm±in L2(Ohm;R3)as h&0;(∞2) for any ^m±2Dom(Φ±AE), there exists a sequence {^m(h)|0&h&1}ΩL2(Ohm;R3), such that ^m(h)!^m±in L2(Ohm;R3)and Φ(h)AE(^m(h))!Φ±AE(^m±),as h&0.This theorem is proved by relying on some classes of open sets, mentioned in thefollowing lemma.Lemma 3.2. (Open coverings for non-degenerate parts) There exists a sequence{?AE±`|`=1,2,3,···} Ω(0,1) and a covering {S`|`=1,2,3,···}ΩS\A±0of S\A±0with smooth boundaries @S`(`=1,2,3,···), such that:8&&&:;6=S1ΩΩ · · · ΩΩ S`ΩΩ ···ΩΩ S\A±0=1[`=0S`,?AE±1&···&?AE±`&···&0= lim`!1 ?AE±`,and AE±∏?AE±`on S`,for`=1,2,3,···.Hence, if we assume the conditions (a1)-(a2) as in introduction, then a sequence{Ohm+`}:= {S`?(0,1) |`=1,2,3,···} turns out to be a covering of an open setOhm+:= Ohm\(A±0?(0,1)), with Lipschitz boundaries @Ohm+`(`=1,2,3,···), such that:8&:;6=Ohm+1Ω···ΩOhm+`Ω···ΩOhm+=1[`=0Ohm+`ΩOhm\A(h)0,L3(Ohm\Ohm+)=0,AE(h)∏AE±∏?AE±`&0on Ohm+`,for`=1,2,3,··· and 0&h&1.(14)Proof of Lemma 3.2.This lemma is a direct consequence of the line of arguments,discussed in [16, Lemma 4.1, Remark 4-5].Proof of Theorem 3.1.First, we verify the assertion (∞1). Then, it is enough toconsider only the case when lim inf h&0Φ(h)AE(ˇm(h))&1, since another case is trivial.On account of (i-3) in Lemma 2.1,wefindasequence{ˇhi|i=1,2,3,···} Ω(0,1)and a sequence {ˇu(i)|i=1,2,3,···} ΩW1,2(Ohm;R3)\L2(Ohm;S2), such that:(ˇhi&0 and ˇu(i)!ˇm±in L2(Ohm;R3), as i!1,limi!1 Φ(ˇhi)0,AE(ˇu(i))=liminfh&0Φ(h)AE(ˇm(h)).Here, in the light of (14) and (i-1) in Lemma 2.1,|@3ˇm±|2L2(Ohm+`;R3)∑lim infi!1 |@3ˇu(i)|2L2(Ohm+`;R3)∑1?AE±`supi∏1Φ(ˇhi)0,AE(ˇu(i))limi!1ˇh2i=0,`=1,2,3,··· ,(15)
8 REJEB HADIJI AND KEN SHIRAKAWAwhich implies @3ˇm±= 0, L3-a.e. in Ohm+. Hence, the limit ˇm±can be regarded tobelong to the class L2(S\A±0;R3)(=L2(S;R3)) of binary functions. Subsequently,let us set:ˇu(i)(x1,x2):= ˇu(i)(x1,x2,ci),for L2-a.e. (x1,x2)2Sand i=1,2,3,···;by using a collection {ci|i=1,2,3,···} Ω(0,1) of constants, such that:ZSAE(ˇhi)(x1,x2,ci)|rPˇu(i)(x1,x2,ci)|2dL2∑ZOhmAE(ˇhi)|rPˇu(i)|2dL3,i=1,2,3,··· .Then, with the helps from (15) and Fubini’s theorem, it is computed that:limi!1 |ˇ√(i)°ˇu(i)|2L2(Ohm;R3)=limi!1 ≥|ˇ√(i)°ˇu(i)|2L2(Ohm+`;R3)+|ˇ√(i)°ˇu(i)|2L2(Ohm\Ohm+`;R3)?∑limi!1 ZOhm+`Z10|@3ˇu(i)|2dL1dL3+4L3(Ohm\Ohm+`)∑limi!1 |@3ˇu(i)|2L2(Ohm+`)+4L3(Ohm\Ohm+`)=4L3(Ohm\Ohm+`),for `=1,2,3,···.It implies that: ˇ√(i)!ˇm±in L2(Ohm;R3), as i!1; (16)since L3(Ohm\Ohm+`)!0 as `!1. Taking into account of (a2),(16), (ii-2) in Lemma2.1 and the lower semi-continuity of Φ±AE, we deduce that:lim infh&0Φ(h)AE(ˇm(h))= limi!1 Φ(ˇhi)AE(ˇu(i))∏lim infi!1 ZOhmAE(ˇhi)|rPˇu(i)|2dL3∏lim infi!1 ZSAE(ˇhi)(x1,x2,ci)|rˇ√(i)(x1,x2)|2dL2∏lim infi!1 Φ±AE(ˇ√(i))∏Φ±AE(ˇm±).Thus, the assertion (∞1) is concluded.Next, we verify the assertion (∞2). Let us take any ^m±2Dom(Φ±AE). Then,construction of the required sequence {^m(h)|0&h&1}will be on the basis of asequence {^u(i)|i=1,2,3,···} ΩW1,2(S;R3)\L2(S;R3) , which will be obtained asthe approximating sequence, as in (ii-3) of Lemma 2.1,whenm=^m±. Here, notingthat AE(h)!AE±in C(Ohm) as h&0, there exists a sequence {^hi|i=1,2,3,···} Ω(0,1), such that:8&&&&&&&:o^h1&···&^hi&···&0= limi!1^hi,o0∑Φ(h)0,AE(^u(i))°Φ±0,AE(^u(i))=ZOhm(AE(h)°AE±)|rP^u(i)|2dL3&12i,for any 0 &h&^hi,i=1,2,3,···.On that basis, the finding sequence {^m(h)}will be constructed by putting:^m(h):= ^u(i)in L2(Ohm;R3), if ^hi+1 ∑h&^hi,i=1,2,3,···;with an optional setting ^m(h):= ^u(1) in L2(Ohm;R3), for ^h1∑h&1.The above Theorem 3.1 actually implies the Γ-convergence of free-energies, statedas follows.Corollary 2. (Γ-convergence from F(h)AEto F±AEas h&0) Let us assume the con-ditions (a1)-(a2). Then, the sequence {F(h)AE|0&h&1}of free-energies convergesto the limiting one F±AE, in the sense of Γ-convergence, as h&0.
3D-2D ASYMPTOTIC OBSERVATION FOR MINIMIZATION PROBLEMS 9Proof of Corollary 2.This corollary is immediately concluded, by taking into ac-count of Theorem 3.1 and (Fact 2) in Section 2.Remark 2. On account of (13) and (Fact 2), we will see that the sequence {^m(h)|0&h&1}as in (∞2) of Theorem 3.1 will realize the convergence:F(h)AE(^m(h))!F±AE(^m±) as h&0.4. Proof of Main Theorem. We divide this section into two subsections, for therespective assertions (I) and (II) of Main Theorem.4.1. Proof of (I) of Main Theorem. The proof of this assertion will be a slightmodification of the argument, discussed in [16, Section 5.1]. In fact, under (a1), andunder the fixed setting of 0 &h&1, we can take the so-called minimizing sequence{m(i)§|i=1,2,3,···} ΩDom(F(h)AE) that is supposed to satisfy:F(h)AE(m(i)§)&F(h)§:= infm2L2(Ohm;R3)F(h)AE(m) as i!1.Here, on account of (13), (Fact 1) and Corollary 1, a convergence subsequence{m(ik)§|k=1,2,3,···} Ω{m(i)§}will be found with the limit m§2L2(Ohm;R3),and it will be seen that:(m(ik)§!m§in L2(Ohm;R3), &(m(ik)§)!&(m§)inL1(Ohm),F(h)mag(m(ik)§)!F(h)mag(m§),as k!1.In response to the above, we infer from the lower semi-continuity of F(h)AEthat thelimit m§is one of minimizers of (MP)(h).4.2. Proof of (II) of Main Theorem. Let us assume the conditions (a1)-(a2),and let us take a sequence {m(h)|0&h&1}of minimizers of F(h)AE,0&h&1.Let us set ∫§:= [1,0,0] 2S2. Then, for the variational identity (10)whenm?∫§, taking the solution itself as the test function vin (10) yields that F(h)mag(∫§)∑1,for any 0 &h&1(see[16, Section 5.2], for details). In view of this,Φ(h)AE(m(h))∑F(h)AE(m(h))∑F(h)AE(∫§)=Φ(h)AE(∫§)+|&(∫§)|L1(Ohm)+F(h)mag(∫§)∑&(∫§)+1,for all 0 &h&1. (17)Since, the above (17) implies that {m(h)}ΩUF(&(∫§) + 1), we can apply Corol-lary 1,tofindasequence{hi|i=1,2,3,···} Ω(0,1) and a limiting functionm±2L2(Ohm;R3), such that:(hi&0,m(hi)!m±in L2(Ohm;R3),&(m(hi))!&(m±)inL1(Ohm), as i!1.Here, taking into account of Theorem 3.1, Corollary 2and (17),F±AE(m±)∑lim infi!1 F(hi)AE(m(hi))∑&(∫§) + 1;and hence m±2Dom(F±AE)ΩL2(S;R3).Next, taking any ^m±2Dom(F±AE) (= Dom(Φ±AE)), and taking the sequence{^m(h)|0&h&1}ΩL2(Ohm;R3), obtained in (∞2) of Theorem 3.1,itwillbeseenfrom Theorem 3.1 Corollary 2and Remark 2that:
10 REJEB HADIJI AND KEN SHIRAKAWAF±AE(m±)∑lim infi!1 F(hi)AE(m(hi))∑lim supi!1F(hi)AE(m(hi))∑limi!1 F(hi)AE(^m(hi))=F±AE(^m±).(18)It implies that m±solves the limiting problem (MP)±. Furthermore, putting ^m±=m±in (18), it is deduced that:F(h)AE(m(h))!F±AE(m±) as h&0.Thus, we conclude the assertion (II).REFERENCES[1]F. Acanfora, G. Cardone and S. Mortola, On the variational convergence of non-coercivequadratic integral functionals and semicontinuity problems, NoDEA Nonlinear Differential Equations Appl., 10 (2003), 347–373.[2]R. Alicandro and C. Leone, 3D-2D asymptotic analysis for micromagnetic thin films,ESAIMControl Optim. Calc. Var., 6(2001), 489–498.[3]H. Ammari, L. Halpern and K. Hamdache, Asymptotic behaviour of thin ferromagnetic films,Asymptot. Anal., 24 (2000), 277–294.[4]J.-F. Babadjian and M. Ba??a, 3D-2D analysis of a thin film with periodic microstructure,Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 223–243.[5]M. Ba??a and E. Zappale, A note on the 3D-2D dimensional reduction of a micromagneticthin film with nonhomogeneous profile, Appl. Anal., 86 (2007), 555–575.[6]A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thinfilms, Indiana Univ. Math. J., 49 (2000), .[7] W. F. Brown, “Micromagnetics,” Interscience Tracts on Physics and Astronomy, no. 18, JohnWiley & Sons, New York, 1963.[8]G. Carbou, Thin layers in micromagnetism, Math. Models Methods Appl. Sci., 11 (2001),.[9] G. Dal Maso, “An introduction to Γ-convergence,” Birkh¨auser, 1993.[10]A. Desimone, R. Kohn, S. M¨uller and F. Otto, A reduced theory for thin-film micromagnetics,Comm. Pure Appl. Math., 55 (2002), .[11]I. Fonseca and G. Francfort, 3D-2D asymptotic analysis of an optimal design problem forthin films, J. Reine Angew. Math., 505 (1998), 173–202.[12]A. Gaudiello and R. Hadiji, Junction of one-dimensional minimization problems involvingS2valued maps, Adv. Differential Equations, 13 (2008), 935–958.[13]A. Gaudiello and R. Hadiji, Asymptotic analysis, in a thin multidomain, of minimizing mapswith values in S2, Ann. Inst. H. Poincar?e Anal. Non Lineaire, 26 (2009), 59–80.[14] A. Gaudiello and R. Hadiji, Junction of ferromagnetic thin films, to appear in Calc. Var. Partial Differential Equations (2010).[15] G. Gioia and R. D. James, Micromagnetics of very thin film,Proc.R.Soc.Lond.A,453(1997), 213–223.[16]R. Hadiji and K. Shirakawa, Asymptotic analysis for micromagnetics of thin films governedby indefinite material coefficients, Commun. Pure Appl. Anal., 9(to appear).[17]O. A. Hafsa, Variational formulations on thin elastic plates with constraints, J. Convex Anal., 12 (2005), 365–382.[18]R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials, Contin. Mech. Thermodyn., 2(1990), 215–239.[19]R. Kohn and V. Slastikov, Another thin-film limit of micromagnetics, Arch. Ration. Mech. Anal., 178 (2005), 227–245.[20]A. Visintin, On Landau-Lifshitz’ equations for ferromagnetism, Japan J. Appl. Math., 2(1985), 69–84.Received xxxx 20 revised xxxx 20xx.E-mail address:hadiji@univ-paris12.frE-mail address:kenboich@kobe-u.ac.jp
ArticleFull-text availableAug 2015ArticleFeb 2014ArticleJun 2015Show moreArticleNovember 2010 · In this paper, starting from the classical 3D micromagnetic energy, we determine, via an asymptotic analysis, the free energy
of two joined ferromagnetic thin films. We distinguish different regimes depending on the limit of the ratio between the small
thicknesses of the two films. ArticleSeptember 2010 · 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... [Show full abstract]ArticleSeptember 2015 · Starting from a 3. D non-convex and nonlocal micromagnetic energy for ferromagnetic materials, we determine, via an asymptotic analysis, the free energy of two joined ferromagnetic thin films. We distinguish different regimes depending on the limit of the ratio between the small thickness of the two films. ArticleFebruary 2014 · In this paper, starting from the classical 3D non-convex and nonlocal micromagnetic energy for ferromagnetic
materials, we determine, via an asymptotic analysis, the free energy of a multi-structure consisting
of a nano-wire in junction with a thin film and of a multi-structure consisting of two joined nano-wires. We
assume that the volumes of the two parts composing each multi-structure... [Show full abstract]Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.This publication is from a journal that may support self archiving.