Modeling thin curved ferromagnetic films
Hamdi Zorgati
1May 24, 2005
Abstract
The behavior of a thin film made of a ferromagnetic material in the ab-sence of an external magnetic field, is described by an energy depending on the magnetization of the film verifying the saturation constraint. The free energy consists of an induced magnetostatic energy and an energy term with density including the exchange energy and the anisotropic energy. We study the behavior of this energy when the thickness of the curved film goes to zero. We show that the minimizers of the free energy converge to the mini-mizers of a local energy depending on a two-dimensional magnetization by Γ-convergence arguments.
1
Introduction
The theory of micromagnetism developed by L.D. Landau, E.M. Lifschitz [9] and W.F. Brown [3], describes the magnetic behavior of ferromagnetic bodies. Ac-cording to this theory, the equilibrium state of a ferromagnetic body is described by its magnetization m, wich is a vector field defined on R3, vanishing outside the body. The magnetization represents the volume density of macroscopic magnetic moment. This means that m induces a magnetic field in all the space. When a ferromagnetic body occupying a domain Ωis submitted to an external magnetic field He, the observed states are the minimizers of an energy Eedepending on the
magnetization m: Ee(m) = Z Ω α|∇m|2+ Qϕ(m) + He.m + Z R3 1 2|H| 2.
1CEREMADE, CNRS UMR 7534, Universit´e Paris-Dauphine, Place du Mar´echal de Lattre
de Tassigny, 75775, Paris, France & Laboratoire Jacques-Louis Lions, Universit´e Pierre et Marie Curie, 75252 Paris Cedex 05, France, email: zorgati@ann.jussieu.fr
The first term in the free energy represents the exchange energy. It penalizes the spatial variations of the magnetization m in order to model the tendency of a spec-imen to exhibit regions of uniform magnetization (magnetic domains) separated by very thin transition layers (domain walls). The constant α is a non-negative constant. The second term is the anisotropic energy term. It models the existence of preferred directions of the magnetization (easy axes), along whichϕ, which is supposed to be a nonnegative, even function exhibiting cristallographic symmetry, vanishes. In general, ϕis supposed to be a polynomial function. The coefficient
Q controls the relative importance of the anisotropic energy compared to the
mag-netostatic energy. The magmag-netostatic energy, that is the fourth term of the free energy, is induced by the magnetization m, where H = −∇u is the magnetic field
induced by m in R3, and u is a scalar potential verifying the magnetostatic equa-tion
div(−∇u + m) = 0 in R3. (1.1)
The third term composing the free energy is the external energy or the Zeeman energy. It is due to the external magnetic field in which the ferromagnetic body is placed. More details can be found in A. De Simone [6]. At constant temperature the magnetization m verifies the saturation condition
|m| = M inΩ,
where M is a nonnegative constant that we suppose equal to 1.
The study of ferromagnetic thin films is of great interest, because this type of material is found in several fields in industry, such as audio and video tapes containing ferromagnetic ribbons used to provide high density in audio or video recording.
The first mathematical works on ferromagnetic films concerned wide films with constant thickness, see A. De Simone [6], B. Dacorogna and I. Fonseca [4]. Then, G. Gioia and R.D. James [8] studied the behavior of a thin ferromagnetic film, with no external magnetic field, when its thickness goes to zero. Gioia and James considered a thin film of thickness h occupying the domain Ωh. The film
has an energy per unit volume with no external energy term. They show that the magnetization minimizing the free energy of the thin film converges, when the thickness goes to zero, to a magnetization minimizing a limit energy. This limit energy is local, which means that there is no magnetostatic equation in the limit model. Gioia and James also show that the limit magnetization is inde-pendent of the direction normal to the film, wich means that the limit model is two-dimentional. In [2], R. Alicandro and C. Leone extended the study of Gioia and James considering a more general density W , depending on the magnetization and its gradient, verifying certain growth hypotheses and including the exchange and anisotropic energies. Alicandro and Leone used the concept of tangential
quasiconvexity introduced by B. Dacorogna, I. Fonseca, J. Maly and K. Trivisa [5] in order to compute the relaxation of a class of energy functionals where the admissible fields are constrained to remain in a C1, q-dimensional manifold of Rd. In this paper, we follow [2] and we use Γ-convergence arguments to study the behavior of the energy of a curved ferromagnetic thin film and its minimiz-ers when the thickness of the film goes to zero. After setting the problem and rescaling the energy, we study the magnetostatic energy term by examining the magnetostatic potential solution of the equation (1.1). We get the behavior of the magnetostatic energy setting a new minimization problem. Then, we recall the notion of tangential quasiconvexity before computing theΓ-limit of the sequence of energies that gives the behavior of almost minimizing sequences. Next, we rewrite the limit model on the curved surface following [11]. Finally, we apply our results to the Gioia and James model and to the model with external magnetic field. These results were announced in [14]
2
Notation and geometrical preliminaries
Throughout this article, we assume the summation convention unless otherwise specified. Greek indices take their values in the set {1, 2} and Latin indices in the set {1, 2, 3}.
Let (e1, e2, e3) be the canonical orthonormal basis of the Euclidean space R3. We denote by |v| the norm of a vector v in R3, by u · v the scalar product of two vec-tors in R3, by u ∧ v their vector product and by u ⊗ v their tensor product. Let M33 be the space of 3 × 3 real matrices endowed with the usual norm |F| =ptr(FTF).
This is a matrix norm in the sense that |AB| ≤ |A||B|. We denote by A = (a1|a2|a3) the matrix in M33whose ith column is ai. We consider a thin curved film of
thick-ness h > 0 occupying at rest an open domain eΩh. This reference configuration of
the film is described as follows. We are thus given a surface eS, called the
midsur-face of the film. This surmidsur-face is a bounded two-dimensional C1-submanifold of R3 and we assume for simplicity that it admits an atlas consisting of one chart. Letψ be this chart, i.e. a C1-diffeomorphism from a bounded open subsetωof R2onto
e S.
Let aα(x) =ψ,α(x) be the vectors of the covariant basis of the tangent plane
Tψ(x)S associated with the charte ψ, whereψ,α denotes the partial derivative ofψ
with respect to xα. We assume that there existsδ> 0 such that |a1(x) ∧ a2(x)| ≥δ on ¯ωand we define the unit normal vector a3(x) = |aa1(x)∧a2(x)
1(x)∧a2(x)|, which belongs to
C1( ¯ω, R3). The vectors a1(x), a2(x) and a3(x) constitute the covariant basis at the point x. We define the contravariant basis by the relations ai(x) · aj(x) =δij, so
Next, we define a mappingΨ: ω× R → R3by
Ψ(x1, x2, x3) =ψ(x1, x2) + x3a3(x1, x2).
It is well known that there exists h∗> 0 such that for all 0 < h < h∗, the restriction ofΨtoΩh=ω× ]−h/2, h/2[ is a C1-diffeomorphism on its image by the tubular
neighborhood theorem. For such values of h, we set eΩh=Ψ(Ωh). Alternatively,
we can write e Ωh= n e x ∈ R3, ∃eπ(ex) ∈ eS,ex =eπ(ex) +ηa3 ψ−1(eπ(ex)) with −h 2 <η< h 2 o ,
whereeπdenotes the orthogonal projection from eΩhonto eS, which is well defined
and of class C1for h < h∗. Equivalently, everyex ∈ eΩhcan be written as
e x =eπ(x) +e h e x −eπ(x)e · a3 ψ −1( e π(x))e i · a3(ψ−1(eπ(x))).e
Thus, we have a curvilinear coordinate system in eΩhnaturally associated with the
chartψby
(x1, x2) =ψ−1(eπ(x)) and xe 3= (ex −eπ(x)) · ae 3(ψ
−1(
e
π(ex))).
Since the scalar potentialuehis defined on R3we suppose that the middle surface
e
Ωh is the image of an open domain ω ⊂ R2 through a C1-diffeomorphism ¯ψ:
R2→ R3extendingψto R2. More generally, we suppose thatΨis the restriction
to R2× (−h2,h2) of a C1-diffeomorphism ¯Ψ of R3. We can also suppose that Ψ is equal to the identity outside a compact set containingΩh. In what follows, we
will keep the notationΨto mean ¯Ψ.
For all x ∈ ¯ω, we let A(x) = (a1(x)|a2(x)|a3(x)). We note that A(x) is an invert-ible matrix on ¯ω, and that its inverse is given by A(x)−1= (a1(x)|a2(x)|a3(x))T. We also note that det A(x) = | cof A(x) · e3| = |a1(x) ∧ a2(x)| ≥δ> 0 on ¯ω. We clearly have
∇Ψ(x1, x2, x3) = A(x1, x2) + x3(a3,1(x1, x2)|a3,2(x1, x2)|0).
The matrix∇Ψ(x1, x2, x3) is thus everywhere invertible in ¯Ωhand its determinant
is strictly positive, and therefore equal to the Jacobian of the change of variables, for h small enough.
In the following, h denotes a generic sequence of real numbers in ]0, h∗[ that
tends to zero. The next convergences are easily established.
Lemma 2.1 We have
(
∇Ψ−1◦Ψ(x
1, x2, hx3) → A(x)−1, det∇Ψ(x1, x2, hx3) → det A(x),
uniformly on ¯Ω1 when h → 0. In particular, infΩ¯1det∇Ψ(x1, x2, hx3) ≥δ/2 > 0
3
The three-dimensional and rescaled problems
The behavior of a curved ferromagnetic thin film of thickness h occupying the domain eΩhis described by a micromagnetic energyeehdepending on itsmagneti-zationmeh: R3→ R3. The vector field mehvanishes outside the film. We refer to
[6] and the references therein for more information about ferromagnetic materials. The magnetization of the film minimizes an energyeehof the form
e eh(meh) = 1 h Z e Ωh h W (meh,∇meh) + 1 2∇ueh.meh i dx,e (3.1)
under the saturation constraint
|meh| = 1 in eΩh.
The free energy is composed of an energy term with density W depending on the magnetization and its gradient, that includes the exchange and anisotropic ener-gies. The density verifies the following growth, coercivity and locally lipschitz dependence assumptions
∃c > 0, ∃p ∈]1, +∞[, ∀y ∈ R3and ∀F ∈ M3, |W (y, F)| ≤ c(1 + |F|p), ∃γ> 0, ∃β≥ 0, ∀y ∈ R3and ∀F ∈ M3,W (y, F) ≥γ|F|p−β,
∀y ∈ R3and ∀F, F0∈ M3, |W (y, F) −W (y, F0)| ≤ c(1 + |F|p−1+ |F0|p−1) |F − F0|,
(3.2) In the case of Gioia and James we have W (y, F) =ϕ(y) +α|F|2and p = 2.
The second term in (3.1) depends on the magnetizationmehand a scalar
poten-tial ueh for the induced magnetic field Hh= −∇euh, solution of the magnetostatic
equation
div(−∇ueh+meh) = 0 in R 3,
which we can also write div Bh= 0, where Bh= Hh+meh in R 3.
The observed states of the ferromagnetic film are the solutions of the mini-mization problem e eh(meh) = min e m∈eVh e eh(m),e (3.3) where e Vh= e m ∈ L∞(R3; R3), such thatme| eΩh ∈ W1,p( eΩh; R3),
|m| = 1, a.e. in ee Ωhandm = 0, a.e. in ee Ω
c h ,
with p > 1. In this article we do not make any convexity assumption on W , con-sequently problem (3.3) may well not possess any solutions. Thus, we consider
a diagonal minimizing sequence mehfor the sequence of energieseeh over the sets e
Vh. More specifically, we assume that
e mh∈Vehandeeh(meh) = inf e m∈eVh e eh(m) + he ε(h),
whereεis a positive function such thatε(h) → 0 when h → 0.
UsingΓ-convergence arguments, we study the behavior of the energyeehand its almost minimizers in the above sense when the thickness of the curved film goes to zero. We begin by flattening and rescaling the minimizing problem in order to work on a fixed cylindrical domain.
In order to flatten the domain, we define mh : R3→ R3 and uh: R3→ R by
setting for all x ∈ R3, mh(x) =meh(Ψ(x)) and uh(x) =ueh(Ψ(x)). This implies that
∇meh(x) =e ∇mh(x) ∇Ψ −1(Ψ(x)) and∇ e uh(x) =e ∇Ψ −1(Ψ(x))T∇ uh(x) withex =
Ψ(x). We thus set eh(mh) =eeh(meh) and we obtain eh(mh) = 1 h Z Ωh h W (mh,∇mh ∇Ψ−1◦Ψ)+ 1 2 ∇Ψ −1◦ΨT∇ uh· mh i det∇Ψdxh.
Then, we rescale the problem by setting m(h)(x) = mh(x1, x2, hx3) and u(h)(x) =
uh(x1, x2, hx3) for all x ∈ R3. This implies that ∇meh(ex) = (m(h),1|m(h),2|
1
hm(h),3)Ahand∇ueh(ex) = A
T
h∇hu(h),
with Ah(x) =∇Ψ−1◦Ψ(x1, x2, hx3) and∇hu(h) =
u(h),1 u(h),2 h−1u(h),3 . These relations
may also be written∇meh=∇m(h)IhAhand∇ueh= A
T hIh∇u(h) with Ih= 1 0 0 0 1 0 0 0 1h . We note that Ah(x) = A(x1, x2) + hx3 a3,1|a3,2|0 −1
, for all x ∈ ¯ω× [−1, 1] (but
not necessarly outside of this domain). Setting e(h)(m(h)) = eh(mh), we obtain
e(h)(m(h)) = Z Ω1 h W (m(h), (m(h),1|m(h),2|h−1m(h),3)Ah) +1 2A T h∇hu(h) · m(h) i dhdx, with dh(x) = det∇Ψ(x1, x2, hx3).
The magnetostatic equation then reads
∇ − ATh(x)Ih∇u(h)(x) + m(h)(x)
and the saturation constraint reads
|m(h)(x)| = 1 for all x ∈Ω1. (3.5)
4
Magnetostatic energy behavior
In this section we study the behavior of the magnetostatic energy term
Emag(h)(m) = Z Ω1 1 2A T h∇hu · m dhdx,
when the thickness of the curved film goes to zero. Let us thus be given m : R3→ R3 such that m|Ω1 ∈ L
∞(Ω
1; S2) and m = 0 onΩc1. We consider the minimization problem: Find u(h, m) ∈ U such that
Im(h)(u(h, m)) = inf v∈UIm(h)(v), (4.1) with Im(h)(v) = 1 2 Z R3 A T h∇hv − m 2 dhdx and U =v ∈ L1loc(R3),∇v ∈ L2(R3; R3), Z B v dx = 0 ,
where B is the unit ball of R3. The condition R
Bv dx = 0 excludes the trivial
translations v → v + c. We endow U with the scalar product
(u, v)U,h=
Z
R3
ATh∇hu · ATh∇hv dhdx.
When endowed with this scalar product, U is a Hilbert space (see [13]). Thus, problem (4.1) has a unique solution u(h, m) ∈ U verifying the following Euler-Lagrange equation
Z
R3
ATh∇hu(h, m) − m · ATh∇hv dhdx = 0, ∀ v ∈ U, (4.2)
that is the weak form of (3.4). Setting v = u(h, m) in (4.2), we obtain
Z R3 ATh∇hu(h, m) 2 dhdx = Z R3 ATh∇hu(h, m) · m dhdx, (4.3)
where the right-hand side is equal to twice the magnetostatic energy term Emag(h) corresponding to a given magnetization m. The following proposition gives the behavior of the magnetostatic energy term under convergence of the magnetiza-tion.
Proposition 4.1 Let ¯m(h) be a sequence of functions in L∞(R3; R3) such that
¯
m(h) = 0 onΩc1and | ¯m(h)| = 1 onΩ1, verifying ¯m(h) → ¯m(0) strongly in L2(R3; R3).
Let ¯u(h, ¯m(h)) be the solution of the minimizing problem (4.1) associated to ¯m(h). Then, we have ∇u(h, ¯¯ m(h)) → 0 in L2(R3; R3) and1 hu(h, ¯¯ m(h)),3→ w in L 2 (R3), where w ∈ L2(R3) verifies Z R3 |wae3|2d0dx = Z Ω1 |a3· ¯m(0)|2d0dx,
andae3 represents the third column vector of AT0(x) =∇Ψ−1 Ψ(x1, x2, 0) which
is equal to a3onΩ1. We have also,
Emag(h)( ¯m(h)) → Emag(0)( ¯m(0)) = 1 2 Z Ω1 |(a3, ¯m(0))|2d0dx.
We recall that a3is the third vector of the covariant basis associated to the diffeo-morphismΨonΩ1.
Proof Since ¯u(h, ¯m(h)) minimizes Im(h)h¯ over U and 0 ∈ U , we have
Im(h)¯ (h)( ¯u(h, ¯m(h))) ≤ Im(h)¯ (0) = 1 2 Z R3 | ¯m(h)|2dhdx = 1 2vol( eΩh) ≤ c. Using the triangle inequality, we obtain that
Z R3 A T h∇hu(h, ¯¯ m(h)) 2 dhdx 12 ≤ c + Z Ω1 dhdx 12 ≤ c0. Since ∇hu(h, ¯¯ m(h)) ≤ A−Th L∞(R3;M3) A T h∇hu(h, ¯¯ m(h)) , we have k∇u(h, ¯¯ m(h))kL2(R3;R3)≤ c and 1 hk ¯u(h, ¯m(h)),3kL2(R3)≤ c.
Thus, for a subsequence still denoted h, there exists H ∈ L2(R3; R3) and w ∈ L2(R3) such that
∇u(h, ¯¯ m(h)) * H weakly in L2(R3; R3) and1
hu(h, ¯¯ m(h)),3* w weakly in L
2 (R3).
The Lpversion of the Poincar´e Lemma implies the existence of u(0) ∈ Hloc1 (R3; R3)
such that H =∇u(0). Thus,
∇u(h, ¯¯ m(h)) *∇u(0) weakly in L2(R3; R3), (4.5) with u(0),3= 0, which implies that∇u(0) = 0 in L2(R3) since∇u(0) ∈ L2(R3; R3).
Consequently,
u(0) = 0, (4.6)
since u(0) ∈ U . Let us show that the convergences in (4.4) and (4.5) are actually strong. Equation (4.3) reads in this case
Z R3 ATh∇hu(h, ¯¯ m(h)) 2 dhdx = Z R3 ATh∇hu(h, ¯¯ m(h)) · ¯m(h) dhdx. (4.7)
The convergences (4.4), (4.5) and the strong convergence of ¯m(h) to ¯m(0) in L2(R3; R3) imply that Z R3 ATh∇hu(h, ¯¯ m(h)) · ¯m(h) dhdx → Z Ω1 AT0(∇pu(0) + we3) · ¯m(0) d0dx, (4.8) where ∇pu = u,1 u,2 0
. Thus, the L2(R3; R3) norm of ATh∇hu(h, ¯¯ m(h)) is
conver-gent. In the next step, we identify the limit of this norm using some test functions constructed on the model of the one used by Gioia and James in [8]. We consider a sequence of functions wε∈ Cc∞(R3) that converges strongly in L2(R3) to w when ε→ 0. Let aε> 0 be such that the projection of the support of wε on the x3axis belongs to [−aε, aε] and we suppose that aε> 1. We set forλ> 0
vεh,λ(x1, x2, x3) = h Z x3 −aε wε(x1, x2, s)ds− h λ Z x3 aε χ[aε,aε+λ] (r)dr Z aε −aε wε(x1, x2, s)ds+cε. The constant cεis chosen in such a way as to satisfy the condition on the ball B and the second term guarantees that ∇vεh,λ ∈ L2(R3; R3). We have that limh→0vεh,1,λ =
limh→0vεh,2,λ = 0 in L2(R3) forε,λfixed and
1 hv ε,λ h,3= wε− 1 λχ[aε,aε+λ](x3) Z aε −aεw ε(x 1, x2, s)ds.
Since ¯u(h, ¯m(h)) is the solution of the minimization problem (4.1) we have that
Z R3 A T h∇hu(h, ¯¯ m(h)) 2 − 2ATh∇hu(h, ¯¯ m(h)) · ¯m(h) dhdx ≤ Z R3 A T h∇hvεh,λ 2 − 2ATh∇hvεh,λ· ¯m(h) dhdx. (4.9)
Using (4.8), (4.6) and the fact that AT0 = ∇Ψ−1◦Ψ(x1, x2, 0)T = (a1|a2|a3) in Ω1, where airepresents the vectors of the contravariant basis associated with the diffeomorphismψwith |a3| = 1, we obtain
Z R3 ATh∇hu(h, ¯¯ m(h))· ¯m(h) dhdx → Z Ω1 AT0(we3)· ¯m(0) d0dx = Z Ω1 wa3· ¯m(0) d0dx. (4.10) On the other hand, we have
Ah(x) = ∇Ψ−1◦Ψ(x1, x2, hx3) T
→ ∇Ψ−1◦Ψ(x1, x2, 0) T
= AT0(x),
in C0(R3). Let eaibe the column vectors of AT0 which are defined in R3. We know that eai= ai, inΩ
1. Passing to the limit in (4.9) when h → 0 and using (4.10), we obtain that lim h→0 Z R3 A T h∇hu(h, ¯¯ m(h)) 2 dhdx ≤ Z R3 wε−1 λχ[aε,aε+λ](x3) Z aε −aεw ε(x 1, x2, s)ds e a3 2 d0dx −2 Z Ω1 ( wε−1 λ χ[aε,aε+λ](x3) Z aε −aε wε(x1, x2, s)ds a3· ¯m(0)+2wa3· ¯m(0) ) d0dx.
Expanding this expression, we obtain
lim h→0 Z R3 A T h∇hu(h, ¯¯ m(h)) 2 dhdx ≤ Z R3 wε 2 + 1 λ2 χ[aε,aε+λ](x3) Z aε −aεw ε(x 1, x2, s)ds 2 −2 λ wεχ[aε,aε+λ](x3) Z aε −aε wε(x1, x2, s)ds ea3 2 d0dx + Z Ω1 n2 λ χ[aε,aε+λ](x3) Z aε 0 wε(x1, x2, s)ds a3· ¯m(0) − 2wεa3· ¯m(0) + 2wa3· ¯m(0) o d0dx. (4.11) Since the projection of the support of wε on the x3-axis belongs to [−aε, aε], we have that wε(x)χ[aε,aε+λ](x3) = 0, a.e. in R3, thus the third term of the right hand side of inequality (4.11) vanishes. We also have thatχ[aε,aε+λ](x3) = 0 inΩ1since
vanishes. Considering all this, inequality (4.11) becomes lim h→0 Z R3 A T h∇hu(h, ¯¯ m(h)) 2 dhdx ≤ Z R3 wε 2 + 1 λ2 χ[aε,aε+λ](x3) Z aε −aεw ε(x 1, x2, s)ds 2 |ae3|2d0dx − 2 Z Ω1 wεa3· ¯m(0) − wa3· ¯m(0) d0dx = Z R3 wεae3 2 d0dx − 2 Z Ω1 wεa3· ¯m(0) − wa3· ¯m(0) d0dx + 1 λ2 Z R2 Z aε −aεw ε(x 1, x2, s)ds 2 |ae3|2d0dσ Z R χ2 [aε,aε+λ](x3)dx3 . Noting that R Rχ 2
[aε,aε+λ](x)dx =λ, and passing to the limit when λ→ +∞ we
obtain lim h→0 Z R3 A T h∇hu(h, ¯¯ m(h)) 2 dhdx ≤ Z R3 wε e a3 2 d0dx − 2 Z Ω1 wεa3· ¯m(0) − wa3· ¯m(0) d0dx. Then we take the limit asε→ 0 and obtain
lim h→0 Z R3 ATh∇hu(h, ¯¯ m(h)) 2 dhdx ≤ Z R3 wae3 2 d0dx. (4.12) Due to the weak convergences in (4.4), (4.5) and since AT0(∇pu(0) + we3) =
AT0(we3) = wae 3in R3, we have Z R3 w e a3 2 d0dx ≤ lim inf Z R3 ATh∇hu(h, ¯¯ m(h)) 2 dhdx. (4.13)
Then, we deduce the convergence of the norm thanks to (4.12) and (4.13) , there-fore
ATh∇hu(h, ¯¯ m(h)) → wae 3, in L2(R3; R3). It follows that
∇hu(h, ¯¯ m(h)) = Ah−TATh∇hu(h, ¯¯ m(h)) → A0−TAT0(we3) = we3,
that gives the strong convergence in (4.4) and (4.5). Let us now identify the norm of wae3in L2(R3). Letting h → 0 in (4.7), we get Z R3 w e a3 2 d0dx = Z Ω1 wa3· ¯m(0) d0dx. (4.14)
We set ξεh= h Z x3 −1(ae 3· ¯ mε(0))(x1, x2, s)ds− h λ Z x3 1 χ[1,1+λ] (r)dr Z 1 −1(ae 3· ¯ mε(0))(x1, x2, s)ds+cε, where ¯mε(0) is a sequence in C0∞(R3; R3) that converges to ¯m(0) in L2(R3; R3)
whenεgoes to zero with supp ¯mε(0) ⊂Ω1 and cε a constant depending onε. We have thatξεh∈ U, with
1 hξ ε h,3=ae 3· ¯mε(0) −1 λχ[1,1+λ](x3) Z 1 −1 (ea3· ¯mε(0))(x1, x2, s)ds. The minimization problem (4.1) implies that
Z R3 A T h∇hu(h, ¯¯ m(h)) − ¯m(h) 2 dhdx ≤ Z R3 A T h∇hξεh− ¯m(h) 2 dhdx.
Letting h go to zero, we see that
Z R3 A T 0(we3) − ¯m(0) 2 d0dx ≤ Z R3 A T 0 e a3· ¯mε(0) −1 λχ[1,1+λ](x3) Z 1 −1(ae 3· ¯mε(0))(x 1, x2, s)dse3 − ¯m(0) 2 d0dx. (4.15) Let us estimate the right-hand side of this enequality, we have that
Z R3 A T 0 e a3· ¯mε(0)−1 λχ[1,1+λ](x3) Z 1 −1(ae 3· ¯mε(0))(x 1, x2, s)dse3 − ¯m(0) 2 d0dx = Z R3 e a3· ¯mε(0) −1 λχ[1,1+λ](x3) Z 1 −1 (ae3· ¯mε(0))(x1, x2, s)ds e a3− ¯m(0) 2 d0dx = Z R3 e a3· ¯mε(0) 2 + 1 λ2χ 2 [1,1+λ](x3) Z 1 −1(ae 3· ¯mε(0))(x 1, x2, s)ds 2 |ae3|2 +m(0)¯ 2 −2 λ e a3· ¯mε(0)χ[1,1+λ](x3) Z 1 −1 (ae3· ¯mε(0))(x1, x2, s)ds −2(ae3· ¯mε(0))ae3· ¯m(0)−2 λ χ[1,1+λ](x3) Z 1 −1(ae 3· ¯mε(0))(x 1, x2, s)dsea 3· ¯m(0)d 0dx (4.16) and since the projection of the support of ¯mε(0) and of ¯m(0) on the x3axis stays in [0, 1], it follows that Z R3 1 λ (ae3· ¯mε(0))χ[1,1+λ](x3) Z 1 −1(ae 3· ¯mε(0))(x 1, x2, s)ds d0dx = Z R3 1 λ χ[1,1+λ](x3) Z 1 −1 (ea3· ¯mε(0))(x1, x2, s)ds e a3· ¯m(0)d0dx = 0.
We also have that 1 λ2 Z R3 χ2 [1,1+λ](x3) Z 1 −1 (ae3· ¯mε(0))(x1, x2, s)ds 2 |ae3|2d0dx = 1 λ Z R2 Z 1 −1(ae 3· ¯ mε(0))(x1, x2, s)ds 2 |ae3|2d0dσ → λ→+∞0
Applying all this in (4.15) we get, lettingλ→∞
Z R3 wae 3− ¯ m(0) 2 d0dx ≤ Z R3 (ae 3· ¯ mε(0))ae3− ¯m(0) 2 d0dx. Then, we letε→ 0 and obtain
Z R3 h |wae3|2− 2w ae3· ¯m(0)id0dx ≤ Z R3 h |(ae3· ¯m(0))|2− 2|(ae3· ¯m(0))|2id0dx, and using (4.14) we get
− Z R3 w ae3· ¯m(0) d0dx ≤ − Z R3 |(ae3· ¯m(0))|2d0dx. Consequently, we have Z R3 |(ae3· ¯m(0))|2d0dx ≤ Z R3 w ae3· ¯m(0) d0dx = Z R3 |wae3|2d0dx. (4.17) Thanks to the Cauchy-Schwartz inequality, equation (4.14) gives
Z R3 w e a3 2 d0dx = Z R3 (wae3· ¯m(0)) d0dx ≤ Z R3 |w|2d0dx 12Z R3 |(ae3· ¯m(0))|2d0dx 12 , thus, Z R3 |wae3|2d0dx 12 ≤ Z R3 |(ea3· ¯m(0))|2d0dx 12 ,
and using (4.17) we obtain
Z R3 |wea3|2d0dx 12 = Z R3 |(ae3· ¯m(0))|2d0dx 12 = Z Ω1 |(a3· ¯m(0))|2d0dx 12 . Finally, we have Emag(h)( ¯m(h)) → Emag(0)( ¯m(0)) = 1 2 Z Ω1 |(a3· ¯m(0))|2d0dx.
Now that we know the behavior of the magnetostatic energy term, we can compute theΓ-limit of the free energy e(h).
5
Γ
-convergence and behavior of minimizers
We are interested in the behavior of the diagonal minimizing sequence m(h) of the energy e(h), that verifies
m(h) ∈ V and e(h)(m(h)) = inf
m∈Ve(h)(m) +ε(h),
withε(h) → 0 when h → 0 and
V =m ∈ L∞(R3; R3), such that m|Ω1 ∈ W1,p(Ω1; R3),
|m| = 1, a.e. inΩ1and m = 0, a.e. inΩc1 . In the following, we will identify functions in V with functions in W1,p(Ω1; S2) while keeping the same notation.
Let us begin by a lemma on deformations with bounded energies.
Lemma 5.1 Let m(h) be a sequence of magnetizations in V verifying e(h)(m(h)) ≤
c. Then, we have
k∇m(h)kLp(Ω
1;R3)≤ c and
1
hkm(h),3kLp(Ω1;R3)≤ c. (5.1)
Moreover, the limit points of this sequence for the weak topology of W1,p belong to the set
VM= {m ∈ V et m,3= 0 inΩ1}.
Proof Since the magnetostatic energy term is positive, we have
Z Ω1 W (m(h), (m(h),1, m(h),2, 1 hm(h),3)Ah) dhdx ≤ c, and (3.2) implies k(m(h),1, m(h),2, 1 hm(h),3)AhkLp(Ω1;R3)≤ c.
Thus, we obtain (5.1). We deduce the existence of a subsequence, still denoted h and m(0) ∈ W1,p(Ω1; R3) satisfying
m(h) * m(0) in W1,p(Ω1; R3)
with m(0),3 = 0. The weak convergence in W1,p(Ω1; R3) implies that for a sub-sequence still denoted h we have m(h) → m(0) in Lp(Ω1; R3), and thus, for a subsequence, we have m(h) → m(0) a.e. in Ω1, which implies that for almost
Before announcing the main result of this section, let us recall the definition of tangential quasiconvexity, a notion introduced by B. Dacorogna, I. Fonseca, J. Maly and K. Trivisa in [5] that gives the relaxation of a class of functionals where admissible deformations are constrained to remain in a C1 submanifold of dimension q of Rd. This definition was generalized by R. Alicandro and C. Leone in [2] in the case of functionals depending on the gradient of the deformation and the deformation itself. We denote by Md×N the set of matrices d × N. Let
f : Md×N → [0, +∞[ be a Borel measurable function and M a C1submanifold of dimension q of Rd. We denote by Ty(M) the tangent space to M at y ∈ M.
Definition 5.1 Let y ∈ M andξ∈Ty M
N
. The tangential quasiconvexification of f inξrelative to y is defined by QN,dT f (y,ξ) := infn Z Q f y,ξ+∇ϕ(x)dx : ϕ∈ W01,∞ Q; Ty(M) o , (5.2) with Q a cube in RN.
In our case M is the unit sphere S2of R3and Ty(S2) = y⊥, the plane orthogonal to
y. We recall two results proved in [5]
Proposition 5.1 Let f : Md×N→ [0, +∞) be a continuous function such that there exists c > 0 and p ≥ 1 verifying 0 ≤ f (ξ) ≤ c(1 + |ξ|p), for everyξ∈ Md×N. We
define the functional J by J(u) :=
Z
Ω1
QN,dT f (u,∇u)dx, u ∈ W1,p(Ω1; M).
Then, J(.) is sequentially weakly lower semicontinuous in W1,p(Ω1; M).
Theorem 5.1 Let f : Md×N→ [0, +∞) a continuous function verifying 0 ≤ f (ξ) ≤ c(1 + |ξ|p), for p ≥ 1, c > 0 and every ξ∈ Md×N. If M is a C1 submanifold of
dimension q, then F(u) = Z Ω1 QN,dT f (u,∇u)dx, where F(u) := inf un n lim inf n→+∞ Z Ω1 f (∇un)dx : un* u in W1,p(Ω1; M) o .
In our case, the magnetization m is constrained to remain in the sphere S2. Conse-quently, m,i will be in m⊥ almost everywhere. As in Acerbi, Buttazzo and
we define W0:Ω1× R3× M3×2→ R by setting W0(x, y, ¯F) := inf z∈y⊥ W (y, ( ¯F|z)A0(x)) = inf z∈R3 ¯ W (y, ( ¯F|z)A0(x)), with A0(x) =∇Ψ−1◦Ψ(x1, x2, 0) and ¯ W (y, ( ¯F|z)A0(x)) = ( W (y, ( ¯F|z)A0(x)), si z ∈ y⊥, +∞, otherwise.
We show as in Le Dret and Raoult [10], that the function W0 is continuous and satisfies properties analogous to those verified by W , that is to say
( ∃c0 > 0, ∀ ¯F ∈ M3,2, ∀y ∈ R3, ∀x ∈ ¯ω, |W0(x, y, ¯F)| ≤ c 0 (1 + | ¯F|p), ∃γ0> 0, ∃β0≥ 0, ∀ ¯F ∈ M3,2, ∀y ∈ R3, ∀x ∈ ¯ω,W0(x, y, ¯F) ≥γ 0 | ¯F|p−β0.
In order to study the behavior of the free energy of the ferromagnetic curved film and its possible minimizers when the thickness of the film goes to zero, we are going to compute its Γ-limit. The natural space in which we would like to com-pute theΓ-limit is W1,p(Ω1; R3). However, W1,p(Ω1; R3) endowed with the weak topology is not metrizable. Thus, we extend the energy e(h) to all Lp(Ω1; R3) by setting
for all m ∈ Lp(Ω1; R3), e∗(h)(m) = (
e(h)(m) if m ∈ V,
+∞ otherwise, We now are in a position to compute theΓ-limit of the free energy.
Theorem 5.2 The sequence of energies e∗(h)Γ-converges for the strong topology of Lp(Ω1; R3) to
e∗(0)(m) = (R
ωQ2,3T W0 x, m(x), (m,1|m,2) + |(a3(x), m)|2d0dx if m ∈ VM
+∞ otherwise.
The proof of the theorem follows from the following propositions.
Proposition 5.2 We have
Preuve We obtain the lower bound of theΓ-limit by showing that for any sequence
mh∈ Lp(Ω
1; R3) converging strongly to m0in Lp(Ω1; R3), we have lim inf e∗(h)(mh) ≥ e∗(0)(m0).
First, if mh∈ V , then e/ ∗(h)(mh) = +∞and the result is trivial. In the same way, if
mh∈ V and lim inf e∗(h)(mh) = +∞, the result is obvious.
Next, we consider mh ∈ V , such that lim inf e∗(h)(mh) = lim inf e(h)(mh) < +∞. This implies that for a subsequence, there exists c > 0 such that e(h)(mh) < c.
Using the lemma 5.1 we get that for a subsequence still denoted h we have that
mh * m0 in W1,p(Ω1; R3). Thanks to the fact that mh is uniformly bounded in
L∞(Ω1; R3) we obtain that mh→ m0in Ls(Ω1; R3), for all s > 1, in particular for
s = 2. We thus deduce from proposition 4.1 that Emag(h)( ¯mh) → Emag(0)( ¯m0) = 1 2 Z Ω1 |(a3, ¯m0)|2d0dx. (5.3) Let us set Eea(h)(m) = Z Ω1 W m, (m,1|m,2| 1 hm,3)Ah dhdx,
which represents the exchange and anisotropic energies. We have
Eea(h)(mh) = Z Ω1 Wmh, (mh,1|mh,2|1 hm h ,3)Ah dhdx = Z Ω1 n W mh, (mh,1|mh,2|1 hm h ,3)A0 + R(x, h, mh) o dhdx, where R(x, h, mh) = Wmh, (mh,1|mh,2|1 hm h ,3)Ah −Wmh, (mh,1|mh,2|1 hm h ,3)A0 .
Using the properties (3.2) of W and those of the diffeomorphismΨ, we show as in [11] that Z Ω1 R(x, h, mh) dhdx → 0 when h → 0. (5.4) Thus, we have Eea(h)(mh) ≥ Z Ω1 n W0 x, mh, (mh,1|mh,2) + R(x, h, mh) o dhdx ≥ Z Ω1 n Q2,3T W0 x, mh, (mh,1|mh,2)+ R(x, h, mh)odhdx.
Taking the limit when h → 0 we obtain using (5.4) and proposition(5.1) that lim inf Eea(h)(mh) ≥ Z Ω1 Q2,3T W0 x, m0, (m0,1|m0,2)d0dx. Using (5.3) and since m0,3= 0 (lemma 5.1), we obtain that
lim inf e(h)(mh) ≥
Z ω n Q2,3T W0 x, m0(x), (m,10|m0,2) + |(a3(x), m0)|2 o d0dx, Thus, lim inf e∗(h)(mh) ≥ e∗(0)(m0) which implies that
Γ− lim inf e∗(h) ≥ e∗(0).
Then we proceed to the computation of the upper bound of theΓ-limit. We will need the following lemma (see [10])
Lemma 5.2 Let X ,→ Y be two Banach spaces such that X is reflexive and
com-pactly embedded in Y . Let G : X → R a function such that ∀v ∈ X, G(v) ≥ g(kvkX),
with g verifying g(t) → +∞ when t → +∞. Let G∗ defined by G∗(v) = G(v) if v ∈ X , G∗(v) = +∞otherwise. LetΓ− G be the lower semicontinuous envelope of G for the weak topology of X andΓ− G∗the lower semicontinuous envelope of G∗for the strong topology of Y . ThenΓ− G∗= (Γ− G)∗.
Proposition 5.3 We have
Γ− lim sup e∗(h) ≤ e∗(0). (5.5)
Preuve In order to find an upper bound for the Γ-limit, we have to construct a
sequence mh∈ Lp(Ω
1; R3) that converges strongly to m0in Lp(Ω1; R3) verifying lim e∗(h)(mh) ≤ e∗(0)(m0). (5.6)
Let m0∈ Lp(Ω
1; R3). First, if m0∈ V/ M, we have e∗(0)(m0) = +∞. Setting, for
all h, mh = m0 we get (5.6). Next, we consider m0 ∈ VM and we set mh(x) =
m0(x1, x2) + hx3ξ(x1, x2), withξ∈ W01,p(ω; R3) to be chosen later. We temporarily do not consider the saturation constraint |mh| = 1, we will treat it later. We have mh → m0 strongly in W1,p(Ω
1; R3). Using the Lebesgue convergence theorem, we have that for a subsequence still denoted h
Eea(h)(mh) = Z Ω1 W (mh, (m0,α+ hx3ξ,α|ξ)Ah)dhdx → Z Ω1 W (m0, (m0,α|ξ)A0)d0dx.
The function µ :ω× R3→ R defined by µ(x, z) = ¯W (m0(x), (m0,α(x)|z)A0(x)) is a Caratheodory function. Thus, the measurable selection lemma implies that, there existsξ:ω→ R3measurable such that
W0(x, m0(x), (m0,1(x)|m0,2(x))) = ¯W (m0(x), (m0,α(x)|ξ(x))A0(x)), (see [7]). If ξ∈ m/ 0⊥, this implies that W
0(x, m0(x), (m0,1(x)|m0,2(x))) = +∞ and the result is obvious. Ifξ∈ m0⊥a.e., this implies that
W0(x, m0(x), (m0,1(x)|m0,2(x))) = ¯W (m0(x), (m0,α(x)|ξ(x))A0(x))
= W (m0(x), (m0,α(x)|ξ(x))A0(x)). (5.7) and thanks to the properties (3.2) of W , we obtain thatξ∈ Lp(ω
; R3). The density
of Cc∞(ω; R3) in Lp(ω; R3) implies the existence of a sequence ξε ∈ Cc∞(ω; R3)
verifying ξε →ξ strongly in Lp(ω; R3). Next, we project ξε on m0⊥ setting ¯
ξε = (I − m0⊗ m0)ξε. We have ¯ξε∈ W1,p
0 (ω; R3) verifying ¯ξε→ξ strongly in
Lp(ω; R3). Thus, using the Lebesgue convergence theorem, we have that for a
subsequence still denotedε
Z ωW (m 0(x), (m0 ,α(x)|¯ξε(x))A0(x))d0dx → Z ωW (m 0(x), (m0 ,α(x)|ξ(x))A0(x))d0dx. This means that for everyη> 0, there exists anε(η) > 0 such that ∀ε≤ε(η) we
have Z ωW (m 0, (m0 ,α|¯ξε)A0)d0dx ≤ Z ωW (m 0, (m0 ,α|ξ)A0)d0dx +η and thanks to (5.7) we obtain that
Z ωW (m 0 , (m0,α|¯ξε)A0)d0dx ≤ Z ωW0(x, m 0 (x), (m0,1(x)|m0,2(x)))d0dx +ηmeas(eS). Let us set ¯mhε = m0+ hx3ξ¯ε. We have that ¯mhε ∈ W1,p(Ω1; R3). We fix ε> 0, for h < 1 2k¯ξεkL∞(ω;R3) and since |m 0| = 1, we have | ¯mh ε| > |m0| − hk¯ξεkL∞(ω;R3)>12, a.e.Then, we set mhε= m¯hε | ¯mhε|. By construction, we have m h
ε∈ V thanks to the algebra
property of W1,p(ω; R3) ∩ L∞(ω; R3). We have ∇mhε= I | ¯mhε|− ¯ mhε⊗ ¯mhε | ¯mhε|3 ∇m¯hε. We also have mhε → h→0m 0strongly in Lp(Ω 1; R3), and mhε,α → h→0(I − m 0⊗ m0)m0 ,α strongly in Lp(Ω1; R3).
On the other hand, we have
(m0⊗ m0)∇m0i,α= m0i(m0, m0,α) = 0,
since |m0| = 1, and we also have
1 hm h ε,3 → h→0(I − m 0⊗ m0)¯ξε strongly in Lp(Ω 1; R3).
Now, ¯ξε∈ m0⊥so that we have (m0⊗ m0)¯ξε= 0 and thus, we obtain
mhε,α → h→0m 0 ,α and 1 hm h ε,3 → h→0 ¯ ξε strongly in Lp(Ω1; R3). Using the Lebesgue theorem, we see that
Eea(h)(mhε) = Z Ω1 W (mhε, (mhε,α|1 hm h ε,3)Ah)dhdx → h→0 Z ωW (m 0, (m0 ,α|¯ξε)A0)d0dx ≤ Z ωW0(x, m 0 (x), (m0,1(x)|m0,2(x)))d0dx +η. Thus, using proposition (4.1), we obtain
lim h→0e(h)(m h ε) ≤ Z Ω1 n W0(x, m0(x), (m0,1(x)|m0,2(x))) + |(a3, m0)|2 o d0dx +η. Let us set G∗(m) = (R ωW0 x, m(x), (m,1|m,2) + |(a3(x), m)|2d0dx if m ∈ VM +∞ otherwise.
We have shown that
lim
h→0e
∗(h)(mh
ε) ≤ G∗(m0) +η,
which implies thatΓ− lim suph→0e∗(h)(m0) ≤ G∗(m0) +η. Since this is true for
everyη> 0, we obtain that
Γ− lim sup
h→0
e∗(h)(m0) ≤ G∗(m0). (5.8) We know that the Γ− lim sup of a function is weakly lower semicontinuous and
that the lsc envelope of the function G defined on VM by
G(m) =
Z
ωW0 x, m(x), (m,1|m,2) + |(a3(x), m)| 2d
is e(0) defined on VM by e(0)(m) = Z ωQ 2,3 T W0 x, m(x), (m,1|m,2) + |(a3(x), m)|2d0dx,
see [5]. Thus, using lemma 5.2, we obtain that the lsc envelope of G∗is e∗(0), and
applying the lsc envelope to both sides of (5.8), we obtain that
Γ− lim sup e∗(h) ≤ e∗(0). (5.9)
The conjunction of Propositions 5.2 and 5.3 gives Theorem 5.2.
Corollary 5.1 The diagonal minimizing sequence m(h) of e(h) is bounded in ¯V and its limit points for the weak topology of W1,p(Ω1, R3) belong to VM and
min-imize the energy e(0) defined by e(0)(m) = Z ω n Q2,3T W0 x, m(x), (m,1|m,2) + |(a3(x), m)|2 o d0dx.
Preuve The proof of the corollary follows from Lemma 5.1 and the standard
Γ-convergence argument.
6
The curved two-dimensional limit model
Since every magnetization m in VM satisfies m,3= 0 inΩ1, we will identify in the following magnetizations m ∈ VM with magnetizations ¯m ∈ ¯VM where
¯
VM= { ¯m ∈ Lp(R2; R3), with ¯m|ω∈ W1,p(ω; R3), | ¯m| = 1 a.e. inω, ¯m = 0 a.e. inωc}.
We consider another chartψ0fromω0, an open set in R2into eS. Working with this
new chart, we obtain exactly the sameΓ-convergence result, but this time, through another diffeomorphismΨ0. This means that the limit model is intrinsic and only depends on the curved surface eS. Let us thus write the limit model on the curved
surface eS. As in [11], for any unit vector e of S2, we consider a bounded open set
Oe ⊂ e⊥ and denote byπe the orthogonal projection on this set. We extend any
functionχ∈ W01,∞(Oe, R3) settingχe(y) =χ(πe(y)) and we define for any y ∈ Oe,
De⊥χ(y) =∇χe(y). To any m ∈ ¯VM we associate a magnetization m : Re 3→ R3 defined bym(e ex) = m(ψ−1(ex)). We have thatm ∈ ee VM with
e
VM= {m ∈ We 1,p(ψ
(R2); R3), |m| = 1 a.e. in ee S, m = 0 a.e. in ee Scet∇meen(ex) = 0},
where n denotes the normal vector to ee S (for a given orientation). All functions
defined on the surface are implicitly assumed to be extended to a tubular neigh-borhood of the surface by being constant on each normal fiber. Partial derivatives are computed on these extensions and then restricted to the surface.
Proposition 6.1 Every magnetizationm associated with a magnetization m mini-e mizing the limit energy e(0), minimizes on eVM the energyee(0) defined by
e e(0)(m) =e Z e S n e W (n(ex),e m,e ∇m(e ex)) + |(en(ex),m(e x))|e 2 o dex, where eW : S2× S2× M3is defined by e W (e, y, F) = inf χ∈W01,∞(Oe;y⊥) h 1 meas(Oe) Z Oe inf z∈y⊥W (y, F + z ⊗ e + De⊥χ(s)) ds i .
Preuve We proceed as in Le Dret and Raoult [11]. For every m ∈ VM we have
e(0)(m) =
Z
ωQ 2,3
T W0 x, m(x), (m,1|m,2) + |(a3(x), m)|2d0dx.
We set ex = ψ(x), m(e x) = m(x). This implies thate ∇m(x) =∇m(e ex)∇ψ(x) and
settingee(0)(m) = e(0)(m) we obtaine
e e(0)(m) =e Z e S n Q2,3T W0 ψ−1(ex),m(e x),e ∇m(e ex)∇ψ(ψ −1( e x)) +(a3(ψ−1(ex)),m(e ex)) 2o dex.
The integral representation of the tangential quasiconvex envelope (5.2) is written in our case Q2,3T W0(x, y,ξ) := inf n 1 meas(O) Z O W0 x, y,ξ+∇χ(s)ds : χ∈ W01,∞ O; y⊥ o ,
with O a bounded open domain in R3 and the infimum does not depend on the choice of this open domain. Thus, we have for allx ∈ ee S
Q2,3T W0 ψ−1(x),e m(e ex),∇m(e ex)∇ψ(ψ −1( e x)) = inf χ∈W01,∞(O;m(e ex) ⊥ ) 1 meas(O) Z O W0 ψ−1(ex),m(e ex),∇m(e x)e∇ψ(ψ −1( e x))+∇χ(s) ds.
The definition of W0implies that
Q2,3T W0 ψ−1(x),e m(e ex),∇m(e ex)∇ψ(ψ −1( e x)) = inf χ∈W01,∞(O;m(e x)e⊥) 1 meas(O) Z O inf z∈m(e ex) ⊥W e m(ex) , ∇m(e ex)∇ψ(ψ−1(x)) +e ∇χ(s)|zA0(ψ−1(x))e ds. (6.1)
Moreover, we have ∇m(e x)e∇ψ(ψ−1(x))+e ∇χ(s)|zA0(ψ−1(x)) =e ∇m(e x)e∇ψ(ψ −1( e x))|0A0(ψ−1(ex)) + (∇χ(s)|0)A0(ψ−1(ex)) + (0|z)A0(ψ −1( e x)). We also have ∇m(e ex)∇ψ(ψ −1 (ex))|0 =∇m(e ex)(∇ψ(ψ−1(ex))|0) =∇m(e ex)(a1(ψ−1(x))|ae 2(ψ −1( e x))|0).
The condition m,3= 0 becomes∇m(e x)ae 3(ψ
−1( e x)) = 0 and so we have ∇m(e x)(e ∇ψ(ψ −1( e x))|0) =∇m(e ex)(a1(ψ−1(x))|ae 2(ψ −1( e x))|a3(ψ−1(x))).e We also have A0(x) =∇Ψ−1(Ψ(x1, x2, 0)) = ∇Ψ(x1, x2, 0) −1 = (a1(ψ−1(x))|ae 2(ψ −1 (ex))|a3(ψ−1(x)))e −1 ,
which implies that
∇m(e ex)(∇ψ(ψ
−1(
e
x))|0)A0(ψ−1(ex)) =∇m(e ex). (6.2)
Next, we set ¯s =∇ψ(ψ−1(ex))s and ¯χ( ¯s) =χ(s). We have ¯χ∈ W01,∞(Oa3(ψ−1(
e x)); R3) and (∇χ(s)|0)A0(ψ−1(x)) = De a3(ψ−1( e x))⊥χ¯( ¯s). (6.3)
Finally, for every z ∈ R3we have
(0|z)A0(ψ−1(ex)) = z ⊗ a3(ψ −1( e x)). (6.4) Choosing O = ∇ψ(ψ−1(x))e −1Oa3(ψ−1( e
x))and replacing (6.2), (6.3) and (6.4) in
(6.1), we obtain Q2,3T W0 ψ−1(x),e m(e ex),∇m(e ex)∇ψ(ψ −1( e x)) = inf χ∈W01,∞(O a3(ψ−1(ex)) ;me ⊥) h 1 meas(Oa3(ψ−1( e x))) Z O a3(ψ−1(ex)) inf z∈me⊥ W (m, F +z⊗ae 3(ψ−1(ex)) + Da 3(ψ−1(ex)) ⊥χ(s)) ds i ,
and thus the result.
6.1
Model with external magnetic field
If the curved film is placed in a uniform external magnetic field He, the free energy
governing the behavior of the film will contain a term of external energy Eext, called the Zeeman energy, depending on the magnetizationmehof the film via
e Eext(meh) = 1 h Z e Ωh(He,meh)dex.
Setting for all x ∈Ω1, m(h)(x) =m(e Ψ(x1, x2, hx3)) and Eext(h)(m(h)(x)) = eEext(meh)
we obtain
Eext(h)(m(h)) =
Z
Ω1
(He, m(h)) dhdx.
When m(h) converges to m(0) strongly in L2(Ω1; R3) with m(0),3 = 0 inΩ1, we have for a subsequence still denoted h
Eext(h)(m(h)) → Eext(0)(m(0))
=
Z
ω(He, m(0)) d0dx,
and theΓ-convergence analysis remains unchanged. The total limit energy will be
e(0)(m(0)) = Z ω n Q2,3T W0 x, m(0)(x), (m(0),1|m(0),2) + |(a3(x), m(0))|2 + (He, m(0)) o d0dx and its expression on the curved surface is
e e(0)(m) =e Z e S n e W (n(ex),e m,e ∇m(e ex)) + |(n(e ex),m(e x))|e 2+ ( eHe,m)e o dex.
6.2
The curved Gioia and James model
We apply our results to the particular case of the Gioia and James model [8], that is with W (y, F) =ϕ(y) +α|F|2. The limit energy reads
e e(0)(m) =e Z e S n e W (n(eex),m,e ∇m(e ex)) + |(n(ex),e m(e x))|e o dex, with e W (n(ex),e m,e ∇m(e ex)) = inf χ∈W01,∞(O e n;me ⊥) h 1 meas(O e n) Z O e n inf z∈me ⊥ ϕ(m)+e α∇m+z⊗e n+De e n⊥χ(s) 2 dsi.
We have that ∇ e m + z ⊗en + D e n⊥χ(s) 2 = |∇m|e 2+ |z ⊗en|2+ |D e n⊥χ(s)|2 + 2∇m : z ⊗e en + 2∇m : De e n⊥χ(s) + 2z ⊗en : D e n⊥χ(s). (6.5) Since∇men = 0, this implies thate ∇m : z⊗e n = 0. We also have, since De
e n⊥χ(s)en = 0, that z ⊗en : D e n⊥χ(s) = 0. Thus, we have ∇m + z ⊗e n + De e n⊥χ(s) 2 = |∇m|e2+ |z ⊗n|e2+ |D e n⊥χ(s)|2+ 2∇m : De e n⊥χ(s).
This means that the infimum for z ∈me⊥ is reached when z belongs tone⊥∩me⊥ in particular for z = 0. Thus, we obtain
e W (n(ex),e m,e ∇m(e ex)) = inf χ∈W01,∞(O e n;me ⊥) h 1 meas(O e n) Z O e n ϕ (m)+e α∇m+De e n⊥χ(s) 2 dsi = inf χ∈W01,∞(O;me⊥) h 1 meas(O) Z O ϕ (m) +e α∇ e m + (∇χ(s)|0)A0(ψ−1(ex)) 2 dsi,
with O an arbitrary bounded open domain of R2. The invertibility of the matrix
A0(ψ−1(ex)) implies that e W (en(ex),m,e ∇m(e x)) =e inf χ∈W01,∞(D;me⊥) h 1 meas(D) Z D ϕ (m) +e α∇m +e ∇χ(s) 2 dsi =ϕ(m) +e αQ3,3T Φ(∇m),e
withΦ: M3→ R defined by Φ(A) = |M|2 and D an arbitrary bounded open do-main of R3. SinceΦis convex, its tangential quasiconvex envelope isΦitself and we have eW (en(ex),m,e ∇m(e x)) =e ϕ(m) +e α|(∇m)|e 2, therefore e e(0)(m) =e Z e S n ϕ(m) +e α|(∇m)|e 2+ |(n(eψ−1(x)),e m(e x))|e 2odx.e
And thus we have a generalization of the results of [8].
Acknowledgements. I wish to thank Professor Herv´e Le Dret for many useful
discussions concerning the subject of this paper.
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