Modeling thin curved ferromagnetic ﬁlms

The behavior of a thin ﬁlm made of a ferromagnetic material in the absence 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 G -convergence arguments.


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.According to this theory, the equilibrium state of a ferromagnetic body is described by its magnetization m, wich is a vector field defined on R 3 , 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 Ω is submitted to an external magnetic field H e , the observed states are the minimizers of an energy E e depending on the magnetization m: 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 specimen 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 which ϕ, which is supposed to be a nonnegative, even function exhibiting cristallographic symmetry, vanishes.In general, ϕ is supposed to be a polynomial function.The coefficient Q controls the relative importance of the anisotropic energy compared to the magnetostatic energy.The magnetostatic 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 R 3 , and u is a scalar potential verifying the magnetostatic equation div(−∇u + m) = 0 in R 3 . (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 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 Ω h .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 independent 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 C 1 , q-dimensional manifold of R d .
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 minimizers 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 the Γ-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 (e 1 , e 2 , e 3 ) be the canonical orthonormal basis of the Euclidean space R 3 .We denote by |v| the norm of a vector v in R 3 , by u•v the scalar product of two vectors in R 3 , by u ∧ v their vector product and by u ⊗ v their tensor product.Let M 33 be the space of 3 × 3 real matrices endowed with the usual norm |F| = tr(F T F).This is a matrix norm in the sense that |AB| ≤ |A||B|.We denote by A = (a 1 |a 2 |a 3 ) the matrix in M 33 whose ith column is a i .We consider a thin curved film of thickness h > 0 occupying at rest an open domain Ω h .This reference configuration of the film is described as follows.We are thus given a surface S, called the midsurface of the film.This surface is a bounded two-dimensional C 1 -submanifold of R 3 and we assume for simplicity that it admits an atlas consisting of one chart.Let ψ be this chart, i.e. a C 1 -diffeomorphism from a bounded open subset ω of R 2 onto S.
Let a α (x) = ψ ,α (x) be the vectors of the covariant basis of the tangent plane T ψ(x) S associated with the chart ψ, where ψ ,α denotes the partial derivative of ψ with respect to x α .We assume that there exists δ > 0 such that |a 1 (x) ∧ a 2 (x)| ≥ δ on ω and we define the unit normal vector a 3 (x) = a 1 (x)∧a 2 (x) |a 1 (x)∧a 2 (x)| , which belongs to C 1 ( ω, R 3 ).The vectors a 1 (x), a 2 (x) and a 3 (x) constitute the covariant basis at the point x.We define the contravariant basis by the relations a i (x) • a j (x) = δ i j , so that a α (x) ∈ T ψ(x) S and a 3 (x) = a 3 (x).

Next, we define a mapping
It is well known that there exists h * > 0 such that for all 0 < h < h * , the restriction of diffeomorphism on its image by the tubular neighborhood theorem.For such values of h, we set Ω h = Ψ(Ω h ).Alternatively, we can write where π denotes the orthogonal projection from Ω h onto S, which is well defined and of class C 1 for h < h * .Equivalently, every x ∈ Ω h can be written as Thus, we have a curvilinear coordinate system in Ω h naturally associated with the chart ψ by Since the scalar potential u h is defined on R 3 we suppose that the middle surface We can also suppose that Ψ is equal to the identity outside a compact set containing Ω h .In what follows, we will keep the notation Ψ to mean Ψ.
We also note that det The matrix ∇Ψ(x 1 , x 2 , x 3 ) is thus everywhere invertible in Ωh and 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.

The three-dimensional and rescaled problems
The behavior of a curved ferromagnetic thin film of thickness h occupying the domain Ω h is described by a micromagnetic energy e h depending on its magnetization m h : R 3 → R 3 .The vector field m h vanishes outside the film.We refer to [6] and the references therein for more information about ferromagnetic materials.
The magnetization of the film minimizes an energy e h of the form under the saturation constraint 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 energies.The density verifies the following growth, coercivity and locally lipschitz dependence assumptions ) In the case of Gioia and James we have W (y, F) = ϕ(y) + α|F| 2 and p = 2.
The second term in (3.1) depends on the magnetization m h and a scalar potential u h for the induced magnetic field which we can also write div B h = 0, where The observed states of the ferromagnetic film are the solutions of the minimization problem where in Ω h and m = 0, a.e. in Ω c h , with p > 1.In this article we do not make any convexity assumption on W , consequently problem (3.3) may well not possess any solutions.Thus, we consider a diagonal minimizing sequence m h for the sequence of energies e h over the sets V h .More specifically, we assume that where ε is a positive function such that ε(h) → 0 when h → 0. Using Γ-convergence arguments, we study the behavior of the energy e h and 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 m h : R 3 → R 3 and u h : R 3 → R by setting for all Then, we rescale the problem by setting m(h

but not necessarly outside of this domain). Setting e(h)(m(h)) = e h (m h ), we obtain e(h)(m(h))
= The magnetostatic equation then reads and the saturation constraint reads

Magnetostatic energy behavior
In this section we study the behavior of the magnetostatic energy term when the thickness of the curved film goes to zero.Let us thus be given m : ) and m = 0 on Ω c 1 .We consider the minimization problem: Find u(h, m) ∈ U such that with where B is the unit ball of R 3 .The condition B v dx = 0 excludes the trivial translations v → v + c.We endow U with the scalar product 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 that is the weak form of (3.4).
2), we obtain where the right-hand side is equal to twice the magnetostatic energy term E mag (h) corresponding to a given magnetization m.The following proposition gives the behavior of the magnetostatic energy term under convergence of the magnetization.

Proposition 4.1 Let m(h) be a sequence of functions in L
be the solution of the minimizing problem (4.1) associated to m(h).Then, we have and a 3 represents the third column vector of We recall that a 3 is the third vector of the covariant basis associated to the diffeomorphism Ψ on Ω 1 .
Proof Since ū(h, m(h)) minimizes I h m(h) over U and 0 ∈ U, we have Using the triangle inequality, we obtain that Thus, for a subsequence still denoted h, there exists The L p version of the Poincaré Lemma implies the existence of u(0 with u(0) ,3 = 0, which implies that ∇u(0 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 The convergences (4.4), (4.5) and the strong convergence of m(h) to m(0) in where 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 w ε ∈ C ∞ c (R 3 ) that converges strongly in L 2 (R 3 ) to w when ε → 0. Let a ε > 0 be such that the projection of the support of w ε on the x 3 axis belongs to [−a ε , a ε ] and we suppose that a ε > 1.We set for λ > 0 The constant c ε is chosen in such a way as to satisfy the condition on the ball B and the second term guarantees that ∇v Since ū(h, m(h)) is the solution of the minimization problem (4.1) we have that Using (4.8), (4.6) and the fact that , where a i represents the vectors of the contravariant basis associated with the diffeomorphism ψ with |a 3 | = 1, we obtain (4.10) On the other hand, we have in C 0 (R 3 ).Let a i be the column vectors of A T 0 which are defined in R 3 .We know that a i = a i , in Ω 1 .Passing to the limit in (4.9) when h → 0 and using (4.10), we obtain that Expanding this expression, we obtain Since the projection of the support of w ε on the x 3 -axis belongs to [−a ε , a ε ], we have that w ε (x)χ [a ε ,a ε +λ] (x 3 ) = 0, a.e. in R 3 , thus the third term of the right hand side of inequality (4.11) vanishes.We also have that χ [a ε ,a ε +λ] (x 3 ) = 0 in Ω 1 since a ε > 1.Thus, the fourth term of the right hand side of inequality (4.11) also vanishes.Considering all this, inequality (4.11) becomes Noting that R χ 2 [a ε ,a ε +λ] (x)dx = λ, and passing to the limit when λ → +∞ we obtain Then we take the limit as ε → 0 and obtain Due to the weak convergences in (4.4), (4.5) and since A T 0 (∇ p u(0) + we 3 ) = A T 0 (we 3 ) = w a 3 in R 3 , we have Then, we deduce the convergence of the norm thanks to (4.12) and (4.13) , therefore that gives the strong convergence in (4.4) and (4.5).Let us now identify the norm of w a 3 in L 2 (R 3 ).Letting h → 0 in (4.7), we get We set where mε (0) is a sequence in C ∞ 0 (R 3 ; R 3 ) that converges to m(0) in L 2 (R 3 ; R 3 ) when ε goes to zero with supp mε (0) ⊂ Ω 1 and c ε a constant depending on ε.We have that The minimization problem (4.1) implies that Letting h go to zero, we see that Let us estimate the right-hand side of this enequality, we have that and since the projection of the support of mε (0) and of m(0) on the x 3 axis stays in [0, 1], it follows that We also have that 1 Applying all this in (4.15) we get, letting λ → ∞ Then, we let ε → 0 and obtain and using (4.14) we get Consequently, we have Thanks to the Cauchy-Schwartz inequality, equation (4.14) gives and using (4.17) we obtain Finally, we have Now that we know the behavior of the magnetostatic energy term, we can compute the Γ-limit of the free energy e(h).

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 with ε(h) → 0 when h → 0 and in Ω 1 and m = 0, a.e. in Ω c 1 .In the following, we will identify functions in V with functions in W 1,p (Ω 1 ; S 2 ) 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
(5.1) Moreover, the limit points of this sequence for the weak topology of W 1,p belong to the set Proof Since the magnetostatic energy term is positive, we have Thus, we obtain (5.1).We deduce the existence of a subsequence, still denoted h and m(0 with m(0) ,3 = 0.The weak convergence in W 1,p (Ω 1 ; R 3 ) implies that for a subsequence still denoted h we have m(h) → m(0) in L p (Ω 1 ; R 3 ), and thus, for a subsequence, we have m(h) → m(0) a.e. in Ω 1 , which implies that for almost every x ∈ Ω 1 we have |m(0)| = 1.
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 C 1 submanifold of dimension q of R d .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 M d×N the set of matrices d × N. Let f : M d×N → [0, +∞[ be a Borel measurable function and M a C 1 submanifold of dimension q of R d .We denote by T y (M) the tangent space to M at y ∈ M.

Definition 5.1 Let y ∈ M and ξ ∈ T y M
N .The tangential quasiconvexification of f in ξ relative to y is defined by In our case M is the unit sphere S 2 of R 3 and T y (S 2 ) = y ⊥ , the plane orthogonal to y.We recall two results proved in [5] Proposition 5.1 Let f : M d×N → [0, +∞) be a continuous function such that there exists c > 0 and p ≥ 1 verifying 0 ≤ f (ξ) ≤ c(1 + |ξ| p ), for every ξ ∈ M d×N .We define the functional J by Then, J(.) is sequentially weakly lower semicontinuous in W 1,p (Ω 1 ; M). where In our case, the magnetization m is constrained to remain in the sphere S 2 .Consequently, m ,i will be in m ⊥ almost everywhere.As in Acerbi, Buttazzo and Percivale [1] for nonlinearly elastic strings and Le Dret and Raoult [10] for membranes, we define W 0 : We show as in Le Dret and Raoult [10], that the function W 0 is continuous and satisfies properties analogous to those verified by W , that is to say 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 compute the Γ-limit is W 1,p (Ω 1 ; R 3 ).However, W 1,p (Ω 1 ; R 3 ) endowed with the weak topology is not metrizable.Thus, we extend the energy e(h) to all L p (Ω 1 ; R 3 ) by setting for all m ∈ L p (Ω 1 ; R 3 ), e * (h We now are in a position to compute the Γ-limit of the free energy.

Theorem 5.2
The sequence of energies e * (h) Γ-converges for the strong topology of L p (Ω The proof of the theorem follows from the following propositions.Preuve We obtain the lower bound of the Γ-limit by showing that for any sequence m h ∈ L p (Ω 1 ; R 3 ) converging strongly to m 0 in L p (Ω 1 ; R 3 ), we have First, if m h / ∈ V , then e * (h)(m h ) = +∞ and the result is trivial.In the same way, if m h ∈ V and lim inf e * (h)(m h ) = +∞, the result is obvious.
Next, we consider m h ∈ V , such that lim inf e * (h)(m h ) = lim inf e(h)(m h ) < +∞.This implies that for a subsequence, there exists c > 0 such that e(h)(m h ) < c.Using the lemma 5.1 we get that for a subsequence still denoted h we have that m h m 0 in W 1,p (Ω 1 ; R 3 ).Thanks to the fact that m h is uniformly bounded in L ∞ (Ω 1 ; R 3 ) we obtain that m h → m 0 in L s (Ω 1 ; R 3 ), for all s > 1, in particular for s = 2.We thus deduce from proposition 4.1 that which represents the exchange and anisotropic energies.We have Using the properties (3.2) of W and those of the diffeomorphism Ψ, we show as in [11] that Thus, we have Taking the limit when h → 0 we obtain using (5.4) and proposition(5.1) that Using (5.3) and since m 0 ,3 = 0 (lemma 5.1), we obtain that Then we proceed to the computation of the upper bound of the Γ-limit.We will need the following lemma (see [10]) (5.5) Preuve In order to find an upper bound for the Γ-limit, we have to construct a sequence Let Setting, for all h, m h = m 0 we get (5.6).Next, we consider m 0 ∈ V M and we set m h (x) = m 0 (x 1 , x 2 ) + hx 3 ξ(x 1 , x 2 ), with ξ ∈ W 1,p 0 (ω; R 3 ) to be chosen later.We temporarily do not consider the saturation constraint |m h | = 1, we will treat it later.We have m h → m 0 strongly in W 1,p (Ω 1 ; R 3 ).Using the Lebesgue convergence theorem, we have that for a subsequence still denoted h ) is a Caratheodory function.Thus, the measurable selection lemma implies that, there exists ξ : ω → R 3 measurable such that W 0 (x, m 0 (x), (m 0 ,1 (x)|m 0 ,2 (x))) = W (m 0 (x), (m 0 ,α (x)|ξ(x))A 0 (x)), (see [7]).If ξ / ∈ m 0 ⊥ , this implies that W 0 (x, m 0 (x), (m 0 ,1 (x)|m 0 ,2 (x))) = +∞ and the result is obvious.If ξ ∈ m 0 ⊥ a.e., this implies that (5.7) and thanks to the properties (3.2) of W , we obtain that ξ ∈ L p (ω; R 3 ).The density of C ∞ c (ω; R 3 ) in L p (ω; R 3 ) implies the existence of a sequence ξ ε ∈ C ∞ c (ω; R 3 ) verifying ξ ε → ξ strongly in L p (ω; R 3 ).Next, we project ξ ε on m 0 ⊥ setting ξε = (I − m 0 ⊗ m 0 )ξ ε .We have ξε ∈ W 1,p 0 (ω; R 3 ) verifying ξε → ξ strongly in L p (ω; R 3 ).Thus, using the Lebesgue convergence theorem, we have that for a subsequence still denoted This means that for every η > 0, there exists an ε(η) > 0 such that ∀ε ≤ ε(η) we have Let us set mh ε = m 0 + hx 3 ξε .We have that mh ε ∈ W 1,p (Ω 1 ; R 3 ).We fix ε > 0, for h < a.e.Then, we set By construction, we have m h ε ∈ V thanks to the algebra property of W 1,p (ω; R 3 ) ∩ L ∞ (ω; R 3 ).We have We also have m h ε → h→0 m 0 strongly in L p (Ω 1 ; R 3 ), and On the other hand, we have since |m 0 | = 1, and we also have Now, ξε ∈ m 0 ⊥ so that we have (m 0 ⊗ m 0 ) ξε = 0 and thus, we obtain Using the Lebesgue theorem, we see that Thus, using proposition (4.1), we obtain Let us set We have shown that lim which implies that Γ − lim sup h→0 e * (h)(m 0 ) ≤ G * (m 0 ) + η.Since this is true for every η > 0, we obtain that 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 V M by 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 W 1,p (Ω 1 , R 3 ) belong to V M and minimize the energy e(0) defined by Preuve The proof of the corollary follows from Lemma 5.1 and the standard Γconvergence argument.
We consider another chart ψ from ω , an open set in R 2 into S. Working with this new chart, we obtain exactly the same Γ-convergence result, but this time, through another diffeomorphism Ψ .This means that the limit model is intrinsic and only depends on the curved surface S. Let us thus write the limit model on the curved surface S. As in [11], for any unit vector e of S 2 , we consider a bounded open set O e ⊂ e ⊥ and denote by π e the orthogonal projection on this set.We extend any function χ ∈ W 1,∞ 0 (O e , R 3 ) setting χ e (y) = χ(π e (y)) and we define for any y ∈ O e , D e ⊥ χ(y) = ∇χ e (y).To any m ∈ VM we associate a magnetization m : R 3 where n denotes the normal vector to S (for a given orientation).All functions defined on the surface are implicitly assumed to be extended to a tubular neighborhood of the surface by being constant on each normal fiber.Partial derivatives are computed on these extensions and then restricted to the surface.Preuve We proceed as in Le Dret and Raoult [11].For every m ∈ V M we have We set x = ψ(x), m( x) = m(x).This implies that ∇m(x) = ∇ m( x)∇ψ(x) and setting e(0)( m) = e(0)(m) we obtain The integral representation of the tangential quasiconvex envelope (5.2) is written in our case with O a bounded open domain in R 3 and the infimum does not depend on the choice of this open domain.Thus, we have for all The definition of W 0 implies that Moreover, we have We also have The condition m ,3 = 0 becomes ∇ m( x)a 3 (ψ −1 ( x)) = 0 and so we have We also have which implies that ∇ m( x)(∇ψ(ψ −1 ( x))|0)A 0 (ψ −1 ( x)) = ∇ m( x).(6.2) Next, we set s = ∇ψ(ψ −1 ( x))s and χ( s) = χ(s).We have χ ∈ W 1,∞ 0 (O a 3 (ψ −1 ( x)) ; R 3 ) and (∇χ(s)|0)A 0 (ψ −1 ( x)) = D a 3 (ψ −1 ( x)) ⊥ χ( s).(6.3) Finally, for every z ∈ R 3 we have (0|z)A 0 (ψ −1 ( x)) = z ⊗ a 3 (ψ −1 ( x)).(6.4) Choosing O = ∇ψ(ψ −1 ( x)) −1 O a 3 (ψ −1 ( x)) and replacing (6.2), (6.3) and (6.4) in (6.1), we obtain + D a 3 (ψ −1 ( x)) ⊥ χ(s)) ds , and thus the result.

Model with external magnetic field
If the curved film is placed in a uniform external magnetic field H e , the free energy governing the behavior of the film will contain a term of external energy E ext , called the Zeeman energy, depending on the magnetization m h of the film via Setting for all x ∈ Ω 1 , m(h)(x) = m(Ψ(x 1 , x 2 , hx 3 )) and E ext (h)(m(h)(x)) = E ext ( m h ) we obtain E ext (h)(m(h)) =

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| Since ∇ m n = 0, this implies that ∇ m : z⊗ n = 0. We also have, since D n ⊥ χ(s) n = 0, that z ⊗ n : D n ⊥ χ(s) = 0. Thus, we have This means that the infimum for z ∈ m ⊥ is reached when z belongs to n ⊥ ∩ m ⊥ in particular for z = 0. Thus, we obtain And thus we have a generalization of the results of [8].

Proposition 5 . 3
the lower semicontinuous envelope of G for the weak topology of X and Γ − G * the lower semicontinuous envelope of G * for the strong topology of Y .Then Γ − G * = (Γ − G) * .We have Γ − lim sup e * (h) ≤ e * (0).