We develop and we explain the two-scale convergence in the co- variant formalism, i.e

AIMS’ Journals

VolumeX, Number0X, XX200X pp.X–XX

GEOMETRIC TWO-SCALE CONVERGENCE ON MANIFOLD AND APPLICATIONS TO THE VLASOV EQUATION.

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Centre de physique th´eorique-CNRS Luminy-Marseille

Fr´enod Emmanuel

LMBA (UMR 6205) Universit´e de Bretagne-Sud Vannes

(Communicated by the associate editor name)

Abstract. We develop and we explain the two-scale convergence in the co- variant formalism, i.e. using differential forms on a Riemannian manifold. For that purpose, we consider two manifoldsM andY, the first one contains the positions and the second one the oscillations. We establish some convergence results working on geodesics on a manifold. Then, we apply this framework on examples.

1. Introduction. The two-scale convergence initiated by Nguetseng [7] and de- veloped by Allaire [6], establishes convergence results for a sequence of functions (u)>0 containing oscillations of periodand defined in an open domainW ofRⁿ to a functionU(x,y) onW ×Rⁿ and periodic in the second variabley.

The principle is:

On the open domain W, we fixe the period and we suppose that the solution of equation: Lu =f isu where L is a differential operator which induces oscilla- tions of period and f is a source term that does not depend on , (we can also put boundary conditions). So, we will consider

• the space of functionsr-integrable inW, denoted byL^r(W) and defined such that the set of all measurable functions fromW to Ror toCwhose absolute value raised to the r-th power has finite integral, i.e

L^r(W) ={f measurable such thatkfk_r= Z

W

|f |^{r 1}_r

<+∞},

for 1 6 r < +∞ and if r = +∞ it is defined as the set of all measurable functions fromW toRor toCwhose their essential supremum is finite,

• the space of functions r-integrable on W, r-integrable on Y and Y-periodic and denoted byL^r(W, L^r_per(Y)) i.e

L^r(W, L^r_per(Y) ={f measurable such that Z

W

Z

Y

|f |^{r 1}_r

<+∞},

2010Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35.

Key words and phrases. Two-scale convergence, Differential geometry, Vlasov equation . The first author is supported by NSF grant xx-xxxx.

1

for 16r <+∞,

• and the space of functions of classC⁰ with a compact support on W and of classC⁰ onY andY-periodic denoted byC_c⁰(W, C_per⁰ (Y)).

In the following, we will say that the sequence of functions (u)>0 in L^r(W) for r∈(1,+∞] two-scale converges to a functionU in the spaceL^r(W, L^r_per(Y)) if for any functionsψinC_c⁰(W, C_per⁰ (Y)), we have

→0lim Z

W

u(x)ψ(x,x )dx=

Z

Y

Z

W

U(x,y)ψ(x,y)dxdy. (1) We call U the two-scale limit ofu in L^r(W, L^r_per(Y)). Nguetseng [7] and Allaire [6] established a two-scale convergence criterion which is very useful to establish an equation verified by the two-scale limit.

Since Physic’s equations can be written using differential forms on manifold, we want to develop the two-scale convergence in this formalism, i.e. on manifold. We must use tools of differential geometry in covariant formalism. This point of view will allow us to work in a larger context, more adapted for differential equations.

First, we detail all important notions and all useful tools to establish geometric two-scale convergence. We can observe that differential forms are not always regular objects. So we remind some notions of functional analysis and we improve them to use them for differential forms. The functional analysis in the context of covariant formalism is developed by Scott [4, 3] and called L^r-cohomology. Then we adapt the two-scale convergence on manifold using the Birkhoff’s theorem and we do asymptotic analysis using geometric objects (differential forms). This study was begun by Pak [2]. We explain notions of strong and weak convergence and in the last section, we apply this new point of view on Vlasov equation, in a context close to Fr´enod, Sonnendr¨ucker [5] and Han-Kwan [13]. In this dimensionless Vlasov equation we observe the apparition of finite Larmor radius and oscillations with a period. So, the dimensionless Vlasov equation can be written asLu= 0.

2. Reminders on differential geometry. In the following, we will consider that M andY are twon-dimensional Riemannian manifolds.

For all pointspin M, there exists a tangent space toM atpdenoted by TpM. It represents the set of tangent vectors toM atp. On tangent spaceTpM, we have a scalar productgp (the Riemannian metric) represented by a symmetric matrix gp, nondegenerate and positive definite, such that for allu, v∈TpM, we have:

g_p(u, v) :=u·(g_pv),

where·is the scalar product. The tangent bundle ofM is the disjoint union of the tangent spaces toM:

T M := G

p∈M

T_pM := [

p∈M

{p} ×T_pM.

With the Riemannian metric we can define, on each tangent spaceT_pM, a Banach norm given by

kvpk=g_p(v_p, v_p),

for allp∈M andvp∈TpM. With the help of this norm, we can define the length of a piecewiseC¹ curveγ: [t1, t2]→M joining p1 top2in M by

L(p1, p2, γ) :=

Z p₂ p1

kγ⁰(t)kdt.

We define also by

d(p1, p2) := infL(p1, p2, γ)

where the infimum is on all the curves which are piecewise C¹ and joiningp₁ to p₂ inM. The curves minimizing the lengthL(p₁, p₂, γ) are called geodesics and a geodesic which length equalsd(p₁, p₂) is called a minimizing geodesic. Forp∈M andvp∈TpM, there exists an unique geodesicγv_p defined in the neighborhood of psuch that

γ_vp(0) =p and γ_v⁰

p(0) =v_p. Let us denote by

V0={tvp∈TpM |vp∈TpM, kvpk= 1 andt∈[0, c(vp)]}, (2) and define onV0the exponential map

exp^M_p : V0 → M

vp 7→ γv_p(1), (3)

where

c(vp) := sup{t >0 :d(p, exp^M_p (t vp) =t}.

Notice thatV0 is the larger open set containing 0 in TpM such as if vp ∈V0 then γv_p is defined fort= 1. The geodesic verifyingγv_p(0) =pandγ_v⁰_p(0) =vp has the form

{γv_p(t) =exp^M_p (tvp), t∈[0, c(vp)]}.

T M is a differential orientable manifold with dimension 2n. A section of T M is a map f : M → T M such that π(f(p)) =pfor all p∈M where πis the natural projection

π:T M →M.

Let us also denote byT_p^?M the dual space ofT_pM. It is also a vectorial space and its elements are called 1-forms. In the same way, we define the cotangent bundle of M byT^?M:

T^?M := G

p∈M

T_p^?M := [

p∈M

{p} ×T_p^?M.

Using the exterior product∧, fromkelementsµ₁, . . . , µ_k ofT_p^?M, we can define the k-formµ₁∧ · · · ∧µ_k so the set of all the k-forms atpbyVk

(T_p^∗M) and the bundle manifold

^_k

(T^∗M) := G

p∈M

^_k

(T_p^∗M) := [

p∈M

{p} ×^_k (T_p^∗M).

Since, as T M, T^?M and Vk

(T^∗M) for k = 1, . . . , n are bundle manifolds with the natural projections π^? : T^?M →M and π^?_k : Vk

(T^∗M) →M, we can define sections inT^?M andVk

(T^∗M) as being the mapsf andf_k from M to T^?M and from M to Vk

(T^∗M) respectively and such that π^?(f(p)) =pand π_k^?(f_k(p)) =p for allpin M. And so a differentialk-form onM is a section of Vk

(T^∗M):

ω^k:M →Vk

(T^?M),

and the set of differentialk-forms onM is denoted by Ω^k(M). In a local coordinate chart (U, ϕ) such thatϕ:U ⊂Rⁿ→ϕ(U)⊂M, we denote by (x₁, . . . , x_n) the local coordinate system ofp∈M. All differentialk-forms have the following expression inU

X

i₁,...,i_k

ω^k_i

1,...,i_k(x)dx_i1∧ · · · ∧dx_ik,

whereω^k_i1,...,ik(x) are functions fromU to R.

We also have some operators acting on differential forms as the exterior product, the exterior derivative, the pull-back, the push-forward, the interior product, the Lie derivative and the Hodge star operator. These operators are useful to translate equations in the language of differential geometry.

• Theexterior product: Letω^k∈Ω^k(M) andη^l∈Ω^l(M). The exterior product ofω^k andη^k is a differential formω^k∧η^l∈Ω^k+l(M) which acts, for all points pinM, and on (k+l) vectorsξ1, . . . , ξk+l ofTpM in the following way:

ω^k∧η^l

p(ξ₁, . . . , ξ_k+l):=X

σ∈Sk+l

(−1)^sgn(σ)

k!l! ω^k_p ξ_σ(1), . . . , ξ_σ(k)

η_p^l ξ_σ(k+1), . . . , ξ_σ(k+l) , withS_k+l the set of all permutations of{1, . . . , k+l}.

• The exterior derivative d is a linear operator from Ω^k(M) to Ω^k+1(M) and we have

dω_p^k(ξ₀, . . . , ξ_k) :=

k

X

i=0

(−1)ⁱξ_i·ω^k_p(ξ₀, . . .ξˆ_i, . . . , ξ_k)

+ X

06i<j6k

(−1)^i+jω_p^k([ξi, ξj], . . . ,ξˆi, . . . ,ξˆj. . . , ξk) with ω_p^k a differential k-form on M at pand [ξi, ξj] = ξiξj −ξjξi is theLie brackets. The symbol ˆξ_i means that we take off the vectorξ_i.

• Thepull-back: Letf :Y →M be a differentiable map andω^kbe a differential k-form on M. The pull-pack of ω^k byf,f^?(ω^k) is the differentialk-form on Y defined by:

(f^?(ω^k))_q(ξ₁, . . . , ξ_k) :=ω_f(q)^k (f_?ξ₁, . . . , f_?ξ_k),

where q∈Y,ξ1, . . . , ξk ∈TqY andf? :T Y −→T M is thepush forward, i.e.

the map which associates to any vectorξofTqY the vectordfq(ξ) inT_f(q)M.

• Theinterior product of a differentialk-form ω^k onM along the vector field τ on M, is a differential (k−1)-form i_τω^k which acts for all p∈ M in the following way:

∀ξ₁, . . . , ξ_k−1∈T_pM i_τω^k_p(ξ₁, . . . , ξ_k−1) :=ω_p^k(τ, ξ₁, . . . , ξ_k−1).

i_τ is a linear operator from Ω^k(M) to Ω^k−1(M).

• TheLie derivative L along the vector field τ is linear operator from Ω^k(M) to Ω^k(M). For any differential k-forms ω^k, for all points p∈ M and for all ξ₁, . . . , ξ_k ∈T_pM:

Lτω^k

p(ξ1, . . . , ξk) = d dt

_t=0

ω^k_p (φ^t_p)?(ξ1), . . . ,(φ^t_p)?(ξk) ,

withφ^t_pthe flow of the vector fieldτ∈T_pM and (φ^t_p)_?the push forward such that (φ^t_p)_?(ξ) = dφ^t_p(ξ) for all ξ ∈ T_pM. We also rewrite the Lie derivative using the homotopy formula:

Lτ =d i_τ+i_τd.

Before defining the Hodge star operator, we notice that the metric defines an inner product onT^?M as well as onT M. The map

ˆ

gp : TpM → T_p^?M vp 7→ g(vp,·)

defines an isomorphism from TpM to T_p^?M, that says that we can identify the tangent TpM and the cotangent spaceT^?M. We may extend this identification to all vector fields onM,T M and the space of all differential forms onM,T^?M. That means we can define an inner product (ωp, ηp) for any two differential 1-formsωand η onM at each pointpand so we have a function (ω, η) onM. We shall generalize it to the case of differentialk-forms. For two elements of the formα1∧ · · · ∧αk and β1∧ · · · ∧βk (αi, βj∈T^?M) the value of inner product is

(α₁∧ · · · ∧α_k, β₁∧ · · · ∧β_k) =det((α_i, β_j)_i,j).

In this way we have the inner product defined for any two differential k-forms on M. For example in the local coordinates ofp, that gives

(dx₁∧ · · · ∧dx_n, dx₁∧ · · · ∧dx_n) =|g_p⁻¹|,

where | g⁻¹_p | is the determinant of the inverse matrix associated with the metric gp. In the local coordinate system ofp, we can write the natural measure onM:

vol_p(g) =vol_p= q

(|g_p|)dx₁∧ · · · ∧dx_n.

Now, we can define correctly the Hodge star operator and the co-exterior deriv- ative.

• The Hodge star operator is a linear isomorphism ? : Ω^k(M) −→ Ω^n−k(M) which associates to each differential k-form ω^k, a differential (n−k)-form ω^n−k (wherenis the dimension ofM). It has the following property that at each point we have

(ω^k, β^k)volp(g) =ω^k∧?β^k,

forω^kandβ^kdifferentialk-forms onM. And for an orthogonal basisdx₁, . . . , dx_n at a pointp, we have that

?(dxi₁∧ · · · ∧dxi_k) = p|g_p|

(n−k)!gⁱ_p^1l1. . . g_p^iklkδl₁...l_kl_k+1...l_ndxl_k+1∧ · · · ∧dxl_n, where x is the local coordinate system of p, gp is the matrix associated to the metric in these chart, |gp| is the determinant of the Riemannian metric g_p, g_p^i1l1 is the component of inverse matrix of g_p and δ is the Levi-Civita permutation symbol.

In the local coordinate system, for a differentialk-form ω^k with the form

ω^k_p = X

(i1<...<i_k)∈{1,...,n}

ω_i1,...,ik(x)dx_i1∧ · · · ∧dx_ik, we have:

?ω_p^k= X

(j1<...<jn−k)∈{1,...,n}

q

|gp|gⁱ_p^1l1. . . g_p^iklkδi₁...i_kl_k+1...l_nωi₁,...,i_k(x)dxj₁∧ · · · ∧dxj_n−k. The Hodge star operator has also the following property:

? ? ω^k = (−1)^k(n−k)ω^k.

• The co-exterior derivative d^∗ : Ω^k(M)→ Ω^k−1(M) is a derivation operator defined byd^∗ω^k := (−1)^n(k−1)+1? d ? ω^k.

3. Geometric tools: TheL^r-cohomology. The reader is referred to [1] to have more details about this section.

3.1. Generalities. To do L^r-cohomology, we must define what it means for dif- ferential k-forms to be r-integrable on a manifold. To do that, we remind some definitions.

We say that a differential k-form ω^k is measurable on M if it is a measurable section of vectorial bundle Vk

(T^?M) and that it isdefined almost everywhere on M if there exists a domain N ⊂ M with measure zero such that the function ω^k:M →Vk

(T^?M) is well-defined on the domain M\N.

Using the definition of a differentialk-form in local coordinate system,ω^k is mea- surable onM if and only if for all charts covering M, the coefficients ω_i^k

1,...,i_k are measurable functions anddefined almost everywhereif for all charts coveringM the coefficientsω^k_i

1,...,i_k are functions defined almost everywhere.

A differentialk-formω^k has ank-dimensional compact support KinM if for all pointpoutside ofKand for all vectorsξ1, . . . , ξk ∈TpM we haveω^k_p(ξ1, . . . , ξk) = 0.

The integral of a differentialk-form is defined as follows: LetM^k be a differential k-dimensional manifold included inM, (Ui⊂R^k, ϕi)_i∈I be a set of charts covering M^k and {λi}_i∈I be a partition of unity subordinate to {ϕ(Ui)} (i.e indexed over the same set I such that supp λi ⊆ϕ(Ui)). So all differential k-forms ω^k on M^k can be written as

ω^k=X

i∈I

λiω^k. So the integral overM^k ofω^k is then defined as

Z

M^k

ω^k =X

i∈I

Z

ϕ_i(U_i)

λiω^k =X

i∈I

Z

U_i

ϕ^?_i(λiω^k).

A differentialk-formω^k is said to beC^sonM, if for allp∈M,ω^k_p ∈Vk

(T_p^?M) andϕ^?(ω^k) is a differentialk-formC^sonU ⊂Rⁿfor all chartsϕcoveringM. The set of differentialk-forms C^s onM is denoted by Ω^k_s(M).

Now, we are ready to define differentialk-formsr-integrable on a manifold.

We denote byL^r(M,Vk

) the space ofr-integrable differentialk-forms. It is defined by

L^r(M,Vk

) ={α∈Ω^k(M) mesurable such thatkαk_Lr(M,Vk)<+∞}, for 16r6+∞, where

kαk_Lr(M,Vk):=

Z

M

|α|^r_pvol_p ¹_r

, for 16r <∞where

kαk_L+∞(M,Vk):= sup ess

p∈M

|α|p, with

|α|^r_p= ?(α^k_p∧?α^k_p)^r₂ .

In local coordinate system, we can observe that|α|^r_p corresponds to

|α|^r_p=



 X

i₁,...,i_k

(αi1,...,i_k(x))²





r 2

.

We can associate a scalar product with normkαk_L2(M,Vk):

< α, β >_L2(M,Vk):=

Z

M

α∧?β ,

withα, β∈Ω^k(M) being measurable. k · kand<·,·>do not depend on charts on M. The function spaceH^1,d(M,Vk) is the completion of Ω^k(M) with the respect of the norm

< α, α >_L2(M,Vk)+< dα, dα >_L2(M,Vk+1)

¹₂

, (4)

and the function spaceH^1,δ(M,Vk) is the completion of Ω^k(M) with the respect of the norm

< α, α >_L2(M,Vk)+< δα, δα >_L2(M,Vk−1)

¹₂

. (5)

If M is a compactn-dimensionnal Riemannian manifold with boundary, we want to define the tangential component and the normal component. These notions are important because they permit to have the Green-Stokes formula and the integration by parts. For doing this, we observe that we can associate forms to vector fields (and vice-versa) and we can choose a local coordinate system such that the Riemannian metric has the form of 2-forms:

g²_p=X

i,j

g_p^i,jdxⁱ⊗dx^j,

where⊗corresponds to the tensor product. For two vectors with the formP

iai ∂

∂xⁱ

andP

ibi ∂

∂xⁱ in the local coordinate system ofp, we have g²_p(X

i

a_i ∂

∂xⁱ,X

j

b_j ∂

∂x^j) =X

i,j

a_ig_p^i,jb_j, and

g²_p(X

i

ai

∂

∂xⁱ,·) =X

i,j

aig_p^i,jdx^j.

That means thatg_p² takes a vector field µ_p to an 1-form denoted by g_p²(µ_p) =µ^]_p. We denote byµ the outgoing unit normal vector to∂M and byµ^] the differential 1-form associated to µ. With these, the tangential component of the differential k-form ω^k corresponds to µ^]∧ω^k and the normal component of ω^k is iµω^k. We reformulate the Green-Stokes formula for the differential forms in two different ways:

• with the exterior derivative:

< dω^k, α^k+1>_L2(M,Vk+1)=< ω^k, d^∗α^k+1>_L2(M,Vk)+< ν^]∧ω^k, α^k+1>_L2(∂M,Vk+1), withα^k+1 a differential (k+ 1)-form,

• and with the co-exterior derivative

< d^∗ω^k, ψ^k−1>_L2(M,Vk−1)=< ω^k, d ψ^k−1>_L2(M,V_k

)−< i_νω^k, ψ^k−1>_L2(M,Vk−1), withψ^k+1 a differential (k−1)-form.

Now we suppose that ¹_r+¹_s = 1 with 16r, s6+∞.

Proposition 1. 1. (H¨older’s Inegality) Ifα^k∈L^r(M,Vk

)andβ^k ∈L^s(M,Vk

), soα^k∧?β^k ∈L¹(M,Vn

)and we have

kα^k∧?β^kk_L1(M,Vn)6kα^kk_Lr(M,Vk)kβ^kk_Ls(M,Vk). 2. (Minkowski’s inegality) If α^k, β^k ∈L^r(M,Vk

)so α^k+β^k ∈L^r(M,Vk

)and we have

kα^k+β^kk_Lr(M,Vk)6kα^kk_Lr(M,Vk)+kβ^kk_Lr(M,Vk). 3. L^r(M,Vk

) is a Banach space.

4. L²(M,Vk

) is a Hilbert space.

5. (Density) C_c²(M,Vk

) The set of differentialk-forms of class C² with a com- pact support inM are dense inL^r(M,Vk

)

Now we can define the Sobolev’s space in these formalism. Since we have three derivative operators, the exterior derivatived, the co-exterior derivatived^∗ and the Laplacian ∆ =d^∗d+d d^∗, we have three types of Sobolev’s space:

W^d,r(M,^_k

) := {ω∈L^r(M,^_k

) :d ω∈L^r(M,^_k+1 )}, W^d∗,r(M,^_k

) := {ω∈L^r(M,^_k

) :d^∗ω∈L^r(M,^_k−1 )}, W^∆,r(M,^_k

) := {ω∈L^r(M,^_k

) : ∆ω∈L^r(M,^_k )}, with the following norms

kωk_Wd,r(M,Vk) :=

kωk^r_Lr(M,Vk)+kd ωk^r_Lr(M,Vk)

¹_r , kωk_Wd∗,r(M,V_k) :=

kωk^r_Lr(M,Vk)+kd^∗ωk^r_Lr(M,Vk)

¹_r , kωk_W∆,r(M,Vk) :=

kωk^r_Lr(M,Vk)+kd ωk^r_Lr(M,Vk)+kd^∗ωk^r_Lr(M,Vk)

_r¹ . Remark 1. IfM is an open subset ofRⁿ, the usual Sobolev’s spaceW^1,r(M), is the intersection betweenW^d,r(M,Vk

) andW^d∗,r(M,Vk

).

We can also define the Dirichlet boundary conditions by W₀^d,r(M,^_k

) := {ω∈C_c²(M,^_k

) :ν^]∧ω= 0 sur ∂M}

k k_{W d,r}

, W₀^d∗,r(M,^_k

) := {ω∈C_c²(M,^_k

) :i_νω= 0 sur ∂M}

k k_{W d}∗,r

.

3.2. The weak convergence. For a differentialk-formω^k∈L²(M,Vk

), we have a linear form onC_c²(M,Vk

):

< ω^k, ψ^k>_L2(M,Vk):=

Z

M

ω_p^k∧?ψ_p^k, with ψ^k ∈ C_c²(M,Vk

) and since L²(M,Vk

) is a Hilbert’s space, we can define a weak convergence.

Definition 3.1. We say that a sequence of differentialk-forms (ω_n^k)_n∈N∈L²(M,Vk

) weakly converges to a differential k-form ω^k ∈L²(M,Vk

) if and only if for a test differentialk-formψ^k∈C_c²(M,Vk

) we have

< ω_n^k, ψ^k >_L2(M,Vk)→< ω^k, ψ^k >_L2(M,Vk), whenntends to infinity.

Moreover the test differential k-forms verify the Dirichlet’s boundary condi- tions, so for all ψ^k+1 ∈ C_c²(M,Vk+1

) we can define the exterior derivative of α^k∈L²(M,Vk

) in the weak sense i.e:

< d ω^k, ψ^k+1>_L2(M,Vk+1)=< ω^k, d^?ψ^k+1>_L2(M,Vk), and its co-exterior derivative for allψ^k−1∈C_c²(M,Vk−1

):

< d^?ω^k, ψ^k−1>_L2(M,Vk−1)=< ω^k, d ψ^k−1>_L2(M,Vk).

4. The geometric two-scale convergence.

4.1. Introduction. For the geometric two-scale convergence, we need to work with two n-dimensional Riemannian manifolds, denoted in the following by M and Y. Later, we will suppose thatM is geodesically complete and possibly with boundary and that Y is compact, without boundary and with ergodic geodesic flow. So we must define what is it to be a differential form onM ×Y. First, we see that, for (p, q)∈M ×Y, a Riemannian metric onM×Y can be defined as follows :

g_(p,q)^M×Y : T_(p,q)(M ×Y)×T_(p,q)(M ×Y) → R

(u, v) 7→ g^M_p (dP_(p,q)^M (u), dP_(p,q)^M (v)) +g^Y_q (dP_(p,q)^Y (u), dP_(p,q)^Y (v)) whereP^M andP^Y are the following natural projections

P^M :M ×Y →M and P^Y :M×Y →Y, and so

dP_(p,q)^M :T_(p,q)(M ×Y)→T_PM(p,q)M anddP_(p,q)^Y :T_(p,q)(M×Y)→T_P(p,q)^Y Y.

We deduce the identification

T_(p,q)(M×Y)∼=T_pM ⊕T_qY,

andT_(p,q)^? (M ×Y)∼=T_p^?M⊕T_q^?Y. With the previous projections P^M andP^Y we have that

T^?(M ×Y) := G

(p,q)∈M×Y

T_(p,q)^? (M×Y) := [

(p,q)∈M×Y

{(p, q)} ×T_(p,q)^? (M×Y)

= [

(p,q)∈M×Y

{(p, q)} ×T_p^?M ×T_q^?Y, We deduce that

^_k

P^MT^?(M ×Y) = [

(p,q)∈M×Y

{(p, q)} ×^_k (T_p^?M) and so

^_k

P^MT^?(M×Y)∧^_l

P^YT^?(M×Y) = [

(p,q)∈M×Y

{(p, q)} ×^_k

(T_p^?M)×^_l (T_q^?Y).

Now we can define differential (k, l)-forms on M ×Y who are k-form on M and l-form onY as:

ω^k,l_(p,q):TpM ∧. . .∧TpM

| {z }

k

∧TqY ∧. . .∧TqY

| {z }

l

−→R,

i.e. ω^k,l_(p,q) ∈Vk

(T_p^?M)∧Vl

(T_q^?Y) and so a differential (k, l)-form onM ×Y is a map such that

ω^k,l: M×Y −→ Vk

P^M(T^∗(M×Y))∧Vl

P^Y(T^∗(M ×Y)) (p, q) 7−→ ((p, q), ω^k,l_(p,q)).

In a local coordinate chart (U_i^M×U_j^Y, ϕ^M_i , ϕ^Y_j ) where (U_i^M, ϕ^M_i ) is a local coordinate chart on M and (U_j^Y, ϕ^Y_j ) is a local coordinate chart onY, a differential formω^k,l onM×Y as the following expression

ω_(p,q)^k,l = X

i1<···<ik j1<···<jl

ωi₁,...,i_k,j₁,...,j_l(x, y)dxⁱ¹∧ · · · ∧dx^ik∧dy^j1∧ · · · ∧dy^jl,

for allp∈M and q∈Y and x, yare respectively the local coordinate system ofp etq.

Now to introduce the spaceL^r(M,Vk

L^s(Y,Vl

)), the set of differential (k, l)-forms M ×Y, r-integrable on M and s-integrable on Y, we denote by {λ_i,j}_(i,j)∈I×J a partition of unity subordinate to {ϕ^M_i (U_i^M)×ϕ^Y_j(U_j^Y)}. So all differential (k, l)- formsω^k,l onM×Y can be written as

ω^k,l= X

(i,j)∈I×J

λi,jω^k,l.

We say thatω^k,l∈L^r(M,Vk

L^s(Y,Vl

)) if all components of (ϕ^M_i )^?(ϕ^Y_j)^?λi,jω^k,l= (ϕ^Y_j)^?(ϕ^M_i )^?λi,jω^k,l are inL^r(U_i^M, L^s(U_j^Y)) and

X

(i,j)∈I×J

k(ϕ^Y_j)^?(ϕ^M_i )^?λi,jω^k,lk_Lr(U_i^M,L^s(U_j^Y))<+∞

Now if we observe (1) we see that we must define what means the evaluation in

x

for differential forms on M ×Y. To explain this, we will use the geodesics on manifoldM andY.

Letp0∈M, q0∈Y and an isomorphism j be such that Tp₀M

∼j

=Tq₀Y. We see in (2) page3that for allp∈exp^M_p0(V0), there existsv∈V0⊂Tp₀M such that

p= exp^M_p0(v), where exp^M_p

0 is defined by (3).

The geometric two-scale convergence results from Birkhoff’s theorem (1931) [9], Hopf’s theorem (1939) [11] and the Mautner’s Theorem (1957) [10]. Birkhoff’s theorem [9] says that for all probability space (χ, µ) and an ergodic flowφ^t, we have forf ∈L^r(χ, µ),

1 T

Z T 0

f(φ^t)dt−→^T

+∞

Z

M

f(x)dµ .

For our concerns, the geodesic flow must be ergodic on Y to develop geometric two-scale convergence issues. The Hopf’s theorem [11] and the Mautner’s theorem [10] give the conditions for the geodesic flow to be ergodic. The Hopf’s theorem [11]

stipulates that in a compact Riemannian manifold with a finite volume and with a negative curvature the geodesic flow is ergodic. Mautner showed that in symmetric Riemannian manifold the geodesic flows are also ergodic.

Torus, projective spaces, hyperbolic spaces, Heisenberg’s space,Sl(n,R)/SO(n,R), the symmetric space of quaternion-K¨ahler are examples of symmetric Riemanian manifold. If the manifoldY satisfies these conditions then it is geodesically complete and so the Hopf-Rinow’s theorem says there exists v∈V₀⊂T_p0M that all qin Y can be written as

q= exp^Y_q0(j(v)), where exp^Y_q

0 is the exponential map on Y (see (3) page 3 for its definition). To use Birkhoff’s theorem with an ergodic flow, we suppose that M and Y are n- dimensional Riemannian manifolds, moreoverM is assumed to be geodesically com- plete and possibly with boundary andY is assumed to be compact, without bound- ary and with ergodic geodesic flow i.e. verify the Mautner’s condition or Hopf’s condition. With the properties ofM andY, for any givenp0∈M and q0∈Y, we define for allp∈M,p∈Y as

p= exp^Y_q0 1

j(v)

,

wherev∈V0is such thatp= exp^M_p0(v).Once we have introduced this notation, we easily see that ifψ^k,0_(p,q)is a differentialk-form onM and a differential 0-form onY at point (p, q) inM×Y with enough regularity, thenψ_(p,p^k,0₎is a differentialk-form onM.

We have defined all the context and all tools to do geometric two-scale conver- gence in the covariant formalism.

4.2. The geometric two-scale convergence.

Definition 4.1. For (α^k)>0a sequence of differentialk-forms inL^r(M,Vk

), we say that it converges tothe two-scale limit α^k₀∈L^r(M,Vk

L^r(Y,V0

)) if for all differential k-form ψ^k∈C_c²(M,Vk

Ω⁰(Y)), we have

< α^k, ψ^k_(p,p)>_Lr(M,Vk)−→< α^k₀, ψ^k>_Lr(M,VkL^r(Y,V0)), when tends to 0. α^k₀ is called the two-scale limit of α^k in L^r(M,Vk

L^r(Y,V0

)).

We also say thatα^k two-scale converges strongly toα^k₀ if

→0limkα^k −(α^k₀)_(p,p)k_Lr(M,Vk)= 0.

We have also the following proposition [1,2]:

Proposition 2. We suppose thatM and Y are n-dimensional Riemannian man- ifolds, and M is complete possibly with boundary and Y is compact and verify the Mautner’s condition or the Hopf ’s condition. For ψ^k ∈L²(M,Vk

Ω⁰(Y,V0

)), we have

→0limkψ_(p,p^k )k_L2(M,Vk)=kψ^kk_L2(M,VkL²(Y,V0)).

Proof. Y is a compact Riemannian manifold so there exists a finite open (Ui)16i6m

covering ofY. We denote by (ψ^k_n)_(p,q)= X

j₁6···6j_k

X

i

ψ_j1,...,jk(x, y_i)χ_i(y)dx_j1∧ · · · ∧dx_jk

forx, ythe local coordinates of pandq,y_i ∈U_i andχ_i the characteristic function onU_i. With the help of the dominate convergence theorem [6],ψ^k_n converges toψ^k whenntends to infinity and so we have

kψ_(p,p^k )k_L2(M,Vk)− kψ^kk_L2(M,VkL²(Y,V0)) = kψ_(p,p^k )k_L2(M,Vk)

− k(ψ^k_n)_(p,p)k_L2(M,Vk)

+ k(ψ^k_n)_(p,p)k_L2(M,Vk)

− kψ_n^kk_L2(M,VkL²(Y,V0))

+ kψ_n^kk_L2(M,VkL²(Y,V0))

− kψ^kk_L2(M,VkL²(Y,V0)). First, we show that

→0limk(ψ_n^k)_(p,p)k_L2(M,Vk)=kψ_n^kk_L2(M,VkL²(Y,V0)). For the left hand side of the equality, we get

→0limk(ψ_n^k)_(p,p)k_L2(M,Vk)= lim

→0

s X

j16···6j_k

X

i

Z

M

(ψj₁,...,j_k(x, yi)χi(x))²volx.

Since Y is geodesically complete and since on Y the geodesic flow is ergodic, we haveχi(x) converges weakly tovol(Ui) . Hence

→0limk(ψ_n^k)_(p,p)k_L2(M,Vk) = s

X

j₁6···6j_k

X

i

Z

M

(ψ_j1,...,jk(x, y_i)vol(U_i))²vol_x

= s

X

j16···6j_k

X

i

Z

M

Z

Y

(ψj₁,...,j_k(x, yi)χi(y))²volxvoly

= kψ^k_nk_L2(M,VkL²(Y,V0)). That implies

→0limkψ_(p,p^k )k_L2(M,Vk)= lim

→0kψ_(p,p^k )k_L2(M,Vk)− k(ψ^k_n)_(p,p)k_L2(M,Vk)

+kψ^k_nk_L2(M,VkL²(Y,V0)). Then whenntends to infinity we found that

→0limkψ_(p,p^k )k_L2(M,Vk)− kψ^kk_L2(M,VkL²(Y,V0)) = 0.

With the help of this proposition, we formulate a geometric two-scale convergence with one parameter with the same conditions forM andY.

Theorem 4.2. For(α^k)a bounded sequence inL²([0,+∞), L²(M,Vk

)), there ex- ists a subsequence (α^k

j)of(α^k)and a differential form α^k₀ ∈L²([0,+∞), L²(M,^_k

L²(Y,^₀ ))) such that for all

ψ^k∈L²([0,+∞), C_c²(M,^_k

Ω⁰(Y))), we have

lim_j→0

Z ∞ 0

< α^k_j(t), ψ^k_(p,p)(t)>_L2(M,Vk)dt= Z ∞

0

< α^k₀(t), ψ^k(t)>_L2(M,VkL²(Y,V0))dt . Proof. We denote by

F(ψ^k) :=

Z ∞ 0

< α^k(t), ψ_(p,p^k )(t)>_L2(M,Vk)dt,

and since (α^k) is a bounded sequence there exists a positive realcsuch that kα^kk_L2([0,+∞),L²(M,Vk)) 6c.

We obtain

|F(ψ^k)| 6 Z ∞

0

kα^kk_L2(M,Vk)kψ^k_(p,p)(t)k_L2(M,Vk)dt 6 c

Z ∞ 0

kψ^k(t)k_L2(M,VkL²(Y,V0))dt and so F ∈

L²([0,+∞), L²(M,Vk

))0

and there exists a subsequence such that F^j converges toF⁰∈

L²([0,+∞), L²(M,Vk

))⁰

. Moreover, kψ^k_(p,p)(t)k_L2(M,Vk)6kψ^k(t)k_L2(M,VkL²(Y,V0)),