On the Pullback Equation <span class="mathjax-formula">${\phi }^{*}\left(g\right)=f$</span>

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On the pullback equation ϕ ^∗ (g) = f

S. Bandyopadhyay, B. Dacorogna

Section de Mathématiques, EPFL, 1015 Lausanne, Switzerland Received 3 July 2008; accepted 22 October 2008

Available online 24 December 2008

Abstract

We discuss the existence of a diffeomorphismϕ:Rⁿ→Rⁿsuch that ϕ^∗(g)=f

wheref, g:Rⁿ→Λ^k are closed differential forms and 2kn. Our main results (the casek=nhaving been handled by Moser [J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965) 286–294] and Dacorogna and Moser [B. Dacorogna, J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990) 1–26]) are that

– whennis even andk=2, under some natural non-degeneracy condition, we can prove the existence of such diffeomorphism satisfying Dirichlet data on the boundary of a bounded open set and the natural Hölder regularity; at the same time we get Darboux theorem with optimal regularity;

– we are also able to handle the degenerate cases whenk=2 (in particular whennis odd),k=n−1 and some cases where 3kn−2.

©2008 Published by Elsevier Masson SAS.

Résumé

Nous montrons l’existence d’un difféomorphismeϕ:Rⁿ→Rⁿsatisfaisant ϕ^∗(g)=f

où f, g:Rⁿ→Λ^k sont des formes différentielles fermées et 2kn. Nos résultats principaux (le cask=na été discuté notamment dans Moser [J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965) 286–294] et Dacorogna et Moser [B. Dacorogna, J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst.

H. Poincaré Anal. Non Linéaire 7 (1990) 1–26]) sont les suivants.

– Sinest pair,k=2 et sous des conditions naturelles de non dégénérescence, nous montrons l’existence et la régularité dans les espaces de Hölder d’un tel difféomorphisme satisfaisant de plus une condition de Dirichlet. On obtient aussi le théorème de Darboux avec la régularité optimale.

– Par ailleurs quandk=2 etnest impair ouk=n−1, ainsi que quelques cas particuliers où 3kn−2, nous montrons l’existence locale d’un tel difféomorphisme satisfaisant, en outre, des conditions de Cauchy.

* Corresponding author.

E-mail addresses:saugata.bandyopadhyay@gmail.com (S. Bandyopadhyay), bernard.dacorogna@epfl.ch (B. Dacorogna).

0294-1449/$ – see front matter ©2008 Published by Elsevier Masson SAS.

doi:10.1016/j.anihpc.2008.10.006

©2008 Published by Elsevier Masson SAS.

Keywords:Darboux theorem; Symplectic forms; Pullback; Hölder regularity

1. Introduction

In this article we discuss the existence of a diffeomorphismϕ:Rⁿ→Rⁿsuch that

ϕ^∗(g)=f (1)

wheref, g:Rⁿ→Λ^kare closed differential forms (i.e.df=dg=0), 2kn(the casek=1 is rather special and will also be discussed)

g=

1i₁<···<i_kn

g_i1···ik(x) dxⁱ¹∧ · · · ∧dx^ik and similarly forf. The meaning of (1) is that

1i1<···<ikn

gi₁···i_k

ϕ(x)

dϕⁱ¹∧ · · · ∧dϕ^ik=

1i1<···<ikn

fi₁···i_k(x) dxⁱ¹∧ · · · ∧dx^ik.

Whenk=2 andn=2mis even, then the celebrated Darboux theorem (cf. for example, Abraham, Marsden and Ratiu [1], McDuff and Salamon [11] or Taylor [17]) states that ifg=ω₀is the standard symplectic form, namely

ω₀=g= m i=1

dxⁱ∧dx^m+i

andf :Rⁿ→Λ²withdf=0 andf (p)=g(p)for a certainp∈Rⁿ, then there exists a diffeomorphismϕ defined in the neighbourhood ofpsuch that

ϕ^∗(g)=f and ϕ(p)=p.

This fundamental result was generalized by Moser in his seminal article [13] for the casek=nandk=2 withn even, obtaining also a global result. He proposed to solve (1) by studying the flow associated to an appropriate vector fieldu_t, namely

_d

dtϕ_t(x)=u_t(ϕ_t(x)), t∈ [0,1], ϕ₀(x)=x.

A solution of (1) is then given byϕ=ϕ₁. Now (for the notations see below) the vector field is recovered through two linear equations, namely

dα=f−g (2)

meaning thatα:Rⁿ→Λ^k−1and for everyt∈ [0,1], ut

tg+(1−t )f

=α. (3)

The first one is a system oflineardifferential equations and is, at least locally, solvable sincedf=dg=0. The second one is just alinearsystem of algebraic equations and Moser observed that it is well posed (in the sense that, whatever αis, there exists a uniqueu_t solving (3)), under some non-degeneracy conditions, only whenk=nork=2 andn even. Of course one could consider, with essentially no change, a more general closed homotopyft, withf0=f and f1=g, and then the two equations read as

dα_t = −d

dtf_t and u_tf_t=α_t

but we will here, for the sake of simplicity, restrict our attention to the homotopyft=tg+(1−t )f.

The casek=nafter the paper of Moser received considerable attention notably by Banyaga [2], Dacorogna [4], Reimann [14], Tartar [16], Zehnder [19]. Eq. (1) takes then the following form

g ϕ(x)

det∇ϕ(x)=f (x).

The next important step, still fork=n, appeared in Dacorogna and Moser [7], where it was shown how to handle both the boundary value and regularity problems. It should be emphasized that the flow method, as elegant as it is, does not allow to handle regularity problems and in [7] it was necessary to combine a fixed point argument and an iteration procedure. The exact statement can be found in Theorem 12 below. Posterior contributions can also be found in Burago and Kleiner [3], McMullen [12], Rivière and Ye [15] and Ye [18].

Our purpose in the present report is twofold.

(1) In the non-degenerate case whenk=2 andneven, the other non-degenerate case beingk=nand already solved in [7], we can handle full boundary condition (meaning Dirichlet condition) as well as regularity in Hölder spaces.

This is a delicate point and requires fine approximations of Hölder functions, a subtle fixed point argument and an iteration scheme. Our main results are Theorems 15 and 18. Although we will treat mainly contractible domains, we point out at each stage how our results can be extended to topologically more complex domains.

(2) We also show how to deal with degenerate problems. There we mostly obtain local results, though, in some particular cases, we can treat global problems. We are, however, able to impose Cauchy data (but, in general, neither Dirichlet data nor regularity) and not just the value at one point as in Darboux theorem. If we impose Cauchy data, then, of course, we need to assume that the tangential parts of f andgcoincide. We achieve this goal by solving Eqs. (2) and (3) simultaneously and not separately as Moser did. This is indeed a more flexible procedure, since (2) is underdetermined while (3) is overdetermined. In particular, only under some minor non- degeneracy conditions, we can also handle the cases:

– k=2 andnodd with maximal rank (cf. Theorem 20), – k=n−1 (cf. Theorem 21),

– we finally suggest through simple examples that our method applies to higher degenerate case whenk=2 or to the case ofkforms with 3kn−2.

A systematic study of these last cases will be undertaken elsewhere.

2. Preliminaries and notations 2.1. Notations

We will denote ak-formg:Rⁿ→Λ^kby

g=

1i1<···<ikn

g_i1···ik(x) dxⁱ¹∧ · · · ∧dx^ik or sometimes by

g=

I∈Tk

g_Idx^I

whereTk= {(i₁, . . . , i_k): 1i₁<· · ·< i_kn}is the set of strictly increasingk-indices. More generally we assign meaning tog_i1···ik for anyk-index by

gi₁···i_k =(sgnσ )gi_{σ (1)}···i_{σ (k)}

whereσ is a permutation of{1, . . . , k}.

(1) Ifg∈Λ^k, 1kn, andu∈Rⁿthenug∈Λ^k−1(also denoted by some authors byi_u(g)) is defined as

ug=

1i₁<···<i_kn

g_i1···ik k r=1

(−1)^r−1u^irdxⁱ¹∧ · · · ∧dx^ir−1∧dx^ir+1∧ · · · ∧dx^ik

=(−1)^k−1

1i₁<···<i_k−1n

n i_k=1

g_i1···iku^ikdxⁱ¹∧ · · · ∧dx^ik−1. Note that whenk=2, we have

ug= − n

i=1

n j=1

g_iju^jdxⁱ.

(2) It will often be convenient to representu→ugas a matrix operating on a vector. We therefore introduce the antisymmetric representation ofg∈Λ^k as the matrixg¯∈R(_k−ⁿ₁)×n

so that, by abuse of notations, ug=(−1)^k−1gu.¯

For example whenk=2, we have

¯

g=(g_ij)∈R^n×n withg_ij= −g_{j i}.

Our terminology “non-degenerate” and “degenerate” refers to the fact thatg¯is invertible or not.

We then consider Λ^k_g⁻¹=

w∈Λ^k−1: ∃u∈Rⁿwithug=w . We easily find that ifg=0, then, for every 2kn,

kdimΛ^k_g⁻¹n

and in general, if 3kn−1, it can take any of the intermediate values, but(k+1). So that whenk=n−1, the dimension cannot be maximal, more precisely, ifg=0 we always have

dimΛⁿ_g⁻²=n−1.

Whenk=2, the dimension is necessarily even, this means that there exists an integer 1l[n/2]such that dimΛ¹_g=2l.

Therefore whenk=2, the matrixg¯∈R^n×nis always singular whennis odd.

It will sometimes be more convenient to express the setΛ¹_gin terms of the kernel ofg¯the antisymmetric represen- tation ofgand its dimension by the rank ofg. Namely¯

Λ¹_g=(kerg)¯ ^⊥ and dimΛ¹_g=rankg.¯ (3) Ifg:Rⁿ→Λ^k is such that

g=

1i1<···<ikn

gi₁···i_kdxⁱ¹∧ · · · ∧dx^ik then

dg=

i₁<···<i_k+1

_k+1

γ=1

(−1)^γ−1∂g_{i1···iγ−1iγ+1···ik+1}

∂x^iγ

dxⁱ¹∧ · · · ∧dx^ik+1 and, forν∈Rⁿ,

g∧ν=

i₁<···<i_k+1

_k+1

γ=1

(−1)^γ−1g_{i1···iγ−1iγ+1···ik+1}ν^iγ

dxⁱ¹∧ · · · ∧dx^ik+1.

Ifνis the outward unit normal to the boundary of a setΩ, we callg∧νthe tangential part andνgthe normal part of the formg.

(4) Forf, g∈Λ^k we write inner product as f;g =

1i1<···<ikn

fi₁···i_kgi₁···i_k∈R.

(5) To avoid burdening the notations, we will sometimes, by abuse of notations, identify a 1-form with a vector field inRⁿand an-form with a function.

We now gather some simple algebraic and analytical formulas that follow from the above definitions and that we will use throughout.

(i) For everyf ∈Λ^k,g∈Λ^k−1andu∈Rⁿ, then f;g∧u =(−1)^k−1uf;g.

(ii) Letf∈Λ^k,g∈Λ^landu∈Rⁿ, then

u(f∧g)=(uf )∧g+(−1)^kl(ug)∧f.

(iii) Letf ∈C¹(Rⁿ;Λ^k)be closed

f =

I∈Tk

fIdx^I andu∈C¹(Rⁿ;Rⁿ), then

d[uf] =

I∈Tk

f_Id

udx^I +

I∈Tk

gradf_I;udx^I. So, in particular, whenk=2

d[uf] =

1i<jn

f_ij

duⁱ∧dx^j+dxⁱ∧du^j

+

1i<jn

gradf_ij;udxⁱ∧dx^j.

(iv) Let 0knbe an integer,Ω⊂Rⁿbe a smooth domain andf, g∈C¹(Ω;Λ^k)satisfyingf=gon∂Ω, then df∧ν=dg∧ν on∂Ω

whereνis a normal to∂Ω.

2.2. Function spaces and Hölder approximations

We will use the following functional notations. LetΩ⊂Rⁿbe an open set andra non-negative integer.

(1) Let 0< α <1, we denote byC^r,α(Ω)the usual set of Hölder functions and byC^r,α(Ω;Λ^k)the set ofk-forms

g=

1i1<···<ikn

gi₁···i_kdxⁱ¹∧ · · · ∧dx^ik so thatgi₁···i_k∈C^r,α(Ω).

(2) The setC^ω(Ω)will denote the set of analytic functions andC^ω(Ω;Λ^k)the set ofk-forms whose components are inC^ω(Ω).

(3) The sets Diff^r(Ω;Rⁿ), Diff^r,α(Ω;Rⁿ)and Diff^ω(Ω;Rⁿ), denote the sets of diffeomorphismsϕ so thatϕ∈ C^r(Ω;Rⁿ)andϕ⁻¹∈C^r(ϕ(Ω);Rⁿ),C^r,α andC^ωrespectively. Whenϕ(Ω)=Ω, we just let Diff^r(Ω), respectively Diff^r,α(Ω), Diff^ω(Ω).

In the sequel we will have to approximate closed forms inC^r,α(Ω;Λ^k)by smooth closed forms in a precise way and we will need the following theorem.

Theorem 1.LetΩ⊂Rⁿ be a bounded smooth contractible domain,0< βα <1,pq r,q2,r1and 1knbe integers andg∈C^q,α(Ω;Λ^k)∩C^p,α(∂Ω;Λ^k)withdg=0. Then for every >0, there existg∈ C^∞(Ω;Λ^k)∩C^p,α(Ω;Λ^k)and a constantγ=γ (p, q, r, α, β, Ω) >0such thatdg=0,g∧ν=g∧νon∂Ωand

g−gC^r,β(Ω)γ ^{q−r+α−β}gC^q,α(Ω), g_C^p,α_(Ω) γ

^p−q

g_C^q,α_(Ω)+^p−qgC^p,α(∂Ω)

.

Before starting the proof of the theorem, we need the equivalent of the theorem but for functions.

Lemma 2.LetΩ⊂Rⁿbe a bounded smooth domain,0< βα <1,pqr2be integers andu∈C^q,α(Ω)∩ C^p,α(∂Ω). Then for every >0, there existu∈C^∞(Ω)∩C^p,α(Ω)and a constantγ=γ (p, q, r, α, β, Ω) >0such thatu=uon∂Ωand

u−u_C^r,β_(Ω)γ ^{q−r+α−β}u_C^q,α_(Ω), u_Cp,α(Ω) γ

^p−q

u_Cq,α(Ω)+^p−quC^p,α(∂Ω)

.

Proof. We first find (see Hörmander [9])v∈C^∞(Ω)and a constantγ₁such that v−u_Cr,β(Ω)γ₁^{q−r+α−β}u_Cq,α(Ω) and v_Cp,α(Ω) γ1

^p−qu_Cq,α(Ω). We then fix the boundary data as follows. Letu∈C^∞(Ω)∩C^p,α(Ω)be the solution of

u=v inΩ,

u=u on∂Ω ⇔

[u−v] =0 inΩ, u−v=u−v on∂Ω.

The solution satisfies Schauder estimates u_C^p,α_(Ω)γ2

v_C^p,α_(Ω)+ uC^p,α(∂Ω)

,

u−v_C^r,β_(Ω)γ2u−vC^r,β(∂Ω)γ2u−v_C^r,β_(Ω). The combination of the estimates onuandvgives the result. 2

We can now go back to the proof of Theorem 1.

Proof. Step 1. It is easy to see that, sinceΩis contractible, we can findG∈C^q+1,α(Ω;Λ^k−1)∩C^p+1,α(∂Ω;Λ^k−1) and a constantγ such thatdG=g,

G_C^q+1,α(Ω)γg_C^q,α_(Ω) and GC^p+1,α(∂Ω)γgC^p,α(∂Ω).

This is easily obtained by an appropriate use of Theorem 3 below and we leave out the details.

Step 2. Applying Lemma 2 on each component ofG, we getGas in the lemma. Settingg=dG, we have the claim. 2

3. Variations on the Poincaré lemma 3.1. The case without constraints

We start with the following theorem (cf. [5]).

Theorem 3(Dacorogna). Letr0,2knbe integers and0< α <1. LetΩ⊂Rⁿ be a bounded smooth con- tractible domain andνdenote the outward unit normal. The following two conditions are then equivalent.

(i) • Either2kn−1andf ∈C^r,α(Ω;Λ^k)with df=0 inΩ and f∧ν=0 on∂Ω;

• ork=nandf ∈C^r,α(Ω)with

Ω

f (x) dx=0.

(ii) There existsw∈C^r+1,α(Ω;Λ^k−1)satisfying dw=f inΩ,

w=0 on∂Ω.

Remark 4.

(i) Whenr=0, the constraintdf=0 has to be interpreted in the sense of distributions.

(ii) We now briefly discuss the case whereΩ is not contractible, but only open and connected. To this aim we first define, for 0kn, the set ofharmonick-formswith Dirichlet boundary condition as the vector space

D_k(Ω):=

ψ∈C⁰ Ω;Λ^k

∩C¹ Ω;Λ^k

: dψ=0, δψ=0 inΩandψ∧ν=0 on∂Ω .

Note that we always have for connectedΩ D0(Ω) {0} and Dn(Ω)R.

Furthermore if 1kn−1 and if the setΩis contractible, then D_k(Ω) {0} ⊂Λ^k,

while for general sets we have dimD_k(Ω)=B_n−k

whereBk are the Betti numbers ofΩ (cf. Duff and Spencer [8] and Kress [10]). Theorem 3 remains valid for such general sets if we add the following necessary condition

Ω

f;ψdx=0, ∀ψ∈D_k(Ω).

Observe finally that whenk=nin Theorem 3 we therefore have no new condition.

(iii) Although there are infinitely many solutionsw, the actual construction singles out a precise one and in fact we can construct a linear isomorphism of Banach spacesL:X→Y where (when 2kn−1 and similar ones whenk=n)

X=

w∈C^r+1,α Ω;Λ^k

:w=0 on∂Ω , Y=

f ∈C^r,α

Ω;Λ^k+1

: df=0 inΩandf ∧ν=0 on∂Ω ,

that associates in an isomorphic way to everyw∈Xa uniquef=Lw∈Y such thatdw=f.

(iv) The theorem, with an easier and more direct proof, is still valid whenk=1 and regularity holds even inC^r spaces.

(v) The theorem is also valid for unbounded smooth domains such as the half plane.

3.2. The case with constraints

We start with the simplest case of constant constraints. In the sequel we let H=

x∈Rⁿ: xⁿ>0

andν= −e_nwill denote the outward unit normal.

Theorem 5.Let2kn,b∈Rⁿwithbⁿ=0andf ∈C^r(H;Λ^k),r0, with df=0 inH and f∧e_n=0 on∂H.

Letw∈C^r(H;Λ^k−1)be defined by w(x)=

xⁿ/bⁿ 0

[bf]

x+b

t−xⁿ bⁿ

dt.

Thenwis the unique solution of

⎧⎨

⎩

dw=f inH, bw=0 inH, w=0 on∂H.

If, moreover,f;c∧b =0for a certainc∈Λ^k−1, thenwalso satisfies w;c =0 inH.

Remark 6.The constraint bw=0 can be seen as _n−1

k−2

independent equations. When k=2, there is only one independent equation and it is of the formw;b =0.

We now consider the case of non-constant constraints and, in general, one can expect only local solutions. We start with the casek=2, which requires less stringent smoothness assumptions than the cases 3kn(cf. Theorem 8 for this last case). Although we will state the following results in the half planeH, both theorems mentioned below can be proved, by flattening out the boundary, for any smooth domainΩ.

Theorem 7.Letr0,a∈C^r+1,α(H;Λ¹)witha_n(x)=0for everyx∈Handf ∈C^r,α(H;Λ²)with df =0 inH and f ∧en=0 on∂H.

Then there exist >0andw∈C^r,α(H∩B(0);Λ¹)satisfying

⎧⎨

⎩

dw=f inH∩B(0), w;a =0 inH∩B(0), w=0 on∂H∩B(0).

Proof. Set w=u+dv

where (this has a solutionu∈C^r+1,α(H;Λ¹)by Theorem 3) du=f inH,

u=0 on∂H

and

dv(x);a(x) =_n

i=1a_i(x)^∂v

∂xⁱ = −u(x);a(x) ifxⁿ>0,

v(x)=0 ifxⁿ=0.

This last problem has a solution,v∈C^r+1,α(H∩B(0)), and thusw∈C^r,α. Note that fromv(x)=0 whenxⁿ=0, we deduce that

∂v

∂xⁱ

x¹, . . . , xⁿ⁻¹,0

=0 for everyi=1, . . . , n−1.

Sinceu=0 on∂Handa_n(x)=0, we deduce from the differential equation and the above identities that

∂v

∂xⁿ

x¹, . . . , xⁿ⁻¹,0

=0

so thatdv=0 whenxⁿ=0. This concludes the proof of the theorem. 2

We now discuss the case 3kn. The result and the proof below are still valid ifk=2.

Theorem 8.Leta^λ∈C^ω(H;Λ^k−1),λ=1, . . . ,_n−1

k−2

, with A=A(x)=

a_j^λ

1···j_k−2n(x)λ=1,···,(ⁿ_k⁻₋¹₂)

1j₁<···<j_k−2n−1∈R(ⁿ_k⁻₋¹₂)×(ⁿ_k⁻₋¹₂) invertible for everyx∈H andf∈C^ω(H;Λ^k)with

df =0 inH and f ∧en=0 on∂H.

Then there exist >0andw∈C^ω(H∩B(0);Λ^k−1)satisfying

⎧⎨

⎩

dw=f inH∩B(0),

w;a^λ =0 inH∩B(0), λ=1, . . . ,_n−1

k−2

, w=0 on∂H∩B(0).

Proof. Set w=u+dv

where (this has a solutionu∈C^ω(H;Λ^k−1)by Theorem 3 or by Theorem 5) du=f inH,

u=0 on∂H.

The formv:Rⁿ→Λ^k−2is defined as follows. Choose first vj₁···j_k−3n=0 for every 1j1<· · ·< jk−3n−1

(if k=3, one just reads v_n=0 and if k=2, the above constraints do not exist) and then solve, using Cauchy–

Kowalewski theorem, the system of equations

dv(x);a^λ(x) = −u(x);a^λ(x) ifxⁿ>0, λ=1, . . . ,_n−1

k−2

,

v(x)=0 ifxⁿ=0,

More precisely first observe that with our choice ofv, we have

v=

1j₁<···<j_k−2n−1

v_{j1···jk−2}dx^j1∧ · · · ∧dx^jk−2

then

dv=

1j1<···<jk−2n−1

_n

i=1

∂v_{j1···jk−2}

∂xⁱ dxⁱ

∧dx^j1∧ · · · ∧dx^jk−2

=

1j₁<···<j_k−2n−1

(−1)^k∂v_{j1···jk−2}

∂xⁿ dx^j1∧ · · · ∧dx^jk−2∧dxⁿ

+

1j₁<···<j_k−2n−1

_n−1

i=1

∂v_{j1···jk−2}

∂xⁱ dxⁱ

∧dx^j1∧ · · · ∧dx^jk−2.

We therefore obtain that dv;a^λ

=(−1)^k

1j1<···<jk−2n−1

a^λ_j

1···j_k−2n

∂vj₁···j_k−2

∂xⁿ +g^λ

=(−1)^k

A ∂v

∂xⁿ λ

+g^λ

whereAis as in the statement of the theorem andg^λ is a term that does not involve derivatives ofvwith respect to the variablexⁿ. Note that the system

dv(x);a^λ(x)

= −

u(x);a^λ(x)

, λ=1, . . . , n−1

k−2 can therefore be seen, sinceAis invertible, as

⎧⎪

⎪⎨

⎪⎪

⎩

∂v

∂xⁿ =(−1)^k+1A⁻¹

⎡

⎣

⎛

⎝ u;a¹ ... u;a(^n−1k−2)

⎞

⎠+

⎛

⎝ g¹

... g(^nk−−¹2)

⎞

⎠

⎤

⎦ ifxⁿ>0,

v=0 ifxⁿ=0

(recall that we are considering only thevj₁···j_k−2 with 1j1<· · ·< jk−2n−1 and thus _∂x^∂vn ∈R(ⁿ_k⁻₋¹₂)). Since all the coefficients are analytic, we have locally a unique analytical solution in the neighbourhood ofx=0.

We also note that _∂x^∂vi =0 atxⁿ=0 for everyi=1, . . . , n−1 and that also _∂x^∂vn =0 atxⁿ=0, since thereu=0 andg^λ=0;so thatdv=0 whenxⁿ=0. 2

4. Abstract results 4.1. Necessary conditions

We give here some elementary necessary conditions, whose proofs are straightforward.

Theorem 9.LetΩ⊂Rⁿbe a smooth domain,1kn,f, g∈C¹(Ω;Λ^k)withdg=0inΩandϕ∈Diff¹(Ω;Rⁿ) be such thatϕ^∗(g)=f inΩ, then

df =0.

Moreover the two following results hold.

(i) IfΩ is bounded,ϕ(Ω)=Ω andn=mkwithman integer, then

Ω

f^m=

Ω

g^m wheref^m=f ∧ · · · ∧f

!" #

m-times

. (ii) Ifϕ(x)=xforx∈∂Ω, then

f ∧ν=g∧ν on∂Ω.

4.2. The flow method

We have the following abstract result, which is the theorem of Moser [13] with the additional consideration on the boundary data. In several textbooks the flow method is also called the Lie transform method.

Theorem 10 (Moser). LetΩ⊂Rⁿ be a bounded smooth domain(ν denoting the outward unit normal),r1and 1knbe integers andf_t∈C¹([0,1];C^r(Ω;Λ^k))(respectivelyf_t∈C¹([0,1];C^r,α)with0< α <1)satisfying, for everyt∈ [0,1],

df_t=0 inΩ and f_t∧ν=f₀∧ν on∂Ω.

Assume that there existsu_t∈C¹([0,1];C^r(Ω;Rⁿ))(respectivelyu_t∈C¹([0,1];C^r,α))verifying, for everyt∈ [0,1], d[utft] = −d

dtft inΩ and ut=0 on∂Ω.

Then there existsϕ∈Diff^r(Ω)(respectivelyϕ∈Diff^r,α(Ω))such that ϕ^∗

f1(x)

=f0(x), x∈Ω. (4)

and

ϕ(x)=x, x∈∂Ω.

Proof. We solve (cf. [13]) _d

dtϕ_t(x)=u_t(ϕ_t(x)), t∈ [0,1], ϕ0(x)=x.

The above system has a solutionϕ_t satisfying, for everyt∈ [0,1], ϕ_t^∗(f_t)=f₀ inΩ and ϕ_t(x)=x on∂Ω.

This concludes the proof. 2

4.3. The fixed point method

The following theorem is particularly useful when dealing with non-linear problems, once good estimates are known for the linearized problem. We give it under a general form, because we will need it this way in Theorem 15.

However in many instances, one can chooseX₁=X₂=XandY₁=Y₂=Y in the statement below (in which case the hypothesis(H_XY)below reduces to requiring thatXis a Banach space andY a normed space). Our theorem will lean on the following hypotheses.

(H_XY) LetX₁⊃X₂be Banach spaces andY₁⊃Y₂be normed spaces such that the following property holds: if u_ν−→^X1 u and u_νX2 r,

thenu∈X₂and uX₂lim inf

ν→∞ u_νX₂.

(H_L) L:X₂→Y₂is a linear isomorphism of Banach spaces and there existk₁, k₂>0 such that for everyf∈Y₂

$$L⁻¹f$$

X_ik_ifYi, i=1,2.

(HQ) Q:X2→Y2is such thatQ(0)=0 and for everyu, v∈X2withuX₁,vX₁1, the following two inequal- ities hold

$$Q(u)−Q(v)$$

Y1c

uX₁,vX₁

u−vX₁, (5)

$$Q(v)$$

Y₂c

vX1,0

vX2, (6)

wherec:R+×R+→R+is continuous, separately increasing andc(0,0)=0.

Theorem 11(Fixed point theorem). LetX₁, X₂, Y₁, Y₂, L, Qsatisfy(H_XY),(H_L)and(H_Q). Then, for everyf ∈Y₂ verifying

2 max{k1, k2}c

2k1fY₁,2k1fY₁

1 and 2k1fY₁1, (7) there existsu∈X2such that

Lu=Q(u)+f and uXi2kifYi, i=1,2. (8)

Proof. We set

N (u)=Q(u)+f.

We next define B=

u∈X₂: uX_i2kifY_i, i=1,2 .

We endowBwith·X₁norm; the property(H_XY)ensures thatBis closed. We now want to show thatL⁻¹N:B→B is a contraction mapping (cf. Claims 1 and 2 below). Applying Banach fixed point theorem we will have indeed found a solution verifying (8) and the proof will be complete.

Claim 1. Let us first show thatL⁻¹Nis a contraction onB. To show this, letu, v∈Band use (5), (7) to get that

$$L⁻¹N (u)−L⁻¹N (v)$$

X1k₁$$N (u)−N (v)$$

Y1=k₁$$Q(u)−Q(v)$$

Y1

k1c

uX₁,vX₁

u−vX₁

k₁c

2k1fY₁,2k1fY₁

u−vX₁

1

2u−vX1.

Claim 2. We next showL⁻¹N:B→Bis well-defined. First, note that

$$L⁻¹N (0)$$

X₁k₁$$N (0)$$

Y₁=k₁fY1. Therefore, using Claim 1, we obtain

$$L⁻¹N (u)$$

X1$$L⁻¹N (u)−L⁻¹N (0)$$

X1+$$L⁻¹N (0)$$

X1

1

2uX₁+k₁fY₁2k1fY₁. It remains to show that

$$L⁻¹N (u)$$

X₂2k2fY2. Using (6), we have

$$L⁻¹N (u)$$

X2k2$$N (u)$$

Y2k2$$Q(u)$$

Y2+k2fY₂

k₂c

uX1,0

uX2+k₂fY2 2k²₂c

uX1,0

fY2+k₂fY2

=k₂ 2k2c

uX₁,0 +1

fY₂

k₂ 2k2c

2k1fY1,2k1fY1

+1 fY2

2k2fY₂.

This concludes the proof of Claim 2 and thus of the theorem. 2

For the sake of illustration we give here an academic example loosely related to our problem.

Example.LetΩ ⊂Rⁿ be a bounded smooth contractible domain and 0< α <1. Letr1 and 1kn−2 be integers. Consider the formw:Rⁿ→Λ^kwhere

w=

I∈Tk

wIdx^I

where Tk is the set of ordered k-indices. Let I1, . . . , Ik+1∈Tk, then there exists >0 such that for every f ∈ C^r,α(Ω;Λ^k+1)with

fC^r,α , df =0 and f ∧ν=0 on∂Ω there existsw∈C^r+1,α(Ω;Λ^k)satisfying

dw+%_k+1

r=1dw_Ir=f inΩ,

w=0 on∂Ω.

The proof is immediate if we set X=X1=X2=

w∈C^r+1,α Ω;Λ^k

: w=0 on∂Ω , Y =Y₁=Y₂=

f ∈C^r,α

Ω;Λ^k+1

: df=0 inΩandf ∧ν=0 on∂Ω ,

Lequal to the operator constructed in Remark 4 (Lw=f being equivalent todw=f) and Q(a)=

k&+1 r=1

daI_r.