New estimates for the div-curl-grad operators and elliptic problems with L1-data in the whole space and in the half-space

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New estimates for the div-curl-grad operators and elliptic problems with L1-data in the whole space and in

the half-space

Chérif Amrouche, Huy Hoang Nguyen

To cite this version:

Chérif Amrouche, Huy Hoang Nguyen. New estimates for the div-curl-grad operators and elliptic

problems with L1-data in the whole space and in the half-space. 2011. �hal-00559615�

New estimates for the div-curl-grad operators and elliptic problems with L

-data

in the whole space and in the half-space

Ch´erif Amrouche^a, Huy Hoang Nguyen^b

aLaboratoire de Math´ematiques Appliqu´ees, UMR CNRS 5142 Universit´e de Pau et des Pays de l’Adour – 64000 Pau – France

bDepartamento de Matematica, IMECC, Universidade Estadual de Campinas Caixa Postal 6065, Campinas, SP 13083-970, Brazil

Abstract

In this paper, we study the div-curl-grad operators and some elliptic problems in the whole spaceRⁿ and in the half-spaceRⁿ+, withn≥2. We consider data in weighted Sobolev spaces and inL¹.

Keywords: div-curl-grad operators, elliptic, half-space, weighted Sobolev spaces

2000 MSC:35J25, 35J47

1. Introduction

The purpose of this paper is to present new results concerning the div-curl- grad operators and some elliptic problems in the whole space and in the half- space with data and solutions which live in L¹ or in weighted Sobolev spaces, expressing at the same time their regularity and their behavior at infinity. Re- cently, new estimates for L¹-vector field have been discovered by Bourgain, Br´ezis and Van Schaftingen (see [23], [10], [11], [12], [13], [15]) which yield in particular improved estimates for the solutions of elliptic systems inRⁿ or in a bounded domain Ω⊂Rⁿ. Our work presented in this paper is naturally based on these very interesting results and our approach rests on the use of weighted Sobolev spaces.

This paper is organised as follows. In this section, we introduce some notations and the functional framework. Some results concerning the weighted Sobolev spaces and the spaces of traces are recalled. In Section 2 and Section 3, our work is focused on the div-grad-curl operators and elliptic problems in the whole space. After the case of the whole space, we then pass to the one of the half- space. Results in the half-space are presented in Section 4 (The div-grad opera-

Email addresses: cherif.amrouche@univ-pau.fr(Ch´erif Amrouche^a), nguyen@ime.unicamp.br(Huy Hoang Nguyen^b)

tors), Section 5 (Vector potentials) and in the last section of this paper (Elliptic problems).

In this paper, we use bold type characters to denote vector distributions or spaces of vector distributions withn components andC >0 usually denotes a generic positive constant that may depends on the dimensionn, the exponentp and possibly other parameters, but never on the functions under consideration.

For any real number 1< p <∞, we takep⁰to be the H¨older conjugate ofp. Let Ω be an open subset in then-dimensional real euclidean space. A typical point x ∈ Rⁿ is denoted byx = (x⁰, x_n), where x⁰ = (x₁, x₂, ..., x_n−1) ∈Rⁿ⁻¹ and x_n ∈R. Its distance to the origine is denoted byr=|x|= (x²₁+...+x²_n)^1/2. LetRⁿ+ denote the closure of the upper half-spaceRⁿ+={x ∈Rⁿ; xn >0}. In the half-space, its boundary is defined by Γ = {x ∈Rⁿ; xn = 0} ≡ Rⁿ⁻¹. In order to control the behavior at infinity of our functions and distributions, we use for basic weight the quantity ρ=ρ(r) = 1 +r, which is equivalent to r at infinity. We define D(Ω) to be the linear space of infinite differentiable functions with compact support on Ω. Now, letD⁰(Ω) denote the dual space of D(Ω), the space of distributions on Ω. For anyq∈N,Pqstands for the space of polynomials of degree≤q. Ifqis strictly negative integer, we set by convention Pq ={0}. Given a Banach spaceB, with dual spaceB⁰ and a closed subspace X ofB, we denote byB⁰ ⊥X (or more simplyX^⊥, if there is no ambiguity as to the duality product) the subspace ofB⁰ orthogonal toX, i.e.

B⁰⊥X=X^⊥={f ∈B⁰| ∀v∈X, < f, v > = 0}= (B/X)⁰.

The spaceX^⊥ is called the polar space ofX in B⁰ and is also denoted by X^◦. We also introduce the space

V(Ω) = {ϕ∈D(Ω), divϕ= 0}.

In this paper, we want to consider some particular weighted Sobolev spaces (see [4], [5]). The open set Ω will be denote the whole space or the half-space.

We begin by defining the space

W₀^1,p(Ω) ={u∈ D⁰(Ω), u w1

∈L^p(Ω),∇u∈L^p(Ω)}, where

w1= 1 +r ifp6=n and w1= (1 +r) ln(2 +r) ifp=n.

This space is a reflexive Banach space when endowed with the norm:

||u||_W^1,p

0 (Ω)= (|| u w1

||^p_Lp(Ω)+||∇u||^p_Lp(Ω))^1/p. We also introduce the space

W₀^2,p(Ω) ={u∈ D⁰(Ω), u w2

∈L^p(Ω), ∇u w1

∈L^p(Ω), D²u∈L^p(Ω)},

where

w₂= (1 +r)² ifp /∈ {n

2, n} and w₂= (1 +r)² ln(2 +r), otherwise, which is a Banach space endowed with its natural norm given by

||u||_W2,p

0 (Ω)= (|| u w2

||^p_Lp(Ω)+||∇u w1

||^p_Lp(Ω)+||D²u||^p_Lp(Ω))^1/p. We need to give also the definition of the following space

W₋₁^1,p(Ω) ={u∈ D⁰(Ω), u

w₃ ∈L^p(Ω), ∇u

1 +r ∈L^p(Ω)}, where

w3= (1 +r)² ifp6=n/2 and w3= (1 +r)² ln(2 +r), otherwise.

This space is also a reflexive Banach space and we can show thatW₀^2,p(Ω) ,→ W₋₁^1,p(Ω). Note that the logarithmic weight only appears if p = n or p = ⁿ₂. From now on, when we writeW_α^m,p(Ω), it means thatm, pand αare taken as in these above definitions of the weighted Sobolev spaces. It is also true for the generalized case of the weighted Sobolev spaces. The weights in these above definitions are chosen so that the corresponding space satisfies two properties.

On the one hand, the spaceD(Ω) is dense inW_α^m,p(Ω). On the other hand, the following Poincar-type inequality holds inW_α^m,p(Ω) (see [4], [5] and [6]). The semi-norm

|.|_Wα^m,p_(Ω) = ( X

|λ|=m

||(1 +r)^α∂^λu||^p_Lp(Ω))^1/p

defines onW_α^m,p(Ω)/Pq^∗ a norm which is equivalent to the quotient norm,

∀u∈W_α^m,p(Ω), ||u||_Wα^m,p_(Ω)/P

q∗ ≤C|u|_Wα^m,p_(Ω), (1) withq^∗ = inf(q, m−1), where qis the highest degree of the polynomials con- tained inW_α^m,p(Ω). We define the space

◦

W^m,p_α (Ω) =D(Ω)^{||.||Wαm,p(Ω)},

and its dual space, W_−α^−m,p0(Ω), is a space of distributions. In addition, the semi-norm|.|_Wα^m,p_(Ω) is a norm onW^{◦ m,p}α (Ω) that is equivalent to the full norm

||.||_Wα^m,p_(Ω):

∀u ∈W^{◦ m,p}_α (Ω), ||u||_Wα^m,p_(Ω)≤C|u|_Wα^m,p_(Ω). (2) When Ω = Rⁿ, we have W_α^m,p(Rⁿ) =W^{◦ m,p}α (Rⁿ). We will now recall some properties of the weighted Sobolev spaces W_α^m,p(Ω). All the local properties of W_α^m,p(Ω) coincide with those of the classical Sobolev space W^m,p(Ω). A quick computation shows that for m ≥ 0 and if n

p +α does not belong to

{i ∈ Z; i ≤m}, then P[m−n/p−α] is the space of all polynomials included in W_α^m,p(Rⁿ) (fors∈R, [s] stands for the integer part ofs). For allλ∈Nⁿ where 0≤ |λ| ≤2m withm= 1 orm= 2, the mapping

u∈W_α^m,p(Ω)→∂^λu∈W_α^m−|λ|,p(Ω) is continuous. Recall the following Sobolev embeddings (see [1]):

W₀^1,p(Ω),→L^p∗(Ω) where p^∗= np

n−p and 1< p < n.

Also recall

W₀^1,n(Rⁿ),→BM O(Rⁿ).

The spaceBM O is defined as follows: A locally integrable function f belongs toBM Oif

||f||BM O =: sup

Q

1

|Q|

Z

Q

|f(x)−fQ|dx <∞, where the supremum is taken on all the cubes andfQ = _|Q|¹ R

Qf(x)dx is the average of f on Q. In the literature, we also find studies using the following spaces:

Wc^1,p(Ω) = D(Ω)^{|| ∇.||Lp} and Wc^2,p(Ω) = D(Ω)^{|| ∇2.||Lp}.

In fact, when Ω =Rⁿ we can prove that cW^1,p(Rⁿ) = W₀^1,p(Rⁿ) if 1 < p <

n and cW^2,p(Rⁿ) = W₀^2,p(Rⁿ) if 1 < p < n/2. When n/p−m ≤ 0 with m= 1 or m= 2,Wc^m,p(Rⁿ) is not a space of distributions. For instance, in J.

Deny, J. L. Lions [16], they show thatcW^1,2(R²) is not a space of distributions.

Without going into details, let (ϕν) be a sequence of functions of D(R²) such that ||∇ϕν||_L2(R²) is a Cauchy sequence. Applying Proposition 9.3 [4], there exists a constantcν such that

c∈infR

||ϕν+c||_W0

−1,−1(R²) = ||ϕν+cν||_W0

−1,−1(R²)

is also a Cauchy sequence and therefore converges. But this does not mean that ϕ_ν alone converges and from [4], ||∇ϕν||L²(R²) tends to zero while < ϕ_ν, ψ >

tends to infinity for manyψofD(R²) instead of converging to a constant times the mean value of ψ. These considerations suggest that the space cW^1,2(R²) lacks the constant functions and is not a space of distributions.

In order to define the traces of functions ofW_α^m,p(Rⁿ+), we introduce for any σ∈(0,1) the space

W₀^σ,p(Rⁿ) = {u∈ D⁰(Rⁿ); w^−σu∈L^p(Rⁿ) and Z

Rⁿ×Rⁿ

|u(x)−u(y)|^p

|x−y|^n+σp dxdy<∞ }, where

w=ρ if n

p 6=σ and w=ρ(ln (1 +ρ))^1/σ if n p =σ.

It is a reflexive Banach space equipped with its natural norm

||u||_W^σ,p

0 (Rⁿ) =

u w^σ

p L^p(Rⁿ)

+ Z

Rⁿ×Rⁿ

|u(x)−u(y)|^p

|x−y|^n+σp dxdy ^1/p

. Similarly, for any real numberα∈R, we define the space

W_α^σ,p(Rⁿ) = {u∈ D⁰(Rⁿ); w^α−σu∈L^p(Rⁿ) and Z

Rⁿ×Rⁿ

|ρ^α(x)u(x)−ρ^α(y)u(y)|^p

|x−y|^n+σp dxdy<∞ }, where

w=ρ if n

p +α6=σ and w=ρ(ln (1 +ρ))^1/(σ−α) if n

p +α=σ.

For anys∈R⁺, we set

W_α^s,p(Rⁿ) = {u∈ D⁰(Rⁿ); 0≤ |λ| ≤k, ρ^α−s+|λ|(ln(1 +ρ))⁻¹∂^λu∈L^p(Rⁿ);

k+ 1≤ |λ| ≤[s]−1, ρ^α−s+|λ|∂^λu∈L^p(Rⁿ); |λ|= [s], ∂^λu∈W_α^σ,p(Rⁿ)}, where

k=k(s, n, p, α) =





 s−n

p−α if n

p+α∈ {σ, ..., σ+ [s]},

−1, otherwise,

withσ=s−[s]. It is a reflexive Banach space equipped with the norm

||u||_Wα^s,p_(Rn) = ( X

0≤|λ|≤k

||ρ^α−s+|λ|(ln(1 +ρ))⁻¹∂^λu||^p_Lp(Rⁿ)

+ X

k+1≤|λ|≤[s]−1

||ρ^α−s+|λ|∂^λu||^p_Lp(Rⁿ))^1/p+ X

|λ|=[s]

||∂^λu||_Wα^σ,p_(Rn).

We notice that this definition coincides with the previous definition of the weighted Sobolev spaces whens=mis a nonnegative integer. Ifuis a function onRⁿ+, we denote its trace of orderj on the hyperplane Γ by

∀j∈N, γju: x⁰7−→ ∂ⁱu

∂x^jn

(x⁰,0).

Finally, we recall the following traces lemma due to Hanouzet [19] and extended by Amrouche-Neˇcasov´a [6] to this class of weighted Sobolev spaces.

Lemma 1.1. The mapping

γ= (γ0, γ1, ..., γ_m−1) : D(Rⁿ+)→

m−1

Y

j=0

D(Rⁿ⁻¹),

can be extended to a linear continuous mapping, still denoted byγ, γ: W_α^m,p(Rⁿ+)→

m−1

Y

j=0

W_α^{m−j−1/p,p}(Rⁿ⁻¹).

Moreover,γ is surjective and Kerγ =W^◦m,pα (Rⁿ+).

In order to finish this section, we recall the definition of curl operator. When n= 2, we define the curl operator for distributionsϕ∈ D⁰(Ω) andv ∈D⁰(Ω) by

curlϕ= (∂ϕ

∂x₂,−∂ϕ

∂x₁) and curlv= ∂v2

∂x₁ − ∂v1

∂x₂.

Whenn= 3, we define the curl operator of a distributionv∈D⁰(Ω) as follows curlv= (∂v₃

∂x2

−∂v₂

∂x3

,∂v₁

∂x3

−∂v₃

∂x1

,∂v₂

∂x1

− ∂v₁

∂x2

).

The following properties are easily obtained.

curl(curlϕ) =−∆ϕ withn= 2,

−∆v+∇(divv) =

(curl(curlv) when n= 3, curl(curlv) when n= 2.

2. The div-grad operators in the whole space

The following proposition was given by J. Bourgain and H. Br´ezis [10]. We give here a detailed proof.

Proposition 2.1. Let f ∈Lⁿ(Rⁿ). Then there existsu∈L^∞(Rⁿ) such that divu=f with ||u||L^∞(Rⁿ)≤Cn||f||Lⁿ(Rⁿ). (3) Proof. We consider the following unbounded operator

A=∇:L^n/(n−1)(Rⁿ)→L¹(Rⁿ), with

D(A) =W₀^1,1(Rⁿ) ={v∈L^n/(n−1)(Rⁿ),∇v∈L¹(Rⁿ)}.

It is easy to see thatAis closed: if vn→v in L^n/(n−1)and∇vn→zinL¹(Rⁿ), then we havez=∇v. On the other hand,D(A) is dense inL^n/(n−1)(Rⁿ). We know that for allu∈D(A),

||u||_Ln/(n−1)(Rⁿ)≤C|| ∇u||_L1(Rⁿ). (4) Thanks to Theorem II.20 [14], the adjoint operator

A^∗ = −div :L^∞(Rⁿ)→Lⁿ(Rⁿ)

is surjective, i.e., for all f ∈ Lⁿ(Rⁿ), there exists u ∈ L^∞(Rⁿ) such that divu =f. Then from Theorem II.5 [14], there existsc >0 such that

divBL^∞(0,1)⊃BLⁿ(0, c), (5) where B_E(0, α) is the open ball in E of radius α >0 centered at origin. Let nowf ∈Lⁿ(Rⁿ) satisfyingf 6= 0 and set

h = c f

||f||_Ln(Rⁿ)

.

Therefore, we can deduce from (5) the existence of vn in L^∞(Rⁿ) satisfying

||vn||_L^∞_(Rn)≤1 such that

divvn→h inLⁿ(Rⁿ).

We can then extract a subsequence (vn_k) such thatvn_k

*∗ v in L^∞(Rⁿ) with

||v||L^∞(Rⁿ) ≤1 and h = divv. Hence, we obtain the property (3) with u = 1

c||f||_Ln(Rⁿ)v.

Remark 1. Proposition 2.1 can be improved by showing thatu actually belongs toW^1,n₀ (Rⁿ)∩L^∞(Rⁿ) (see Theorem 2.3). Note thatW₀^1,n(Rⁿ),→BM O(Rⁿ), but the corresponding embedding inL^∞(Rⁿ) does not take place.

Recall now De Rham’s Theorem: let Ω be any open subset ofRⁿ and letf be a distribution ofD⁰(Ω) that satisfies:

∀v ∈V(Ω), <f,v >_D⁰_{(Ω)×D(Ω)}= 0. (6) Then there existsπinD⁰(Ω) such thatf =∇π. In particular, iff ∈W^−1,p₀ (Rⁿ) with 1< p <∞and satisfies

∀v ∈V(Rⁿ), <f,v >_W−1,p

0 (Rⁿ)×W^1,p₀ ⁰(Rⁿ) = 0, (7) then there exists a unique π ∈ L^p(Rⁿ) such that f = ∇π and the following estimate holds

||π||L^p(Rⁿ) ≤ C||f ||_W^−1,p

0 (Rⁿ). Similarly, iff ∈L^p(Rⁿ), with 1< p <∞, and satisfies

∀v ∈V(Rⁿ), Z

Ω

f ·v = 0, (8)

thenf =∇πwithπ∈W₀^1,p(Rⁿ). Note that V(Rⁿ) is dense inHp(Rⁿ) for all 1 ≤p <∞ (see Alliot-Amrouche [2] for p >1 and Miyakawa [20] for p= 1), but is not dense inH_∞(Rⁿ), where for any 1≤p≤ ∞,

Hp(Rⁿ) ={v ∈L^p(Rⁿ); divv = 0}.

AsV(Rⁿ) is dense in

V^1,p₀ ⁰(Rⁿ) =n

v∈W₀^1,p0(Rⁿ); divv= 0o , then (7) is equivalent to

∀v ∈ V₀^1,p0(Rⁿ), < f,v >

W^−1,p₀ (Rⁿ)×W₀^1,p0(Rⁿ)= 0. (9) The same property holds for the relation (8) withV(Rⁿ) replaced byHp⁰(Rⁿ).

The following corollary gives an answer whenf ∈L¹(Rⁿ).

Corollary 2.2. Assume that f∈L¹(Rⁿ)satisfying

∀v∈H_∞(Rⁿ), Z

Rⁿ

f ·vdx= 0.

Then there exists a uniqueπ∈L^n/(n−1)(Rⁿ)such thatf=∇πwith the estimate

||π||_Ln/(n−1)(Rⁿ)≤C||f||_L1(Rⁿ). (10) Proof. This corollary can be proved because from (4), we have Im∇is closed subspace ofL¹(Rⁿ), so that [H∞(Rⁿ)]^o= (Ker div)^o= Im∇, where

[H_∞(Rⁿ)]^o =

f ∈L¹(Rⁿ), Z

Rⁿ

f ·vdx = 0, ∀v ∈H_∞(Rⁿ)

. Recall that ifE is a Banach space andM a subspace of the dualE⁰, then the polar (or the orthogonal) ofM is defined as follows

M^o = {f ∈E; < f, v >= 0, ∀v∈M}.

Remark 2. Observe first that the hypothesis of Corollary 2.2 implies that f ∈ L¹₀(Rⁿ), i.e., f ∈ L¹(Rⁿ) and

Z

Rⁿ

f = 0. Next, note that the conclusion of the above corollary shows that f ∈ W₀^{−1,n/(n−1)}(Rⁿ) and then divf ∈ W₀^{−2,n/(n−1)}(Rⁿ). Moreover, we have

∀λ∈P1, <divf, λ >

W₀^{−2,n/(n−1)}(Rⁿ)×W₀^2,n(Rⁿ) = 0. (11) Now recall a result in J. Bourgain - H. Br´ezis (cf. [11] or [12]).

Theorem 2.3. For all f ∈Lⁿ(Rⁿ), there existsw∈W^1,n₀ (Rⁿ)∩L^∞(Rⁿ)such thatdiv w = f and

||w||_W1,n

0 (Rⁿ)+||w||_L^∞_(Rn) ≤ C||f||_Ln(Rⁿ).

Remark 3. Note that J. Bourgain and H. Br´ezis use the space cW^1,n(Rⁿ) (re- spectively,Wc^2,n(Rⁿ)), which is defined by the adherence ofD(Rⁿ) for the norm

|| ∇.||_Ln(Rⁿ) (respectively, the adherence ofD(Rⁿ) for the norm || ∇².||_Ln(Rⁿ)) as their functional framework. Our choice is the weighted Sobolev space and it seems to us more adaptive (see the introduction in Section 1 for the explana- tion).

The second point of the following theorem is an extension of Corollary 2.2 and (7) withp=n/(n−1).

Corollary 2.4. i) There exists C >0 such that for all u∈L^n/(n−1)(Rⁿ), we have the following inequality

||u||_Ln/(n−1)(Rⁿ) ≤ C inf

g+h=∇u(||g||_L1(Rⁿ)+||h ||_W−1,n/(n−1)

0 (Rⁿ)) (12)

withg∈L¹(Rⁿ)andh∈W^{−1,n/(n−1)}₀ (Rⁿ).

ii) Let f ∈ L¹(Rⁿ) +W^{−1,n/(n−1)}₀ (Rⁿ) satisfying the following compatibility condition

∀v∈V^1,n₀ (Rⁿ)∩L^∞(Rⁿ), < f,v >= 0. (13) Then there exists a uniqueπ∈L^n/(n−1)(Rⁿ) such thatf=∇π.

Proof. i) We consider two following operators

A=−∇:L^n/(n−1)(Rⁿ)→L¹(Rⁿ) +W^{−1,n/(n−1)}₀ (Rⁿ), A^∗= div :W₀^1,n(Rⁿ)∩L^∞(Rⁿ)→Lⁿ(Rⁿ).

The rest of this proof is similar to the one of Proposition 2.1.

ii) The second point is a consequence of the first one.

Remark 4. Remark that for all u∈W₀^1,1(Rⁿ),

||u||_Ln/(n−1)(Rⁿ) ≤ C|| ∇u||L¹(Rⁿ), (14) and for all u∈L^n/(n−1)(Rⁿ),

||u||_Ln/(n−1)(Rⁿ) ≤ C|| ∇u||_W−1,n/(n−1)

0 (Rⁿ). (15)

The inequality (14) is well-known. We now consider (15). It is shown in [4] that

∆ is an isomorphism fromW₀^2,n(Rⁿ)/P1into Lⁿ(Rⁿ). By duality, we have

∆ : L^n/(n−1)(Rⁿ)→W₀^{−2,n/(n−1)}(Rⁿ)⊥P1

is also an isomorphism. Then for all u∈L^n/(n−1)(Rⁿ), we can deduce

||u||_Ln/(n−1)(Rⁿ) ≤ C||∆u||_W−2,n/(n−1)

0 (Rⁿ). (16)

Moreover, we can also see immediately that

||∆u||_W−2,n/(n−1)

0 (Rⁿ) = ||div∇u||_W−2,n/(n−1)

0 (Rⁿ)

and the following operator

div : W^{−1,n/(n−1)}₀ (Rⁿ)→W₀^{−2,n/(n−1)}(Rⁿ) is continuous. Then

||∆u||_W−2,n/(n−1)

0 (Rⁿ) ≤ C|| ∇u||_W−1,n/(n−1)

0 (Rⁿ)

and we deduce easily (15). The inequality (12), stronger than (14) or (15) is especially interesting if u∈L^n/(n−1)(Rⁿ) and∇u /∈L¹(Rⁿ).

Recall now the definition of Riesz transforms Rjf = cnp.v.

f∗ xj

|x|ⁿ⁺¹

, j= 1, ..., n, where cn = Γ ⁿ⁺¹₂

/πⁿ⁺¹² . Also recall that their Fourier transforms satisfy Rd_jf = iξj

|ξ|fb.

The following corollary is proved in [11]. We give here a little different proof.

Corollary 2.5. Assume thatF∈W₀^1,n(Rⁿ). Then there existsY∈W₀^1,n(Rⁿ)∩

L^∞(Rⁿ) such that

F =

n

X

j=1

RjYj.

Proof. Let F ∈ W₀^1,n(Rⁿ) and define f byfb(ξ) = |ξ|Fb(ξ). Then we have Rdjf(ξ) = d∂F

∂xj

(ξ).Therefore, Rjf = ∂F

∂xj

∈Lⁿ(Rⁿ) for all j = 1, ..., n. Hence, R_jR_jf ∈ Lⁿ(Rⁿ) and we deduce f = −

n

X

j=1

R_jR_jf ∈ Lⁿ(Rⁿ). Thanks to Theorem 2.3, there existsY∈W^1,n₀ (Rⁿ)∩L^∞(Rⁿ) such that f = divY,i.e., fb=

n

X

j=1

iξjYj. Then,Fb=

n

X

j=1

iξj

|ξ|cYj, that means thatF =

n

X

j=1

RjYj. An another result was established by J. Bourgain and H. Br´ezis [11] (see also H. Br´ezis and J. Van Schaftingen [15]).

Theorem 2.6. Let Ωbe a Lipschitz bounded open domain inRⁿ.

i) For all ϕ∈ W^1,n₀ (Ω), there exist ψ ∈ W₀^1,n(Ω)∩L^∞(Ω) andη ∈W₀^2,n(Ω) such that

ϕ=ψ+∇η, (17)

with the following estimate

|| ∇ψ||_Ln(Ω)+||ψ||_L∞(Ω)+||D²η||_Ln(Ω) ≤ C|| ∇ϕ||_Ln(Ω), (18) whereC only depends on Ω.

ii) For all ϕ∈W^1,n(Ω), there existψ ∈W^1,n(Ω)∩L^∞(Ω) andη ∈W^2,n(Ω) such that (17) holds withψ·n= 0 onΓ and satisfying the following estimate

||ψ||W^1,n(Ω)+||ψ||L^∞(Ω)+||η||W^2,n(Ω) ≤ C||ϕ||W^1,n(Ω). (19) In the above theorem,W^1,n₀ (Ω) is the classical Sobolev space of functions in W^1,n(Ω) vanishing on the boundary of Ω andW^2,n₀ (Ω) is the one of functions in W^2,n(Ω) whose traces and the normal derivative are vanished on the boundary of Ω.

We now prove a similar result corresponding to weighted Sobolev spaces.

Corollary 2.7. For all ϕ ∈ W₀^1,n(Rⁿ), there exist ψ ∈ W₀^1,n(Rⁿ)∩L^∞(Rⁿ) andη∈W₀^2,n(Rⁿ) such that

ϕ=ψ+∇η, with the following estimate

||ψ||_W1,n

0 (Rⁿ)+||ψ||_L^∞_(Rn)+||D²η||_Ln(Rⁿ) ≤ C|| ∇ϕ||_Ln(Rⁿ). (20) Proof. Thanks to the density ofD(Rⁿ) inW^1,n₀ (Rⁿ), there exists a sequence (ϕ_k)k∈N^∗ in D(Rⁿ) that converges toward ϕ in W^1,n₀ (Rⁿ). Let Br_k be a ball such that suppϕ_k ⊂Br_k and we setϕ⁰_k(x) =ϕ_k(rkx).Then we deduceϕ⁰_k ∈ W₀^1,n(B₁). Applying Theorem 2.6, there exist ψ⁰_k ∈W^1,n₀ (B₁)∩L^∞(B₁) and η_k⁰ ∈W₀^2,n(B1) such that

ϕ⁰_k=ψ⁰_k+∇η⁰_k, with the following estimate

|| ∇ψ⁰_k||_Ln(B₁)+||ψ⁰_k||_L∞(B₁)+||D²η_k⁰ ||_Ln(B₁) ≤ C|| ∇ϕ⁰_k||_Ln(B₁). We now set

ψ_k(x) = ψ⁰_k(x rk

) and η_k(x) = r_kη_k⁰(x rk

).

Then we have

ϕ_k=ψ_k+∇ηk. (21) Moreover, since

|| ∇ψ_k||_Ln(Rⁿ) = || ∇ψ⁰_k||_Ln(B1),

||ψ_k||_L^∞_(Rn) = ||ψ⁰_k||_L^∞_(B1),

|| ∇ϕ_k||_Ln(Rⁿ) = || ∇ϕ⁰_k||_Ln(B1), we then have that

|| ∇ψ_k||_Ln(Rⁿ)+||ψ_k||_L^∞_(Rn) ≤ C|| ∇ϕ_k||_Ln(Rⁿ) ≤ C|| ∇ϕ||_Ln(Rⁿ). (22)

Then there exists a sequence (ak) in Rⁿ such that ψ_k +ak is bounded in W₀^1,n(Rⁿ) and

||ψ_k+ak||_W^1,n

0 (Rⁿ) ≤ C|| ∇ϕ||Lⁿ(Rⁿ). (23) As||ψ_k||_L^∞_(Rn)is also bounded, then the sequence (ak) is bounded inRⁿ. Then we can extract a subsequence, again denoted by (ak), such that lim

k→∞ak =a.

We know that there existsψ₀inW^1,n₀ (Rⁿ) such thatψ_k+a_k*ψ₀inW^1,n₀ (Rⁿ) and

||ψ₀||_W^1,n

0 (Rⁿ) ≤ C|| ∇ϕ||Lⁿ(Rⁿ).

Then, we have ψ_k * ψ₀−a in W^1,n₀ (Rⁿ). Moreover, ψ_k *^∗ ψ in L^∞(Rⁿ), then it implies thatψ=ψ₀−a. In addition, we have the following estimate

|| ∇ψ||Lⁿ(Rⁿ)+||ψ||L^∞(Rⁿ) ≤ C|| ∇ϕ||Lⁿ(Rⁿ). From (21), we can deduce that

||D²ηk||_Ln(Rⁿ) ≤ C|| ∇ϕ||_Ln(Rⁿ), i.e., there existsα_k ∈P1 such that

ηk+αk is bounded in W₀^2,n(Rⁿ), (24) and there existsη₀inW₀^2,n(Rⁿ) such thatη_k+α_k * η₀inW₀^2,n(Rⁿ). Asϕ_kand ψ_kare bounded inW^1,n₀ (Rⁿ), it is also true for∇ηk, then from (24), we deduce that ∇α_k is bounded inW^1,n₀ (Rⁿ). Therefore, there exists a real sequenceb_k such thatα_k+b_k is bounded inW₀^2,n(Rⁿ). Consequently, the sequenceη_k−b_k is bounded inW₀^2,n(Rⁿ) and we can extract a subsequence, denoted in the same way, such thatηk−bk * ηinW₀^2,n(Rⁿ). Then we have

ϕ_k =ψ_k+∇(ηk−bk) with the estimate

||D²(ηk−bk)||_Ln(Rⁿ) ≤ C|| ∇ϕ||_Ln(Rⁿ).

We pass to limite in the above decomposition, we shall obtainϕ=ψ+∇ηwith

|| ∇ψ||_Ln(Rⁿ)+||ψ||_L^∞_(Rn)+||D²η||_Ln(Rⁿ) ≤ C|| ∇ϕ||_Ln(Rⁿ),

and then we deduce (20).

We have another version of Corollary 2.7.

Theorem 2.8. For all ϕ ∈ W₀^1,n(Rⁿ), there exist ψ ∈ W^1,n₀ (Rⁿ)∩L^∞(Rⁿ) andη∈W₀^2,n(Rⁿ) such that

ϕ=ψ+∇η, with the following estimate

||ψ||_W1,n

0 (Rⁿ)+||ψ||_L^∞_(Rn)+||η||_W2,n

0 (Rⁿ) ≤ C||ϕ||_W1,n 0 (Rⁿ).

Proof. Letϕ∈W^1,n₀ (Rⁿ). We resume the obtained functionsak,ψ_k,a,ψ₀, ψ andη in the proof of Corollary 2.7. We have

|ak|= 1

|B1| Z

B₁

|ak| ≤ 1

|B1| Z

B₁

|ak|ⁿ 1/n

|B1|^(n−1)/n. Then we deduce from (22) and (23) that

|ak| ≤ C||ak||_Ln(B1) ≤ C||ak+ψ_k||_Ln(B1)+C||ψ_k||_Ln(B1)

≤ C|| ∇ϕ||_Ln(Rⁿ). and

||a||_L^∞_(Rn) ≤ C|| ∇ϕ||_Ln(Rⁿ). Asψ = ψ₀−a, then

||ψ||_W1,n

0 (Rⁿ) ≤ C|| ∇ϕ||_Ln(Rⁿ) ≤ C||ϕ||_W1,n 0 (Rⁿ). Asϕ=ψ+∇η, then we have

|| ∇η||_W^1,n

0 (Rⁿ) ≤C||ϕ||_W^1,n

0 (Rⁿ). Therefore, there existsb∈Rsuch that

||η+b||_W2,n

0 (Rⁿ)≤C|| ∇η||_W1,n

0 (Rⁿ)≤C||ϕ||_W1,n 0 (Rⁿ),

and the proof is complete.

Remark 5. This theorem suggests this open question: if ϕ∈W^2,n/2₀ (Rⁿ), are there a function ψ ∈ W^2,n/2₀ (Rⁿ)∩L^∞(Rⁿ) and a function η ∈ W₀^3,n/2(Rⁿ) such that

ϕ=ψ+∇η, with the corresponding estimate ?

We define now the space

X(Rⁿ) ={f ∈L¹(Rⁿ), divf ∈W₀^{−2,n/(n−1)}(Rⁿ)}, which is Banach space endowed with the following norm

||f ||X(Rⁿ) = ||f ||L¹(Rⁿ)+||divf ||_W−2,n/(n−1)

0 (Rⁿ). (25)

Theorem 2.9. Let f ∈X(Rⁿ). Then f ∈ W^{−1,n/(n−1)}₀ (Rⁿ) and we have the following inequality

∀ϕ ∈ W^1,n₀ (Rⁿ)∩L^∞(Rⁿ), | Z

Rⁿ

f·ϕ| ≤ C||f||_X(Rn)||ϕ||_W1,n

0 (Rⁿ). (26)

Proof. We consider the following linear operatorϕ−→^F Z

Rⁿ

f ·ϕ defined on D(Rⁿ). Thanks to Theorem 2.8, we have

| <F,ϕ> | = |R

Rⁿf ·(ψ+∇η)|

= |R

Rⁿf ·ψ− <divf, η >_W−2,n/(n−1)

0 (Rⁿ)×W₀^2,n(Rⁿ) |

≤ ||f ||L¹(Rⁿ)||ψ||L^∞(Rⁿ) +||divf ||_W−2,n/(n−1) 0

||η||_W^2,n

0

≤ C||f ||_X(Rn)||ϕ||_W1,n 0 (Rⁿ).

As D(Rⁿ) is dense in W^1,n₀ (Rⁿ) and by applying Hahn-Banach Theorem, we can uniquely extendF by an elementFe ∈W^{−1,n/(n−1)}₀ (Rⁿ) satisfying

||Fe||_W−1,n/(n−1)

0 (Rⁿ) ≤ C||f ||X(Rⁿ).

Besides, the linear operator f −→ Fe from X(Rⁿ) into W^{−1,n/(n−1)}₀ (Rⁿ) is continuous and injective. Therefore,X(Rⁿ) can be identified to a subspace of W₀^{−1,n/(n−1)}(Rⁿ) with continuous and dense embedding.

Remark 6. i) Letf ∈X(Rⁿ). Then Z

Rⁿ

f = 0if and only if

∀λ∈P1, < divf , λ >_W−2,n/(n−1)

0 (Rⁿ)×W₀^2,n(Rⁿ)= 0.

Note that Z

Rⁿ

f_i =< f_i,1 >_W−1,n/(n−1)

0 (Rⁿ)×W₀^1,n(Rⁿ). ii)Letf ∈X(Rⁿ) satisfying

Z

Rⁿ

f = 0. Then we have the following inequality:

for everyϕ∈W^1,n₀ (Rⁿ),

|<f,ϕ>_W−1,n/(n−1)

0 (Rⁿ)×W₀^1,n(Rⁿ)| ≤ C||f ||_X(Rn)||∇ϕ||_Ln(Rⁿ). Actually, we observe that for anya∈Rⁿ,

|<f,ϕ>| = |<f,ϕ+a>| ≤ ||f ||_X(Rn)||ϕ+a||_W1,n 0 (Rⁿ). Consequently, taking the infinum, we have for everyϕ ∈ W^1,n₀ (Rⁿ) (see [4]):

|<f,ϕ>| ≤ C||f ||_X(Rn)||∇ϕ||_Ln(Rⁿ). (27) It is then easy to deduce the following corollary.

Corollary 2.10. Let f∈L¹(Rⁿ)anddivf= 0. Then Z

Rⁿ

f=0and for every ϕ∈W^1,n₀ (Rⁿ)∩L^∞(Rⁿ), we have

| Z

Rⁿ

f·ϕ| ≤ C||f||_L1(Rⁿ)||∇ϕ||_Ln(Rⁿ).

Corollary 2.11. Let f ∈ L¹(R³) and curlf ∈ W^−2,3/2₀ (R³). Then we have f∈W^−1,3/2₀ (R³)and the following estimate: for everyϕ ∈ W^1,3₀ (R³)∩L^∞(R³),

| Z

R³

f·ϕ| ≤ C(||f||_L1(R³)+||curlf||_W−2,3/2

0 (R³))||ϕ||_W1,3 0 (R³). If moreover

Z

R³

f = 0, then for every ϕ ∈ W^1,3₀ (R³)∩L^∞(R³), we have the following estimate

| Z

R³

f·ϕ| ≤ C(||f ||_L1(R³)+||curlf||_W−2,3/2

0 (R³))||∇ϕ||_L3(R³). Proof. Using Proposition 2.13, the proof is similar as the one of Theorem 2.9

and Remark 6.

Whenf ∈X(Rⁿ), we can improve Corollary 2.2 as follows (recall thatV(Rⁿ) is not dense inH_∞(Rⁿ)).

Proposition 2.12. Assume that f∈X(Rⁿ)satisfying

∀v∈V(Rⁿ), Z

Rⁿ

f·vdx= 0. (28)

Then there exists a uniqueπ∈L^n/(n−1)(Rⁿ)such thatf=∇πand the following estimate holds

||π||_Ln/(n−1)(Rⁿ) ≤ C||f||_W−1,n/(n−1) 0 (Rⁿ).

Proof. This proposition is an immediate consequence of the embeddingX(Rⁿ) ,→W^{−1,n/(n−1)}₀ (Rⁿ) and De Rham’s Theorem (see Alliot - Amrouche [2]).

Remark 7. i) First remark that the hypothesis divf ∈ W₀^{−2,n/(n−1)}(Rⁿ) of Proposition 2.12 is necessary because if f =∇π withπ∈L^n/(n−1)(Rⁿ), then f ∈W^{−1,n/(n−1)}₀ (Rⁿ) and divf = ∆π∈W₀^{−2,n/(n−1)}(Rⁿ).

ii)Also note that (28) is equivalent to

∀v ∈V(Rⁿ), < f,v >_W−1,n/(n−1)

0 (Rⁿ)×W^1,n₀ (Rⁿ)= 0. (29) AsV(Rⁿ) is dense inV^1,n₀ (Rⁿ), (29) is also equivalent to

∀v∈V₀^1,n(Rⁿ), < f,v >_W−1,n/(n−1)

0 (Rⁿ)×W^1,n₀ (Rⁿ)= 0. (30) Since vectors of the canonical basis ofRⁿ belong toV₀^1,n(Rⁿ), we deduce that if (28) holds, then

Z

Rⁿ

f = 0.