Isotropically and Anisotropically Weighted Sobolev Spaces for the Oseen Equation

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Isotropically and Anisotropically Weighted Sobolev Spaces for the Oseen Equation

Chérif Amrouche, Ulrich Razafison

To cite this version:

Chérif Amrouche, Ulrich Razafison. Isotropically and Anisotropically Weighted Sobolev Spaces for the Oseen Equation. Advances in Mathematical Fluid Mechanics, Birkhauser Verlag, 2010, pp.1-24.

�10.1007/978-3-642-04068-9_1�. �hal-00335163�

Spaces for the Oseen Equation

Ch´ erif Amrouche and Ulrich Razafison

Dedicated to Giovanni Paolo Galdi at the occasion of his 60th Birthday

Abstract. This contribution is devoted to the Oseen equations, a linearized form of the Navier-Stokes equations. We give here some results concerning the scalar Oseen operator and we prove Hardy inequalities concerning functions in Sobolev spaces with anisotropic weights that appear in the investigation of the Oseen equations.

Mathematics Subject Classification (2000).76D05, 35Q30, 26D15, 46D35.

Keywords. Oseen equations, anisotropic weights, Hardy inequality, Sobolev weighted spaces.

1. Introduction

In an exterior domain Ω of R³, the Oseen system is obtained by linearizing the Navier-Stokes equations, describing the flow of a viscous fluid past the obstacle R³\Ω, around a nonzero constant vector which is the velocity at infinity. When Ω =R³, the system can be written as follow:

−ν∆u+k∂u

∂x1

+∇π=f in R³, divu=g inR³,

(1.1) where we add the condition at infinity

lim

|x|→∞u(x) =u_∞. (1.2)

The data are the viscosity of the fluidν, the external forces acting on the fluidf, a functiong, a constant vectoru∞and a realk >0. The unknowns are the velocity

of the fluid u and the pressure function π. Let us now notice that the pressure satisfies the Laplace equation

∆π= div f+ν∆g−k ∂g

∂x₁, (1.3)

and each componentui of the velocity satisfies

−ν∆ui+k∂ui

∂x1

=fi− ∂π

∂xi

. (1.4)

Hence we see that the Oseen problem (1.1) is related to the following equation:

−ν∆u+k∂u

∂x₁ =f inR³. (1.5)

Therefore, the results arising from the analysis of (1.5) can be used for the investi- gation of the Oseen problem (1.1). To prescribe the growth or the decay properties of functions at infinity, we consider here weighted Sobolev spaces where the weight reflects the decay properties of the fundamental solutionOof (1.5) defined by

O(x) = 1

4πν|x|e^{−k(|x|−x1)/2ν}. (1.6) Note now that, at infnity, O has the same following decay properties than the fundamental solution of Oseen

O(x) =O(η⁻¹₋₁(x)), ∇O(x) =O(η_−3/2^−3/2(x)), ∂²O(x) =O(η₋₂⁻²(x)), ....

whereη^α_β(x)≡η^α_β = (1 +|x|)^α(1 +|x|−x1)^βwill be the weight function considered.

Equation (1.5) has been investigated by Farwig (see[6]) in weightedL²-spaces, with the weightη_β^α.

Furthermore, forr=|x|sufficiently large, we obtain the following anisotropic es- timates:

|O(x)| ≤C r⁻¹(1 +s)⁻², |_∂x^∂O

1(x)| ≤C r⁻²(1 +s)⁻³²,

|_∂x^∂O

j(x)| ≤C r⁻³²(1 +s)⁻³²(1 +²_r), j= 2,3, ifn= 3,

(1.7)

|O(x)| ≤C r⁻¹²(1 +s)⁻¹, |_∂x^∂O

1(x)| ≤C r⁻³²(1 +s)⁻¹,

|_∂x^∂O

2(x)| ≤C r⁻¹(1 +s)⁻¹, ifn= 2.

(1.8) Note also the following properties:

∀p >3, O ∈L^p(R³) and ∀p∈]3

2,2[, ∇ O ∈L^p(R³), (1.9)

∀p∈]2,3[, O ∈L^p(R³) and ∀p∈]4 3,3

2[, ∇ O ∈L^p(R³), (1.10) O ∈L¹_loc(Rⁿ) and ∇ O ∈L¹_loc(Rⁿ), for n= 2,3. (1.11)

Observe that whenf∈ D(R³), thenu=O∗f is a solution of (1.5). We have also u=F⁻¹(m₀(ξ)Ff),withm₀(ξ) = (|ξ|²+ikξ₁)⁻¹and _∂x^∂u

j =F⁻¹(m₁(ξ)Ff),with m1(ξ) =iξj(|ξ|²+ikξ1)⁻¹. Here Ff is the Fourier transform off.

2. Scalar Oseen Potential in three dimensional space

This section is devoted to the L^p estimates of convolutions with Oseen kernels.

Before that, we introduce some basic weighted Sobolev spaces. We first setρ(x) = (1+|x|²)¹²,lgρ=ln(1 +ρ) and we define

W₀^1,p(R³) =

v∈ D⁰(R³); v

ω₁ ∈L^p,∇v∈L^p(R³)

,

withω1=ρifp6= 3,ω1=ρlgρifp= 3 andW₀^−1,p0(R³) = (W₀^1,p(R³))⁰. We recall that D(R³) is dense inW₀^1,p(R³) and the constant functions belong to W₀^1,p(R³) ifp≥3. We now introduce a second family of weighted spaces:

Wf₀^1,p(Rⁿ) =

v∈W₀^1,p(Rⁿ), ∂v

∂x1

∈W₀^−1,p(Rⁿ)

and we can prove that

D(Rⁿ) is dense inWf₀^1,p(Rⁿ).

Theorem 2.1. Let f ∈L^p(R³). Then _∂x^∂2O

j∂x_k ∗f ∈ L^p(R³) (in the sense of principal value), _∂x^∂O

1 ∗f ∈L^p(R³)and the following estimate holds k ∂²O

∂x_j∂x_k ∗fk_Lp(R³)+k∂O

∂x₁ ∗fk_Lp(R³)≤Ckfk_Lp(R³). (2.1) Moreover,

1) if1< p <2, thenO ∗f ∈L^2−p2p (R³)and satisfies kO ∗fk

L

2p

2−p(R³)≤CkfkL^p(R³). (2.2) 2) If 1< p <4, then _∂x^∂O

j ∗f ∈L^4−p4p (R³)and verifies the estimate k∂O

∂xj

∗fk

L

4p

4−p(R³)≤Ckfk_Lp(R³). (2.3) Proof.By Fourier’s transform, from Equation (1.5) we obtain:

F( ∂²O

∂xj∂xk

∗f) = −ξjξk

ξ²+iξ1

F(f).

Now, the function ξ7→ m(ξ) = _ξ^−ξ2+iξ^jξk1 is of class C² in R³\ {0} and satisfies for everyα= (α₁, α₂, α₃)∈N³

|∂^|α|m

∂ξ^α (ξ)| ≤C|ξ|^−α,

where, |α| = α1+α2+α3 and C is a constant not depending on ξ. Then, the linear operator

A:f 7→ ∂²O

∂x_j∂x_k ∗f(x) = Z

R³

e^ixξ −ξjξk

ξ²+iξ1

Ff(ξ)dξ

is continuous fromL^p(R³) into L^p(R³) (see E. Stein [20], Thm 3.2, p.96). There- fore, _∂x^∂2O

j∂xk∗f ∈L^p(R³) and satisfies k ∂²O

∂x_j∂x_k ∗fk_Lp(R³)≤Ckfk_Lp(R³). We also have

F(∂O

∂x1

∗f) = iξ₁

ξ²+iξ₁F(f)

and since the functionξ7→m1(ξ) = _ξ2^iξ+iξ¹ 1 have the same properties than m(ξ), it follows that _∂x^∂O

1 ∗f ∈L^p(R³) and satisfies the estimate k∂O

∂x1

∗fkL^p(R³)≤CkfkL^p(R³),

which proves the first part of the proposition and Estimate (2.1). Next, to prove inequalities (2.2) and (2.3), we adapt the technique used by Stein in [20] who studied the convolution off ∈L^p(Rⁿ) with the kernel |x|^α−n. Let us decompose the functionK asK1+K_∞where,

K1(x) =K(x) if |x| ≤µ and K1(x) = 0 if |x|> µ, K_∞(x) = 0 if |x| ≤µ and K_∞(x) =K(x) if |x|> µ.

(2.4) The function K will denote successively O and _∂x^∂O

j and µ is a fixed positive constant which need not be specified at this instance. Next, we shall show that the mappingf 7→K∗f is ofweak-type(p, q), withq=_2−p^2p whenK=Oandq=_4−p^4p whenK=_∂x^∂O

j, in the sense that:

for allλ >0, mes{x;|(K∗f)(x)|> λ} ≤

Cp,q

kfk_Lp(R³)

λ ^q

. (2.5) SinceK∗f = K1∗f +K∞∗f, we have now:

mes{x;|K∗f|>2λ} ≤mes{x;|K1∗f|> λ}+ mes{x;|K_∞∗f|> λ}. (2.6) Note that it is enough to prove inequality (2.5) withkfkL^p(R³)= 1. We have also:

mes{x;|(K₁∗f)(x)|> λ} ≤ kK1∗fk^p_Lp(R³)

λ^p ≤ kK1k^p_L1(R³)

λ^p , (2.7)

and

kK_∞∗fk_L∞(R³)≤ kK_∞k_Lp0(R³). (2.8) 1) Estimate (2.2). Observe that O₁∈L¹(R³) and O_∞∈L^p0(R³) for 1≤p <2.

Then, the integral O₁∗f converges almost everywhere and O_∞∗f converges everywhere. Thus,O ∗f converges almost everywhere. But

∀µ >0, kO1kL¹(R³) ≤ Cµ. (2.9) Next, by using (1.7), we have for anyp⁰ >2:

∀µ >0, kO∞k_Lp0

(R³)≤Cµ^2−p

0

p0 . (2.10)

Choosing now λ = Cµ

2−p0

p0 or equivalently µ = C⁰λ^p−2p . Then from (2.10) and (2.8) we havekO∞∗fkL^∞(R³)< λ and so mes{x; |O∞∗f|> λ}= 0. Finally, for 1≤p <2, we get from inequalities (2.9), (2.6) and (2.7):

mes{x∈R³; |(O ∗f)(x)|> λ} ≤

Cp

1 λ

2−p^2p

. (2.11)

Therefore, for 1≤p <2, the operatorR:f 7→ O ∗f is of weak-type (p,_2−p^2p ).

2) Estimate (2.3). Here we take K = _∂x^∂O

j. First, according to (2.1), _∂x^∂O

1 ∗f ∈ W^1,p(R³) then, by the Sobolev embedding results, we have in particular, _∂x^∂O

1∗f ∈ L^4−p4p (R³). It remains to prove Estimate (2.3) forj= 2,3. First we have:

k∂O

∂xj

k_L1(R³)≤cµ, if µ≤1 andk∂O

∂xj

k_L1(R³)≤cµ¹², if µ >1.

Furthermore, we have forp⁰> ⁴₃: Z

|x|>µ

∂O

∂xj

(x)

p⁰

dx ≤ Cµ^4−3p0, if µ≤1,

Z

|x|>µ

∂O

∂x_j(x)

p⁰

dx ≤Cµ^4−3p

0

2 , if µ >1.

Summarising we obtain:

a) If 0< µ <1, Z

|x|<µ

|∂O

∂xj

(x)|dx ≤cµ and Z

|x|>µ

|∂O

∂xj

(x)|^p0dx ≤Cµ^4−3p0, b) ifµ≥1,

Z

|x|<µ

|∂O

∂xj

(x)|dx ≤cµ¹² and Z

|x|>µ

|∂O

∂xj

(x)|^p0dx ≤Cµ^4−3p

0 2 .

Setting λ=Cµ

4−3p0

p0 in the case a) orλ=Cµ

4−3p0

2p0 in the case b), we get in both cases:

mes{x∈R³; |K∗f(x)|> λ} ≤

C_p1 λ

_4−p^4p

. (2.12)

Thus, for 1 ≤ p < 4, the operator R_j : f 7→ _∂x^∂O

j ∗f is of weak-type (p,_4−p^4p ).

Applying now the Marcinkiewicz interpolation’s theorem, we deduce that, for 1<

p < 2, the linear operator R is continuous from L^p(R³) into L^2−p2p (R³) and for 1< p <4,R_j is continuous fromL^p(R³) intoL^4−p4p (R³).

Remark 2.2. Another proof of Theorem 2.1 consists in using Fourier’s approach.

Let (f_j)_j∈N ⊂ D(R³) be a sequence which converges to f ∈ L^p(R³). Then the sequence (u_j)_j∈Ngiven by:

uj =F⁻¹(m0(ξ)Ffj), m0(ξ) = (|ξ|²+iξ1)⁻¹, (2.13) satisfies the equation−∆uj+^∂u_∂x^j

1 =f_j. Let us recall now the:

Lizorkin Theorem. Let D ={ξ∈R³; |ξ|>0} and m :D −→ C, a continuous function such that its derivatives ^∂km

∂ξ₁^k1∂ξ₂^k2∂ξ₃^k3 are continuous and verify

|ξ1|^k1+β|ξ2|^k2+β|ξ3|^k3+β

∂^km

∂ξ₁^k1∂ξ₂^k2∂ξ₃^k3

≤M, (2.14)

wherek1, k2, k3∈ {0,1}, k=k1+k2+k3 and0≤β <1. Then, the operator A:g7−→ F⁻¹(m0Fg),

is continuous from L^p(R³)intoL^r(R³)with ¹_r= ¹_p−β.

Applying this continuity property with f_j ∈ L^p(R³) and β = ¹₂, we show that (u_j) is bounded inL^2−p2p (R³) if 1< p <2. Thus, this sequence has a subsequence still denoted by (uj) which converges weakly to u and which satisfies T u = f. For the derivative ofuj with respect tox1, the corresponding multiplier is of the form m(ξ) = iξ1(|ξ|²+iξ1)⁻¹. It follows that (2.14) is satisfied for β = 0 and

∂u

∂x1 ∈L^p(R³). The same property takes place for the derivatives of second order withm(ξ) =ξkξl(|ξ|²+iξ1)⁻¹. Finally, we verify withβ = ¹₄, that the derivative of (uj) with respect toxk is bounded inL^4−p4p (R³), which implies _∂x^∂u

k ∈L^4−p4p (R³).

Theorem 2.1 states that _∂x^∂2O

j∂xk ∗f ∈L^p(R³) and under some conditions on p,

∂O

∂x_j ∗f ∈ L^4−p4p (R³) and O ∗f ∈ L^2−p2p (R³). Now, using these results and the classical Sobolev embedding results, we have the following:

Theorem 2.3. Letf ∈L^p(R³).

1) Assume that 1 < p < 4. Then ∇O ∗f ∈ L^4−p4p (R³) with the estimate (2.3).

Moreover,

i) if1< p <3, then ∇O ∗f ∈L^3−p3p (R³)with the estimate k∇O ∗fk

L

3p 3−p(R³)

≤Ckfk_Lp(R³). (2.15) ii) Ifp= 3, then ∇O ∗f ∈L^r(R³)for anyr≥12 and satisfies

k∇O ∗fk_L^r_(R3)≤Ckfk_Lp(R³). (2.16) iii) If 3< p <4, then ∇O ∗f ∈L^∞(R³)and verifies the estimate

k∇O ∗fkL^∞(R³)≤CkfkL^p(R³). (2.17) 2) Assume that 1 < p < 2. Then O ∗f ∈ L^2−p2p (R³) with the estimate (2.2).

Moreover,

i) if1< p < ³₂, then O ∗f ∈L^3−2p3p (R³)and satisfies kO ∗fk

L

3p

3−2p(R³)≤CkfkL^p(R³). (2.18) ii) Ifp= ³₂, then O ∗f ∈L^r(R³)for anyr≥6 and

kO ∗fk_Lr(R³)≤Ckfk_Lp(R³). (2.19) iii) If ³₂ < p <2, then O ∗f ∈L^∞(R³)and the following estimate holds

kO ∗fk_L^∞_(R3)≤Ckfk_Lp(R³). (2.20) Proof.1)If 1< p <4, the previous theorem asserts that _∂x^∂O

j ∗f ∈L^4−p4p (R³) and _∂x^∂2O

j∂x_k∗f ∈L^p(R³). If 1< p <3, there exists a unique constantk(f)∈Rsuch thatv=_∂x^∂O

j∗f+k(f)∈W₀^1,p(R³). Thenk(f) =v−_∂x^∂O

j∗f ∈W₀^1,p(R³)+L^4−p4p (R³).

As none of both spaces contains constants then k(f) = 0, which implies that

∂O

∂xj∗f ∈W₀^1,p(R³). Now, the Sobolev embedding results yield _∂x^∂O

j∗f ∈L^3−p3p (R³) and Estimate (2.15). Ifp≥3, again by the previous theorem, we have _∂x^∂O

j ∗f ∈ W₀^1,p(R³). Then _∂x^∂O

j ∗f ∈ BM O(R³) if p = 3. Applying now the interpolation theorem betweenBM O(R³) andL^p(R³), we get _∂x^∂O

j ∗f ∈L^r(R³) for anyr≥12.

By Sobolev embedding results, if 3< p <4, we have _∂x^∂O

j ∗f ∈L^∞(R³), ) and the case 1) is proved.

2)By the previous theorem, if 1< p <2, we haveO ∗f ∈L^2−p2p (R³) and∇O ∗f ∈ L^3−p3p (R³). Now by Sobolev embedding results, O ∗f ∈ L^p∗(R³), where _p¹∗ =

3−p

3p −¹₃= ¹_p−²₃ if 1< p < ³₂, which yields (2.15). For the remainder of the proof, we use the same arguments that in the previous case withO ∗f instead of _∂x^∂O

j∗f and _∂x^∂O

j ∗f instead of _∂x^∂2O

j∂x_k∗f.

Remark 2.4. In Farwig and Sohr [8], Theorem 2.3 proves existence of solutions to the Oseen equations with forces in L^p, thanks to the Lizorkin theorem’s. These solutions, which are not explicit, belong to homogeneous Sobolev spaces. Here, in

Theorem 2.1, we prove some continuity properties for the Oseen potential, without using Lizorkin theorem’s, and in Theorem 2.3, we complete thoses properties, thanks to Sobolev embeddings and we find the same results as the ones given in [8].

Remark 2.5. i) We can also have the result given by Theorem 2.3 2), by showing thatO ∈L^2,∞(R³),i.e.

sup

µ>0

µ²mes{x∈R³; O(x)> µ}<+∞. (2.21) So that, for any 1 < q < 2, according to weak Young inequality (cf. [19], chap.

IX.4), we obtain:

kO ∗fk

L

2q 2−q,∞

(R³)≤CkOk_L^2,∞_(R3)kfk_Lq(R³). (2.22) Let nowp∈]1,2[. There existp0 andp1 such that 1< p0< p < p1<2 and such that the operator R : f 7−→ O ∗f is continuous from L^p0(R³) into L

2p0 2−p0,∞

(R³) and fromL^p1(R³) into L

2p1 2−p1,∞

(R³). The Marcinkiewicz theorem allows again to conclude that the operatorRis continuous fromL^p(R³) intoL^2−p2p (R³)

ii) The same remark remains valid for∇ Othat belongs to L^43,∞(R³).

Using the Young inequality with the relations (1.10) and (1.11), we get the follow- ing result:

Proposition 2.6. Let f ∈L¹(R³). Then

1) O ∗f ∈L^p(R³)for anyp∈]2,3[and satisfies the estimate

kO ∗fk_Lp(R³)≤Ckfk_L1(R³), (2.23) 2) ∇O ∗f ∈L^p(R³)for any p∈]⁴₃,³₂[and the following estimate holds

k∇O ∗fk_L^p_(R3)≤Ckfk_L1(R³). (2.24) Remark 2.7. Taking ”formally”p= 1 in Theorem 2.3, we find thatO ∗f ∈L^q(R³) for anyq∈]2,3[ and∇O ∗f ∈L^q(R³) for anyq∈]⁴₃,³₂[. We notice that they are the same results obtained in Theorem 2.6 by using the Young inequality.

Now, we are going to study the Oseen potentialO∗fwhenfbelongs toW₀^−1,p(R³).

For that purpose, we give the following definition of the convolution off with the fundamental solutionO:

∀ϕ∈ D(R³), hO ∗f, ϕi=:hf,O ∗˘ ϕi_W−1,p

0 (R³)×W₀^1,p0(R³), (2.25) where ˘O(x) =O(−x). With theL^∞weighted estimates obtained in [14] (Thms 3.1 and 3.2), we get an estimate on the convolution of ˘Owith a functionϕ∈ D(R³) which we shall use afterward as follow

Lemma 2.8. For anyϕ∈ D(R³)we have the estimates

|O ∗˘ ϕ(x)| ≤ Cϕ

1

|x|(1 +|x|+x1), (2.26)

|∇O ∗˘ ϕ(x)| ≤ Cϕ

1

|x|³²(1 +|x|+x1)³², (2.27) whereCϕ depends on the support ofϕ.

Remark 2.9. 1)The behaviour on|x|of ˘O ∗ϕand its first derivatives is the same that of ˘O, but the behaviour on 1 +s⁰ is slightly different (see (1.7).

2)From estimates (2.26), (2.27) we find that

∀q > 4

3, O ∗˘ ϕ∈W₀^1,q(R³). (2.28) 3)In (2.26) and (2.27), when ϕtends to zero inD(R³), thenCϕ tends to zero in R.

The next theorem studies the continuity of the operatorsR andRj when f belongs toW₀^−1,p(R³).

Theorem 2.10. Assume that 1 < p < 4 and let f ∈ W₀^−1,p(R³) satisfying the compatibility condition

hf,1i_W−1,p

0 (R³)×W₀^1,p0(R³) = 0, when 1< p≤ 3

2. (2.29)

ThenO ∗f ∈L^4−p4p (R³)and∇O ∗f ∈L^p(R³)with the following estimate kO ∗fk

L

4p

4−p(R³)+k∇O ∗fkL^p(R³)≤Ckfk_W^−1,p

0 (R³). (2.30) Moreover,

i)if 1< p <3, thenO ∗f ∈L^3−p3p (R³) and the following estimate holds kO ∗fk

L

3p

3−p(R³)≤Ckfk_W^−1,p

0 (R³). (2.31)

ii) Ifp= 3, thenO ∗f ∈L^r(R³)for any r≥12and satisfies kO ∗fk_Lr(R³)≤Ckfk_W−1,p

0 (R³). (2.32)

iii) If3< p <4, thenO ∗f ∈L^∞(R³)and we have the estimate kO ∗fk_L∞(R³)≤Ckfk_W−1,p

0 (R³). (2.33)

Proof.Let 1< p <4. By Lemma 2.8 and Remark 2.9 point 3), ifϕ→0 in D(R³), thenCϕ→0 whereCϕ is defined by (2.26). Thus, ˘O ∗ϕ→0 inW₀^1,p0(R³) for allp∈]1,4[, which implies thatO ∗f ∈ D⁰(R³). Next, there existsF∈L^p(R³) such that

f = divF and kFk_Lp(R³)≤Ckfk_W−1,p

0 (R³). (2.34) According to (2.1), we have for anyϕ∈ D(R³),

|h∂O

∂xj

∗f, ϕi_D⁰_(R3)×D(R³)| = |hF,∇ ∂

∂xj

O ∗˘ ϕi_Lp(R³)×L^p0(R³)|

≤ Ckfk_W^−1,p

0 (R³)kϕk_Lp0

(R³).

Then we deduce the second part of (2.30). We also have for allϕ∈ D(R³):

hO ∗f, ϕi_D0(R³)×D(R³)=−hF,∇O ∗˘ ϕi_Lp(R³)×L^p0(R³), and by (2.3):

|hO ∗f, ϕi_D⁰(R³)×D(R³)| ≤Ckfk_W^−1,p

0 (R³)kϕk

L

4p 5p−4(R³).

Note that 1 < p <4 ⇐⇒1 < _5p−4^4p <4. Consequently, we have the first part of (2.30). Moreover, by Sobolev embeddings, O ∗f ∈L^3−p3p (R³) if 1< p <3,O ∗f belongs to L^r(R³) for all r ≥ 12 ifp = 3 and belongs to L^∞(R³) if 3 < p < 4.

Thus, we showed that if 1< p <4, the operatorsRandRj are continuous.

Corollary 2.11. Assume that 1 < p < 4. If u is a distribution such that ∇u ∈ L^p(R³) and _∂x^∂u

1 ∈W₀^−1,p(R³), then there exists a unique constantk(u)such that u+k(u)∈L^4−p4p (R³)and

ku+k(u)k

L

4p

4−p(R³) ≤C(k ∇ukL^p(R³)+k ∂u

∂x1

k_W^−1,p

0 (R³)). (2.35) Moreover, if 1< p <3, thenu+k(u)∈L^3−p3p (R³), wherek(u)is defined by:

k(u) =− lim

|x|→∞

1 ω3

Z

S₂

u(σ|x|)dσ, (2.36)

where, ω3 denotes the area of the sphereS2 andutends to the constant−k(u)as xtends to infinity in the following sense:

lim

|x|→∞

Z

S₂

|u(σ|x|) +k(u)|dσ= 0. (2.37) If p = 3, then u+k(u) belongs to L^r(R³) for any r ≥12. If 3 < p < 4, then u belongs toL^∞(R³), is continuous in R³ and tends to −k(u)pointwise.

Proof.We set g=−∆u+_∂x^∂u

1 ∈W₀^−1,p(R³). SinceP_[1−3

p0] contains at most constants and according to the density ofD(R³) inWf₀^1,p(R³), theng satisfies the compatibility Condition (2.29). By the previous theorem, there exists a unique v = O ∗g ∈ L^4−p4p (R³) such that ∇v ∈ L^p(R³) and _∂x^∂v

1 ∈ L^p(R³), satisfying T(u−v) = 0, whereT is the Oseen operator, with the estimate:

kvk

L

4p

4−p(R³)≤C(k ∇uk_L^p_(R3)+k ∂u

∂x₁ k_W−1,p

0 (R³)). (2.38) Settingw=u−v, we have for alli= 1,2,3,_∂x^∂w

i ∈L^p(R³) and satisfiesT(_∂x^∂w

i) = 0.

Then by an uniqueness argument, we deduce that ∇u = ∇v and consequently there exists a unique constantk(u), defined by (2.36), such thatu+k(u) =v. The last properties are consequences of Sobolev embeddings.

Remark 2.12. Letu∈ D⁰(R³) such that∇u∈L^p(R³).

i) If 1 < p < 3, we know that there exists a unique constant k(u) such that u+k(u) ∈ L^3−p3p (R³). Here, the fact that in addition _∂x^∂u

1 ∈ W₀^−1,p(R³) we also haveu+k(u)∈L^4−p4p (R³), with _4−p^4p <_3−p^3p .

ii) If 3 ≤p <4, for any constant k, u+k belongs only to W₀^1,p(R³) but not to the space L^r(R³). But, if moreover _∂x^∂u

1 ∈W₀^−1,p(R³) then, u+k(u)∈L^4−p4p (R³) for some unique constantk(u). Moreoveru+k(u)∈L^r(R³) for any r≥ _4−p^4p and u∈L^∞(R³) ifp >3.

3. Weighted Hardy inequalities

In this section, our aim is to give some weighted anisotropic Hardy inequalities in Rⁿ withn≥2.

Forα, β∈R, we consider the anisotropic weight functions η_β^α= (1 +r)^α(1 +s)^β, with

s=s(x) =r−x1. We define the weighted space

L^p_α,β(Rⁿ) ={v∈ D⁰(Rⁿ), η_β^αv∈L^p(Rⁿ)}, which is a Banach space for its natural norm given by

kvk_L^p

α,β(Rⁿ)=kη^α_βvkL^p(Rⁿ). We introduce the first family of weighted Sobolev spaces,

W_α,β^1,p(R³) =n v∈L^p

α−¹₂,β(Rⁿ),∇v∈L^p_α,β(Rⁿ)o , X_α,β^1,p(R³) =n

v∈L^p

α−¹₂,β−¹₂(Rⁿ),∇v∈L^p_α,β(Rⁿ)o , Y_α,β^1,p(R³) =n

v∈L^p_α−1,β(Rⁿ),∇v∈L^p_α,β(Rⁿ)o . These are Banach spaces for their natural norms. Observe that

W_α,β^1,p(R³)⊂X_α,β^1,p(R³)⊂Y_α,β^1,p(R³).

All the local properties of the spaces W_α,β^1,p(R³),X_α,β^1,p(R³) and Y_α,β^1,p(R³) coincide with those of classical Sobolev spacesW^1,p(Rⁿ). Moreover, we have the following properties:

Proposition 3.1.

The space D(Rⁿ)is dense inW_α,β^1,p(R³)

resp. in X_α,β^1,p(R³)and inY_α,β^1,p(R³) .

Proof.It relies on a truncation procedure. Letu∈W_α,β^1,p(R³),ϕ∈ D(Rⁿ), with 0 ≤ ϕ(x) ≤ 1, ϕ(x) = 1 ifr ≤ 1, ϕ(x) = 0 if r ≥ 2, and setϕ_k(x) = ϕ(x/k), uk=uϕk. We have

ku_k−uk^p

W_α,β^1,p(R³)=ku_k−uk^p_Lp α−1

2,β(Rⁿ)+k∇(u_k−u)k^p_Lp α,β(Rⁿ)

≤ k(ϕk−1)uk^p_Lp α−1

2,β(Rⁿ)+Ck(ϕk−1)∇uk^p_Lp α,β(Rⁿ)

+Cku∇ϕkk^p_Lp

α,β(Rⁿ), (3.1)

whereC is a positive real. Sinceu∈W_α,β^1,p(R³), it is clear that the first two terms of the right hand side of (3.1) tend to zero, whenktends to∞. Now, the last term of (3.1) can be written,

ku∇ϕ_kk^p_Lp

α,β(Rⁿ)= Z

{k≤r≤2k}

η^αp_βp|u∇ϕ_k|^pdx and, since|∇ϕk(x)| ≤ 1

k|∇ϕ(x/k)|, we arrive at ku∇ϕ_kk^p_Lp

α,β(Rⁿ)≤C Z

{k≤r≤2k}

η_βp^(α−1)p|u|^pdx.

Recalling that u∈ W_α,β^1,p(R³), this last quantity tends to zero as k tends to ∞.

Then, since each uk has a compact support and the topologies of W_α,β^1,p(R³) and W^1,p(Rⁿ) coincide on this support, the statement of the proposition follows from the density ofD(Rⁿ) inW^1,p(Rⁿ). The proof is the same for the two other spaces.

The previous proposition implies that the dual spaces respectively denotedW_−α,−β^−1,p0 (Rⁿ), X_−α,−β^−1,p0 (Rⁿ), Y_−α,−β^−1,p0 (Rⁿ) are subspaces of D⁰(Rⁿ). Letρ be the weight function ρ= 1 +r=η¹₀ and lgr= ln (1 +ρ). For α∈R, we recall the following weighted Sobolev spaces

W_α^0,p(Rⁿ) ={u∈ D⁰(Rⁿ), ρ^αu∈L^p(Rⁿ)}=L^p_α,0(Rⁿ), (3.2) W_α^1,p(Rⁿ) ={u∈W_α−1^0,p (Rⁿ),∇u∈W_α^0,p(Rⁿ)}, if n

p +α6= 1, (3.3) W_α^1,p(Rⁿ) ={(lgr)⁻¹u∈W_α−1^0,p(Rⁿ),∇u∈W^0,p_α (Rⁿ)}, if n

p +α= 1. (3.4) We have the following identity:

W_α^1,p(Rⁿ) =Y_α,0^1,p(Rⁿ) if n

p +α6= 1.

We will now prove some one-dimensional inequalities.

Lemma 3.2. Let γ ∈ R satisfy γ+ n−1

2 > 0 and θ^∗ ∈]0, π/2[. Then for any positive measurable functionf defined on ]0, θ^∗[, such that

Z θ^∗

0

(1−cosθ)^γ+p2(sinθ)ⁿ⁻²[f(θ)]^pdθ <+∞, one has

Z θ^∗

0

(1−cosθ)^γ(sinθ)ⁿ⁻²[F(θ)]^pdθ≤C Z θ^∗

0

(1−cosθ)^γ+p2(sinθ)ⁿ⁻²[f(θ)]^pdθ, (3.5) with

F(θ) = Z θ^∗

θ

f(t)dt. (3.6)

Proof.Let us first notice that on ]−^π₂,^π₂[, the following inequality holds 1

2sin²θ≤1−cosθ≤sin²θ. (3.7) We now set

J = Z θ^∗

0

(1−cosθ)^γ(sinθ)ⁿ⁻²(F(θ))^pdθ.

In view of Inequality (3.7), we find J =

Z θ^∗

0

(1−cosθ)^γ(sinθ)ⁿ⁻³sinθ(F(θ))^pdθ

≤2^(n−3)/2 Z θ^∗

0

(1−cosθ)^γ+n−32 sinθ(F(θ))^pdθ.

From (3.6) and sinceγ+ⁿ⁻¹₂ >0, an integration by parts yields J ≤C

Z θ^∗

0

(1−cosθ)^γ+n−12 f(θ)(F(θ))^p−1dθ.

Using the H¨older inequality, we obtain J ≤C

Z θ^∗

0

(1−cosθ)^{γ+n−12 p}(sinθ)−(n−2)(p−1)(f(θ))^pdθ and from (3.7), we prove (3.5).

Remark 3.3. By the same way, we can prove that, ifγ∈R, satisfyγ+¹₂ >0 and θ^∗∈]0, π/2[, then for any positive measurable functionf defined on ]−θ^∗,0[, such that

Z 0

−θ^∗

(1−cosθ)^γ+p2[f(θ)]^pdθ <+∞,

one has Z 0

−θ^∗

(1−cosθ)^γ[F(θ)]^pdθ≤C Z 0

−θ^∗

(1−cosθ)^γ+p2[f(θ)]^pdθ, (3.8) with

F(θ) = Z θ

−θ^∗

f(t)dt.

Remark 3.4. (i) As a consequence of Inequality (3.5) for n = 2 and Inequality (3.8), for anyw∈ D(]−θ^∗, θ^∗[) withγ+¹₂ >0, one has

Z θ^∗

−θ^∗

(1−cosθ)^γ|w(θ)|^pdθ≤C Z θ^∗

−θ^∗

(1−cosθ)^γ+p2|w⁰(θ)|^pdθ. (3.9) (ii) Inequality (3.5) also implies that for anyw∈ D([0, θ^∗[),γ+ⁿ⁻¹₂ >0, one has

Z θ^∗

0

(1−cosθ)^γ(sinθ)ⁿ⁻²|w(θ)|^pdθ≤C Z θ^∗

0

(1−cosθ)^γ+p2(sinθ)ⁿ⁻²|w⁰(θ)|^pdθ.

(3.10) We now consider the sector

S=SR,λ={x∈Rⁿ, r > R, 0< s < λr}, withR >0 and 0< λ <1. (3.11) We start to prove a Hardy-type inequality in the sector S.

Lemma 3.5. Let α, β∈R such thatβ >max(0,(1−n+p)/2p). Then we have

∀u∈ D(S), kuk_L^p

α−1 2,β−1

2

(S)≤Ck∇uk_L^p

α,β(S). (3.12) Proof.Letube inD(S). Sinceβ >0, it is enough to prove

I= Z

S

(1 +r)^(α−12)ps^(β−12)p|u|^pdx≤C Z

S

(1 +r)^αps^βp|∇u|^pdx. (3.13) Indeed, let us assume that Inequality (3.13) holds. Then, if 0< β < 1

2, thanks to (3.13), we have

Z

S

(1 +r)^(α−12)p(1 +s)^(β−12)p|u|^pdx≤ Z

S

(1 +r)^(α−12)ps^(β−12)p|u|^pdx

≤C Z

S

(1 +r)^αps^βp|∇u|^pdx

≤C Z

S

(1 +r)^αp(1 +s)^βp|∇u|^pdx.

Now, ifβ≥ 1 2, Z

S

(1 +r)^(α−12)p(1 +s)^(β−12)p|u|^pdx≤C Z

S

(1 +r)^(α−12)p(1 +s^(β−12)p)|u|^pdx

≤C Z

S

(1 +r)^αp(s^p/2+s^βp)|∇u|^pdx and we obtain (3.12). First, we prove Inequality (3.13) for the case n ≥ 3. Let θ= (θ1, θ2, ..., θ_n−1)∈]0, π[ⁿ⁻²×]0,2π[,R >0,θ^∗∈]0,^π₂[ fixed and consider

∆ ={(r, θ)∈R⁺×]0, π[ⁿ⁻²×]0,2π[, r > R, θ₁∈]0, θ^∗[}. (3.14) To establish (3.13), we introduce the generalized spherical coordinates

x1=rcosθ1, x2=rsinθ1cosθ2, ..., x_n−1=rsinθ1...sinθ_n−2cosθ_n−1,

x_n=rsinθ₁...sinθ_n−2sinθ_n−1, (3.15)

where (r, θ)∈∆. Now takingu(x) =v(r, θ) and observing that

∂v

∂θ₁

≤r|∇u|, it is sufficient to prove that

I= Z

∆

(1 +r)^(α−12)p(r−rcosθ1)^(β−12)prⁿ⁻¹(sinθ1)ⁿ⁻²|v|^pdrdθ

≤C Z

∆

(1 +r)^αp(r−rcosθ1)^βprⁿ⁻¹(sinθ1)ⁿ⁻²r^−p

∂v

∂θ1

p

drdθ.

(3.16)

We immediately have I≤

Z

∆

(1 +r)^αpr^βp(1−cosθ₁)^(β−12)prⁿ⁻¹(sinθ₁)ⁿ⁻²r^−p|v|^pdrdθ. (3.17) We now set

J = Z θ^∗

0

(1−cosθ1)^(β−12)p(sinθ1)ⁿ⁻²|v|^pdθ1.

Sinceβ >(1−n+p)/2p, we have (β−¹₂)p+ⁿ⁻¹₂ >0. Moreoveru∈ D(S) implies that, for (r, θ)∈∆, the functionθ₁→v(r, θ) belongs toD([0, θ^∗[). Therefore from (3.10), we get

J ≤C Z θ^∗

0

(1−cosθ₁)^βp(sinθ₁)ⁿ⁻²

∂v

∂θ₁

p

dθ₁. (3.18) In view of inequalities (3.17) and (3.18), we obtain (3.16).

We now continue the proof of (3.13) for the casen= 2. We define

∆ ={(r, θ)∈R⁺×]−π, π[, r > R, θ∈]−θ^∗, θ^∗[} (3.19) and we introduce the polar coordinates

x1=rcosθ , x2=rsinθ, (3.20)