Exterior Stokes problem in the half-space

(1)

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Exterior Stokes problem in the half-space

Chérif Amrouche, Florian Bonzom

To cite this version:

(2)

Chérif Amrou he

∗

, Florian Bonzom

LaboratoiredeMathématiquesetdeleursAppli ations,UMRCNRS5142,Universitéde PauetdesPaysdel'Adour,IPRA,Avenuedel'Université,64000Pau edex,Fran e

Abstra tThe purposeof this work isto solvethe exteriorStokesproblemin the half-spa e

R

n

+

. We study the existen e and the uniqueness of generalized solutions in weighted

L

p

theory with

1 < p < ∞

. Moreover,we onsider the aseofstrongsolutionsandveryweaksolutions.Thispaperextendsthestudies donein[3℄foranexteriorStokesproblem inthewholespa eand in[5℄forthe studyoftheLapla eequationinthesamegeometryashere.

Key words Weighted Sobolev spa es; Stokes operator; Diri hlet boundary onditions;Exteriorproblem;Half-spa e.

AMS Classi ation35D05;35D10;35J50;35J55;35Q30;76D07;76N10.

1 Introdu tion and preliminaries Consider

ω

0

a ompa tregionof

R

n

+

= {x ∈ R

n

, x

n

> 0}

,

Γ

0

theboundary of

ω

0

and

Ω

the omplementof

ω

0

in

R

n

+

.Thispaperisdevotedtotheresolution oftheStokessystem

(S

D

)











−∆u + ∇π = f

in

Ω,

div

u

= h

in

Ω,

u

= g

0

on

Γ

0 ,

u

= g

1

on

R

n−1

_.

We noti ethat this problem is anextension of theStokessystem foran exte-rior domain in the whole spa e, studied in several works where some of them introdu ethehomogeneousSobolevspa es

H

1,p

0 (

c

ω

0 )

(where

c

_ω

0

isthe omple-mentin

R

n

of

ω

0

)obtainedas the losureof

D(

c

_ω

0 )

with respe tto thenorm

k∇ · k

L

p

₍

c

_ω

₀

₎

.Theexisten eandtheuniquenessofasolutionofsu haproblem with homogeneousboundary onditionsin

H

1,p

0 (

c

ω

0 ) × L

p

(

c

ω

0 )

has been stu-died by Kozonoand Sohr ([22℄, [23℄) and Galdi and Simader ([18℄). Another pointof view,whi his ours,is tosear hasolutionin weightedSobolevspa es

W

m,p

α

(

c

ω

0 )

(seedenitionbelow).Thesespa esarewell-adaptedtotheLapla e andStokesequationsbe ausetheysatisfyanoptimalPoin aré-typeinequality. They alsoprovidesomepre ise information on thebehaviourof thefun tions at innity, whi h isnotobviousfrom thedenition of

H

1,p

0 (

c

ω

0 )

. Forthis ap-proa h, we refer to Girault and Sequeira [19℄ (when

n = 2

or

n = 3

,

p = 2

∗

(3)

and

α = 0

),Spe ovius-Neugebauer([26℄when

n ≥ 3

and

n

p

+ α /

∈ Z

forstrong solutions and when

n = 2

and

2 p

+ α /

∈ Z

for weak solutions in [27℄) and to AlliotandAmrou he[3℄.

Here,ouroriginalityistosolvetheexteriorStokesprobleminthehalf-spa e andnotanymoreinthewholespa e.Thatimpliesanadditionaldi ultydueto thenatureoftheboundarywhi hisnotboundedsin eit ontains

R

n−1

.So,we havetointrodu eweightseveninthespa esoftra es.We an iteHanouzet[21℄ whohasgiventherstresultsforsu hspa esin1971andAmrou he,Ne£asovà [9℄ whohaveextendedthese resultsin 2001 toweightedSobolevspa es whi h possesslogarithmi weights(wejust re allthatlogarithmi weightsallowusto haveaPoin aré-typeinequalityevenin the riti al ases;seebelowformore details). We remindtheworks ofsomeauthors whi h havestudied theStokes problem in the half-spa e.The rstones are due to Cattabriga[14℄ whohave hosenthesettingofhomogeneousSobolevspa es.Similarresultsaregivenby FarwigandSohr[16℄ andGaldi[17℄. Ontheotherhand Maz'ya,Plamenevskii and Stupyalis [24℄ have studied the problem in weighted Sobolev spa es, but onlyinthedimension3.Finally,we itetheworksofAmrou he,Ne£asováand Raudin [10℄ who onsider weak solutions for any dimension. Nevertheless, we noti e that all these works on ern only the Stokes system in the half-spa e whereas in this paper, we deal with the exterior Stokes problem in the half-spa e.We ansummarizeourworksayingthatitisanextensionoftheexterior problemin thewholespa eandoftheprobleminthehalf-spa e.

Westate that, here,wewill on entrate only on thebasi weightsfor the sakeofsimpli ityandbe ausetheyarethemostusual.Thepaperisorganized asfollows.Se tions2and3aredevotedtothe aseofgeneralizedsolutions res-pe tivelywhen

p = 2

and

p 6= 2

.InSe tion 4,we onsiderstrongsolutionsand giveregularityresultsa ordingtothedata.Finally,inSe tion5,wendvery weaksolutionstothehomogeneousproblemwithsingularboundary onditions. ThemainresultsofthisworkareTheorems2.5and3.6forgeneralizedsolutions, Theorems 4.2 and4.4 forstrong solutionsand Corollary5.4and Theorem 5.5 forveryweaksolutions.

We ompletethisintrodu tionwithashortreviewof theweightedSobolev spa es and theirtra espa es. Foranyinteger

q

wedenote by

P

q

thespa e of polynomialsin

n

variables,ofdegreelessthanorequalto

q

,withthe onvention that

P

q

isredu edto

{0}

when

q

isnegative.

Foranyrealnumber

p ∈ ]1, +∞[

,wedenoteby

p

′

thedualexponentof

p

:

1 p

+

1 p

′

= 1.

Let

x

= (x

1 , . . . , x

n

)

be a typi al point of

R

n

,

x

′

_{= (x}

1 , . . . , x

n−1

)

and let

r = |x| = (x

2

1 + · · · + x

2 n

)

1/2

denoteitsdistan etotheorigin.Weshallusetwo basi weights:

ρ(r) = (1 + r

2 )

1/2

and

lg r = ln(2 + r

2 _).

Asusual,

D(Ω)

isthespa eofindenitelydierentiablefun tionswith ompa t support,

D

′

_(Ω)

(4)

spa eofrestri tionsto

Ω

offun tionsin

D(R

n

₎

.

Then,foranynonnegativeintegers

n

and

m

andrealnumbers

p > 1

and

α

, setting

k = k(m, n, p, α) =











−1

if

n

p

+ α /

∈ {1, . . . , m},

m −

n

p

− α

if

n

p

+ α ∈ {1, . . . , m},

wedenethefollowingspa e:

W

m,p

α

(Ω) = {u ∈ D

′

(Ω);

∀λ ∈ N

n

: 0 6 |λ| 6 k, ρ

α−m+|λ|

(lg r)

−1

D

λ

u ∈ L

p

(Ω);

∀λ ∈ N

n

: k + 1 6 |λ| 6 m, ρ

α−m+|λ|

D

λ

u ∈ L

p

(Ω)}.

ItisareexiveBana hspa eequippedwithitsnaturalnorm:

kuk

W

m,p

α

(Ω)

= (

X

06|λ|6k

kρ

α−m+|λ|

(lg r)

−1

D

λ

uk

p

_L

p

_(Ω)

+

X

k+16|λ|6m

kρ

α−m+|λ|

D

λ

uk

p

_L

p

_(Ω)

)

1/p

_.

Wealsodenethesemi-norm:

|u|

W

m,p

α

(Ω)

= (

X

|λ|=m

kρ

α

D

λ

uk

p

_L

p

_(Ω)

)

1/p

_.

The weightsdened previously are hosenso that the spa e

D(Ω)

is densein

W

m,p

α

(Ω)

and sothat thefollowingPoin aré-typeinequalityholdsin the follo-wingspa es:let

α

bearealnumber,

m ≥ 1

anintegerand

q

′

_{= min(q, m − 1)}

, where

q

isthehighestdegreeofthepolynomials ontainedin

W

m,p

α

(Ω)

.Then:

∀u ∈ W

m,p

α

(Ω),

inf

k∈P

q′

ku + kk

W

m,p

α

(Ω)

≤ C |u|

W

m,p

α

(Ω)

,

and

∀u ∈

W

◦

m,p

α

(Ω) = D(Ω)

k.k

_W

m,p

α

(Ω)

_{, kuk}

W

m,p

α

(Ω)

≤ C |u|

W

m,p

α

(Ω)

.

ThistheoremisprovedbyAmrou he,GiraultandGiroire[8℄in anexterior domainand byAmrou heandNe£asovà[9℄in thehalf-spa e.It isextendedto this domain by an adequatepartition of unity. We denote by

W

−m,p

′

−α

(Ω)

the dualspa eof

◦

W

m,p

α

(Ω)

andwenoti ethat itisaspa eofdistributions.

Now, we want to dene the tra es of fun tions of

W

m,p

α

(Ω)

. These tra es havea omponenton

Γ

0

andanother omponenton

R

n−1

.Forthetra eson

Γ

0

, wereturntoAdams [1℄ orNe£as [25℄for thedenition of

W

m−j−

1 p

,p

(Γ

0 )

with

(5)

offun tions on

R

n−1

,weintodu e,forany

σ ∈ ]0, 1[

,thespa e

W

σ,p

₀

(R

n

_{) = {u ∈ D}

′

_(R

n

_{), ω}

−σ

_{u ∈ L}

p

_(R

n

_),

Z

R

n

_×R

n

|u(x) − u(y)|

p

|x − y|

n+σp

dxdy < ∞},

where

ω =







ρ

if

n

p

6= σ,

ρ(lgρ)

1/σ

if

n

p

= σ.

ItisareexiveBana hspa eequippedwithitsnaturalnorm

(k

u

ω

σ

k

p

L

p

_(R

n

₎

+

Z

R

n

_×R

n

|u(x) − u(y)|

p

|x − y|

n+σp

dxdy)

1/p

_.

Forany

s ∈ R

+

and

α ∈ R

,weset

W

s,p

α

(R

n

) = {u ∈ W

[s],p

[s]+α−s

(R

n

_{), ∀|λ| = [s], ρ}

α

_D

λ

_{u ∈ W}

s−[s],p

0 (R

n

)}.

ItisareexiveBana hspa eequippedwithitsnaturalnorm

kuk

W

s,p

α

(R

n

)

= kuk

W

[s],p

_[s]+α−s

(R

n

₎

+

X

|λ|=s

kρ

α

_D

λ

_uk

W

s−[s],p

0 (R

n

)

.

Wenoti ethatthisdenition oin ideswiththedenitiongivenatthebeginning of this paper when

s = m

is a nonnegative integer. As in [9℄, we have the followinglemma:

Lemma1.1. Foranyinteger

m ≥ 1

andrealnumber

α

,wedene themapping

γ : D(R

n

+

) → (D(R

n−1

))

m

u 7→ (γ

0 u, . . . , γ

m−1

u),

where for any

k = 0, . . . , m − 1

,

γ

k

u =

∂

k

_u

∂x

k

n

. Then,

γ

an be extended by ontinuitytoalinearand ontinuousmappingstilldenotedby

γ

from

W

m,p

α

(R

n

+

)

to

m−1

_Y

j=0

W

m−j−

1 p

,p

α

(R

n−1

)

.Moreover,

γ

isontoand

Ker

γ =

◦

W

m,p

α

(R

n

+

).

Inall this arti le,wesuppose that

Γ

0

is of lass

C

1,1

,ex ept when

p = 2

, where

Γ

0

anbe onsideredto beLips hitz- ontinuousonly.

Wewill denoteby

C

apositiveandreal onstantwhi h mayvaryfromline toline andweset

E

= E

n

(6)

2 Study of the problem

(S

D

)

when

p

= 2

.

First,wenoti ethatitisequivalenttosolvetheproblemwithhomogeneous boundary onditions. Indeed,the fun tion

g

1

is in

W

1−

1

2 ,2

0 (R

n−1

)

, so,thanks toLemma 1.1,thereexists

u

1 ∈ W

1,2

0 (R

n

+

)

su hthat

u

1 = g

1

on

R

n−1

and

ku

1 k

W

1,2

0 (R

n

+

)

≤ C kg

1 k

W

1− 1

2 ,2

0 (R

n−1

)

.

Now,let

η

bethetra eof

u

1

on

Γ

0

,

g

= g

0 − η ∈ H

1

2 _(Γ

0 )

andlet

R > 0

be su hthat

ω

0 ⊂ B

R

⊂ R

n

+

.Itis learthatthefun tion

h

0

dened by

h

0 = g on Γ

0 ,

h

0 = 0 on ∂B

R

,

belongs to

H

1

2 (Γ

0 ∪ ∂B

R

)

. We know that there exists an extension

u

h

0 ∈

H

1 (Ω

R

)

,where

Ω

R

= Ω ∩ B

R

,su hthat

u

h

0 = h

0

on

Γ

0 ∪ ∂B

R

andsu hthat

ku

h

0 k

H

1 (Ω

_R

)

≤ C kh

0 k

_H

1 ₂

(Γ

0 ∪∂B

R

)

.Weset

u

0 = u

h

0 in Ω

R

,

u

0 = 0 in Ω \ Ω

R

.

Wehave

u

0 ∈ H

1 _(Ω)

,

u

0 = g

on

Γ

0

,

u

0 = 0

on

R

n−1

and

ku

0 k

H

1 _(Ω)

≤ C kgk

H

1

2 (Γ

0 )

.

Thusthefun tion

u

0 + u

1 |Ω

isin

W

1,2

0 (Ω)

anditstra esare

g

0

on

Γ

0

and

g

1

on

Γ

1

.Thisallowsustosolveonlythefollowingproblem:let

f

bein

W

−1,2

0 (Ω)

and

h

bein

L

2 _(Ω)

,wewanttond

(u, π) ∈ W

1,2

0 (Ω) × L

2 (Ω)

solutionof

(S

0 )

−∆u + ∇π = f

in Ω,

div

u

= h

in Ω,

u

= 0

on

Γ

0 ,

u

= 0

on

R

n−1

_.

Now,wewanttoestablishLemma2.2tohaveadataforthedivergen eredu ed tozero.Forthis,weusethispreliminarylemma :

Lemma2.1. Thereexistsareal onstant

C > 0

dependingonlyon

ω

0

su hthat the following holds. For any

h ∈ L

2 _(Ω)

,there exists aunique

ϕ ∈ W

2,2

0 (Ω)/R

solution of

∆ϕ = h in Ω

and

∂ϕ

∂n

= 0 on Γ

0 ∪ R

n−1

_.

Moreover,

ϕ

satises

kϕk

_W

2,2

0 (Ω)/R

≤ C khk

L

2 _(Ω)

.

Proof.First,wedene

Ω

′

thesymmetri regionof

Ω

withrespe tto

R

n−1

,

e

Ω = Ω ∪ Ω

′

_{∪ R}

n−1

and

Γ

e

0 = ∂ e

Ω

.Let

h

bein

L

2 _(Ω)

andletthefun tion

h

∗

be dened,foralmostany

(x

(7)

So,

eh ∈ L

2 _(R

n

₎

and, supposingrstthat

n > 2

,as[7℄allowsus tosaythat

∆ : W

2,2

0 (R

n

) −→ L

2 (R

n

)

isonto,wededu ethat thereexists

e

u ∈ W

2,2

0 (R

n

)

su hthat

∆e

u = e

h

in

R

n

and

ke

uk

_W

2,2

0 (R

n

)

≤ C khk

L

2 _(Ω)

. Wedenote by

u ∈ W

2,2

0 (e

Ω)

the restri tionof

u

e

to

e

Ω

. Wenoti e that we have

∆u = h

∗

in

Ω

e

andthat

∂u

∂n

∈ H

1

2 (e

Γ

0 )

. Thanksto Proposition 3.12in [8℄, (there isno onditionof ompatibilitybe ause

n > 2

), thereexists

z ∈ W

2,2

1 (e

Ω) ⊂ W

2,2

0 (e

Ω)

su hthat

∆z = 0

in

Ω

e

and

∂z

∂n

=

∂u

∂n

on

e

Γ

0 ,

he king

kzk

_W

2,2

0 (e

Ω)

≤ C kuk

W

2,2

0 (e

Ω)

.

Now,weset

w = u − z

.Then

w ∈ W

2,2

0 (e

Ω)

satises

∆w = h

∗

in

Ω

e

and

∂w

∂n

= 0

on

Γ

e

0 ,

(1) andwehave

kwk

_W

2,2

0 (e

Ω)

≤ C khk

L

2 _(Ω)

.

If

n = 2

, we annotapplythis reasoningbe ausea onditionof ompatibility appearswhen wewanttouseProposition3.12of[8℄.Nevertheless,we annd dire tly

w ∈ W

2,2

0 (e

Ω)

, solutionof(1), withoutneedingthespa e

W

2,2

1 (e

Ω)

(see Theorem7.13in[20℄).Then, weset,foralmostany

(x

′

_{, x}

n

) ∈ e

Ω

,

v(x

′

, x

n

) = w(x

′

, −x

n

).

As

h

∗

isevenwithrespe tto

x

n

,weeasily he kthat

v

issolutionofthesame problem that

w

satises. So,noti ing that the kernelof this problemis

R

, we dedu e that

v = w + c

in

Ω

e

, with

c ∈ R

, and onsequently,

∂w

∂n

= 0

on

R

n−1

. Thus, the fun tion

w

|Ω

∈ W

2,2

0 (Ω)

is solution of our problem. Moreover, this solutionisuniqueuptoareal onstant.Indeed,if

z ∈ W

2,2

0 (Ω)

isin thekernel ofthis problem,

z

∗

∈ W

2,2

0 (e

Ω)

isin

R

,thekerneloftheproblem(1), so

z ∈ R

.

Lemma 2.2. There exists a real onstant

C > 0

depending only on

ω

0

su h that for any

h ∈ L

2 _(Ω)

,there exists

w

∈

◦

W

1,2

₀

(Ω)

he king

div w = h in Ω

and

kwk

W

1,2

0 (Ω)

≤ C khk

L

2 _(Ω)

.

Proof.Let

h

be in

L

2 _(Ω)

. We know,thanks to theprevious lemma, that thereexists aunique

ϕ ∈ W

(8)

We set

v

= ∇ϕ ∈ W

1,2

0 (Ω)

. So

kvk

W

1,2

0 (Ω)

≤ C khk

L

2 _(Ω)

. Moreover, we set

g

0 = v

_|Γ

₀

∈ H

1

2 _(Γ

0 )

and

g

1 = v

|R

n−1

∈ W

1−

1

2 ,2

0 (R

n−1

)

. Thanksto Theorem 4.2in [10℄,thereexists

(z, θ) ∈ W

1,2

0 (R

n

+

) × L

2 (R

n

+

)

solutionof

−∆z + ∇θ = 0

in

R

n

+

,

div

z

= 0

in

R

n

+

,

z

= g

1

on

R

n−1

_,

satisfying

kzk

W

1,2

0 (R

n

+

)

≤ C kg

1 k

W

1− 1

2 ,2

0 (R

n−1

)

.

Wedenote again by

z

the restri tionof

z

to

Ω

and

g

= g

0 − z

|Γ

0 ∈ H

1

2 (Γ

0 )

. Weobservethat

Z

Γ

0 g

· n dσ =

Z

Γ

0 v

· n dσ −

Z

Γ

0 z

· n dσ =

Z

Γ

0 ∂ϕ

∂n

dσ −

Z

ω

0

div

z

dx = 0.

Now, let

R > 0

be su h that

ω

0 ⊂ B

R

⊂ R

n

+

and

Ω

R

= B

R

∩ Ω

. Then, the previous onditionbeing he ked,wehavethefollowingresult(see [6℄): there exists

y

∈ H

1 _(Ω

R

)

su h that div

y

= 0

in

Ω

R

,

y

= g

on

Γ

0 ,

y

= 0

on

∂B

R

,

and

kyk

H

1 _(Ω

R

)

≤ C

R

(kg

0 k

H

1

2 (Γ

0 )

+ kg

1 k

W

1− 1

2 ,2

0 (R

n−1

)

).

Wedenoteagainby

y

itsextensionby

0

in

Ω

. So

y

∈ W

1,2

0 (Ω)

and div

y

= 0

in

Ω,

y

= g

on

Γ

0 ,

y

= 0

on

R

n−1

_,

Finally,weset

u

= z

|Ω

+ y ∈ W

1,2

0 (Ω)

. Thefun tion

u

saties

div

u

= 0

in

Ω,

u

= g

0

on

Γ

0 ,

u

= g

1

on

R

n−1

_,

andtheestimate

kuk

W

1,2

0 (Ω)

≤ C kvk

W

1,2

0 (Ω)

.

Finallythefun tion

w

= v − u

is solutionofthesettedproblem.

Soto solve

(S

0 )

, it is su ient to solvethe followingproblem

(S

00 )

: nd

(u, π) ∈ W

1,2

0 (Ω) × L

2 (Ω)

solutionof

(S

00 )

−∆u + ∇π = f

in Ω,

div

u

= 0

in Ω,

u

= 0

on

Γ

0 , u = 0

on

R

n−1

_.

Forthis,asanimmediate onsequen eofthepreviouslemma,wederiverst thefollowingBabu²ka-Brezzi ondition(see[12℄and[13℄).

(9)

Weintrodu ethe ontinuousbilinearformdened on

◦

W

1,2

0 (Ω) × L

2 (Ω)

by

b(w, q) = −

Z

Ω

q

div

w

dx.

Let

B ∈ L (

◦

W

1,2

₀

(Ω), L

2 (Ω))

be the asso iatedlinear operator and let

B

′

_∈

L (L

2 (Ω), W

−1,2

0 (Ω))

thedual operatorof

B

,i.e

b(w, q) = < Bw, q >

L

2 _(Ω)×L

2 _(Ω)

= < w, B

′

q >

◦

W

1,2

0 (Ω),W

−1,2

0 (Ω)

.

It is learthat

B = −

div andthat

B

′

_{= ∇}

.Asa onsequen eoftheinf-sup ondition(2),weknowthat

B

isanisomorphismfrom

◦

W

1,2

0 (Ω)/V

onto

L

2 _(Ω)

and

B

′

isanisomorphismfrom

L

2 _(Ω)

onto

V

◦

with

V

= {v ∈

W

◦

1,2

0 (Ω),

div

v

= 0

in

Ω},

whi hisanHilbertspa eand

V

◦

= {f ∈ W

−1,2

0 (Ω), ∀w ∈ V , < f , w >

_W

−1,2

0 (Ω),

◦

W

1,2

0 (Ω)

= 0}.

Thus,wehavethefollowingDeRham's theorem:

Corollary 2.4. The operator

∇

isan isomorphism from

L

2 _(Ω)

to

V

◦

. Now,wedenetheproblem:nd

u

∈ V

su hthat

(F V) ∀v ∈ V ,

Z

Ω

∇u · ∇v dx = < f , v >

W

−1,2

0 (Ω),

◦

W

1,2

0 (Ω)

.

Usingthese ondPoin aré-typeinequalitygivenintheintrodu tionforthe equi-valen eofthenormandthesemi-normin

◦

W

1,2

₀

(Ω)

andapplyingLax-Milgram theorem, we he k that

(F V)

hasa uniquesolution

u

∈ V

. Finally, wenoti e that problems

(S

00 )

and

(F V)

areequivalent,obtainingthepressurethanksto Corollary2.4.Thus,thereexistsaunique

(u, π) ∈ W

1,2

0 (Ω) × L

2 (Ω)

solutionof

(S

00 )

.

In onsequen e,wehavethefollowingtheorem: Theorem 2.5. For any

f

∈ W

−1,2

0 (Ω)

,

h ∈ L

2 _(Ω)

,

g

0 ∈ H

1

2 (Γ

0 )

and

g

1 ∈

W

1−

1

2 ,2

0 (R

n−1

)

,thereexistsaunique

(u, π) ∈ W

1,2

0 (Ω) × L

2 (Ω)

solution ofthe problem

(S

D

)

−∆u + ∇π = f

in Ω,

div u = h

in Ω,

u

= g

0 on Γ

0 , u = g

1 on R

n−1

.

Moreover,

(u, π)

satises

kuk

W

1,2

₀

(Ω)

+ kπk

L

2 _(Ω)

≤ C (kf k

_W

−1,2

0 (Ω)

+ khk

L

2 _(Ω)

+ kg

0 k

H

1

2 (Γ

0 )

+ kg

1 k

W

1− 1

2 ,2

0 (R

n−1

)

),

(10)

3 Study of the problem

(S

D

)

when

p

6= 2

.

First,wesupposethat

p > 2

andwewanttostudythekerneloftheStokes system.Weset:

D

p

₀

(Ω) = {(z, η) ∈

W

◦

1,p

₀

(Ω) × L

p

_{(Ω), −∆z + ∇η = 0 and div z = 0 in Ω}.}

To hara terizethisspa e,itisusefultoshowthefollowinglemma: Lemma 3.1. Let

p > 2

,

f

be in

W

−1,p

0 (R

n

+

)

and

h

be in

L

p

_(R

n

+

)

, both with ompa tsupportin

R

n

+

,and

(v, η) ∈ W

1,2

0 (R

n

+

) × L

2 (R

n

+

)

theuniquesolutionof

(S

+

)







−∆v + ∇η = f

in R

n

+

,

div v = h

in R

n

+

,

v

= 0

on R

n−1

_.

Then, wehave

(v, η) ∈ W

1,p

0 (R

n

+

) × L

p

(R

n

+

)

and

(v, η)

satises

kvk

W

1,p

0 (R

n

+

)

+ kηk

L

p

_(R

n

+

)

+ kvk

W

1,2

0 (R

n

+

)

+ kηk

L

2 _(R

n

+

)

≤ C (kf k

W

−1,p

0 (R

n

+

)

+ khk

L

p

_(R

n

+

)

),

where

C

isareal positive onstantwhi h dependsonlyon

p

,

ω

0

andthesupport of

f

and

h

.

Proof.Let

f

bein

W

−1,p

0 (R

n

+

)

and

h

in

L

p

_(R

n

+

)

with ompa tsupportin

R

n

₊

; weeasily he k that

f

∈ W

−1,2

0 (R

n

+

)

and

h ∈ L

2 _(R

n

+

)

be ause

p > 2

and let

(v, η) ∈ W

1,2

0 (R

n

+

) × L

2 (R

n

+

)

bethesolutionof

(S

+

)

satisfying

kvk

W

1,2

0 (R

n

+

)

+ kηk

L

2 _(R

n

+

)

≤ C (kf k

W

−1,2

0 (R

n

+

)

+ khk

L

2 _(R

n

+

)

).

(3)

Thanksto[10℄,thereexists

(u, π) ∈ W

1,p

0 (R

n

+

) × L

p

(R

n

+

)

solutionof

(S

+

)

su h that

kuk

W

1,p

0 (R

n

+

)

+ kπk

L

p

_(R

n

+

)

≤ C (kf k

W

−1,p

0 (R

n

+

)

+ khk

L

p

_(R

n

+

)

).

(4) Weset

(w, τ ) = (u − v, π − η)

whi hsatises

−∆w + ∇τ = 0 in R

n

+

,

div w = 0 in R

n

+

,

w

= 0 on R

n−1

,

andwewanttoprovethat

(w, τ ) = (0, 0)

.Weeasilyshow(seeProposition 4.1 in [10℄) that

w

n

,the

n

th omponentof

w

, whi his in

W

1,p

0 (R

n

+

) + W

1,2

0 (R

n

+

)

, satises

∆

2 w

n

= 0 in R

n

+

,

w

n

= 0 on R

n−1

,

∂w

n

∂x

n

= 0 on R

n−1

.

Here, the dis ussion splits into three steps : rst, if

p 6= n

and

n 6= 2

, then

w

n

∈ W

0,p

−1

(R

n

+

) + W

0,2

−1

(R

n

+

)

.Foralmostall

(x

(11)

and we he k(see [11℄, [15℄) that

w

e

n

is theunique extension of

w

n

su h that

∆

2 e

w

n

= 0

in

R

n

.Moreover,forany

ϕ ∈ D(R

n

₎

, wehave

< e

w

n

, ϕ >

D

′

_(R

n

_),D(R

n

₎

=

Z

R

n

+

w

n

[ϕ − 5ψ − 6x

n

∂ψ

∂x

n

− x

2 n

∆ψ] dx

where

ψ ∈ D(R

n

₎

is dened by

ψ(x

′

_{, x}

n

) = ϕ(x

′

, −x

n

)

, whi h allows us to provethat

w

e

n

is in

W

−2,p

−3

(R

n

) + W

−2,2

−3

(R

n

)

.So

w

e

n

is abiharmoni tempered distribution and onsequently a biharmoni polynomial. Finally, as the spa e

W

−2,p

−3

(R

n

) + W

−2,2

−3

(R

n

)

doesnot ontainpolynomial,wededu efromthisthat

e

w

n

= 0

in

R

n

and so

w

n

= 0

in

R

n

+

. Now, if

n = p

, we have

W

1,p

0 (R

n

+

) ⊂

W

0,p

−1,−1

(R

n

+

)

, and we may pro eed with the same reasoning sin e the loga-rithmi fa tor does not hange the proof. When

n = 2

, wehave

W

1,2

0 (R

n

+

) ⊂

W

0,2

−1,−1

(R

n

+

)

andgetthesameresultwiththesamearguments,simplynoti ing that

w

n

ould be equal to a onstantin

R

n

+

but that this onstant would be ne essaryequaltozerobe ause

w

n

= 0

on

R

n−1

. Consequently,inany ase,wehave

w

n

= 0

in

R

n

+

.Wededu efromthis(see Proposition 4.1,[10℄)that

τ ∈ L

p

_(R

n

+

) + L

2 (R

n

+

)

satises

∆τ = 0 in R

n

+

,

∂τ

∂n

= 0 on R

n−1

_.

Now,weset foralmostany

(x

′

_{, x}

n

) ∈ R

n

,

τ

∗

(x

′

, x

n

) =

τ (x

′

_{, x}

n

)

if x

n

≥ 0,

τ (x

′

_{, −x}

n

)

if x

n

< 0,

and weeasily he k that

τ

∗

is aharmoni tempereddistribution, soa harmo-ni polynomial, in ludedin

L

p

_(R

n

_{) + L}

2 _(R

n

₎

,a spa ewhi hdoesnot ontain polynomial. Thus, we on lude that

τ = 0

in

R

n

+

. Then, weshow that

w

′

₌

(w

1 , . . . w

n−1

) ∈ W

1,p

0 (R

n

+

) + W

1,2

0 (R

n

+

)

satises

∆w

′

= 0 in R

n

+

,

w

′

= 0 on R

n−1

.

Wesetforalmostany

(x

′

_{, x}

n

) ∈ R

n

,

w

′∗

(x

′

_{, x}

n

) =

w

′

(x

′

_{, x}

n

)

if x

n

≥ 0,

−w

′

_(x

′

_{, −x}

n

) if x

n

< 0,

andweeasily he kthat

w

′∗

_{∈ W}

1,p

0 (R

n

) + W

1,2

0 (R

n

)

isaharmoni tempered distribution,soaharmoni polynomialin

R

n

.Thus,

w

′

isaharmoni polyno-mial in

R

n

+

and

∇w

′

isanharmoni polynomialin

L

p

_(R

n

+

) + L

2 (R

n

+

)

,aspa e whi hdoesnot ontainpolynomial.So

∇w

′

_{= 0}

in

R

n

+

andlike

w

′

_{= 0}

in

R

n−1

, wehave

w

′

_{= 0}

in

R

n

+

.Finally,wededu efromthisthat

(w, τ ) = (0, 0)

.

Now,wehavethefollowingtheorem: Theorem3.2. The kernel

D

p

0 (Ω)

isredu edto

{(0, 0)}

when

p > 2

. Proof. Let

(z, π)

be in

D

p

0 (Ω)

. We denote by

z

e

and

e

π

the extensions by

(12)

−∆e

z

+ ∇e

π ∈ W

−1,p

₀

(R

n

+

)

andweeasily he kthat

h

e

hasa ompa tsupportin

R

n

₊

.Thus,we an applythepreviouslemmawhi h assuresusthatthere exists aunique

(v, η) ∈ (W

1,p

0 (R

n

+

) ∩ W

1,2

0 (R

n

+

)) × (L

p

(R

+

n

) ∩ L

2 (R

n

+

))

solutionof

−∆v + ∇η = e

h

in R

n

+

,

div v = 0 in R

n

+

,

v

= 0 on R

n−1

.

Noti ing that div

z

e

= 0

in

R

n

+

, we see that

(e

z

, e

π)

and

(v, η)

are solutions of thesameproblem, whi h,thanksto[10℄, hasauniquesolutionin

W

1,p

0 (R

n

+

) ×

L

p

(R

n

+

)

.So

(e

z

, e

π) = (v, η)

in

R

n

+

and,settingagain

v

and

η

therestri tionsof

v

and

η

to

Ω

,wededu ethat

v

= z,

η = π

in

Ω.

So,

(v, η) ∈ W

1,p

0 (Ω) × L

p

(Ω)

satises

−∆v + ∇η = 0 in Ω,

div v = 0 in Ω,

v

= 0 on Γ

0 ∪ R

n−1

.

But,

(v, η) ∈ W

1,2

0 (Ω) × L

2 (Ω)

and in thisspa e, thereis, thanksto Theorem 2.5, a unique solution to the above problem, whi h is

(0, 0)

. Thus,

D

p

0 (Ω) =

{(0, 0)}

.

Now,supposingthat

p > 2

,wewanttosolvetheStokessystemwith homo-gemeousboundary onditions,that is to say: let

f

bein

W

−1,p

0 (Ω)

and

h

be in

L

p

_(Ω)

,wewanttond

(u, π) ∈ W

1,p

0 (Ω) × L

p

(Ω)

solutionoftheproblem

(S

0 )

−∆u + ∇π = f

in Ω,

div u = h

in Ω,

u

= 0

on Γ

0 , u = 0

on R

n−1

.

First,weestablishthefollowinglemma:

Lemma3.3. For ea h

p > 2

andfor any

f

∈ W

−1,p

0 (Ω)

and

h ∈ L

p

_(Ω)

,both with ompa tsupportin

Ω

,thereexistsaunique

(u, π) ∈ (W

1,p

0 (Ω)∩W

1,2

0 (Ω))×

(L

p

_{(Ω) ∩ L}

2 _(Ω))

solutionof

(S

0 )

. Proof.Let

f

bein

W

−1,p

0 (Ω)

and

h

bein

L

p

_(Ω)

with ompa tsupportin

Ω

. Then, like

p > 2

, we easily he ksthat

f

∈ W

−1,2

0 (Ω)

and

h ∈ L

2 _(Ω)

and that

kfk

W

−1,2

0 (Ω)

+ khk

L

2 _(Ω)

≤ C (kf k

_W

−1,p

0 (Ω)

+ khk

L

p

_(Ω)

),

where

C

isarealpositive onstantwhi hdependsonlyon

p

,

ω

0

andthesupports of

f

and

h

. Wededu e from Theorem 2.5 that there exists aunique

(u, π) ∈

W

1,2

₀

(Ω) × L

2 _(Ω)

solutionof

(S

0 )

. It stays to show that

(u, π) ∈ W

1,p

0 (Ω) ×

L

p

_(Ω)

.Wedenoteby

u

e

∈ W

1,2

0 (R

n

+

)

and

π ∈ L

e

2 _(R

n

+

)

theextensionsby

0

in

R

n

+

of

u

and

π

and weset

e

f

= −∆e

u

+ ∇e

π

and

eh =

div

u.

e

Letusshownowthat

f

e

∈ W

−1,p

0 (R

n

+

)

and

eh ∈ L

p

_(R

n

+

)

.Wedenethefun tion

χ ∈ D(Ω)

su hthat

χ = 1

in

θ

where

θ

isanopenboundedsubsetof

Ω

su hthat supp

f

⊂ θ

. We denoteby

χ

e

theextension of

χ

by

0

in

R

(13)

andfor

ϕ ∈ D(R

n

+

)

, wehave

< e

h, ϕ >

D

′

_(R

n

+

),D(R

n

+

)

=

Z

Ω

hϕ dx.

So,

f

e

∈ W

−1,p

0 (R

n

+

)

and

eh ∈ L

p

_(R

n

+

)

. Finally, we an apply Lemma 3.1 to on lude that

(e

u

, e

π) ∈ W

1,p

0 (R

n

+

) × L

p

(R

n

+

)

. Thus, by restri tion,

(u, π) ∈

W

1,p

₀

(Ω) × L

p

_(Ω)

.

Now,weestablishthefollowingtheorem:

Theorem 3.4. For ea h

p > 2

,there exists a real onstant

C > 0

depending only on

ω

0

and

p

su hthat the following holds. Forany

g

0 ∈ W

1−

1 p

,p

(Γ

0 )

and

g

1 ∈ W

1−

1 p

,p

0 (R

n−1

)

,there existsaunique

(u, π) ∈ W

1,p

0 (Ω) × L

p

(Ω)

solution of

(S

′

)

−∆u + ∇π = 0 in Ω,

div u = 0

in Ω,

u

= g

0 on Γ

₀

, u = g

1 on R

n−1

.

Moreover,

(u, π)

satises

kuk

W

1,p

0 (Ω)

+ kπk

L

p

_(Ω)

≤ C (kg

0 k

W

1− 1

p

,p

_(Γ

₀

₎

+ kg

1 k

_W

1− 1

p

,p

0 (R

n−1

)

).

Proof.The uniqueness omes from Theorem 3.2. Then, thanks to Propo-sition 4.1 of [10℄, there exists aunique

(w, τ ) ∈ W

1,p

0 (R

n

+

) × L

p

(R

n

+

)

solution of

−∆w + ∇τ = 0 in R

n

+

,

div w = 0 in R

n

+

,

w

= g

1 on R

n−1

.

We denote again by

w

and

τ

the restri tions of

w

and

τ

to

Ω

and we set

g

= g

0 − w

_|Γ

₀

∈ W

1−

1 p

,p

(Γ

0 )

. Thus, it remains to show that there exists

(y, λ) ∈ W

1,p

₀

(Ω) × L

p

_(Ω)

solutionof

(S

′′

₎

(S

′′

₎

−∆y + ∇λ = 0

in Ω,

div y = 0

in Ω,

y

= g

on Γ

0 , y = 0

on R

n−1

.

Forthis,let

R > 0

besu hthat

w

0 ⊂ B

R

⊂ R

n

+

,

Ω

R

= B

R

∩ Ω

and

ψ ∈ D(R

n

₎

withsupportin ludedin

Ω

R

su hthat

Z

Ω

R

ψ(x) dx +

Z

Γ

0 g

· n dσ = 0.

Thankstoresultsinboundeddomains(see[6℄),thereexists

(v, η) ∈ W

1,p

(Ω

R

)×

L

p

_(Ω

R

)

su hthat

−∆v + ∇η = 0

in Ω

R

, div v = ψ

in Ω

R

,

v

= g

on Γ

0 ,

v

= 0

on ∂B

R

.

Next,weextend

(v, η)

by

(0, 0)

in

Ω

andwedenoteby

(e

v

, e

η) ∈ W

1,p

0 (Ω)×L

p

(Ω)

thisextensionwhi hsatises

−∆e

v

+ ∇e

η = ξ

in Ω,

div e

v

= ψ

in Ω,

e

(14)

where

ξ

∈ W

−1,p

0 (Ω)

.Wenoti ethat

ξ

and

ψ

havea ompa tsupportin

Ω

R

so that bythepreviouslemma, thereexists

(z, ν) ∈ W

1,p

0 (Ω) × L

p

(Ω)

solutionof

−∆z + ∇ν = −ξ

in

Ω,

div

z

= −ψ

in

Ω,

z

= 0

on

Γ

0 , z = 0

on

R

n−1

_.

Finally,

(y, λ) = (e

v

+ z, e

η + ν) ∈ W

1,p

0 (Ω) × L

p

(Ω)

is solution of

(S

′′

₎

, so

(u, π) = (w + y, µ + λ) ∈ W

1,p

₀

(Ω) × L

p

_(Ω)

issolutionof

(S

′

₎

andtheestimate followsimmediately.

Now, we an solvethe problem with homogeneousboundary onditionsin the ase

p > 2

.

Theorem 3.5. For any

p > 2

,

f

∈ W

−1,p

0 (Ω)

and

h ∈ L

p

_(Ω)

, there exists

(u, π) ∈ W

1,p

₀

(Ω) × L

p

_(Ω)

solutionof

(S

0 )

.Moreover,

(u, π)

satises

kuk

W

1,p

0 (Ω)

+ kπk

L

p

_(Ω)

≤ C (kf k

_W

−1,p

0 (Ω)

+ khk

L

p

_(Ω)

),

where

C

isa real positive onstantwhi h dependsonly on

p

and

ω

0

.

Proof.Theuniqueness omesfromTheorem3.2.Then,asa onsequen eof the se ond Poin aré-typeinequality given in the introdu tion, weknow there exists atensorof se ond order

F ∈ L

p

_(Ω)

su h that div

F = f

. Weextend

F

(respe tivelyh)by

0

in

R

n

,andwedenoteby

F

e

(respe tively

eh

)thisextension. Then,weset

f

e

=

div

F

e

andwenoti ethat

f

e

|Ω

= f

.Wehave

f

e

∈ W

−1,p

0 (R

n

)

and

eh ∈ L

p

_(R

n

₎

.Thanksto[2℄,thereexists

(v, η) ∈ W

1,p

0 (R

n

)×L

p

(R

n

)

solution of

−∆v + ∇η = e

f

and div

v

= e

h

in

R

n

_.

Wedenoteagainby

v

∈ W

1,p

0 (Ω)

and

η ∈ L

p

_(Ω)

therestri tionsof

v

and

η

to

Ω

. Wehave

v

|Γ

0 ∈ W

1−

1 p

,p

(Γ

₀

)

and

v

|R

n−1

∈ W

1−

1 p

,p

0 (R

n−1

)

, thus, thanksto Theorem3.4,there exists

(w, τ ) ∈ W

1,p

0 (Ω) × L

p

(Ω)

solutionof

−∆w + ∇τ = 0

in

Ω,

div

w

= 0

in

Ω,

w

= −v

|Γ

0

on

Γ

0 , w = −v

|R

n−1

on

R

n−1

_.

So,

(u, π) = (v + w, η + τ ) ∈ W

1,p

0 (Ω) × L

p

(Ω)

is solution of

(S

0 )

and the estimatefollowsimmediately.

Now,wesupposethat

1 < p < 2

.Thanksto theprevioustheorem,

T :

W

◦

1,p

₀

′

(Ω) × L

p

′

(Ω) −→ W

−1,p

0 ′

(Ω) × L

p

′

(Ω),

(u, π) −→ (−∆u + ∇π,

div

u),

isanisomorphism.So,byduality,

T

∗

:

W

◦

1,p

₀

(Ω) × L

p

(Ω) −→ W

−1,p

₀

(Ω) × L

p

(Ω),

isalsoanisomorphismand,asitisstandardto he kthat

T

∗

_{(u, π) = (−∆u −}

(15)

Finally,itremainstoreturntothegeneralproblemwith

p 6= 2

and nonhomo-geneousboundary onditions.Forthis,likeforthe ase

p = 2

,weshowthatthere exists afun tion

w

∈ W

1,p

0 (Ω)

su h that

w

= g

0

in

Γ

0

and

w

= g

1

in

R

n−1

. Then, wehavejust seen that there exists aunique

(v, π) ∈ W

1,p

0 (Ω) × L

p

(Ω)

solutionof

−∆v + ∇π = f + ∆w

in

Ω, v = 0

on

Γ

0 ,

div

v

= h −

div

w

in

Ω, v = 0

on

R

n−1

_.

In onsequen e,the fun tion

(u = v + w, π) ∈ W

1,p

0 (Ω) × L

p

(Ω)

isasolution oftheproblem

(S

D

)

andwehavethefollowingtheorem :

Theorem3.6. For any

p 6= 2

,

f

∈ W

−1,p

0 (Ω)

,

h ∈ L

p

_(Ω)

,

g

0 ∈ W

1−

1 p

,p

(Γ

0 )

and

g

1 ∈ W

1−

1 p

,p

0 (R

n−1

)

, there exists a unique

(u, π) ∈ W

1,p

0 (Ω) × L

p

(Ω)

(S

D

)

(S

D

)

−∆u + ∇π = f

in Ω,

div u = h

in Ω,

u

= g

0 on Γ

0 , u = g

1 on R

n−1

.

Moreover,

(u, π)

satises

kuk

W

1,p

0 (Ω)

+ kπk

L

p

_(Ω)

≤ C (kf k

_W

−1,p

0 (Ω)

+ khk

L

p

_(Ω)

+ kg

0 k

W

1− 1

p

,p

_(Γ

₀

₎

+ kg

1 k

_W

1− 1

p

,p

0 (R

n−1

)

),

where

C

p

and

ω

0

.

4 Strong solutions and regularity for the Stokes system

(S

D

)

.

Inthisse tion, weare interestedin theexisten eofstrong solutionsofthe Stokes system

(S

D

)

, i.e. of solutions

(u, π) ∈ W

2,p

ℓ+1

(Ω) × W

1,p

ℓ+1

(Ω)

. Here, we limitourselvestothetwo ases

ℓ = 0

and

ℓ = −1

.

First,wegiveresultsforthe ase

ℓ = 0

.Wenoti ethatinthis ase,wehave the ontinuousinje tions

W

2,p

1 (Ω) ֒→ W

1,p

0 (Ω)

and

W

1,p

1 (Ω) ֒→ L

p

(Ω)

.So,the twotheoremswhi hfollowshowthatgeneralizedsolutionsofTheorems2.5and 3.6,withastrongerhypothesisonthedata,arein fa tstrongsolutions. Theorem 4.1. For any

p > 1

satisfying

n

p

′

6= 1

,

f

∈ W

0,p

1 (Ω)

and

h ∈

W

1,p

1 (Ω)

, there exists a unique

(u, π) ∈ W

2,p

1 (Ω) × W

1,p

1 (Ω)

solution of

(S

0 )

. Moreover,

(u, π)

satises

kuk

W

2,p

1 (Ω)

+ kπk

W

1,p

1 (Ω)

≤ C (kf k

W

0,p

1 (Ω)

+ khk

W

1,p

1 (Ω)

),

where

C

p

and

ω

0

. Proof.First,wenoti ethatwehavethe ontinuousinje tions

W

0,p

1 (Ω) ֒→

W

−1,p

0 (Ω)

be ause

n

p

′

6= 1

and

W

1,p

(16)

2.5 (

p = 2

) and 3.6 (

p 6= 2

), there exists a unique

(u, π) ∈ W

1,p

0 (Ω) × L

p

(Ω)

solutionof

(S

0 )

.Itremainstoshowthat

(u, π) ∈ W

2,p

1 (Ω) × W

1,p

1 (Ω)

.Forthis, weintrodu ethefollowingpartition ofunity:

ψ

1 , ψ

2 ∈ C

∞

(R

n

), 0 ≤ ψ

1 , ψ

2 ≤ 1, ψ

1 + ψ

2 = 1

in

R

n

_,

ψ

1 = 1

in

B

R

,

supp

ψ

1 ⊂ B

R+1

,

whith

0 < R < R

′

_{< ∞}

su hthat

ω

0 ⊂ B

R

⊂ B

R

′

⊂ R

n

+

.Weset

Ω

R

= Ω ∩ B

R

,

Ω

R+1

= Ω ∩ B

R+1

,

u

i

= ψ

i

u

∈

◦

W

1,p

₀

(Ω)

and

π

i

= ψ

i

π ∈ L

p

_(Ω)

for

i = 1

or

2

. Wenoti ethat supp

(u

1 , π

1 ) ⊂ Ω

R+1

andwedenoteby

(e

u

1 , e

π

1 )

theextension by

(0, 0)

of

(u

1 , π

1 )

in

c

_ω

0

.Finally,weset

e

f

1 = −∆e

u

1 + ∇e

π

1 ,

eh

1 =

div

u

e

1

and

(f

1 , h

1 )

theirrestri tionto

Ω

.Wehavein

Ω

:

f

1 = −∆u

1 + ∇π

1 = ψ

1 f

− 2∇ψ

1 · ∇u − ∆ψ

1 u

+ π∇ψ

1

and

h

1 =

div

u

1 = ψ

1 h +

div

ψ

1 u.

As

u

∈ W

1,p

0 (Ω)

andsupp

ψ

1 ⊂ Ω

R+1

,then

f

1 ∈ W

0,p

1 (Ω)

and

h

1 ∈ W

1,p

1 (Ω)

. Thus

f

e

1 ∈ W

0,p

1 (

c

ω

0 )

,

eh

1 ∈ W

1,p

1 (

c

ω

0 )

and

(e

u

1 , e

π

1 )

satises







−∆e

u

1 + ∇e

π

1 = e

f

1 in

c

ω

0 ,

div

u

e

1 = e

h

1 in

c

_ω

0 ,

e

u

1 = 0

on Γ

0 .

So, thanks to regularity results in a lassi al exterior domain (see [3℄), we have

(e

u

1 , e

π

1 ) ∈ W

2,p

1 (

c

ω

0 ) × W

1,p

1 (

c

ω

0 )

and onsequently

(u

1 , π

1 ) ∈ W

2,p

1 (Ω) ×

W

1,p

₁

(Ω)

.

Now,wedenoteby

(e

u

2 , e

π

2 )

theextensionby

(0, 0)

of

(u

2 , π

2 )

in

R

n

+

and

e

f

2 = −∆e

u

2 + ∇e

π

2 ,

eh

2 =

div

u

e

2 .

As supp

( e

f

2 , e

h

2 ) ⊂ Ω

and as

f

e

2 |Ω

= f − f

1 ∈ W

0,p

1 (Ω)

and

eh

2|Ω

= h − h

1 ∈

W

1,p

1 (Ω)

,wehave

e

f

2 ∈ W

0,p

₁

(R

n

₊

),

and

eh

2 ∈ W

1,p

1 (R

n

+

).

Thus,thankstoTheorem5.2of[10℄,wededu efromthisthat

u

e

2 ∈ W

2,p

1 (R

n

+

),

and

π

e

2 ∈ W

1,p

1 (R

n

+

)

. Byrestri tion,wehave

u

2 ∈ W

2,p

1 (Ω)

,

π

2 ∈ W

1,p

1 (Ω)

and so

(u, π) ∈ W

2,p

1 (Ω) × W

1,p

1 (Ω)

.Theestimatefollowsimmediately.

Now,asat the end ofthe previousse tion, we ansolvetheproblem with nonhomogeneousboundary onditions.

Theorem 4.2. For any

p > 1

satisfying

n

p

′

6= 1

,

f

∈ W

0,p

1 (Ω)

,

h ∈ W

1,p

1 (Ω)

,

g

0 ∈ W

2−

1 p

,p

(Γ

0 )

and

g

1 ∈ W

2−

1 p

,p

(17)

W

2,p

₁

(Ω) × W

1,p

₁

(Ω)

(S

D

)

.Moreover,

(u, π)

satises

kuk

W

2,p

1 (Ω)

+ kπk

W

1,p

1 (Ω)

≤ C (kf k

W

0,p

1 (Ω)

+ khk

W

1,p

1 (Ω)

+ kg

0 k

W

2− 1

p

,p

_(Γ

₀

₎

+ kg

1 k

_W

2− 1

p

,p

1 (R

n−1

)

),

where

C

p

and

ω

0

. Now,weexaminethebasi ase

ℓ = −1

, orrespondingto

f

∈ L

p

_(Ω)

.First, westudythekernelofsu haproblem.Weset

S

p

0 (Ω) = {(z, π) ∈ W

2,p

0 (Ω) × W

1,p

0 (Ω), −∆z + ∇π = 0

in

Ω,

div

z

= 0

in

Ω

and

z

= 0

on

Γ

0 ∪ R

n−1

_}

The hara terizationofthiskernelisgivenbythisproposition: Proposition 4.3. For ea h