FIRST ORDER APPROACH TO LP ESTIMATES

FOR THE STOKES OPERATOR IN LIPSCHITZ DOMAINS

Sylvie MONNIAUX - Univ . Aix Marseille ( FRANCE )

joint work with Alan McIntosh - Canberra ( Australia )

MAMFLOWS2017 - Bfdlewo - January 617

VERY WEAKLY LIPSCHITZ Dontiws

rc

itsa Sounded

open set sit .

N

r = Upj (B) pj

: B → pjks ) unif locally Lifschitzj=i

&1 = j⇐yj with XJECYN , suppnxj C- gjks )

Examples. bounded strongly Lipchitz domains

• bounded weakly Lipchitz domains ( two bricks )

" a ball cut in the middle

:ay

} xttyi -7<1 } 1 { 2=0, xzo

, ity 's 1) .

>

x

Derham complex on R

1<p<A

,R E 1123 bounded

opensubset

The exterior algebra on R'

is

11 = A @ A'

@ 112 @A⇒ '

�1� ¢3 €3 ¢

The exterior derivative d on r is

d : 0 → LPG,

e) I Elrii ) -4 Ethel DUG ,Q→o .

Note that d is an unbounded operator on LPCRM ) with domain

DPCD ) = { he Ear ,M :

duE LPA ,n ) }

RK : d'

= 0

The dual De 12ham complex1 1

The dual of the exterior derivative d : Dpld ) → LP lr,

^ ) isI :DYI )→Mr,D/ §: 0 ←

ear,¢)¥ Each) ¥ Ein,¢3¥ Licr,

¢ ) c- 0

( {+ pt,= 1 ) where the

-

inch des boundary conditions : the domain of1

[ in LPG ,n ) is the completion of Earn ) in the graph norm Hullp , +11 fully ,

Again : I2

- O

Remak If r is a bounded strongly Lipson :lz domain,

then

DPLJ ) = } ue LP'

Cr,

N : JUEECR,

n ) & vi u = 0 on Or }N ,B

. Integration by parts formula :

< du,

v }~ = < n,

dv >-

- ( Ju,

v >n .

The Hodge - Dirac operator

Define Dµ=d to on li ( rm with domain DYD ) NDYI ) :

DH is self - adjoint ,bi sectorial of angle w=o

zC2I*5'

bdd

So Dti admits a bounded hobmorphc% '%,Y

" Csi

functional calculus,

i.e.,

Hmo Hf : Sj → e hobmorphic st .

zszpg. ( 12454,24!! ) < °

11-4%7all -< Ko Hfks,q )

Hull f 0<04

where

fan = of ; ↳ fk ) ( at - Dthludz figs ;) O

the Hodge decomposition ( in LYR ,m )

1- M :

nulspace221 r,

m = N2( d) +0 RYE ) R : rangeU +

n

= R2 ( d) +0 WYE )

= ticd )

toRYII

towYDµ)MTDµ)=NTd)nwi#

is finite dimensional

In particular , restricting to A- forms , i.e. ,on li ( r

,Al = Lyrics ) :

44,

it ) = RYP ) &NYd¥12

= } u.cl?lrid):dvu=0 in r

f0 .u=0 on or }

if R is smooth enough

The Hodge Laplacian

- Dµ = dotted with domain DYDI ) nD21§d ) C ice ,m

is non negative , self adjoint

d : 0 → an ,e)I>L4r,e 's I icriddvnllr ,e)→o0 ← c- c- c- ←

:[- div and - D

.. .

- Dµ= - dirt @ - Pkvt afar @ uhad - Idiv +0 - dvI11 bdy and : *bdryond : v .u=0

u×u= 0- Dµ ( Neumann °×adu=0

di✓u=o- Dp ( Dirichlet

Lcplaian ) Laplacian )on 0 - forms on 1 . forms on 2- forms on 3 - forms

The Hodge - Stokes operator2

In 14,

the Stokes operator with Hodge boundary conditions is defined by2

St u = - Atgu = at una,

he H with wlu E D'

/ at )

9defined on 2 - forms

Sµ is non negative self - adjoint in712,

N2lSµ ) = N21Dµ ) nihfr ,t) = } a Eilr,

¢'

) : Eu . -0 inn,

uh .=o in r }

2

Sµ has resilient bounds in H : 40£10,

I )

H z ( ZI -Stp'

Hy,µz,s Co f a e a \ stO

H 2 KI- 5+5'

H E Co

Sµ admits also a bounded holomorphic ) o sot

functional calculus in

742

.

LP questions for Dti, Dh , 5h

( Hp ) Dµ has an LP Hodge decomposition :

LPCRM) = RVD ) to RMI ) @ NPCD ,_ ) .

(Rp ) Dµ is bi sectorial in LPCR ,n ).

In portion : It it%5uHp E C Hull,,

FEEIR

(Fp ) Dµ has a bounded holomoiphic functional calculus in LPGHR ) :

11fcdthullp 5 fnllflb Hull,, ffetiolsl

Remark

IFp ) ⇒ ( tip

Hodge - Dirac operators on very weekly Lipchitz domains

Theorem 1 ( Hodge decomposition )

R E IR"

a very weakly Lipchitz domain.

There exist Hodge exponents

pm , p"

= pj with 1 fpµ C 2 < Ptt f no such that the decompositionMr ,m= RMDIQRPLJIE NMI )

holds if ,and only if , Pµ Cpc pt . For

p in thisrange ,

.MP( d) n NME ) = NPCD

,,) = N2( Dti )

is finite dimensional

- Dµ with domain $(d) n DME ) is a

closed operator

Hint : Snciberg 's thm + potential maps( see later )

Theorem 2

I ER"

a very weshly Lipschutz domain,

1 - pcx .

1 i ) Ifpt < pcptt ,then Dti is b. sectorial of angle O in LPCR

, A).

and for all µE6 , E) , Dn admits a bounded fi holomorphic functionalcalculus in LPCR

,1) .

( ii )

fonresely,

if Dn is bisect aid with a bounded holomophic functionalcalculus in LPIR

,^ )

,then pµ < pcph .

1 iii ) For all re ( max 1 ^, lpty ,

} , pt ) [ qs=I÷ ] and all OEG , 'E )

.

there exists Cro >o sit . ( I+zDµ ). '

: Rrld ) + RYE ) +NYDµ7→[G,n )

with the estimate ,

t 2€ elso

sup | (I+zDµ )' '

all E go Hull fu ERTD )+RT=r )2€ Else + Wr1D+ )

and 2 Kyo st. f FE Hogg )

,

Hf(Dµ ) ullr f Kyollfll .Hull f UERYD ) + RYI )+NYdµ )

.

Off diagonal bounds

PropositionOn a very weekly Lipschutz domain r

,let QE ( pµ , ph ) such

that Dt is b : sectorial in L9( rgn ) .

Let- Elqtk )

then the families { ( ItzD→T'

,a EE is } , { ad ( I+zDµI

'

.

< E Eis }

and / z -8 ( ItzDµI'

,ZEE is } admit the following off . diagonal bounds

-c

dist EL,F )

$ HE Rztttu "Lqgr ,^ )

f C e# " 411<91 , ,n ) fzE¢\Sy

where Rz = /I+zDµ5'

or zd ( I+zDµ5l or a _J ( I+zDµ5 .

Idea of the poofJE Lip ( IR

" ) NEAR " ), 5=1 on E

, 5 :O on F,117311*1^25 distlttj

y :EECR ,N yv=vt[y ,lItzD*5

'] ( Ifu )

v= ( ItzDµ5' ( Itu ) 1- optimal choice of 2

Potential operators on the unit ball

13=136,1 ) E IR"

, YE CEGB ), fy - 1

Rtsukk) : = § yly ). G- y ) ] { tk ' ' uk ( yttlx.gl ) dtdy ,

leken

LPG ,n ) su = heuk,

uk € LPLR.

nk )no if p >n

Then : RB : LPG ,^ ) → W' ' Pcr

,^ ) G E%n ) [ ps = I if pan ]

h - p

and dR.su + Radu + Kan = u where Ktgu = ( fyu ) I KB :L 's→ E

compact

.

On b : kpschitz transformations of the unit ball

p : B → PB uniformly locally bitpschita,

( p*u ) G) = Jaap k7.

u ( pg ) put . back

4,3=45 'd.se

't: define Reis -6*552,3 e* and Kee .

- le's"

kop"

→ same mapping properties & dpstfp + Rppdpis 't KEB = ±.

Potential maps on ray weakly Lipschutz domains ( VWL )

PropositionOn a uwl domain r

,there are potential maps for all PEH ,x )

R : LPCR ,m → LPYRMNDPCD ) S : LPCR ,m -UH ,n ) NICE )

K : LPG,

^ ) → Plr ,^ ) compact L : LPCR,

n ) → EH, n ) Compact

sit ,

dR+Rd + K= I. §S+S[ ÷ L = I dRu= u if a ERMd)

dK=0 [ L = 0 -

K=O on RPK ) L=O on RPCE ) [Su=u if he RPLJ )

. Corollaryon a vwl domain -

-d and § are closed ( unbdd ) operators in LPCR

, 17 ,1 Cpc no

. { RPH ). 1<p< x } and { RPCE)

,^< pcx } are complex interpolation scales

Idea of the proof of bisectoriahty ( dim 3)

Start from p=2 and extrapolate

Pa 2s = E u E R%( d) : u = d Ru

5

°

1212did "

I:(±+ - Dµ5'

ullggA (I+aDµTdRuH% I kttt 2%5 'd Ru "z

E I ,

HRU 112 E Hulls,

,

. Oc

lzktcdiamr: CE( KEJ ) cubes in

Rswith side . length t and corners at

points in TZ" which intersect r

Qkt = 4Gt nr : r= U QI ykeci 14£ , tai ] ) :

LEEJ

Egg = 1 on r 1117411.

-< heTen = dwkt 'EVk

www.than ) : Hwuk ," vk.IE " ±#uBg

+ off diagonal bounds

Estimates on the Hodge exponents on strongly Lipchitz domains

Theorem : et r C R"

be a bounded strongly Lipchitz domain . Then the

tlolge exponents associated with the Hodge decomposition

Mr,

^ ) =RP 1 d) +0 RPCE ) +0 or ( Dm )

are estimated as follows :

pµ =⇐ " "

< In < In,

< High Epa .

h ( It E) + I

In particular , if n .

- 2 pµ < § < 4 < p"

and if n= JPte < 22 <3 < p4

THANK You FOR YOUR ATTENTION