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
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