35
VII.1 Hille-Yosida Theor em
VII.1 Hille-Yosida Theorem
VII.1 Definition and Elementary Properties of maximal monotone operators
Maximal Monotone
HHADA )(:
)(0),( ADvvAv
Let H be a real Hilbert space and let
be an unbounded
linear operator . A is called monotone if
A is called maximal monotone if furthermore
HAIR )( i.e.
fAuutsADuHf ..)(,
Proposition VII.1
1)()(
1
HLAI
Let A be maximal monotone. Then
(a) D(A) is dense in H
AI is a bijection from D(A) onto H
(b) A is closed.
is a bounded operator with1)( AI
(c) For every 0
.)(
)()()(
0)(
00,0),(
),(),(),(0
..)(,)(
)(0),(,)()(
000
002
00000
000
HindenseisADTherefore
ADADADHHence
ADthen
fsoandvvAvSince
vAvvvAvvvfthen
AvvftsADvHAIRSince
ADvvfthenADfLeta
1)(
)(
),(
),(),(),(
)(
0)),((sin,
)),((),(0
0
,)(
:mindet,.
)(,)(
1
1
2
2
1
2
AIHence
ffAIfu
ufufu
ufuAuuuAuu
ufAIbydenotedisuThis
vuvuAcevuthen
vuvuAvuvuAvAuvu
soandAvAuvu
thenfAvvandADvIf
ederuniquelyisuNotefAuu
thatsuchADuisthereHfeveryForb
fAu
fuuAIandADu
fuAIu
fuAIAuuAIu
fuAuuthen
HinfAuanduuthen
HHinfuAuu
closedisAshowTo
nnn
nn
nn
nn
)()(
)()(
)()()()(
),(),(
:
1
11
ufAIuei
ufuAIei
ufAuuei
HfeachforfAuuSolve
ifHAIRClaim
AIandHLAI
andAIbijectionaisAIhaveWe
monotoneimalisANotebofproofthe
inparagraphfirsttheinresulttheapplyingthen
someforRAIRthatSupposec
HL
)1()(..
)1()(..
)1(..
[2
)(:
1)()()(
)()(
)max()(
,0)()(
0010
000
000
0
)(
10
10
10
10
0
00
)(
.11
02,2
,
1
)())(1(
)1()(
0
00
0
10
0
0010
ADuTu
tsHusoandthen
thenifBut
vv
vvAIvTTv
HtoHfromvfAITv
bydefinedTmaptheConsider
Yosida Regularization of A
0
A
0J
Let A be maximal monotone, for each
let )(1
)( 1
JIAandAIJ
is called a resolvent of A and
(by Prop.VII.1 ) 1)()(
HLJandHLJ
is called Yosida regularization of A
322
2
21
1
1
1)(
AAAJI
AAJI
AAIA
AIJ
Proposition VII. 2 p.1
0,)( HvvJAvA
0,)( HvAvJvA
Let A be maximal monotone, Then
(a1)
(c)
(a2)
(b)
HvvvJ
0lim
0),( ADvAvvA
Proposition VII. 2 p.2
)(lim0
ADvAvvA
0,0),( HvvvA
(d)
(e)
(f)
)1
(
0,1
A
HvvvA
)(
)()(1
)()(
)(
)(
0,
0,)()1(
1
vJAvA
vJAvJI
vJAvJI
vvJAI
vAIvJ
VvFor
VvvJAvAshowToa
)()(1
)()(1
)()()(1
)(1
0),(
0),()()2(
1
11
AvJAvJ
vAvvAI
vAIvAIAIvJIvA
ADvanyFor
ADvforAvJvAthatshowToa
AvAvJ
abyAvJvA
ADvFor
ADvAvvAshowTob
)2()(
0),(
0),()(
HvvvJHence
vJv
vJvthen
vJv
vvvJvvv
vJvJvJvvvvJv
denseisAD
vvtsHDvisthere
GivenHvnowLet
asAvvAvJv
ADvthatfirstAssume
HvvvJthatshowToc
bby
0
0
0
11
1111
1111
11
)(
0
lim
0lim
2suplim
2
))((
..)(
0.
00
)(
lim)(
vvA
vvAvvAvA
vAvvA
eofprooftheFrom
HvvvAthatshowTof
vA
vJvJAvAvA
vJvAvJvvAvvA
HvvvAthatshowToe
AvAvJvA
ADvAvvAthatshowTod
cbyaby
1
1),(
1
),(
)(
0,1
)(
0
)),((),(
),(),(),(
0,0),()(
)(limlim
0),(lim)(
2
2
2
)(
0
)2(
0
0
VII.2 Solution of problem of evolution
0)0(
0
uu
Audt
du
Theorem VII.3 Cauchy, Lipschitz. Picard
0)0( uu
Fudt
du
Let E be a Banach space and F be a mapping
From E to E such that
EvuvuLFvFu ,
such that
then for all
));,0([1 ECu
there is a unique Eu 0
));,0([.
],0[)()(
,)(
)(,)(
)()(
)()(sup
.)1(
)(sup
:1
)(sup));,0([
)mindet(0
))(()()0(
0
0
0
00
0
ECuThenT
giveneachforTtforuniformlytutu
becausecontinuousistutthatObverve
EintutoconvergeswhichEinCauchyistu
tgiveneveryFor
uuetutu
tutueuu
XinsequenceCauchyabeuLet
tueu
normwithspaceBanachisXClaim
tueECuXLet
laterederbetokkGiven
dssuFutuuu
Fudt
du
n
n
mnkt
mn
mnkt
tmn
n
kt
tX
kt
t
t
XnXnXnXnnX
Xn
nkt
mnkt
mnkt
tmn
kt
t
uuuuuuuu
nnifuu
nnttutue
havewemtakingBy
nmnttutue
nmniftutueuu
tuethatshowTo
000001
1
,01)()(
,
,,01)()(
,1)()(sup
)(sup)2(
0
0
0
00
0
0
0
0
0
0
0
,0)()(
,0
,,0)()()()(
,
,0
)()(
)()()()()()(
)()(sup
00)3(
nnifuuHence
nntiftutue
havewemtakingBy
nmntiftutuetutuethen
nmnifuu
thatsuchNnisthereGiven
tutueuu
tutuetutuetutue
tutueuu
nasuuthatshowTo
Xn
nkt
mkt
nkt
Xmn
mkt
Xmn
mkt
mnkt
nkt
nkt
tn
n
XuHence
tue
tuk
LuFLu
keu
euk
LetuFLuutue
euk
LtuFLuu
dsesueLtuFLtuu
tuFdsuLdssuLu
tuFdsusuLu
dsuFdsuFsuFu
dssuFutu
ECu
XtobelongstdssuFutu
functiontheXuallForClaim
kt
t
X
ktX
ktkt
ktX
t ksks
tt
t
tt
t
t
))((sup
0)(1
)1()())((
)1()(
)()(
)()(
)()(
)()())((
))(())((
));,0([
.),0[,))(())((
,:2
0
000
000
000
0000
00 000
00 00
0 00 00
00
00
XX
X
ktX
ktX
kt
t kskskt
tkt
tktkt
t
XX
vuk
LvuHence
vuk
L
evuk
L
evuek
L
dsesvsueLe
dssvsuLe
dssvFsuFetvtue
dssvFsuFtvtu
Xvuvuk
LvuClaim
)1(
)1(
)()(
)()(
))(())(()()(
))(())(()()(
,:3
0
0
0
0
tallfortthenL
tfortthen
sssLt
sssLt
tssLttthenL
tIf
dssLdssusuL
dssuFsuFtutu
dssuFutudssuFutu
tututLet
solutionsareuanduthatSuppose
Uniqueness
dssuFutuuu
upo
fixeduniqueahasincipleBanachnContractio
byandcontrationaisthenLkTaking
nnnn
tt
t
tt
t
0)(
100)(
0)(
0)(
0)()(,1
0
)()(ˆ))(
))(ˆ())(()(ˆ)(
))(ˆ()(ˆ,))(()(
)(ˆ)()(
,ˆ
:
))(()()(
,int
,Pr
,2
1
1222
11
00
0
0000
00
Lemma VII.1
));,0([1 HCw
)(twx
If is a function satisfing
0 wAdt
dw
, then the functions
are decreasing on
and )()( twAt
dt
dwt
),0[
.sin)(,
0
)()(
)()()()(
sin)(sin)(
0)(,max
))(),(()(),((2)(
2
2
2
gdecreaistdt
dwtproofpreviousfromthen
dt
dwA
dt
dw
dt
d
t
twttwA
t
twAttwA
t
tdtdw
ttdtdw
gdecreaistwsoandgdecreaistw
twdt
dmonotoneimalisASince
twtwAtwtwdt
dtw
dt
d
Theorem VII.4 (Hille-Yosida) p.1
),0[0)0(
)6(
0
onAu
uudt
du
Let A be a maximal monotone operator in
a Hilbert space H then for all Hu 0there is a unique
))();,0([));,0([1 ADCHCu s.t.
Theorem VII.4(Hille-Yosida)
0)()( AutAutdt
du
where D(A) is equipped with graph norm
i.e. for 22
,),( AuuuorAuuuADuGG
Furthermore, 0)( utu and
))(,)((2)(
))(,)((2)()(
)())(,)((2
))(),(())(,)((
))()(),(())(),()((
))(),(())(),(())(),(())(),((
))(),(())(),(()()(
:
))(,)((2)(:
ˆ)6(ˆ
:1
2
22
22
2
ttdt
dt
dt
d
ttt
tdt
d
t
ttt
twherettttdt
d
ttdt
dttttt
dt
d
ttttttttt
tttttttttttt
ttttttttt
proof
ttdt
dt
dt
dClaim
uuletandofsolutionsbeuanduLet
UniquenessStep
),0[)(ˆ)(..
),0[0)(
0ˆ)0(ˆ)0()0(
.sin)(
,0))(),(())(,)((2)(
,max
2
200
22
2
2
ttutuei
ttthen
uuuuBut
toffunctiongdecreaaistthen
ttAttdt
dt
dt
d
monotoneimalisASince
00
0
)0()()(
),0[sin)(,1.
.
mindet
),0[)0(
0
:2
AuuAuAtuAtdt
duthen
ongdecreistdt
dutVIILemmaBy
TheoremPicardLipschitzCauchy
byederuniquelyisu
onuu
uAdt
du
problemtheofsolutionthebeuLet
Step
Lemma VI.1 (Riesz-Lemma)
boundeddomainCCu ,:, 12 B\
Let
For any
),( xv
fixed , apply Green’s second
identity to u and in the domain
and then let 0 we have
dsn
xu
n
uxudxxu
xx
),(),(),(
