SLIDE 1
AAin + Vlx )U* If For some f- 70 - All * in @ . Ev (4) x S 5- - - PowerPoint PPT Presentation
AAin + Vlx )U* If For some f- 70 - All * in @ . Ev (4) x S 5- - - PowerPoint PPT Presentation
2. Energy space DIED . Assume I uxi > 0 : AAin + Vlx )U* If For some f- 70 - All * in @ . Ev (4) x S 5- fleece ? (d) 42 Then - k . 52 ake k*=foglxD on IR2 ) Example C- is T . = 'T q =G , - but 52=1177 BT . Skate 542
SLIDE 2
SLIDE 3
- 2. Energy space DIED
AAin
Assume I uxi > 0 :
- All *
+ Vlx)U* If
in @
For some f- 70
.5-
Then
Ev (4) x S
- 42
fleece? (d)
.52
k¥
Example C- is
- n IR2)
T ake k*=foglxD±
- but
= 'T µq¥µ⇒
=G ,
52=1177 BT
.42
Skate 5¥42 =§t5④ '
si
s -
F⇒TD
SLIDE 4
Theorem
.Assume 7 a E L
'
eoe (D), A > 0
a.e. :
G) Eu (a) = 9 aG) 42 dx
tee CIGS
.- ¥
Then the completion of CI (D) w.int .
1144
= LEAD't
is the Hilbert space ② (D),
(ie , 4) v
= Soda 4
+ SV44
- scalar prod ,
D
D
@i (D) a. L4D , a
dx
)
.- (*)
- Lambda property
SLIDE 5
c) Heller is
a norm
- n CE1R) ( hehe ⇒ ⇐ 4=0
)
2)
ten
)
is
a Cauchy sequence
w.int
. tohellu
⇒
ten
) is a Cauchy seq
. in L4D, adx
)
⇒
an
a
⇒
Ev ( ie) :
= him Elan)
h -Ooo
)
⇐DIED a L theHDD K
(§
an
- and
"
- so
SLIDE 6
Examples:
1)
- D
- u
IRN ,
N73
.T ake
U*
=
(1+1×12)
- ¥
- T
alent function
- All*
=
e (1+1×12)
- = ↳ 1¥
in IRN
a
= ¥
a
* *⇒
= llelxif
"
⇒ 0! LIRN) as L2 4RN ,
lHxtjtdxjs@i.la
)
- completion of CE4R ') w.int
SLIDE 7
②. LIRN)
a LIE IN) , 0! I 4 L4H)
~ ⑥! ( RN) ⇒ H
'
( Rn) !
2)
- D
- n IR2
- does not satisfy A-property
(
const
is the only positive superharmonic !)
⇒ ②
e)
is not weel defined !
SLIDE 8
3)
- is
- n
IR2 1BT
T ake
U *
= fog 1×1)'t
> 0
in IR7B ,
- a. G) = ¥
= I i×¥g
°②!
7 BY
a LY1R? By + G) dx)
. SLIDE 9
Remark
:
Assume
Ev
so teethe)
but Ev
does not satisfy
a-property
.Then
→ + V has just
- ne luptoseaear)
positive
supersolution
= solution
and
:b TV
is eaeeed critical operator
.Ev satisfies A
- property ⇒
- At V
is
suberitio.ae operator
[ Pinchover, Tintarev ,
JFA 2006]
SLIDE 10
Examples :
i)
- b
- n IR2
- critical
2) D- bounded ,
ai
- Ist Dirichlet eigenvalue
- b- Ii
is critical
- ud
( Qi
is the only positive solution !)
Sleep
= aisle '
tee case f- b- A) 4 ,
= 0
SLIDE 11
Theorem
.ttfe@tlsl.bt 7 use@HR)
,
- D Ust Vas
- 9-
in 52
.&
Minimize the functional
{Heli
- (fie)
→ min
- n ② (d)
- B
Iz Ev
SLIDE 12
Theorem
(weak Maximum principle)
A.
assume UE8Y (D) is a supersolution,
- i. e.
⑦
- DUTVU
= f- 70
in 52
, for some fe②dR))*
Then
U70
.u=u+
- u
- ,
let , u
- e- DIED .
¥-1
4
=
U
- OE5 ounce t9Vuy=fµ
- aw
)nitfVluhi)u=
- Strewth - SV4Y
- Evti) so
⇒ Ev (a)
= 0
⇒ u
- -0k¥
SLIDE 13
Corollary (Weak comparison principle)
Let
u,VeH'eoelsD.nl'ealR,KDdx) are
sub and super solutions :
- but Vlx)u=0 ,
- AV1-4DV SO
in@
and
(u - v)
- e ② (D)
I
U7U
- n 052
Then
U > V
in 52
.- o Luv)
+ KD4V) > 0
in@
.T ake Que CE (D),
Osten → HID'v
F-
repeat previous argument
③
SLIDE 14
Three possibilities for
- Dtv
in 52
:1) 74=10 : Efa)
a 0 ⇒ no positive superset
2)
Er (4) 30 and
→ +V is critical
⇒ exactly
- ne positive (
super
)solution
3) Ev (e) 70
④ a- property to-N is subcriticaei
⇒ large
cone of positive supersolutions
④ variational principle in @I (D) ④
comparison principle
SLIDE 15
Exercise . ( Hardy operator
- D
- Ep in RN)
Show that
- D- ¥2 satisfies A-property in IRN
t
ee ↳ etc) , eH=f¥)
' and
N =3 .
&
4×1=1×11
- but
- u*
It
¥2
> 0
- "⇒¥A¥Io
t.at -1
- roots of skin
- D=,
the
C- EL , Lt)
Ht E- HI,
n)
SLIDE 16
- All + IT U =§g
1×1+2
xx)
= Eu = IT
②
'
⇐ ( Rn)
a
LZGRN , II. dx) ,
tea CH
t
N73
.