PEIpogor - math.ucsd.edu
Transcript of PEIpogor - math.ucsd.edu
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Regularity of dutiful f in r
fIsnterion
regularity
elatedtotheLU zr G boundary value
is very tricky Seeexample in Exercise 31 This is drastically
Regularity is dual toLiouville type theorem different.fm
Bernstein type theoremthe linear case
Theorem to'rgeus Calabi pogorelon
U ECT IR dit 17541 1 TTU o
ther U is a quadratic polynomial
I shall present a proof usingJohn's Lemma Ev 1982
interior C estimate The key is a d estimate
on calabi.sc shya k 1958of Pogo rein tused ag.me
icarguemeattifIIPPEIpogorelou's originalproof of TCP theorembe
Thema Pomelo U C Ehr nc I
is a convex solution of dit 4 9 0 in r
nineThen Hillel Fff Helo Halo 1171net 7l4n5lo dx.smMaxlul
l
f f a 121 1 direction consider W toilet UK o on 2hi say co o
It attains its maximum somewhere Xo Weestimate who
gatXo o
first pick a framerttu U U.j Uiifij at Ko i a
unit'h o I t Y o
0 17login t_ fbad
no TH logue Y Ha aauii iuitkik.it Yi
Imind
Using it U
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I't Es't E Tfi
Now we use n'in.j He µiiidish
uii Closet
n Eu4 i Ese EY uit unenclosedis
yBad
Hui E.euiii soBad
E Eui E Ii o
w i't
Uit n a Undoge 1 dose I fo
TMultiplying n'e we Aw Bso A ne
A B depends on u U doggy doggytudughue
Hence weAt It D UHogy e
T LU 2e42
Namely at themaximum who E C
ul et unlocole C
une et Iulxxiec t.ec
simplyusingthecomexityHukxxkdiThis is similar to Aubin Yan's C estimate on
a compact manifold which was provedlater fpsahpe.mgr2aoYfpIasJeMpwog
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hem Estimatesfor tr Can always be doneby affinetransformation
Assume Bf C R C Bhomex via John's Lemma
Then c Cz o Ci Ciadet72u i inn i
tube h constant IUtd 1 0 fr some doC r
fUe E h in Bath applyCaleb
x C r l Ub t liiilc.I pusc.ee
t.tn Ic i E.nThe above is the key ingredient 47.4 of Gilbey Trudingertobe proved in next Leiture
Undertheassumptionof ICP if C I he c I
dsj U.joxidxJ is a complete metric on IR
Calabi's result 1958 paper Corollary on P 108
his quadrateor is Kuba
It x y E B
Thnx duh I
t.geI pin Hussle CH o as
R R o
7in is constant
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We shall first show how to use Them to jcp situation
Then we prove Them
By translation.ro yikiiiEEiiiEIiauoTheCole I U
Now we consider R x Ubc Ea
which is a comex body in IR by John's theorem
7 E centered at x er do is the center of gravityoff
E f x l life E Crc ENK
This is due to that
Iet 174 I does not change unden an
affine transformation T K A etc will let A
Namely if 4 Cx U Aetc
3 Ena 3gyeafa
Cake Iff af Ii CA5 air A
delta detCA dutCA detain
This has been used when we assume Tin lok id
clearly using orthogonaltransformation E can be put into
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the firm we claimed But it does not changeInfo I
Now let Y fci biyi.sci B
E B badr ri
Ent B smaller ball
hk
Tiff Xu By 1 do
172 13 d 7in B detail ib d
i bij
Nour a vbi t.LIThi is the h in Thm a stake
c a eds Cza
on Siya vs he c due he
c Iue E
I us GI lc.xtebi.ec fa is iM
In particular c Ift B B f e I
at K oor To fur
C E X bi E C
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a'Ie Tu e Ca't in Ius ElH
S y also contains a smaller ball aZ
with radius Ea
This put us intothe situation of applying Calabi's
theorem
precisely d shows that E contains
z E
B Glo lxi.xi.llxi.sc 1ara E b.ie
E I
Also since Be.GE C SE B lo
na
fully Else C SE c sa
fxllisiyet.SC SE tin n'Ent
of aY Bbe I
E Byte Beo C S any
I B do a
Namely Then provide the normalized Situtation to
Hain the needed estimates for JCP
Now we focus on the proof of Thin
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Proof of ThmU the one in Ttm not in Jcp
We use the comparison principle 2 the Alexandrov maximum4 Cpg principle
pogorelorestimate
Ford Apply Cpa to Ichi 4th d u
U
on Ir U E u Yin Lu Ui o
dutiful IU Ui 2 n n
ditchUbc z iz bei hth
UGH he Ici batte T
YApply Cpr to txt th n
TUz
4 In 341 7 u u in r2h
I h Uco on't
no
Fort We need thegradient estimate fr convex functions
9 91,5yay Z 41Mt 71964 Y n
unmeetIf we pick y xtYp9gY
lD9k Gf
As long as such y E N
by taking y tothe limit
tR of such possibility
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The key is still A maximum principle
Eun
the
theine C da.mku
C desc arDfl
Hence Exe Sy f Shm
H eda SETIdist Sk 254 3 Ct hz is ccn
Similarlydist Shg 25 3 Can
Note that xoESn
BK.sc f Bapu isaway from 22
In fact Bc du will not tonnch the boundaryof 25hLhence utockth
If want to be sure
lunate
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To get Hit the Hessianestimate We apply to
ughh
Then 17M ftp.fdistchsIasaT E C 9
cents distCSzh SaNow Pogorehr Estimate
Snp Ih u U.IE C
In particular is BaGlo E S y2
sup un E zE C
4
Since Ui I the minimum one Upp 7 C
app I
This proves the uniformestimate in
The Hidden estimate how followsfrom Evans estimate
G T 17.4
Next We show how to prove Evansestimate The key is
a Harnack estimate of Krylov Safanor Only need the
weaker form
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