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

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

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

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

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

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

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

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

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