Lecture 5 Moser
Statement
strong maximal principle
Theorem 1 (Moser) Let u be a weak solution to
n
Di (aij (x) Dj u) = 0
in B1 Rn
i,j =1
with
def
I (aij ) 1 I
when (aij ) = A = AT ,
I (aij ) and |aij | < 1
when A = AT .
(*)
Suppose u satises
u 0 in B1 (
Lecture 9 Krylov-Safonov
decay estimate
- weak Harnack C Liouville
- Harnack
- L bound in terms of L /Lp
Theorem 1 (Krylov-Safonov) Let u C 0 be a viscosity solution of S (, 0) = 0.
Then u is Hlder continuous and
o
u
C (B1/2 )
C (n, ) u
L (B1 )
with (sm
Lecture 10 Evans-Krylov-(Safonov)
skip C 1,
C 2, estimate
Recall Krylov-Safonov for C 0
u S (, 0) . Now for -elliptic equation
F D2 u = 0
we have
i) u C ;
ii) u C 1, ;
smooth version:
Fij Dij ue = 0.
C 0 version:
u (x + e) u (x)
S (, 0) .
The strong ar
Lecture 11 Dirichlet problem for special Lagrangian equationsa model case
continuity method
a priori estimate
We have answered Dirichlet problem for minimal surface equation with smooth
boundary data. Now we solve Monge-Ampere equations and special Lagr
Lecture 8 Alexandrov
C 1,1 /W 2,n version
viscosity version statement
Alexandrov-Bakelman-Pucci maximum principle:
Let u C 1,1 be a solution to
aij (x) Dij u = f
u 0 on B1 ,
where
I (aij ) 1 I.
Then
1/n
sup u C (n, )
B1
f+
n
.
B1
Proof.
cone gure
Consid
Lecture 7 Minimal Surface equations
non-solvability
strongly convex functional
further regularity
Consider minimal surface equation
div Du 2 = 0 in
.
1+|Du|
u = on
The solution is a critical point or the minimizer of
1 + |Du|2 .
inf
u| =
But the int
Lecture 2 Harmonic functions
invariance
mean value
maximum principle,
(higher order) derivative estimates,
Harnack
weak formulations
mean value
weak/Weyl
viscosity
Invariance for Harmonic functions, solutions to u = 0
u (x + x0 )
u (Rx)
u (tx)
RMK.
Lecture 1 Introduction
equations
source for equations
explicit solutions
reason for better behavior (than waves)
nonlinear theory
De Giorgi-Nash
Krylov-Safonov
The equations
1
u rst derivatives Du and double derivatives D2 u
n
Algebraically
Laplace
u = 1
Lecture 3 Schauder
Calderon-Zygmund
Statement
Examples and counterexamples
C Schauder 1930s
Lp Calderon-Zygmund 1950s
Interior u = f (x)
+ Positive
f C (B1 ) D2 u C B1/2
f Lp (B1 ) D2 u Lp B1/2
1<p<
f BM O D2 u BM O
f H 1 D2 u H 1
- Negative
f C0
D2 f C
Lecture 4 De GiorgiNash
Statement
motivation
Proof
Liouville
Theorem 1 Let u be a weak solution to
n
Di (aij (x) Dj u) = 0
in B1 Rn
i,j =1
with
def
I (aij ) 1 I
when (aij ) = A = AT ,
I (aij ) and |aij | < 1
when A = AT .
(*)
1
Namely for all v H0 (B1 )
a
Lecture 6 Quick applications of Harnack
minimal graph cone
codimension 1
3-d and high codimension
estimates for Greens function
Application 1. Minimal graph cones of codimension 1 must be planes.
cone gure
Analytically
Theorem 1 Any homogeneous order on