We discuss the following equations:
|,| |m
D (a D u) =
| |m
D f
(1)
Here f L2 (), Rn bounded.
The weak formulation of the equation is as follows:
(1)| | a D uD =
(1) f D , C0 () (2)
|,| |m
| |m
And the uniformly ellipticity holds. That is:
2 , x , cfw_
Lecture notes on Green function on a Remannian Manifold
Nov. 26th
Let (M , g ) be a compact Riemannian manifold possibly with boundary. M M with M, M oriented.
We dene the following dieretial operator on the Riemannian manifold:
L = g + aI
Here a L (M ) a
Bochner Technique:
Most of this heavily references Peter Petersen's Riemannian Geometry book. [Left to put in: Proof of
Killing's Equation, Relationship of Lie algebra of Killing elds to Lie algebra of the isometry group of M ]
A vector eld X is Killing i
Notes on Maximal Principles for Second Order
Equations and Greens function
November 17, 2013
Contents
1 Maximal Principal
2
2 Greens function
4
1
1
Maximal Principal
We consider the general form of elliptic operator of the form
n
Lu =
aij Di Dj +
i,j =1
b
MATH 742 THE MOSER-HARNACK INEQUALITY
BO TIAN
This note is typed for Math 742 Geometric Analysis during the fall semester of 2013 at
UMD and is based on Josts Pde book.
.
1. Some Preliminary Results
In this note, we will present a proof of Moser-Harnack i
MATH 742 HEAT EQUATION AND KERNEL
BO TIAN
This note is typed for Math 742 Geometric Analysis during the fall semester of
2013 at UMD and is based on Simons Stanford PDE notes and Evans PDE book.
.
1. Second-Order Parabolic Equations
Second-order parabolic
Geometric Analysis: Lectures 1-4
1
Basics of Dierentiable Manifolds
A dierentiable manifold M is a topological space which is locally homeomorphic to an open subset of
Rn and whose transition functions are smooth maps between these open subsets. Using the
Math 742: Geometric Analysis
Lecture 5 and 6 Notes
Jacky Chong
[email protected]
The following notes are based upon Professor Yanir Rubensteins lectures with reference to
Variational Methods 4th edition by Struwe and A Users Guide to Optimal Transport