FALL 2013
MATH 6326
PRACTICE EXAM
n
1. Let Ba (0) be the unit ball in Rn centered at 0 with radius a > 0.
(a) Let > 0. Show that
n
Ba (0)
1
dx <
|x|
if and only if < n. In particular, evaluate the integral for n > > 0.
(b) Give conditions on for which
n
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The Variational Formulation
(Poisson Equation)
Lecture 09
September 24, 2013
Lecture 09
The Variational Formulation
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Variational Formulation
Here we discuss how PDEs can be reformulated in terms of
variational problems. For this rec
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Heat Equation
Lecture 10
September 26, 2013
Lecture 10
Heat Equation
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Heat Equation
The Heat Equation is
u
+
t
u=0
First, we consider an IVP:
u
+ u = 0,
t > 0, x Rn
t
u (0, x ) = f (x ), x Rn
Recall, in the case of Laplace equation
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Regularity, Self-Similar Behavior
Heat Equation
Lecture 13
October 08, 2013
Lecture 13
Regularity, Self-Similar Behavior
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Regularity
We now discuss how regular solutions of the heat equation are.
For simplicity, we consider only sol
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The Maximum Principle, Uniqueness
Heat Equation
Lecture 12
October 03, 2013
Lecture 12
The Maximum Principle, Uniqueness
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The Maximum Principle for Heat Equation
Suppose Rn be a bounded domain and T > 0. Dene
T = cfw_(t , x ) : x ,
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Greens Function for Half Space, Poisson Kernel
(Poisson Equation)
Lecture 08
September 19, 2013
Lecture 08
Greens Function for Half Space, Poisson Kernel
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Reciprocity of the Greens Function
Theorem
For all x , y , x = y we have
G (x
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Greens Identities, Greens Function
(Poisson Equation)
Lecture 07
September 17, 2013
Lecture 07
Greens Identities, Greens Function
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Greens Identities
Theorem (Greens identities)
Suppose u , v C 2 () in a open, bounded, smooth set . T
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Preliminaries
Lecture 02
August 29, 2013
Lecture 02
Preliminaries
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Compactness
A subset F of a metric space X is precompact if the closure F of F
is compact, or equivalently, if every sequence in F has a
subsequence that converges i
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Some Inportant PDEs
Lecture 03
September 03, 2013
Lecture 03
Some Inportant PDEs
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Elementary PDE
Let u (t , x ) where t 0 and x Rn play the role of time and
space, respectively.
Consider
u
=0
t
Initial conditions are prescribed at t
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The Fundamental Solution
Laplace Equation
Lecture 04
September 05, 2013
Lecture 04
The Fundamental Solution
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Laplace and Poisson Equations
Laplace Equation
=0
(1)
Poisson Equation
= f (x )
where the Laplacian operator is
=
2
2
+ .
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Regularity of harmonic functions
(Laplace Equation)
Lecture 06
September 12, 2013
Lecture 06
Regularity of harmonic functions
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Estimates on Derivatives of Harmonic Functions
As before, let Rn be an open set, and Br (x0 ) a ball of
r
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The Mean-Value Property
Laplace Equation
Lecture 05
September 10, 2013
Lecture 05
The Mean-Value Property
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The mean value property
Example
Any harmonic function in 1-dim is linear u(x ) = ax + b .
Then, for any x R and any > 0:
u(x
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Wave Equation, Conservation of the Energy
1-dim case
Lecture 14
October 10, 2013
Lecture 14
Wave Equation, Conservation of the Energy
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1-dim Wave Equation
The wave equation describes propagation of a disturbance u (t , x )
that move
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Finite Speed of Propagation
(Wave Equation)
Lecture 15
October 15, 2013
Lecture 15
Finite Speed of Propagation
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Consider the case of c = const:
1 2u
u = 0,
t 0, x Rn
c 2 t 2
u (0, x ) = p (x ), x Rn
ut (0, x ) = q (x ), x Rn
(1)
Fi
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Entropy Solution and Uniqueness
of Solution to Burgers Equation
Introduction to 1-dim Conservation Laws
Lecture 24
November 19, 2013
Lecture 24
Entropy Solution and Uniqueness of Solution to Burgers Equatio
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Entropy Solution of Solu
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Weak Derivatives
Lecture 26
November 26, 2013
Lecture 26
Weak Derivatives
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Weak Derivatives
Suppose Rn is an open set.
Denition
A function f L1 () is weakly dierentiable w.r.t. xi if there exists a function
loc
gi L1 () s.t.
loc
f
d
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Properties of Weak Derivatives, Sobolev Spaces
Lecture 28
December 05, 2013
Lecture 28
Properties of Weak Derivatives, Sobolev Spaces
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Properties of Weak Derivatives
Let Rn be an open set.
Lemma
If f L1 () has weak partial derivativ
Distributions
Lecture 27
December 03, 2013
Lecture 27
Distributions
Distributions,
Let Rn be an open set.
Denition
A sequence cfw_n nN of functions n Cc () converges
to Cc () in the sense of test functions if:
1
there exists s.t. supp n for every n N;
2
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Lax-Oleinik Formula
for Solution of Burgers Equation
Introduction to 1-dim Conservation Laws
Lecture 23
November 14, 2013
Lecture 23
Lax-Oleinik Formula for Solution of Burgers Equation
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Lax-Oleinik Formula for Burgers eq.
Consider
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Solving the Wave Equation
Using Spherical Averages (n = 2, 3)
Lecture 17
October 22, 2013
Lecture 17
Solving the Wave Equation Using Spherical Averages (n = 2, 3)
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Method of Spherical Averages
Suppose C m ([0, ) Rn ) with m 2 solves
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Reection method, Spherical averages
(Wave Equation)
Lecture 16
October 17, 2013
Lecture 16
Reection method, Spherical averages
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1-dim Wave Equation
Recall that the solution to 1-dim Cauchy problem with c = const:
is given by
1 2 2
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Nonhomogeneous Problem for Wave Equation
(Wave Equation)
Lecture 18
October 29, 2013
Lecture 18
Nonhomogeneous Problem for Wave Equation
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Nonhomogeneous Problem
For n = 2, 3 consider the Cauchy problem (with c = const):
1
n
2 utt u
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Method of Characteristics
(Nonlinear) 1st Order PDEs
Lecture 19
October 31, 2013
Lecture 19
Method of Characteristics
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Linear Equations
Consider a linear 1st order equation
a(x )
u = c (x )u + f (x ),
x Rn
(1)
where a(x ) = (a1 (x
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Integral Solution, Rankine-Hugoniot Condition
Introduction to 1-dim Conservation Laws
Lecture 21
November 07, 2013
Lecture 21
Integral Solution, Rankine-Hugoniot Condition
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Linear Equations
Consider the IVP for the 1-dim scalar cons
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Method of Characteristics for Fully Nonlinear Case
Nonlinear 1st Order PDEs
Lecture 20
November 05, 2013
Lecture 20
Method of Characteristics for Fully Nonlinear Case
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Nonlinear Equations
Consider a linear 1st order nonlinear equati