Mean Values
for solutions of the heat equation
John McCuan
October 29, 2013
The following notes are intended to address certain problems with the
change of variables and other unclear points (and points simply not covered)
from the lecture.
1
Heat Ball
Th
Bounded Linear Operators
John McCuan
February 27, 2014
Here is a bit more careful treatment of the inequalities/estimates leading
to the conclusion that a second order linear partial dierential operator in
divergence form has associated with it a bounded
Math 6342 A Hamilton-Jacobi Problem
John McCuan
January 9, 2014
Let
u0 (x) =
1 |x|, if |x| 1
0,
otherwise.
Consider the following variational problem: Minimize
b
F [w] =
a
w(t)
2
2
w3 (t) dt + u0 (w(a)
over the admissible class
A = w = (w1 , w2 , w3 ) C
Geometric PDE
and
The Magic of Maximum Principles
Alexandrovs Theorem
John McCuan
December 12, 2013
Outline
1. Youngs PDE
2. Geometric Interpretation
3. Maximum and Comparison Principles
4. Alexandrovs Theorem
1
Variation of Area
G[u] =
1 + |Du|2
(Nonpara
Greens Function for a Domain
John McCuan
September 26, 2013
In view of the importance of football at Georgia Tech, I have decided to
compose the following discussion as a follow-up to Tuesdays lecture. There
were a couple sign errors in my notes. Hopefull
Math 6341, Final Exam
Name and section:
1. (25 points) Solve the initial value problem
(1 + u2 )(ut + ux ) = 1 on
u(x, 0) = u0 (x).
R (0, )
Solution: Dividing both sides of the equation by the positive quantity (1 + u2 ),
the characteristic equation reads
Main existence and uniqueness theorem
John McCuan
April 17, 2014
Given
Lu =
Di (aij Dju) +
bj Dj u + cu
j
with the coecients aij , bj , c bounded and measurable, we are looking for
1
u H0 (U) such that
B(u, v) = u, v
1
for all v H0 (U)
L2
where
B(u, v) =
Math 6342, Exam 1 (practice)
Name and section:
1. (25 points) (Hamilton-Jacobi Equation) Consider the initial value problem
ut + |ux |3 = 0 on (0, )
u(x, 0) = |x|.
(i) Write down the Hopf-Lax formula for a solution of this IVP.
(ii) Evaluate the Hopf-Lax
Math 6342, Exam 2 (practice)
Name and section:
1. (20 points) (Weak/Strong Solutions) Let cfw_aij be a collection of bounded coecients,
1
f L2 (), and u H0 (). Show that if
aij Dj uDi =
f
Cc ()
aij Dj uDi v =
fv
1
v H0 ().
Then
1
Solution: Since H0 is
Math 6342, Final Exam (practice)
Name and section:
The problems on this exam are related to the notes on the main existence and uniqueness
theorem for linear elliptic PDE. These notes are posted at
http:/www.math.gatech.edu/ mccuan/courses/6342/existence.
1
Some comments about the trace in H0
John McCuan
April 30, 2014
Let us imagine that we wish to consider the infemum of the set
1
cfw_M : (u+ M)+ H0 (U)
1
for some function u H0 (U) dened on an open bounded set U. It might
be convenient to note that only
Coercivity and the Poincar inequality
e
John McCuan
April 8, 2014
Coercivity for the bilinear form
B(u, v) =
aij Dj uDi v +
i,j
bj vDj u +
cuv.
j
associated with the linear partial dierential operator
Lu =
Di (aij Dju) +
bj Dj u + cu
j
i,j
is the requir