Nov. 05, 2010
Math 527 A1 Fall 2010
NAME: _
Quiz 4
1 x<0
. Prove using deniProblem 1. Consider the Burgers equation with initial data u0(x) =
0 x>0
1 x < t/2
tion that u(x, t) =
is a weak solution. Sh
M ath 5 2 7 Fall 2 0 0 9 L ecture 2 2 ( N ov. 2 5 , 2 0 0 9 )
S econdOrder Parabolic Equations
In this lecture we study the initial/ boundaryvalue
ut + L u = f
u = 0
u = g
n
Lu =
in UT
on U [ 0, T
M ath 5 2 7 Fall 2 0 0 9 L ecture 2 1 ( N ov. 2 3 , 2 0 0 9 )
S econdOrder Elliptic Equations: Eigenvalues and Eigenfunctions
In this lecture we study the boundaryvalue problem
in U;
Lw = w
on U.
w=
M ath 5 2 7 Fall 2 0 0 9 L ecture 2 0 ( N ov. 1 8 )
S econdOrder Elliptic Equations: Maximum P rinciples
Recall that, for Poisson equation, we have
Weak maximum principle.
and
u
max U in U
(1)
U
Stro
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 4 ( O ct. 2 6 , 2 0 0 9 )
S imilarity S olutions
In previous lectures, we have derived solution formulas for quite a few linear and nonlinear PDEs. We
have also see
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 2 ( O ct. 1 9 , 2 0 0 9 )
Discontinuous S olutions of C onservation L aws
In this lecture we study 1 D conservation laws:
un
,
f ( u) =
f1 ( u)
fn ( u)
.
Such a sy
M ath 5 2 7 Fall 2 0 09 L ecture 7 ( S ep. 2 8, 2 00 9)
Wave Equations: Explicit Formulas
In this lecture we derive the representation formulas for the wave equation in the whole space:
u ut t
Rn ( 0
M ath 5 2 7 Fall 2 0 0 9 L ecture 4 ( S ep. 1 6 , 2 0 0 9 )
P roperties and Estimates of L aplace s and P oisson s Equations
In our last lecture we derived the formulas for the solutions of Poisson s
M ath 5 2 7 Fall 2 0 0 9 L ecture 5 ( S ep. 2 1 , 2 0 0 9 )
Heat Equation: Explicit Formulas
( F irst 2 5 minutes: Q uiz 1 )
We now turn to the heat equation
ut
where
in UT ;
u = f,
on UT
u= g
(1)
U
M ath 5 2 7 Fall 2 0 0 9 L ecture 3 ( S ep. 1 4 , 2 0 0 9 )
L aplace s Equation: Explicit Formulas
In the following two lectures, we will consider the Laplace s equation
u=0
(1)
u = f.
( 2)
and Poisso
M ath 5 2 7 Fall 2 0 0 9 L ecture 6 ( S ep. 2 3 , 2 0 0 9 )
Heat Equation: M aximum P rinciples and Energy Method
We continue the discussion of the heat equation
where
UT
with U R .
n
in UT ;
u= f
UT
M ath 5 2 7 Fall 2 0 0 9 L ecture 8 ( S ep. 3 0 , 2 0 0 9 )
Wave Equations: Uniqueness and Asymptotics
In this lecture we prove the uniqueness for the wave equations. We also prove some asymptotic dec
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 1 ( O ct. 1 4 , 2 0 0 9 )
HamiltonJ acobi Equation: Weak S olution
We continue the study of the HamiltonJacobi equation:
u t + H( D u) = 0
We have shown that
Rn (
M ath 5 2 7 Fall 2 0 0 9 L ecture 9 ( O c t. 5 , 2 0 0 9 )
Method of C haracteristics
In this lecture we try to solve the rst order equation
F( D u , u , x) = 0
in U;
u= g
on U.
(1)
Recall that, when
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 0 ( O ct. 7 , 2 0 0 9 )
HamiltonJ acobi Equation: Explicit Formulas
In this lecture we try to apply the method of characteristics to the HamiltonJacobi equation:
M ath 5 2 7 Fall 2 0 0 9 L ecture 2 3 ( N ov. 3 0 , 2 0 0 9 )
S econdOrder Hyperbolic Equations
We consider the initial/ boundaryvalue problem
u t t + L u =
u =
u =
ut =
f
0
g
h
in UT
on U [ 0, T]
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 9 ( N ov. 1 6 , 2 0 0 9 )
S econdOrder Elliptic Equations: Weak S olutions
1 . Denitions.
In this and the following two lectures we will study the boundary value p
O ct. 1 0
Viscosity S olutions
In this lecture we take a glimpse of the viscosity solution theory for linear and nonlinear PDEs. From our
experience we know that even for linear equations, the existen
Math 527 A1 Homework 2 (Due Oct. 8 in Class)
Exercise 1. (6 pts) Prove the mean value formula for harmonic functions using Poissons formula for the ball (see
Evans 2.2.4c for the formula).
Proof. The
Oct. 22, 2010
Math 527 A1 Fall 2010
NAME: _
Quiz 3
Problem 1. Consider the HamiltonJacobi equation in 1D
ut +
u2
x
= 0,
2
u(x, 0) = g(x) =
x x<0
.
x
x>0
(1)
a) (9 pts) Solve the problem using HopfLa
( Nov. 3, 3pm 4: 30pm, CAB457)
Math 5 2 7 ( 2 008) Midterm
f( x 1 , x 2 ) =
R we dene a function f: R2
1 x 2 > g( x 1 )
.
0 x 2 < g( x 1 )
Problem 1 . For any bounded, continuous function g: R
R by
(1
Math 527 A1 Homework 6 (Due Dec. 8 in Class)
Exercise 1. (10 pts) (5.10.9) Integrate by parts to prove the interpolation inequality
Du2 dx
u2 dx
C
U
1/2
1/2
2
D 2u dx
U
(1)
U
1
for all u Cc (U ). As
Math 527 A1 Homework 4 (Due Nov. 5 in Class)
Exercise 1. (6 pts) Let u be a weak solution of the scalar conservation law. Show that if u C 1()
for some domain , then it is a classical solution in , th
Math 527 A1 Homework 5 (Due Nov. 26 in Class)
Exercise 1. (15 pts) (Evans 4.7.7) Consider the viscous conservation law
ut + F (u)x a ux x = 0
in R (0, )
(1)
where a > 0 and F is uniformly convex.
i. (
Math 527 B1 Homework 1 (Due Sep. 24 in Class)
Sep. 17, 2010
Exercise 1. (5 pts) (1.5.5) Assume that f : Rn
f (x) =

k
R is smooth. Prove
1
D f (0) x + O xk+1
!
as x 0
for each k = 1, 2, . This is
Math 527 A1 Homework 3 (Due Oct. 22 in Class)
Exercise 1. (4 pts) (Evans 3.5.5 c) Solve using characteristics:
u ux1 + ux2 = 1,
u(x1, x1) =
1
x .
2 1
(1)
Solution. Using the method of characteristics,
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 7 ( N ov. 2 , 2 0 0 9 )
Extensions and Traces
1 . Extensions.
In this lecture we rst consider the problem of extending u W k , p( U) to u W k , p( Rn ) . The motiva
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 6 ( N ov. 2 , 2 0 0 9 )
S obolev S paces: Definitions and B asic P roperties
1 . Motivation.
The invention and development of Sobolev spaces are motivated by the st
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 5 ( O ct. 2 8 , 2 0 0 9 )
Asymptotics
1 . Introduction.
Asymptotics studies the behavior of a function at/ near a given point. The simplest asymptotics is the
Taylo
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 3 ( O ct. 2 1 , 2 0 0 9 )
The S ingle C onservation L aw: Existence, Uniqueness, Asymptotics
In the last lecture we see that an appropriate notion of solutions to t