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. Show that it is furthermore an entropy solution.
0 x > t/
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]
on U cfw_ t = 0
(1)
U ( 0, T] for some xed T > 0.
Her
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=0
(1)
Such problem is called eigenvalue problem. When i
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
Strong maximum principle ( assuming U is connected)
u
0 in
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 seen that this can only be done for simple equations. Furt
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 system is called conservation laws as it is derived from
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, ) ;
u = 0,
u( x , 0) = g( x) ,
u t ( x , 0) = h( x) .
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 equation through Green s function:
G( x , y)
u( x) =
g(
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)
UT U cfw_ 0 ( U [ 0, T] ) .
UT U [ 0, T) ;
( 2)
with U R
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 Poisson s equation
Here x U, u: U
R, and U R is a given open
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
U [ 0, T) ;
u= g
on UT
U cfw_ 0 ( U [ 0, T] ) ,
ut
(1
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 decay
results.
Recall that to prove uniqueness for the Lap
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 ( 0, ) ;
u= g
Rn cfw_ t = 0 .
(1)
1 . In general we cann
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 F is quasilinear,
F( D u , u, x) = b( u , x) D u f( u
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:
u t + H( D u , x) = 0
u = g
in Rn ( 0, )
on Rn cfw_ t =
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]
on U cfw_ t = 0
on U cfw_ t = 0
(1)
Here L is again the
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 problem
in U;
Lu = f
Here
or
a i j ( x) u x i
i, j
xj
+
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 existence of solutions is not easy to establish,
where the bes
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 Poisson formula reads (p.41)
u(x) =
r 2 x2
n (n) r
g(
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 HopfLax formula.
b) (1 pts) For a xed t > 0, plot the graph o
( 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)
Let Tf D R2 be the distribution corresponding to f.
a
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 ). Assume U is smooth, and prove this inequality if u H 2(U
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 , that is
ut + f (u)x = 0 for (x, t) ,
u(x, 0) = u0 when (x
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. (5 pts) Show u solves (1) if u(x, t) = v(x t) and v is d
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 Taylors formula in multiindex notation.
(Hint: Fix x Rn
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, we have
F (x, z, p) = z p1 + p2 1.
(2)
p2
1
p1 p2
p =
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 motivation for doing so is that in many cases the boundary U
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 study of elliptic PDEs, for
example the Poisson equation
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
Taylor expansion:
f( x) = f( x 0 ) + f ( x 0 ) ( x x 0 ) +
(
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 the single conservation law
u t + f( u) x = 0,
(1)
u( x