MATH 644, FALL 2011, HOMEWORK 1
Exercise 1. (Duhamels principle) Consider the dierential operator L =
c R are constants. Suppose that the solution to:
c Dx
on Rn . Here,
t u + L(u) = 0, on Rn Rt
x
u|t=0 = , on Rn
x
(1)
is given by u(x, t) = S (t)(x).
Sho
MATH 644, FALL 2011, HOMEWORK 3
Exercise 1. (Weyls lemma) [10 points, or 15 points (5 points extra credit)] In this exercise, we
outline the proof of Weyls lemma, which is a generalization of the Theorem we proved in class that
states that all (C 2 ) harm
MATH 644, FALL 2011, HOMEWORK 6
Exercise 1. (Reection of traveling waves) [5 points] Solve the equation:
(1)
utt c2 uxx = 0 on (0, L) (0, +)
u = g, ut = 0 on (0, L) cfw_t = 0
u = 0 on cfw_0 (0, +) cfw_L (0, +) .
by converting it to a problem on R.
Exercis
MATH 644, FALL 2011, HOMEWORK 5
Exercise 1. (An alternative derivation of the heat kernel in one dimension) [Evans, Problem 11
in Chapter 2; 5 points]
2
Assume n = 1 and u = v ( xt ).
a) Show that ut = uxx if and only if:
(1)
4zv (z ) + (2 + z )v (z ) = 0
MATH 644, FALL 2011, HOMEWORK 8
Exercise 1. (The adjoint of S (t) [5 points]
Let S (t) denote the linear Schrdinger propagator dened in class. Show that:
o
(S (t) = S (t)
where denotes the adjoint with respect to the L2 (Rn ) inner product.
Exercise 2. (S
MATH 644, FALL 2011, HOMEWORK 2
Exercise 1. (A mean value formula for the Poisson equation on B (0, r) [Evans, Problem 3 in
Chapter 2] Suppose that r > 0, and that n 3. Consider B (0, r) Rn and
u C 2 (B (0, r) C (B (0, r)
such that:
u = f, on B (0, r)
u =
MATH 644, FALL 2011, HOMEWORK 7
Exercise 1. (An interpolation inequality) [5 points]
Suppose that 1 p, q and suppose that [0, 1] is given. Dene r by:
1
1
:= +
r
p
q
Show that:
f Lr f p f 1 .
L
Lq
Exercise 2. (More on the Hausdor-Young Inequality) [15 poi
MATH 644, FALL 2011, HOMEWORK 4
Exercise 1. (An explicit form of Harnacks inequality; Evans, Problem 6 from Chapter 2)
a) Use Poissons formula for the ball to prove the inequality:
rn2
r | x|
r + |x|
u(0) u(x) rn2
u(0)
(r + |x|)n1
(r |x|)n1
whenever u is