Math 126 - Fall 2014 - Homework 5 Hard copy due:

Math 126 - Fall 2014 - Homework 5
Hard copy due: Wednesday 10/15/2014
at 4:10pm
Problem 1. Let Ω ⊂ Rd be an open, bounded
smooth solution to

 ut − ∆u = 0
u(0, x) = f (x)

∇u(t, x) · n(x) = 0
Show that
Z
set with smooth boundary ∂Ω. Suppose u is a
(t, x) ∈ (0, ∞) × Ω,
x∈Ω
x ∈ ∂Ω.
Z
u(t, x) dx ≡
Ω
f (x) dx.
Ω
Problem 2. Let Ω ⊂ Rd be an open, bounded set with smooth boundary. Consider the eigenvalue
problem
−∆q(x) = λq(x),
q:Ω→C
with either Dirichlet or Neumann boundary conditions, that is,
q(x) = 0 for x ∈ ∂Ω
∇q(x) · n(x) = 0 for x ∈ ∂Ω.
or
(i) Show that all eigenvalues are real.
(ii) Show that for any eigenvalue we can find a real-valued eigenfunction.
(iii) Suppose that q1 is an eigenfunction with eigenvalue λ1 and q2 is an eigenfunction with eigenvalue
λ2 . Show that if λ1 6= λ2 then q1 and q2 are orthogonal, that is,
Z
q1 (x)q2 (x) dx = 0.
Ω
Problem 3. Consider the eigenvalue problem with homogeneous Dirichlet boundary conditions:
−q 00 (x) = λq(x), x ∈ (0, L)
q(0) = q(L) = 0.
(i) Show that 0 is not an eigenvalue.
(ii) Show that there are no negative eigenvalues.
(iii) Find the eigenvalues and associated eigenfunctions.
Problem 4. Consider the eigenvalue problem with homogeneous Neumann boundary conditions:
−q 00 (x) = λq(x), x ∈ (0, L)
q 0 (0) = q 0 (L) = 0.
(i) Show that there are no negative eigenvalues.
(ii) Find the eigenvalues and associated eigenfunctions.
Problem 5. Suppose that for any smooth F : [−L, L] → C satisfying F (L) = F (−L) we can write
F (x) =
∞
X
cn e
inπ
x
L
for some
cn ∈ C.
n=−∞
(i) Show that if f : (0, L) → R is smooth and satisfies f (0) = f (L) = 0 then we can write
f (x) =
∞
X
an sin( nπ
L x)
for some
an ∈ R.
n=1
(ii) Show that if f : (0, L) → R is smooth and satisfies f 0 (0) = f 0 (L) = 0 then we can write
f (x) =
∞
X
bn cos( nπ
L x)
n=0
1
for some
bn ∈ R.
2
Hints.
To show that a number z is real, it suffices to show z = z.
For problem 2, show that if f and g are a pair of functions satisfying either of the boundary
conditions then
Z
Z
− ∆f (x)g(x) dx = − f (x)∆g(x) dx.
Ω
Ω
For problem 5 you will need to use the identity eiθ = cos θ + i sin θ. Also, follow the hints I gave
in class.