FALL 2010 PDE PRELIMINARY EXAM
Work six of the following eight problems.
1. Let D ⊂ Rd be a bounded domain with smooth boundary ∂D. Assume that the complete,
orthonormal set of eigenfunctions {φn } for the negative Dirichlet Laplacian on D is known,
in other words,
−∆φn = λn φn , in D, n = 1, 2, 3, . . . ,
with φ = 0 on ∂D, and
R
D
φ2n dx = 1.
a. Given f ∈ L2 (D), find a closed-form series solution u(x, t) for the initial boundary
value problem for the heat equation
ut − ∆u = 0, for t > 0, x ∈ D,
u(x, t) = 0, x ∈ ∂D,
u(x, 0) = f x ∈ D.
b. Show that there exists a constant C such that |
R
D
u(x, t)φn (x)dx| ≤ Ce−λn t , for all
n.
c. Give an example (the d=1 case will suffice) which shows that the heat equation does
not exhibit finite propagation speed.
2. For each n ∈ N, consider the Cauchy problem
−∆un = 0,
un (x) =
in U ,
1
sin(nx), on {(x, y) : y = 0},
n2
∂un
1
= ,
∂y
n
on {(x, y) : y = 0},
where U = {(x, y) : 0 < y < 1, x ∈ R}. Find a sequence {un } of solutions to these
problems and prove that {un } does not tend to zero. Explain why this implies that the
Cauchy problem above is not “well posed”.
3.
a. Let G ⊂ Rn be an open set. Define what is meant by a distribution on G. Let
f ∈ L2 (G) be a real valued function. Show how f may be identified with a distribution
Tf .
b. let T be a distribution on G and let α = (α1 , . . . , αn ) be a multiindex. Define the
concept of the α-th distributional derivative of T .
c. Let
r : Rn → R
be defined by


 1 for |x| < 1
r(x) =

 0 for |x| ≥ 1,
where | · | is the Euclidean norm in Rn . Show r may be thought of as a distribution
on Rn and compute its distributional gradient ∇r.
4. Let Ω ∈ Rn be a bounded domain with smooth boundary ∂Ω and unit outward normal η.
Consider the problem
∆u − u = f, on Ω
∂u
= 0, on ∂Ω.
∂η
where f ∈ L2 (Ω) and
R
Ω
f = 0.
a. Prove that for each real > 0, there exists a unique weak solution u ∈ H 1 (Ω).
b. In the case of = 0, find an additional condition on u which guarantees the existence
of a unique solution u0 , such that the solutions u from part (a) converge to u0 as
→ 0+ . Prove existence and uniqueness of u0 (with the extra condition), and prove
that u → u0 in H 1 (Ω) as → 0+ .
5. Consider the first order equation
ut + tan(u)ux = 0, x ∈ R, t ≥ 0,
u(x, 0) = φ(x), x ∈ R.
a. Find a solution to this problem where φ(x) = arctan(x).
b. For


 0 for x ≤ 0
φ(x) =
 π for x > 0,

4
find two weak solutions to the above problem. Prove that one of them is a weak
solution.
6. Let Ω ⊂ Rn be a bounded smooth domain, 0 < T < ∞ be a constant, c ∈ C ∞ (Ω̄) such
that c(x) ≤ 0 for all x ∈ Ω and ψ ∈ C ∞ (Ω̄ × [0, T ]). Let u ∈ C 2,1 (Ω̄ × [0, T ]) be a solution
to
ut = ∆u + c(x)u,
u(x, t) = ψ(x, t),
if (x, t) ∈ Ω × [0, T ],
if (x, t) ∈ Ω × {0}
or if (x, t) ∈ ∂Ω × [0, T ].
a. Prove the Maximum Principle: If ψ(x, t) ≥ 0 then the solution satisfies u(x, t) ≥ 0.
b. Show that solutions to the above initial-boundary value problem are unique.
7. Use Duhamel’s principle to find an explicit solution of
utt (x, t) = uxx (x, t) + sin(x − t), x ∈ R, t > 0,
u(x, 0) = 0, ut (x, 0) = 0.
8. Consider the Dirichlet problem
∆u = f,
u(x) = 0,
in Ω,
on ∂Ω,
where Ω ⊂ Rn is a bounded domain and f ∈ L2 (Ω).
a. Define the concept of a weak H01 (Ω) solution of the above problem.
b. Use the Poincaré inequality:
Z
2
Z
u dx ≤ c
Ω
|∇u|2 dx
for all u ∈ H01 (Ω).
Ω
to prove the existence of a weak solution in H01 (Ω)