2.5. PROBLEMS 85
1. Write down an explicit formula for a function u solving the initial-
value problem
ut + b · Du + cu = 0 in Rn × (0, ∞)
u = g on
Rn
× {t = 0}.
Here c ∈ R and b ∈
Rn
are constants.
2. Prove that Laplace’s equation Δu = 0 is rotation invariant; that is, if
O is an orthogonal n × n matrix and we define
v(x) := u(Ox) (x ∈
Rn),
then Δv = 0.
3. Modify the proof of the mean-value formulas to show for n ≥ 3 that
u(0) = −
∂B(0,r)
g dS +
1
n(n − 2)α(n)
B(0,r)
1
|x|n−2
−
1
rn−2
f dx,
provided
−Δu = f in
B0(0,r)
u = g on ∂B(0,r).
4. Give a direct proof that if u ∈
C2(U)
∩ C(
¯
U ) is harmonic within a
bounded open set U, then
max
¯
U
u = max
∂U
u.
(Hint: Define uε := u +
ε|x|2
for ε 0, and show uε cannot attain its
maximum over
¯
U at an interior point.)
5. We say v ∈
C2(
¯
U ) is subharmonic if
−Δv ≤ 0 in U.
(a) Prove for subharmonic v that
v(x) ≤ −
B(x,r)
v dy for all B(x, r) ⊂ U.
(b) Prove that therefore max
¯
U
v = max∂U v.
(c) Let φ : R → R be smooth and convex. Assume u is harmonic
and v := φ(u). Prove v is subharmonic.
(d) Prove v :=
|Du|2
is subharmonic, whenever u is harmonic.
