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 := u +
ε|x|2
for ε 0, and show 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.
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