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.