2.5. PROBLEMS 87
(b) Now assume only that u ∈
C2(U +)
∩ C(U +). Show that v is
harmonic within U. (Hint: Use Poisson’s formula for the ball.)
11. (Kelvin transform for Laplace’s equation) The Kelvin transform Ku =
¯ u of a function u :
Rn
→ R is
¯(x) u :=
u(¯)|¯|n−2
x x =
u(x/|x|2)|x|2−n
(x = 0),
where ¯ x =
x/|x|2.
Show that if u is harmonic, then so is ¯. u
(Hint: First show that
Dx¯(Dx¯)T
x x =
|¯|4I.
x The mapping x → ¯ x is
conformal, meaning angle preserving.)
12. Suppose u is smooth and solves ut − Δu = 0 in
Rn
× (0, ∞).
(a) Show uλ(x, t) := u(λx,
λ2t)
also solves the heat equation for
each λ ∈ R.
(b) Use (a) to show v(x, t) := x·Du(x, t)+2tut(x, t) solves the heat
equation as well.
13. Assume n = 1 and u(x, t) = v(
x
√
t
).
(a) Show
ut = uxx
if and only if
(∗) v +
z
2
v = 0.
Show that the general solution of (∗) is
v(z) = c
z
0
e−s2/4
ds + d.
(b) Differentiate u(x, t) = v(
x
√
t
) with respect to x and select the
constant c properly, to obtain the fundamental solution Φ for
n = 1. Explain why this procedure produces the fundamental
solution. (Hint: What is the initial condition for u?)
14. Write down an explicit formula for a solution of
ut − Δu + cu = f in
Rn
× (0, ∞)
u = g on
Rn
× {t = 0},
where c ∈ R.
15. Given g : [0, ∞) → R, with g(0) = 0, derive the formula
u(x, t) =
x
√
4π
t
0
1
(t − s)3/2
e
−x
2
4(t−s)
g(s) ds
