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