2.2. LAPLACE’S EQUATION 41

for r 0. Then ˜(x) u = u(rx) solves (42), with ˜(x) g = g(rx) replacing g.

We change variables to obtain Poisson’s formula

(45) u(x) =

r2

−

|x|2

nα(n)r

∂B(0,r)

g(y)

|x − y|n

dS(y) (x ∈

B0(0,r)).

The function

K(x, y) :=

r2

−

|x|2

nα(n)r

1

|x − y|n

(x ∈

B0(0,r),

y ∈ ∂B(0,r))

is Poisson’s kernel for the ball B(0,r).

We have established (45) under the assumption that a smooth solution

of (44) exists. We next assert that this formula in fact gives a solution:

THEOREM 15 (Poisson’s formula for ball). Assume g ∈ C(∂B(0,r)) and

deﬁne u by (45). Then

(i) u ∈

C∞(B0(0,r)),

(ii) Δu = 0 in

B0(0,r),

and

(iii) lim

x→x0

x∈B0(0,r)

u(x) =

g(x0)

for each point

x0

∈ ∂B(0,r).

The proof is similar to that for Theorem 14 and is left as an exercise.

2.2.5. Energy methods.

Most of our analysis of harmonic functions thus far has depended upon

fairly explicit representation formulas entailing the fundamental solution,

Green’s functions, etc. In this concluding subsection we illustrate some

“energy” methods, which is to say techniques involving the

L2-norms

of

various expressions. These ideas foreshadow later theoretical developments

in Parts II and III.

a. Uniqueness. Consider ﬁrst the boundary-value problem

(46)

−Δu = f in U

u = g on ∂U.

We have already employed the maximum principle in §2.2.3 to show

uniqueness, but now we set forth a simple alternative proof. Assume U is

open, bounded, and ∂U is

C1.