2.2. LAPLACE’S EQUATION 31

Proof. Since Φ(x) → 0 as |x| → ∞ for n ≥ 3, ˜(x) u :=

Rn

Φ(x − y)f(y) dy

is a bounded solution of −Δu = f in

Rn.

If u is another solution, w := u− ˜ u

is constant, according to Liouville’s Theorem.

Remark. If n = 2, Φ(x) = −

1

2π

log |x| is unbounded as |x| → ∞ and so

may be

R2

Φ(x − y)f(y) dy.

e. Analyticity. We reﬁne Theorem 6:

THEOREM 10 (Analyticity). Assume u is harmonic in U. Then u is

analytic in U.

Proof. 1. Fix any point x0 ∈ U. We must show u can be represented by a

convergent power series in some neighborhood of x0.

Let r :=

1

4

dist(x0,∂U). Then M :=

1

α(n)rn

u

L1(B(x0,2r))

∞.

2. Since B(x, r) ⊂ B(x0, 2r) ⊂ U for each x ∈ B(x0,r), Theorem 7

provides the bound

Dαu

L∞(B(x0,r))

≤ M

2n+1n

r

|α|

|α||α|.

Now

kk

k!

ek

for all positive integers k, and hence

|α||α|

≤

e|α||α|!

for all multiindices α. Furthermore, the Multinomial Theorem (§1.5) implies

nk

= (1 + · · · +

1)k

=

|α|=k

|α|!

α!

,

whence

|α|! ≤

n|α|α!.

Combining the previous inequalities yields the estimate

(22)

Dαu

L∞(B(x0,r))

≤ M

2n+1n2e

r

|α|

α!.

3. The Taylor series for u at x0 is

α

Dαu(x0)

α!

(x −

x0)α,