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

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,
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