30 2. FOUR IMPORTANT LINEAR PDE
be a multiindex with |α| = k. Then
Dαu
=
(Dβu)xi
for some i ∈ {1, . . . , n},
|β| = k − 1. By calculations similar to those in (20), we establish that
|Dαu(x0)|
≤
nk
r
Dβu
L∞(∂B(x0,
r
k
))
.
If x ∈ ∂B(x0,
r
k
), then B(x,
k−1
k
r) ⊂ B(x0,r) ⊂ U. Thus (18), (19) for
k − 1 imply
|Dβu(x)|
≤
(2n+1n(k
−
1))k−1
α(n)
(
k−1
k
r
)n+k−1
u
L1(B(x0,r))
.
Combining the two previous estimates yields the bound
(21)
|Dαu(x0)|
≤
(2n+1nk)k
α(n)rn+k
u L1(B(x0,r)).
This confirms (18), (19) for |α| = k.
d. Liouville’s Theorem. We assert now that there are no nontrivial
bounded harmonic functions on all of
Rn.
THEOREM 8 (Liouville’s Theorem). Suppose u :
Rn
→ R is harmonic
and bounded. Then u is constant.
Proof. Fix x0 ∈
Rn,
r 0, and apply Theorem 7 on B(x0,r):
|Du(x0)| ≤
√
nC1
rn+1
u
L1(B(x0,r))
≤
√
nC1α(n)
r
u
L∞(Rn)
→ 0,
as r → ∞. Thus Du ≡ 0, and so u is constant.
THEOREM 9 (Representation formula). Let f ∈ Cc
2(Rn),
n ≥ 3. Then
any bounded solution of
−Δu = f in
Rn
has the form
u(x) =
Rn
Φ(x − y)f(y) dy + C (x ∈
Rn)
for some constant C.
