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.