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.
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