56 2 The Dirichlet Problem
|h(x) − h(y)| =
1
ν
n
ρ
n
B
x,ρ
h(z) dz −
1
ν
n
ρ
n
B
y,ρ
h(z) dz
≤
m
ν
n
ρ
n
B
x,δ
B
y,ρ
dz.
Since the volume of B
x,ρ
B
y,ρ
can be made arbitrarily small by taking |x−y|
small, given >0, there is a δ>0 such that vol(B
x,ρ
B
y,ρ
) < (ν
n
ρ
n
/m)
whenever |x−y| <δ.Thus,|h(x)−h(y)| <whenever h ∈F, |x−y| <δ.This
shows that the family F is uniformly equicontinuous on K and, in particular,
equicontinuous on K. Consider now any net {h
α
; α ∈A}in F. Since the net
{h
α
; α ∈A}is uniformly bounded and equicontinuous on K, there is a subnet
that converges uniformly on K by the Arzel`a-Ascoli theorem, Theorem 0.2.4.
Now let {h
α
; α ∈A}be a convergent net of uniformly bounded harmonic
functions on Ω and let h = lim
A
h
α
.LetU be an open subset of Ω with
compact closure U
−
⊂ Ω. Then a subnet {h
α
β
} converges uniformly on U
−
to h. By Lemma 2.2.3, h is harmonic on U.SinceU is arbitrary, h is harmonic
on Ω.
Theorem 2.2.5 If {h
k
} is a monotone increasing (decreasing) sequence of
harmonic functions on an open connected set Ω ⊂ R
n
,thenh = lim
k→∞
h
k
is either identically +∞(−∞) or harmonic on Ω.
Proof: Suppose there is a point x ∈ Ω such that h(x) = lim
k→∞
h
k
(x) <
+∞.LetU be any open set containing x with compact closure U
−
⊂ Ω.
For each k ≥ 1, the nonnegative harmonic function h
k
− h
1
is either strictly
positive or identically zero on Ω by the minimum principle, Theorem 1.5.10.
By Harnack’s Inequality, Theorem 2.2.2, there is a positive constant M such
that
h
k
(y) − h
1
(y) ≤ M(h
k
(x) − h
1
(x)) ≤ M (h(x) − h
1
(x))
for all y ∈ U
−
provided h
k
− h
1
is strictly positive; since this inequality is
trivially true if h
k
−h
1
is identically zero on Ω,itholdsforallk ≥ 1andall
y ∈ U
−
. This implies that the sequence {h
k
} is uniformly bounded on U.By
Theorem 2.2.4, h is harmonic on U and therefore on Ω.
Henceforth, “function” will mean “extended real-valued function.” Har-
monic functions are always real-valued. For each of the following lemmas
stated for l.s.c. (u.s.c.) functions, there is a corresponding result for u.s.c.
(l.s.c.) functions.
Definition 2.2.6 The family F of functions defined on Ω is right-directed
(left-directed)ifforeachpairu, v ∈Fthere is a w ∈Fsuch that u ≤ w
and v ≤ w (w ≤ u and w ≤ v).
Lemma 2.2.7 If H = {h
i
; i ∈ I} is a right-directed (left-directed) fam-
ily of harmonic functions on an open connected set Ω ⊂ R
n
,thenh =
sup
i∈I
h
i
(h =inf
i∈I
h
i
) is either identically +∞(−∞) or harmonic on Ω.