26 2. FOUR IMPORTANT LINEAR PDE
and consequently, using Green’s formulas from §C.2, we compute
φ (r) =
∂B(x,r)
Du(y) ·
y x
r
dS(y)
=
∂B(x,r)
∂u
∂ν
dS(y)
=
r
n

B(x,r)
Δu(y) dy = 0.
Hence φ is constant, and so
φ(r) = lim
t→0
φ(t) = lim
t→0

∂B(x,t)
u(y) dS(y) = u(x).
2. Observe next that our employing polar coordinates, as in §C.3, gives
B(x,r)
u dy =
r
0 ∂B(x,s)
u dS ds
= u(x)
r
0
nα(n)sn−1ds
=
α(n)rnu(x).
THEOREM 3 (Converse to mean-value property). If u
C2(U)
satisfies
u(x) =
∂B(x,r)
u dS
for each ball B(x, r) U, then u is harmonic.
Proof. If Δu 0, there exists some ball B(x, r) U such that, say, Δu 0
within B(x, r). But then for φ as above,
0 = φ (r) =
r
n

B(x,r)
Δu(y) dy 0,
a contradiction.
2.2.3. Properties of harmonic functions.
We now present a sequence of interesting deductions about harmonic
functions, all based upon the mean-value formulas. Assume for the following
that U
Rn
is open and bounded.
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