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.