28 2. FOUR IMPORTANT LINEAR PDE
THEOREM 5 (Uniqueness). Let g ∈ C(∂U), f ∈ C(U). Then there exists
at most one solution u ∈
C2(U)
∩ C(
¯
U ) of the boundary-value problem
(17)
−Δu = f in U
u = g on ∂U.
Proof. If u and ˜ u both satisfy (17), apply Theorem 4 to the harmonic
functions w := ±(u − ˜). u
b. Regularity. Next we prove that if u ∈ C2 is harmonic, then necessarily
u ∈
C∞.
Thus harmonic functions are automatically infinitely differentiable.
This sort of assertion is called a regularity theorem. The interesting point
is that the algebraic structure of Laplace’s equation Δu =
∑n
i=1
uxixi = 0
leads to the analytic deduction that all the partial derivatives of u exist,
even those which do not appear in the PDE.
THEOREM 6 (Smoothness). If u ∈ C(U) satisfies the mean-value prop-
erty (16) for each ball B(x, r) ⊂ U, then
u ∈
C∞(U).
Note carefully that u may not be smooth, or even continuous, up to ∂U.
Proof. Let η be a standard mollifier, as described in §C.5, and recall that
η is a radial function. Set
uε
:= ηε ∗ u in Uε = {x ∈ U | dist(x, ∂U) ε}.
As shown in §C.5,
uε
∈
C∞(Uε).
We will prove u is smooth by demonstrating that in fact u ≡
uε
on Uε.
Indeed if x ∈ Uε, then
uε(x)
=
U
ηε(x − y)u(y) dy
=
1
εn
B(x,ε)
η
|x − y|
ε
u(y) dy
=
1
εn
ε
0
η
r
ε
∂B(x,r)
u dS dr
=
1
εn
u(x)
ε
0
η
r
ε
nα(n)rn−1dr
by (16)
= u(x)
B(0,ε)
ηε dy = u(x).
Thus
uε
≡ u in Uε, and so u ∈
C∞(Uε)
for each ε 0.
