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.