2.2. LAPLACE’S EQUATION 27

a. Strong maximum principle, uniqueness. We begin with the asser-

tion that a harmonic function must attain its maximum on the boundary

and cannot attain its maximum in the interior of a connected region unless

it is constant.

THEOREM 4 (Strong maximum principle). Suppose u ∈

C2(U)

∩ C(

¯

U )

is harmonic within U.

(i) Then

max

¯

U

u = max

∂U

u.

(ii) Furthermore, if U is connected and there exists a point x0 ∈ U such

that

u(x0) = max

¯

U

u,

then

u is constant within U.

Assertion (i) is the maximum principle for Laplace’s equation and (ii) is

the strong maximum principle. Replacing u by −u, we recover also similar

assertions with “min” replacing “max”.

Proof. Suppose there exists a point x0 ∈ U with u(x0) = M := max

¯

U

u.

Then for 0 r dist(x0,∂U), the mean-value property asserts

M = u(x0) = −

B(x0,r)

u dy ≤ M.

As equality holds only if u ≡ M within B(x0,r), we see u(y) = M for all

y ∈ B(x0,r). Hence the set {x ∈ U | u(x) = M} is both open and relatively

closed in U and thus equals U if U is connected. This proves assertion (ii),

from which (i) follows.

Positivity. The strong maximum principle asserts in particular that if U

is connected and u ∈

C2(U)

∩ C(

¯

U ) satisﬁes

Δu = 0 in U

u = g on ∂U,

where g ≥ 0, then u is positive everywhere in U if g is positive somewhere

on ∂U.

An important application of the maximum principle is establishing the

uniqueness of solutions to certain boundary-value problems for Poisson’s

equation.