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.
