32 2. FOUR IMPORTANT LINEAR PDE

the sum taken over all multiindices. We assert this power series converges,

provided

(23) |x − x0|

r

2n+2n3e

.

To verify this, let us compute for each N the remainder term:

RN (x) := u(x) −

N−1

k=0 |α|=k

Dαu(x0)(x

−

x0)α

α!

=

|α|=N

Dαu(x0

+ t(x − x0))(x −

x0)α

α!

for some 0 ≤ t ≤ 1, t depending on x. We establish this formula by writing

out the ﬁrst N terms and the error in the Taylor expansion about 0 for the

function of one variable g(t) := u(x0 + t(x − x0)), at t = 1. Employing (22),

(23), we can estimate

|RN (x)| ≤ M

|α|=N

2n+1n2e

r

N

r

2n+2n3e

N

≤

MnN

1

(2n)N

=

M

2N

→ 0 as N → ∞.

See §4.6.2 for more on analytic functions and partial diﬀerential equa-

tions.

f. Harnack’s inequality. Recall from §A.2 that we write V ⊂⊂ U to

mean V ⊂

¯

V ⊂ U and

¯

V is compact.

THEOREM 11 (Harnack’s inequality). For each connected open set V

⊂⊂ U, there exists a positive constant C, depending only on V , such that

sup

V

u ≤ C inf

V

u

for all nonnegative harmonic functions u in U.

Thus in particular

1

C

u(y) ≤ u(x) ≤ Cu(y)

for all points x, y ∈ V . These inequalities assert that the values of a non-

negative harmonic function within V are all comparable: u cannot be very

small (or very large) at any point of V unless u is very small (or very large)

everywhere in V . The intuitive idea is that since V is a positive distance

away from ∂U, there is “room for the averaging eﬀects of Laplace’s equation

to occur”.