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”.
