2.2. LAPLACE’S EQUATION 41
for r 0. Then ˜(x) u = u(rx) solves (42), with ˜(x) g = g(rx) replacing g.
We change variables to obtain Poisson’s formula
(45) u(x) =
r2
−
|x|2
nα(n)r
∂B(0,r)
g(y)
|x − y|n
dS(y) (x ∈
B0(0,r)).
The function
K(x, y) :=
r2
−
|x|2
nα(n)r
1
|x − y|n
(x ∈
B0(0,r),
y ∈ ∂B(0,r))
is Poisson’s kernel for the ball B(0,r).
We have established (45) under the assumption that a smooth solution
of (44) exists. We next assert that this formula in fact gives a solution:
THEOREM 15 (Poisson’s formula for ball). Assume g ∈ C(∂B(0,r)) and
define u by (45). Then
(i) u ∈
C∞(B0(0,r)),
(ii) Δu = 0 in
B0(0,r),
and
(iii) lim
x→x0
x∈B0(0,r)
u(x) =
g(x0)
for each point
x0
∈ ∂B(0,r).
The proof is similar to that for Theorem 14 and is left as an exercise.
2.2.5. Energy methods.
Most of our analysis of harmonic functions thus far has depended upon
fairly explicit representation formulas entailing the fundamental solution,
Green’s functions, etc. In this concluding subsection we illustrate some
“energy” methods, which is to say techniques involving the
L2-norms
of
various expressions. These ideas foreshadow later theoretical developments
in Parts II and III.
a. Uniqueness. Consider first the boundary-value problem
(46)
−Δu = f in U
u = g on ∂U.
We have already employed the maximum principle in §2.2.3 to show
uniqueness, but now we set forth a simple alternative proof. Assume U is
open, bounded, and ∂U is
C1.
