10 1 Laplace’s Equation
Remark 1.3.4 Note that directional derivatives are preserved under transla-
tion since this is true of ordinary derivatives. Using properties of orthogonal
transformations, as in the above remark, it is easy to see that directional
derivatives are preserved under orthogonal transformations and, in fact, that
inner products are also preserved.
Suppose u is harmonic on the open set Ω and y ∈ Ω.Thenu(x)canbe
regarded as a function of the spherical coordinates (r, θ)relativetoy where
r = |x −y| and θ is the point of intersection of the line segment joining x and
y and a unit sphere with center at y. Suppose that u is a function of r only.
Then Δu, as a function of spherical coordinates, is easily seen to be given by
Δu =
d
2
u
dr
2
+
(n − 1)
r
du
dr
,r=0.
The only functions that are harmonic on R
n
∼{y} and a function of r only
are those that satisfy the equation
d
2
u
dr
2
+
(n − 1)
r
du
dr
=0
on R
n
∼{y}.Ifn = 2, the general solution of this equation is A log r +
B,whereA and B are arbitrary constants. The particular solution u(r)=
−log r is harmonic on R
n
∼{y} andiscalledthefundamental harmonic
function for R
2
with pole y.Ifn ≥ 3, the general solution is Ar
−n+2
+ B;
the particular solution u(r)=r
−n+2
is called the fundamental harmonic
function for R
n
with pole y. The notations u
y
(x)andu(|x−y|) will be used
interchangeably.
1.4 The Mean Value Property
Most of the theorems about harmonic functions have their origin in the works
of Gauss and Green concerning electromagnetic and gravitational forces.
Theorem 1.4.1 (Gauss’ Integral Theorem) If u is harmonic on the ball
B and u ∈ C
1
(B
−
),then
∂B
D
n
udσ =0.
Proof: Put v = 1 in Green’s Identity, Theorem 1.2.2.
Much more can be obtained from Green’s Identity.
Theorem 1.4.2 (Green’s Representation Theorem) If u has continu-
ous second partial derivatives on a closed ball B
−
= B
−
y,ρ
,then