Page 75 - 6637

Solved Problems

We do not need to compute the general solution of Equation 10.1. We only
need the Green function at the point x = p. We know that the general solution of

the equation ∆G = δ(x − ξ) is

1
G(x|ξ) = − + v(x),
4π|x − ξ|

where v(x) is an arbitrary harmonic function. The Green function at the point
x = p is
1
G(p|ξ) = − + const.
4π|p − ξ|

We add the constraint that the Green function vanishes at ρ = R. This determines
the constant.

1 1
G(p|ξ) = − +
4π|p − ξ| 4πR
1 1
G(p|ξ) = − +
4πρ 4πR
1
G (p|ξ) =
ρ
4πρ 2

Now we are prepared to write u(p) in terms of the Green function.

Z
1
u(p) = u(ξ) 2 dA ξ
∂S 4πρ
Z
1
u(p) = u(ξ) dA ξ
4πR 2 ∂S

This is the Mean Value Theorem for harmonic functions.

Exercise 202. Use the fundamental solutions for the modified Helmholz equation

2
∇ u − λu = δ(x − ξ)

in three dimensions
−1 √
u (x|ξ) = e ± λ|x−ξ| ,
±
4π|x − ξ|
to derive a “generalized” mean value theorem:

√

sinh λR 1 Z
√ u(p) = u(x) dA
λR 4πR 2 ∂S

that relates the value of any solution u(x) at a point P to the average of its value
on the sphere of radius R (∂S) with center at P.

71