
864 SECOND-ORDER ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
◮ Particular solutions:
w(r) = A +
B
r
,
w(r, θ) =
Ar
n
+
B
r
n+1
P
n
(cos θ),
w(r, θ, ϕ) =
Ar
n
+
B
r
n+1
P
m
n
(cos θ)(C cos mϕ + D sin mϕ),
where n = 0, 1, 2, . . . ; m = 0, 1, 2, . . . , n; A, B, C, D are arbitrary constants; the P
n
(ξ)
are the Legendre polynomials; and the P
m
n
(ξ) are associated Legendre functions, which are
expressed as
P
n
(x) =
1
n! 2
n
d
n
dx
n
(x
2
− 1)
n
, P
m
n
(x) = (1 − x
2
)
m/2
d
m
dx
m
P
n
(x).
◮ Domain: 0 ≤ r ≤ R or R ≤ r < ∞. First bound ary value problem.
A boundary condition at the sphere surface is prescribed:
w = f (θ, ϕ) at r = R.
1
◦
. Solution of the inner problem (for r ≤ R):
w(r, θ, ϕ)