
826 SECOND-ORDER ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES
◮ Domain: 0 ≤ r ≤ R. Third boundary value problem.
A circle is considered. A boundary condition is prescribed:
∂
r
w + kw = 0 at r = R.
Eigenvalues:
λ
nm
=
µ
2
nm
R
2
; n = 0, 1, 2, . . . ; m = 1, 2, 3, . . .
Here, the µ
nm
is the mth root of the transcendental equation µJ
′
n
(µ) + kRJ
n
(µ) = 0.
Eigenfunctions:
w
(1)
nm
= J
n
r
p
λ
nm
cos nϕ, w
(2)
nm
= J
n
r
p
λ
nm
sin nϕ.
The square of the norm of an eigenfunction is given by
kw
(1)
nm
k
2
=kw
(2)
nm
k
2
=
πR
2
(1+δ
n0
)
2µ
2
nm
(k
2
R
2
+µ
2
nm
−n
2
)[J
n
(µ
nm
)]
2
, δ
ij
=
(
1 for i=j,
0 for i6=j.
⊙ Literature: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin et al. (1964), B. M. Budak, A. A. Samarskii,
and ...