
894 SECOND-ORDER ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
is a necessary condition for a solution of the nonhomogeneous problem to exist. The solu-
tion is then given by
w =
m−1
X
n=1
A
n
λ
n
− λ
m
w
n
+
∞
X
n=m+1
A
n
λ
n
− λ
m
w
n
+ Cw
m
, A
n
=
1
kw
n
k
2
Z
V
Φw
n
dV,
where C is an arbitrary constant and kw
n
k
2
=
R
V
w
2
n
dV .
3
◦
. If λ = λ
m
and
R
V
Φw
m
dV 6= 0, then the boundary value problem for the nonhomoge-
neous equation has no solution.
Remark 10.4. If to each eigenvalue λ
n
there are cor responding p
n
mutually orthogonal eigen-
functions w
(s)
n
(s = 1, . . . , p
n
), then the solution is written as
w =
∞
X
n=1
p
n
X
s=1
A
(s)
n
λ
n
− λ
w
(s)
n
, where A
(s)
n
=
1
kw
(s)
n
k
2
Z
V
Φw
(s)
n
dV, kw
(s)
n
k
2
=
Z
V
w
(s)
n
2
dV