
796 SECOND-ORDER ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES
◮ Third boundary value problem.
The solution of the third boundary value problem for the Poisson equation (1) in the do-
main S with the nonhomogeneous boundary condition
∂w
∂N
+ kw = f(r) for r ∈ L
is given by formula (5) with C = 0, where G = G(r, ρ) is the Green’s function of the third
boundary value problem and is determined by the following conditions:
1
◦
. The function G satisfies the Laplace equation in x, y in the domain S everywhere
except for the point (ξ, η), at which G has a singularity of the form
1
2π
ln
1
|r−ρ|
.
2
◦
. With respect to x, y, the function G satisfies the homogeneous bound