
12.5. Simplest Systems Containing Vector Functions and Operators div and curl 1069
1
◦
. We seek a solution of problem (13) in the form (6), (8). For the function ϕ, we obtain
the Poisson equation
∆ϕ + g(x) = f(x), (14)
where the function g(x) is determined by formula (12).
2
◦
. The solution of problem (13) can be represented in the form
u = ∇ϕ + curl ψ + ∇θ,
ϕ(x) = −
1
4π
Z
V
f(x
′
)
|x −x
′
|
dV
′
, ψ(x) =
1
4π
Z
V
A(x
′
)
|x −x
′
|
dV
′
,
where the function θ satisfies the Laplace equation, ∆θ = 0. The surface V may be un-
bounded if both integrals converge and decay as |x| → ∞ at least at the rate of |x|
−(1+ε)
,
where ε > 0.
3
◦
. Equations (13) are often supplemented w ith the