
934 SECOND-ORDER ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
10.4.2 Equations of the F orm
div [a(x, y, z)∇w] − q(x, y, z)w = −Φ(x, y, z)
Equations of this sort are often encountered in heat and mass transfer theory. For brevity,
the equation is written using the notation
div[a(r)∇w] =
∂
∂x
a(r)
∂w
∂x
+
∂
∂y
a(r)
∂w
∂y
+
∂
∂z
a(r)
∂w
∂z
, r = {x, y, z}.
In what follows, the problems for the equation in question will be considered in a
bounded domain V with a sufficiently smooth surface S. It is assumed that a(r) > 0
and q(r) ≥ 0.
◮ First boundary value problem.
The following boundary condition of the first kind is imposed:
w = f (r) for r ∈ S.
Solution:
w(