358
5 Variational Problems of Conditional Extrema
is the unknown function, u N is the normal derivative of u on the surface S. For the
mixed functional
J [u] =
Ω
F(x 1 , x 2 , . . . , x n , u, u x1 , u x2 , . . . , u xn )d x 1 d x 2 · · · d x n +
S2
G(x 1 , x 2 , . . . , x n , u, u N )d s
(5.4.29)
the corresponding Euler equation and the boundary conditions are as follows
F u −
n
i=1
∂ F u x i
∂ x i
= 0 (in Ω)
(5.4.30)
G u +
n
i=1
F u x i n x i = 0 (on S 2 or S)
(5.4.31)
where, n x i = cos(n, x i ) are the direction cosines of exterior normal on the boundary
surface.
It is thus clear that the Euler equations of two-dimensional and three-dimensional
problem both are the special cases of the Euler equation of the n-dimensional problem.
Example 5.4.3 Find the Euler equation and the natural boundary condition of the
mixed type functional
J [u] =
˚
V
[k(x, y, z)(u
2
x + u
2
y + u
2
z ) + 2u f (x, y, z)]dV +
¨
S
u
2 h(x, y, z)d S
Solution According to Eq. (5.4.24), the Euler equation of the functional is
f (x, y, z) −
∂ku x
∂ x
−
∂ku y
∂ y
−
∂ku z
∂z
= 0 (in V )
or
∂ku x
∂ x
+
∂ku y
∂ y
+
∂ku z
∂z
= f (x, y, z) (in V )
According to the expression (5.4.25), the boundary condition can be written as
hu + k(u x n x + u y n y + u z n z ) = 0 or hu + k∇u · n = 0 (on S)
According the expression (1.3.24) in Chap. 1, there is
∂u
∂n
= |∇u| = ∇u · n =
∂u
∂ x
i +
∂u
∂ x
j +
∂u
∂ x
k
· (n x i + n y j + n z k) = u x n x + u y n y + u z n z
Thus, the boundary condition can also be written as
k
∂u
∂n
+ hu = 0 or k|∇u| + hu = 0 (on S)