5.4 Extremal Problems of Mixed Type Functionals
357
unknown function, u N is the normal derivative of u on the surface S 2 . For the mixed
type functional
J [u] =
˚
V
F(x, y, z, u, u x , u y , u z )dxdydz +
¨
S 2
G(x, y, z, u, u N )dS (5.4.23)
making use of the method deriving the Euler equation of the two-dimensional
problem, the Euler equation and the boundary condition of the three-dimensional
problem can be obtained as follows
F u −
∂ F u x
∂ x
−
∂ F u y
∂ y
−
∂ F u z
∂z
= 0 (in V )
(5.4.24)
G u + F u x n x + F u y n y + F u z n z = 0 (on S 2 or S)
(5.4.25)
where, n x = cos(n, x), n y = cos(n, y), n z = cos(n, z) are the direction cosines of
exterior normal on the boundary surface.
Furthermore, Let the function u(x, y, z) be the continuously differential function
of second order in the spatial domain V (x, y, z), S = S 1 + S 2 =
m
i=1
S 1i +
n
k=1
S 2k is
the surface boundary of V, and one the boundary S 1i u = u 1i is the known function,
on the boundary S 2k u is the known function, u N is the normal derivative of u on the
surface S 2 . For the mixed functional
J [u] =
˚
V
F(x, y, z, u, u x , u y , u z )dxdydz +
n
k=1
¨
S 2k
G k (x, y, z, u, u N )dS
(5.4.26)
making use of the method deriving the Euler equation of the two-dimensional
problem, the Euler equation and the boundary conditions of the three-dimensional
problem can be obtained as follows
F u −
∂ F u x
∂ x
−
∂ F u y
∂ y
−
∂ F u z
∂z
= 0 (in V )
(5.4.27)
G ku + F u x n x + F u y n y + F u z n z = 0 (on S 2k , k = 1, 2, . . . , n)
(5.4.28)
where, n x = cos(n, x), n y = cos(n, y), n z = cos(n, z) are the direction cosines of
exterior normal on the boundary surface.
It can lead to the conclusion from which that if the functional given by the expression (5.4.26) obtains extremum on the function u(x, y, z), then u(x, y, z) must satisfy
the Euler Eq. (5.4.27) and the boundary conditions (5.4.28).
The preceding conclusion drawn can be generalized to the n-dimensional space.
Let u(x 1 , x 2 , . . . , x n ) be the continuously differential function of second order in
the n-dimensional spatial domain Ω(x 1 , x 2 , . . . , x n ), S = S 1 + S 2 is the boundary
of Ω, and on the boundary S 1 u = u 1 is the known function, on the boundary S 2 u
357
unknown function, u N is the normal derivative of u on the surface S 2 . For the mixed
type functional
J [u] =
˚
V
F(x, y, z, u, u x , u y , u z )dxdydz +
¨
S 2
G(x, y, z, u, u N )dS (5.4.23)
making use of the method deriving the Euler equation of the two-dimensional
problem, the Euler equation and the boundary condition of the three-dimensional
problem can be obtained as follows
F u −
∂ F u x
∂ x
−
∂ F u y
∂ y
−
∂ F u z
∂z
= 0 (in V )
(5.4.24)
G u + F u x n x + F u y n y + F u z n z = 0 (on S 2 or S)
(5.4.25)
where, n x = cos(n, x), n y = cos(n, y), n z = cos(n, z) are the direction cosines of
exterior normal on the boundary surface.
Furthermore, Let the function u(x, y, z) be the continuously differential function
of second order in the spatial domain V (x, y, z), S = S 1 + S 2 =
m
i=1
S 1i +
n
k=1
S 2k is
the surface boundary of V, and one the boundary S 1i u = u 1i is the known function,
on the boundary S 2k u is the known function, u N is the normal derivative of u on the
surface S 2 . For the mixed functional
J [u] =
˚
V
F(x, y, z, u, u x , u y , u z )dxdydz +
n
k=1
¨
S 2k
G k (x, y, z, u, u N )dS
(5.4.26)
making use of the method deriving the Euler equation of the two-dimensional
problem, the Euler equation and the boundary conditions of the three-dimensional
problem can be obtained as follows
F u −
∂ F u x
∂ x
−
∂ F u y
∂ y
−
∂ F u z
∂z
= 0 (in V )
(5.4.27)
G ku + F u x n x + F u y n y + F u z n z = 0 (on S 2k , k = 1, 2, . . . , n)
(5.4.28)
where, n x = cos(n, x), n y = cos(n, y), n z = cos(n, z) are the direction cosines of
exterior normal on the boundary surface.
It can lead to the conclusion from which that if the functional given by the expression (5.4.26) obtains extremum on the function u(x, y, z), then u(x, y, z) must satisfy
the Euler Eq. (5.4.27) and the boundary conditions (5.4.28).
The preceding conclusion drawn can be generalized to the n-dimensional space.
Let u(x 1 , x 2 , . . . , x n ) be the continuously differential function of second order in
the n-dimensional spatial domain Ω(x 1 , x 2 , . . . , x n ), S = S 1 + S 2 is the boundary
of Ω, and on the boundary S 1 u = u 1 is the known function, on the boundary S 2 u