164
2 Variational Problems with Fixed Boundaries
Since
F uxx δu xx + F uxy δu xy + F u yy δu yy =
∂
∂ x
(F uxx δu x ) +
∂
∂ y
(F uxy δu x ) +
∂
∂ y
(F u yy δu y )
− δu x
∂
∂ x
F uxx − δu x
∂
∂ y
F uxy − δu y
∂
∂ y
F u yy =
∂
∂ x
(F uxx δu x ) +
∂
∂ y
(F uxy δu x ) +
∂
∂ y
(F u yy δu y )
−
∂
∂ x
δu
∂
∂ x
F uxx
−
∂
∂ x
δu
∂
∂ y
F uxy
−
∂
∂ y
δu
∂
∂ y
F u yy
+ δu
∂ 2
∂ x 2 F uxx +
∂ 2
∂ x∂ y
F uxy +
∂ 2
∂ y 2 F u yy
Under the fixed boundary condition, there is
δu| L = δu x | L = δu y
L
= 0
Applying Green theorem, we get
¨
D
∂
∂ x
F u xx δu x − δu
∂
∂ x
F u xx − δu
∂
∂ y
F u xy
−
∂
∂ y
δu
∂
∂ y
F u yy − F u xy δu x − F u yy δu y
dxdy
=
L
F u xx δu x − δu
∂
∂ x
F u xx − δu
∂
∂ y
F u xy
dy +
δu
∂
∂ y
F u yy − F u xy δu x − F u yy δu y
dx
= 0
so
¨
D
(F u xx δu xx + F u xy δu xy + F u yy δu yy )dxdy
=
¨
D
∂
2
∂ x 2 F u xx +
∂
2
∂ x∂ y
F u xy +
∂
2
∂ y 2 F u yy
δudxdy
(2.8.17)
Substituting Eqs. (2.8.16) and (2.8.17) into Eqs. (2.8.15), (2.8.14) can be obtained.
Quod erat demonstrandum.
Example 2.8.4 Write the Ostrogradsky equation of the functional
J [u] =
¨
D
(u
2
xx + u
2
yy + 2u
2
xy )dxdy
Solution Applying Corollary 2.8.1, the Ostrogradsky equation can be written as
∂
2
∂ x 2 (2u xx ) +
∂
2
∂ y 2 (2u yy ) +
∂
2
∂ x∂ y
(4u xy ) = 0
or
∂
4 u
∂ x 4 + 2
∂
4 u
∂ x 2 ∂ y 2 +
∂
4 u
∂ y 4 = 0