288
4 Problems with Variable Boundaries
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
˜
D
F u −
∂
∂ x
F u x −
∂
∂ y
F u y
ξ dxdy +
Γ ξ(−F u y dx + F u x dy + λΦ u dΓ ) = 0
˜
D
F v −
∂
∂ x
F v x −
∂
∂ y
F v y
ηdxdy +
Γ η(−F v y dx + F v x dy + λΦ v dΓ ) = 0
˜
D
F w −
∂
∂ x
F w x −
∂
∂ y
F w y
ζ dxdy +
Γ ζ(−F w y dx + F w x dy + λΦ w dΓ ) = 0
(4.4.6)
Since ξ , η, ζ are arbitrary second order differentiable functions of x and y, and
the Euler equations hold in D, therefore there are
⎧
⎨
⎩
−F u y dx + F u x dy + λΦ u dΓ = 0
−F v y dx + F v x dy + λΦ v dΓ = 0
−F w y dx + F w x dy + λΦ w dΓ = 0
(4.4.7)
or
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−F u y
dx
dΓ
+ F u x
dy
dΓ
+ Φ u λ = 0
−F v y
dx
dΓ
+ F v x
dy
dΓ
+ Φ v λ = 0
−F w y
dx
dΓ
+ F w x
dy
dΓ
+ Φ w λ = 0
(4.4.8)
According to the theory of linear equations, if the Eqs. (4.4.8) has a nonzero solution, the necessary condition is that the determinant constructed by the coefficients
of
dx
dΓ
,
dy
dΓ
and λ is zero, namely the determinant (4.4.3) should hold. Quod erat
demonstrandum.
Theorem 4.4.1 Let the functional
J =
¨
D
F(x, y, u, u x , u y )dxdy
(4.4.9)
obtain extremum, the boundary curve is on the known surface, namely
ϕ = ϕ(x, y)
(4.4.10)
then there is
F + (ϕ x − u x )F u x + (ϕ y − u y )F u y = 0
(4.4.11)
The expression (4.4.11) is called the condition of transversality or transversality condition at the line of intersection of the extremal surface u = u(x, y) and
known surface ϕ = ϕ(x, y) on the variable boundary.