2.8 Variational Problems Depending on Functions of Several Variables
161
δ J =
¨
D
F u −
∂
∂ x
F u x −
∂
∂ y
F u y
δudxdy = 0
Due to the arbitrariness of δu = εη(x, y), according to the fundamental lemma
of the calculus of variations Lemma 1.5.3, we obtain
F u −
∂
∂ x
F u x −
∂
∂ y
F u y = 0
Quod erat demonstrandum.
Example 2.8.1 Known (x, y) ∈ D, find the Ostrogradsky equation of the functional
J [u] =
1
2
¨
D
(u
2
x + u
2
y )dxdy
The functional expresses the deformation energy of an elastic membrane, it is
called the Dirichlet functional or Dirichlet integral.
Solution 1 By formula (2.8.2), the Ostrogradsky equation can be written as
∂
2 u
∂ x 2 +
∂
2 u
∂ y 2 = 0
This is a two-dimensional Laplace equation. It is one of the basic problems of
equation of mathematical physics, usually it is called the Dirichlet problem. This
problem was posed originally by Green in the form of a conjecture, in 1833 Dirichlet
formally posed this problem on the study of the gravity problem of the variable density
ellipsoid, but at that time which was not caused enough attention.
Solution 2 The original functional can be written in the form of the vector
J [u] =
¨
D
∇u · ∇udxdy
Taking the first variation to the above functional, and using the Green’s theorem,
there is
δ J [u] = 2
¨
D
∇δu · ∇udxdy = 2
¨
D
[∇ · (δu∇u) − uδu]dxdy
= 2
L
∂u
∂n
δud L − 2
¨
D
uδudxdy = −2
¨
D
uδudxdy = 0
For the fixed boundary problem, δu| L = 0, so the integral of the above expression
on the closed curve L vanishes, and in the domain D, δu is arbitrary, if the first variation
of the functional vanishes, there must be