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10 Variational Problems of Functionals with Vector, Tensor …
They are just fundamental equations. Quod erat demonstrandum.
Example 10.2.7 The image denoising functional on the total variation is J [u] =
D
|∇u| +
λ
2
(u − u 0 )
2
dx dy, where, u is the original image; u 0 is the noising
image; λ is the regularity parameter. Find the Euler equation.
Solution Let the integrand F = |∇u| +
λ
2
(u − u 0 )
2 , calculating the various partial
derivatives on u
F u = λ(u − u 0 ), ∇ ·
∂ F
∂|∇u|
∇u
|∇u|
= ∇ ·
∇u
|∇u|
(1)
Substituting the above two equations into the Euler equation, we obtain
∇ ·
∇u
|∇u|
= λ(u − u 0 )
(2)
This equation is one of the most important equations in information processing,
it has very extensive application in many fields.
Example 10.2.8 The functional of time-harmonic acoustic field problem is
J [w, u] =
V
(∇w · ∇u − k
2 wu − w f )dV +
S
(βwu − wg)dS
Find the Euler equation and boundary condition expressed by u.
Solution Putting F = ∇w · ∇u − k
2 wu − w f , the various partial derivatives are
∂ F
∂∇w
= ∇u,
∂ F
∂w
= −k
2 u − f , the Euler equation of the functional is
(( + k
2
)u + f = 0
( 1 )
This equation is called the Helmholtz equation.
The boundary condition is
∂u
∂n
+ βu = g (on S)
(2)
Example 10.2.9 Find the Euler equation of the functional J
=
V
v
2
(∇ × a) · (∇ × a) − J · a
dV , where, v is a constant.
Solution Putting F =
v
2
(∇ × a) · (∇ × a) − J · a, determine the various partial
derivatives
∂ F
∂ a
= − J,
∂ F
∂∇ × a
= v∇ × a
(1)