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Appendix: All Solutions to the Problems
10.28 Find the Euler equation of the functional J [ f ]
=
V
1
2
f
2
+
b
2
6
∇ f · ∇g + ug f
dV .
Solution Putting F =
1
2
f
2
+
b
2
6
∇ f · ∇g + u f g, the various partial derivatives are
∂ F
∂ f
= f + ug,
∂ F
∂∇ f
=
b
2
6
∇g, the Euler equation of the functional is
b
2
6
g − f − ug = 0
10.29 Find the Euler equation of the functional
J [u]
=
S
g
2
|∇u|
2
+ (1 − g)|∇u| +
λ
2
(I − u)
2
dS.
Solution Putting F = g|∇u|
2
+ (1 − g)|∇u| +
λ
2
(I − u)
2 , the various partial
derivatives are
∂ F
∂|∇u|
= g|∇u| + (1 − g),
The Euler equation of the functional is
∇ ·
g∇u + (1 − g)
∇u
|∇u|
+ λ(I − u) = 0
10.30 Find the Euler equation and boundary condition of the functional J [T ] =
V
k
2
∇T · ∇T − q v T + ρc p
∂ T
∂t
T
dV +
S α
1
2
T
2
− T f T
dS.
Solution Putting F =
k
2
∇T · ∇T − q v T + ρc p
∂ T
∂t
T , the various partial derivatives
are
∂ F
∂ T
= −q v + ρc p
∂ T
∂t
,
∂ F
∂∇T
= k∇T
The Euler equation of the functional is
∇ · k∇T + q v − ρc p
∂ T
∂t
= 0
( 1 )
The natural boundary condition is
[k∇T · n + α(T − T f )]
S
= 0 or
k
∂ T
∂ N
+ α(T − T f )
S
= 0
(2)
10.31 The functional for the optimal membrane shape of the cardiac valve prostheses
is J =
˜
D (
√
1 + ∇u · ∇u + λu)dxdy +
C udl, determine its Euler equation and
boundary condition.