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Shape Optimization Problems with Neumann Boundary Condition 243
By redenoting p ε = ˜p δ (1 − H(g) +δ) −1 ∈ L 2 (D) (which conceptually may
depend on δ>0) in (32) we have proved the existence for (27). The uniqueness
of p ε may be shown by contradiction, directly in (27), as the factor multiplying
p ε in the left-hand side of (27) “generates” the whole L 2 (D) whenz ∈ W is
arbitrary.
Coming back to the equation in variations (28), (29) and to the definition of
the control problem (19)–(21), the directional derivative of the cost functional
(19) is given by
∫
1
lim
λ→0 λ [J(u ε + λv) − J(u ε )] = u ε vdx+ 1 ∫
F (y ε )F (z) dσ (33)
D ε [g≡0]
and the Euler equation is
∫
0= u ε vdx+ 1 ∫
F (y ε )F (z) dσ ∀v ∈ L 2 (D) (34)
D ε [g≡0]
with z defined by (28), (29). By using (27) in (34), since z given by (28), (29)
is an admissible test function, we get
⎡
∫ ∫
d∑
0= u ε vdx+ p ε
⎣−
D
D
i,j=1
( )
∂ ∂z
a ij
∂x j ∂x i
∫
=
D
∫
u ε vdx+
⎤
+ a 0 z⎦ dx
D
p ε (1 − H(g))vdx. (35)
This proves (26) and ends the argument.
Remark 4. Theorem 2 may be applied for any control u ∈ L 2 (D). For the
optimal control u ε , the directional derivative (and the gradient) is null and
we obtain u ε = −p ε (1−H(g)), that is, u ε has support in D \Ω g . This relation
is the maximum (Pontryagin) principle applied to the control problem (19)–
(21). Moreover, one can eliminate u ε and write the following system of two
elliptic equations:
d∑
( )
∂ ∂y ε
− a ij + a 0 y ε = f − (1 − H(g)) 2 p ε in D, (36)
∂x j ∂x i
∫
D
p ε
⎡
⎣−
i,j=1
d∑
i,j=1
∂y ε
=0
∂n A
⎤
( )
∂ ∂z
a ij + a 0 z⎦ = 1 ∫
∂x j ∂x i ε
[g≡0]
on∂D,
F (y ε )F (z) dσ ∀z ∈ W,
(37)
which constructs in an explicit manner the extension of the Neumann boundary
value problem from Ω g to D, modulo the approximation discussed in
Proposition 1.