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22
TopologicalderivativesandLevelsetmethodinshapeoptimization
Wenowintroducethefollowingassumption:
(H5)J∈C0,α(Ω×R),J′
v∈C0,α(Ω×R)
Letp∈C2,α(Ω)beasolutionoftheproblem:
{−∆xp(x)−F
v(x,v(x))p(x)=J′
′
p(x)=0,
v(x,v(x)),x∈Ω,
x∈aΩ.
IntegratingbypartsinΩ\Bδ={x∈Ω:|x|>δ}yields:
(1.32)
∫
Ω
(v′(x)−ao(x))J′
v(x,v(x))dx
=−lim
δ→0∫
Ω\Bδ
(∆xp(x)+F
v(x,v(x))p(x))(v
′
′(x)−ao(x))dx
=−lim
δ→0∫
Ω\Bδ
p(x)(∆x+F
v(x,v(x)))(v
′
′(x)−ao(x))dx
−lim
δ→0∫
aΩ
anp(x)(v
′(x)−ao(x))dx
+lim
δ→0∫
aBδ
(a|x|p(x)(v
′(x)−ao(x))−p(x)a
|x|(v′(x)−ao(x)))dx.
By(1.30),wehavev′(x)−ao(x)=0forx∈Ωand:
(∆x+F′
v(x,v(x)))(v′(x)−ao(x))=∆xv′(x)+v′(x)F′
v(x,v(x))−ao(x)F′
v(x,v(x))=0.
Ontheotherhand,a|x|p(x)(v
′(x)−ao(x))=O(δ−1)and,hence:
∫
(v′(x)−ao(x))J′
v(x,v(x))dx
Ω
−lim
δ→0∫
(a|x|p(x)(v
′(x)−ao(x))−p(x)a
|x|(v′(x)−ao(x)))dx
aBδ
=−alim
δ→0∫
p(0)(4π|x|2)−1dsx
aBδ
=−ap(0)=−4πv(O)p(0)cap(ω).
Thus,
J(uE;Ω(E))=J(v;Ω)−E4πv(O)p(0)cap(ω)+...
(1.33)
Letsimilarlytothefirstinequalityin(H4)thefollowingassumptionbevalid:
(H6)Withσ∈(0,1),
|J(x,v(x)+V(x))−J(x,v(x))−V(x)J′
v(x,v(x))|≤c|V(x)|1+σ.
