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26
TopologicalderivativesandLevelsetmethodinshapeoptimization
10402Theformalasymptoticoftheshapefunctional
Weintroducethefollowinghypotheses:
(H7)F∈C0,α(Ω×R),F′
v∈C0,α(Ω×R)forsomeα∈(0,1)andF′
v≤0.
(H8)J∈C0,α(Ω×R),J′
v∈C0,α(Ω×R)
BythemonotonicityofF,theLax-MilgramLemmaandtheregularityofJ,
theproblem:
{−∆xp(x)−F
p(x)=0,
v(x,v(x))p(x)=J′
′
v(x,v(x)),x∈Ω,
x∈aΩ.
(1.46)
admitsauniquesolutionp∈C2,α(Ω).
WereplacethesolutionuEbyitsasymptoticrepresentation(1.41).Asaresultwe
obtainthefirstasymptotictermoforderE2fortheshapefunctional
J(uE;Ω(E))=∫
J(x,v(x))dx
Ω(E)
+∫
(Ew1(E
−1x)+E2w2(E−1x)+E2v′(x))J′
v(x,v(x))dx+···
Ω(E)
=J(v;Ω(E))+E2∫
(−
2π
1
|x|2
xT
m(ω)∇xv(O)
Ω(E)
−
2π
1
ln
|x|
E
mes2ωF(O;v(O))+v′(x))J′
v(x,v(x))dx+···
=J(v;Ω)−E2mes2ωJ(O;v(O))+E2∫
(−
2π
1
|x|2
xT
m(ω)∇xv(O)
Ω
−
2π
1
ln
|x|
E
mes2ωF(O;v(O))+v′(x))J′
v(x,v(x))dx+...
(1.47)
Nowwereplacetheright-handsideof(1.46)accordingtotheequationandtwice
