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18
TopologicalderivativesandLevelsetmethodinshapeoptimization
andthesecondlimitproblemisthelinearexteriorproblem(1.12)withu(O):=
v(O)givenbythesolutionto(1.13).
Ouraimisnowtheconstructionofasymptoticapproximationsforsolutions
to(1.9)insuchawaythatweareabletoobtainanexpansionofagivenshape
functional:
J(uE;Ω(E))=∫
J(x,uE(x))dx,
(1.14)
Ω(E)
ofthefirstorderwithrespecttoE,namely:
J(uE;Ω(E))=J(v;Ω)+ETΩ(O)+o(E),
(cf.(1.7)),where:
(1.15)
J(v;Ω)=∫
Ω
J(x,v(x))dx,
(1.16)
andTΩisthetopologicalderivativeofthefunctionalJ.
Besidethatweneedthelinearizedproblem(1.13),whichgivesustheregular
termsintheasymptoticapproximation:
{−∆xV(x)−F
V(x)=g(x),
v(x,v(x))V(x)=F(x),x∈Ω,
′
x∈aΩ.
(1.17)
ThesolutionVisbuttheso-calledadjointstate.Theadjointstateisintroducedin
ordertosimplifytheexpressionforthetopologicalderivative.
Appropriatefunctionspacesareemployedtoanalyzethesolvabilityofall
boundaryvalueproblemsintroducedabove.TheweightedHölderspacesΛ
l,α
β(Ω)
aredefined[19]astheclosureofC∞
c(Ω\O)(smoothfunctionsvanishinginthe
vicinityofO)inthenorm:
∥Z;Λ
l,α
β(Ω)∥=
∑
l
sup
|x|β−l−α+k|∇k
xZ(x)|+
k=0
x∈Ω
+
x,y∈Ω,|x−y|<
sup
|x|
2
|x|β|x−y|−α|∇l
xZ(x)−∇l
yZ(y)|.
ThestandardnormintheHölderspaceCl,α(Ω)looksasfollows:
∥Z;Cl,α(Ω)∥=
∑
l
sup
|∇k
xZ(x)|+
x,y∈Ω,|x−y|<
sup
|x|
2
|x−y|−α|∇l
xZ(x)−∇l
yZ(y)|.
k=0
x∈Ω
Herel∈{0,1,...},α∈(0,1)andβ∈R.
Nowweintroduceseveralassumptionswhicharerequiredtodefinethetopo-
logicalderivatives:
