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20
TopologicalderivativesandLevelsetmethodinshapeoptimization
isuniquelysolvableandthesolutionoperator:
SE:{F
E,gE}l−→vE
isboundedintheweightedHölderspaces:
SE:Λ
0,α
β(Ω(E))×Λ
2,α
β(aΩ(E))l−→Λ
2,α
β(Ω(E)).
Moreover,inthecaseβ−α∈(2,3)theestimate:
∥vE;Λ2,α
β(Ω(E))∥≤cβ(∥FE;Λ0,α
β(Ω(E))∥+∥gE;Λ2,α
β(aΩ(E))∥)
isvalid,wheretheconstantcβisindependentofE∈(0,E0].
(1.22)
(1.23)
Forthenonlinearproblem(1.9),weshallusetheclassicalsolutionstobound-
aryvalueproblem(1.9),whichmeansthatforgivenF∈C0,α(Ω×R),α∈(0,1),
thesolutionslivesinC2,α(Ω),wereferto[13]and[8]foraresultontheexistence
anduniquenessofsolutionstosemilinearellipticboundaryvalue-problems.It
means,inparticular,thatproblem(1.9)admitstheuniquesolutionuE∈C2,α(Ω(E))
forsome0<α<1andforallE∈[0,E0).
103Topologicalderivativeforsemilinearproblems
in3D
Wepresentherethecompleteanalysisofthesemilinearellipticprobleminthree
spatialdimensions.Suchananalysisisinterestingonitsown,sinceintheexisting
literaturethereisnoelementaryderivationoftheformoftopologicalderivatives
fornonlinearproblemsbesides[18],(seealso[20],Ch.5.7)i.e.,usingtheasymp-
toticapproximationsofsolutionstononlinearPDE’s.Therearesomeresultson
topologicalderivativesofshapefunctionalfornonlinearproblems,seee.g.,[1],
howeversuchresultsaregivenintermsoftheonetermexteriorapproximationof
thesolutionsandwithoutasymtoticallysharpestimate.
10301Formalasymptoticanalysis
Referringto[20],weset:
uE(x)=v(x)+w(E−1x)+Ev′(x)+...,
(1.24)
wherev,v′andwarecomponentsofregularandboundarylayertypes,respec-
tively.Thus:
−∆xv(x)−E−2∆
ęw(ę)−E∆xv′(x)+···
=F(x,v(x)+w(E−1x)+Ev′(x)+···)
=F(x,v(x))+(w(E−1x)+Ev′(x))F′
v(x,v(x))+···
(1.25)
