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103Topologicalderivativeforsemilinearproblemsin3D
21
Inviewof(1.13),thefirsttermsontheleftandtherightarecancelledand,more-
over,wsatisfiestheproblem(1.12)withu(O)=v(O),
{−∆ęw(ę)=0,
w(ę)=−v(O),ę∈aω,
ę∈R3\ω,
whiletheboundarydatumcomesfromtherelation:
v(x)+w(E−1x)+Ev′(x)=v(O)+w(E−1x)+O(E),x∈aωE.
Wehave:
w(ę)=−v(O)P(ę)
(1.26)
(1.27)
wherePisthecapacitypotential[14,31],e.g.,aharmonicfunctioninR3\ωsuch
thatP(ę)=1onaωand:
P(ę)=|ę|−1cap(ω)+O(|ę|−2),
wherecap(ω)isthecapacityofthesetω.Since:
w(E−1x)=−|x|−1Ev(O)cap(ω)+O(E2|x|−2),
wecollectcoefficientsonEin(1.25)andobtain:
{−∆xv
′(x)−v′(x)F′
v(x,v(x))=−ao(x)F′
v′(x)=ao(x),
v(x,v(x)),x∈Ω,
x∈aΩ,
(1.28)
(1.29)
(1.30)
wherea=4πv(O)cap(w)ando(x)=(4π|x|)−1isthefundamentalsolutionof
theLaplaceequationinR3.
SinceadirectcalculationyieldsF′(·,v)o∈Λ0,α
γ
(Ω)withanyγ>1+α,we
obtainthesolutionv′∈Λ2,α
β(Ω)ofproblem(1.30)suchthatv′−v′(O)∈Λ2,α
γ
(Ω)
whereβ−α∈(2,3)andγ−α∈(1,2)canbetakenarbitrarilyintheprescribed
intervals.
Wereferthereaderto[9]forjustificationofasymptotic.
10302Theformalasymptoticoftheshapefunctional
Wehave:
J(uE;Ω(E))=∫
Ω(E)
J(x,v(x))dx+∫
(w(E−1x)+Ev′(x))J′
v(x,v(x))dx+···
Ω(E)
=∫
J(x,v(x))dx+E∫
(v′(x)−ao(x))J′
v(x,v(x))dx+...(1.31)
Ω
Ω
