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102Topologicalderivativesforsemilinearproblems
19
(H1)Limitproblem(1.13)hasasolutionv∈C2,α(Ω)andF∈C0,1(Ω×R)with
acertainα∈(0,1).
(H2)Linearproblem(1.17)withF∈C0,α(Ω),g∈C2,α(aΩ)hasauniquesolution
V∈C2,α(Ω),
∥V;C2,α(Ω)∥≤c(∥F;C0,α(Ω)∥+∥g;C2,α(aΩ)∥).
(1.18)
Hereandinthesequelcstandforapositiveconstantthatmaychangefromplace
toplacebutneverdependsonE.
(H3)F′
v∈C0,α(Ω×R).
If(H3)holdstrueandF′
v(x,v(x))≤0forx∈Ω,(H2)isalsosatisfied.Hy-
pothesis(H2)meanstheexistenceanduniquenessofclassicalsolutionstothe
linearizedprobleminHölderspacesC2,α(Ω)withtheaprioriestimate(1.18).It
turnsoutthatthelinearmappingforproblem(1.17):
S:{F,g}l−→V,
(1.19)
isanisomorphismintheHölderspacesC0,α(Ω)×C2,α(aΩ)→C2,α(Ω).Bya
generalresultin[19],(seealso[21];Ch.3,4)theoperatorremainstobeaniso-
morphisminweightedHölderspacesundertheproperchoiceofindices.
Theorem10Underassumptions(H2)and(H3),themapping(1.19)considered
intheweightedHölderspaces:
S:Λ
0,α
β(Ω)×C2,α(aΩ)l−→Λ2,α
β(Ω)
isanisomorphismifandonlyifβ−α∈(2,3).
Thefollowingresultonasymptoticsisdueto[12,19](seealso[17]and,e.g.,
[21];Ch.3,4).
Theorem20Iftherighthandsidein(1.17)F∈Λ
0,α
γ
(Ω)andγ−α∈(1,2),then
thesolutionVto(1.17)canbedecomposedV(x)=~
V(x)+V(O)andthefollowing
estimateholds:
|V(O)|+∥~
V;Λ
2,α
γ
(Ω)∥≤c(∥F;Λ
0,α
γ
(Ω)∥+∥g;C2,α(aΩ)∥).
(1.20)
Anassertion,similartoTheorem1,isvalidfortheperforateddomainΩ(E)
aswell.Thefollowingresultisdueto[18](seealso[20],Ch.2.4,[21],Ch.6).
Theorem30Underassumptions(H2)and(H3),thelinearizedproblem:
{−∆xv
vE(x)=gE(x),
E(x)−F′
v(x,v(x))vE(x)=FE(x),x∈Ω(E),
x∈aΩ(E)
(1.21)
