Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
24
TopologicalderivativesandLevelsetmethodinshapeoptimization
Sincetheproofusesthesameargumentsasinthreespatialdimensions(note
thatweusetheHöldernormswhichareinsensitivetothespacedimension),we
provideonlytheformalanalysisandimposetheNeumannboundaryconditionson
theholeboundaryaωE.NotethattheDirichletconditiononaωEchangescrutially
theformofasymptoticexpansions(cf.[10]and[18],[[20];Ch.5.7]).
10401Formalasymptoticanalysis
LetΩandωbeboundeddomainsintheplaneR2.Weconsiderthenonlinear
mixedprobleminthesingularlyperturbeddomainΩ(E),definedin(1.8):
(
−∆xuE(x)=F(x,uE(x)),x∈Ω(E),
4
l
anuE(x)=0,
uE(x)=0,
x∈aΩ,
x∈aωE.
Wesearchforasolutionofproblem(1.9)intheform:
uE(x)=v(x)+w(E−1x)+Ev′(x)+ˆ
uE(x),
whereˆ
uEisasmallremainder,satisfyingtheproblem:
(
−∆xˆ
uE(x)=ˆ
FE(x;ˆ
u),x∈Ω(E),
4
uE(x)=ˆ
uE(x)=ˆ
ˆ
ˆ
gE
gE
Ω(x),
ω(x),
x∈aΩ,
l
x∈aω(E).
Here:
FE(x;ˆ
ˆ
u)=F(x,v(x)+w(E−1x)+Ev′(x)
+ˆ
uE(x))−F(x,v(x))
−E(v′(x)−ao(x))F′
v(x,v(x)),
gE
ˆ
Ω(x)=−w(E−1x)−aEo(x),
gE
ˆ
ω(x)=−v(x)+v(O)−Ev′(x).
Referringto[10],[20],especiallyto[18]and[[20]§5.7],weset:
uE(x)=v(x)+Ew1(E−1x)+E2w2(E−1(x)+E2v′(x)+...,
(1.34)
(1.35)
(1.36)
(1.37)
(1.38)
wherev,v′andw1,w2arecomponentofregularandboundarylayertypes,respec-
tively.Precisely,visasmoothsolutionofproblem(1.13)inthetwodimensional
entiredomainΩ.TheTaylorformulayields:
v(x)=v(O)+xT∇xv(O)+
1
2
xT∇2
xv(O)x+O(|x|3).
