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102Topologicalderivativesforsemilinearproblems
17
10202Semilinearellipticequation
LetΩandωbeboundeddomainsinR3withthesmoothboundariesaΩandaω
andthecompactclosuresΩandω,respectively.TheoriginOofthecoordinate
systemisassumedtobelongtothedomainsΩandω.Thefollowingsetsare
introduced:
ωE={x∈R3:ę:=E−1x∈ω},Ω(E):=Ω\ωE,
(1.8)
wherex=(x1,x2,x3)areCartesiancoordinatesinthedomainΩandE>0is
asmallparameter.TheupperboundE0>0ischoseninsuchawaythatfor
E∈(0,E0]thesetωEbelongstothedomainΩ.Wecandiminishthevalueof
E0>0inthesequel,ifnecessary,however,thenotationfortheboundE0remains
unchanged.ThesetωEiscalledhole,oropening,inthedomainΩ(E).
Inthissection,weconsideranonlinearellipticprobleminthesingularlyper-
turbeddomainΩ(E):
{−∆xu
uE(x)
E(x)=F(x,uE(x)),x∈Ω(E),
=0,
x∈aΩ(E).
(1.9)
HereF∈C0,α(Ω×R)andf∈C0,α(Ω)aregivenfunctions,independentofthe
parameterE.Theasymptoticanalysisinthelinearcaseiswellknown(seemono-
graphs[10],[20]),e.g.fortheDirichletboundaryvalueproblemforthePoisson
equation:
{−∆xu
uE(x)=0,
E(x)=f(x),x∈Ω(E),
x∈aΩ(E).
(1.10)
Accordingtothemethodofcompoundasymptoticexpansions[20],inasymp-
toticanalysisof(1.10)thereappeartwolimitproblems.Thefirstoneisobtained
byformallytakingE=0,e.g.fillingtheholeωE:
{−∆xu(x)=f(x),x∈Ω,
u(x)=0,
x∈aΩ,
(1.11)
andthesecondoneistheboundaryvalueproblemwhichfurnishestheleading
boundarylayersterm:
{−∆ęw(ę)=0,
w(ę)=−u(O),ę∈aω,
ę∈R3\ω,
(1.12)
whereu(O)isthevalueattheoriginofthesolutionof(1.11).
Asin[18](seealso[20];§5.7),forthenonlinearproblem(1.9)weobtainalso
twolimitproblems,thefirstlimitproblemisnonlinear:
{−∆xv(x)=F(x,v(x)),x∈Ω,
v(x)=0,
x∈aΩ,
(1.13)
