Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
104Topologicalderivativeformixedsemilinearellipticproblemintwo...
23
Usingthisassumptionleadstotherelation:
|J(uE;Ω(E))−J(v;Ω(E))−∫
(w(E−1x)+Ev′(x)+ˆ
uE(x))J′
v(x,v(x))dx|
Ω(E)
≤c∫
|w(E−1x)+Ev′(x)+ˆ
uE(x)|1+σdx
Ω(E)
≤c∫
(|
|
|
E
x
|
|
|
−1−σ
+|x|−(1+σ)(β−2−α)
Ω(E)
×(E1+σ∥v′;Λ2,α
β(Ω)∥1+σ+∥ˆ
uE(x);Λ2,α
β(Ω)∥1+σ))dx
≤cE1+σ(
\
∫
E
1
r−1−σr2dr+
∫
E
1
r−(1+σ)(β−2−α)r2dr(E1+σ+E(1+K)(1+σ)\
)
≤cE1+σ.
Herewehavetakenintoaccountthat1+σ≤2,(1+σ)(β−2−α)≤2,andboth
theintegrals,extendedontheinterval(0,1),doconverge.
Itsufficestomentionthefollowinginequalities:
|J(uE;Ω(E))−J(v;Ω)|≤cmes3(ωE)≤cE3,
1
∫
|w(E−1x)+aEo(x)||J′
v(x,v(x))|dx≤c
∫
(
E)
r
−2
rdr≤cE2,
Ω(E)
E
∫
|ˆ
uE||J′
v(x,v(x))|dx≤cE1+K∫
|x|−(β−2−α)dx≤cE1+K.
Ω(E)
Ω(E)
Thisconfirmstheformalcalculationsperformedabove.Letusformulatethemain
resultinthreedimensions.
Theorem40Undertheassumptionslistedabove,wehave:
|J(uE;Ω(E))−J(v;Ω)+E4πv(O)p(0)cap(ω)|≤cE1+min(σ,K).
104Topologicalderivativeformixedsemilinearel-
lipticproblemintwospatialdimensions
Thenumericalanalysisisperformedintwospatialdimensions.Therefore,we
introduceamixedsemilinearproblemandanalyzetheasymptoticinsuchacase.
