Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
104Topologicalderivativeformixedsemilinearellipticproblemintwo...
27
integratebypartsinthedomainΩ\Bδ={x∈Ω:|x|>δ}.Wehave:
∫
(−
2π
1
|x|2
xT
m(ω)∇xv(O)−
2π
1
ln
|x|
E
mes2ωF(O;v(O))+v
′(x))J′
v(x,v(x))dx
Ω
=−lim
δ→0∫
(∆xp(x)+F
v(x,v(x))p(x))
′
Ω\Bδ
×(−
2π
1
|x|2
xT
m(ω)∇xv(O)−
2π
1
ln
|x|
E
mes2ωF(O;v(O))+v′(x))dx
=−lim
δ→0∫
p(x)(∆x+F
v(x,v(x)))
′
Ω\Bδ
×(−
2π
1
|x|2
xT
m(ω)∇xv(O)−
2π
1
ln
|x|
E
mes2ωF(O;v(O))+v′(x))dx
−lim
δ→0∫
anp(x)(−
2π
1
|x|2
xT
m(ω)∇xv(O)−
2π
1
ln
|x|
E
mes2ωF(O;v(O))+v
′(x))dx
aΩ
−lim
δ→0∫
[a|x|p(x)(−
2π
1
|x|2
xT
m(ω)∇xv(O)−
2π
1
ln
|x|
E
mes2ωF(O;v(O))+v
′(x))
aBδ
−p(x)a|x|(−
2π
1
|x|2
xT
m(ω)∇xv(O)−
2π
1
ln
|x|
E
mes2ωF(O;v(O))+v′(x))]dx.
Ontheotherhand,theboundarycondition(1.45)impliesthat:
−
2π
1
|x|2
xT
m(ω)∇xv(O)−
2π
1
ln
|x|
E
mes2ωF(O;v(O))+v
′(x)=0.
Furthermore,forthelinearizedoperator∆x+F′
v,theformula:
(∆x+F′
v(x,v(x)))+(−
2π
1
|x|2
xT
m(ω)∇xv(O)−
2π
1
ln
|x|
E
mes2ωF(O;v(O))+v′(x))
=∆xv′(x)+v′(x)F′
v(x,v(x))
+(−
2π
1
|x|2
xT
m(ω)∇xv(O)−
2π
1
ln
|x|
E
mes2ωF(O;v(O)))F′
v(x,v(x))
=0,
isvalidbecausethefunction:
x→(−
2π
1
|x|2
xT
m(ω)∇xv(O)−
2π
1
ln
|x|
E
mes2ωF(O;v(O)),
