Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
101Shapederivativesinsmoothdomains
15
TheenergyfunctionalforthedomainΩtisgivenbytheformula:
J(Ωt)=π(Ωt;u
t)
=
1
2∫
Ωt
∥∇ut∥2
R2dx−∫
Ωt
futdx
=
ψ∈H1
inf
0(Ωt)
π(Ωt;ψ).
10103
Thefirstordershapederivative
Atfirst,letusrecallsomeresultsaboutthefirstshapederivative,orEuleriansemi-
derivative,ofafunctionalJ.ThisshapederivativeindirectionVisdefinedbythe
followinglimit:
t→0
lim
J(Ωt)−J(Ω)
t
,
ifitexistsofcourse.Inourcase,fortheenergyfunctional,wederivethefirst
shapederivativebydifferentiationofthevolumeintegral:
∫
Ω
FΩdy,
whereFΩisafunctiondependingonΩ.Indeed,wehave[35]:
dt(∫
d
Ωt
FΩ
tdx)
|t=0
=∫
Ω
F′
Ωdy+∫
F
FΩ⟨V,ν⟩R2dF(y),
(1.4)
whereF′
ΩdenotestheshapederivativeofFΩindirectionVandνistheexterior
normalvectortoF.Moreover,accordingtostructuretheorem[35],weknowthat
thereexistsadistributiongaΩ∈D
′
1(Ω),supportedbyF,suchthat:
dt(∫
d
Ωt
FΩ
tdx)
|t=0
=⟨gaΩ,⟨V,ν⟩R2⟩,
(1.5)
where⟨·,·⟩denotesdedualitybracket.Inourcase,weobtain,byanintegration
byparts:
gaΩ=−
1
2(
auΩ
aν)
2
∈L1(aΩ).
Consequently,theEuleriansemiderivativeofthefunctionalJindirectionVis
givenby:
dJ(Ω;V)=−
1
2∫
F(
auΩ
aν)
2
⟨V,ν⟩
R2dF(y).
(1.6)
