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14
TopologicalderivativesandLevelsetmethodinshapeoptimization
where⟨·,·⟩R2isthescalarproductinR2.Solvingthisvariationalequationisequiv-
alenttotheminimizationofthefunctionalπ(Ω;·)whichrepresentsthepotential
energyassociatedwith(1.1):
π(Ω;φ)=
1
2∫
Ω
∥∇φ∥2
R2dy−∫
Ω
fφdy,
∀φ∈H1
0(Ω),
where∥·∥R2istheeuclidiannorminR2.Weknowthatthevariationnalequation
(1.2)hasauniquesolutionu=uΩ∈H1
0(Ω).Andso,theenergyfunctionalrelative
tothedomainΩisgivenbytheformula:
J(Ω)=π(Ω;uΩ)
=
1
2∫
Ω
∥∇uΩ∥
R2dy−∫
2
Ω
fuΩdy
=
φ∈H1
inf
0(Ω)
π(Ω;φ).
LetV∈W1,∞(R2;R2)∩C1(R2;R2)beavectorfieldandTt(V)(t≥0)betheap-
plicationdefinedby:
Tt(V):R2−→R2
Xl−→x(t)
wherex(·)isthesolutionoftheordinarydifferentialequation:
(
4
l
dx
dt
x(0)=X.
(t)=V(x(t)),
t>0,
WedenoteΩt=Ωt(V)theimageofthedomainΩbytheapplicationTt(V),i.e.
Ωt=Tt(V)(Ω),andinthesamewaytheboundaryofΩtisobtainedbyFt=
Ft(V)=Tt(V)(F).Thecoordinatesofagivenpointwhichbelongstothedomains
Ω=Ω0,Ωt(t>0),arerespectivelydenotedbyy=(y1,y2)∈Ω,x=(x1,x2)∈Ωt,
moreover,weknowthattheapplicationTt(V)isbijective.Inthesamewayas
previously,thereexistsauniquefunctionut∈H1
0(Ωt)whichissolutionofthe
variationalequality:
∫
Ωt
⟨∇u
t,∇v⟩
R2dx=∫
Ωt
fvdx,
∀v∈H1
0(Ωt).
(1.3)
Thisweaksolutionutisalsoobtainedbyminimizingthepotentialenergyassoci-
atedwith(1.3):
π(Ωt;ψ)=
1
2∫
Ωt
∥∇ψ∥2
R2dx−∫
Ωt
fψdx,
∀ψ∈H1
0(Ωt).
