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fhape optimization

ESAIM: Control, Optimisation and Calculus of Variations (1)

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On shape optimization problems involving the fractionallaplacian
Anne-Laure Dalibard, David Gérard-Varet
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 19 / Issue 4 / October 2013

Published online by Cambridge University Press:

01 August 2013, pp. 976-1013

Print publication:

October 2013
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Our concern is the computation of optimal shapes in problems involving(−Δ)1/2. We focus on the energyJ(Ω) associated to the solution uΩ of thebasic Dirichlet problem( − Δ)1/2uΩ = 1in Ω, u = 0 in Ωc. We show that regularminimizers Ω of this energy under a volume constraint are disks. Our proof goes throughthe explicit computation of the shape derivative (that seems to be completely new in thefractional context), and a refined adaptation of the moving plane method.