On

shape optimization problems involving the fractional laplacian ∗Anne-Laure Dalibard and †David Gérard-Varet Our concern is the computation of optimal shapes in problems involving (−∆)1/2. We focus on the energy J(Ω) associated to the solution uΩ of the basic Dirichlet problem (−∆)1/2uΩ = 1 in Ω, u = 0 in Ωc. We show that regular minimizers Ω of this energy under a volume constraint are disks. Our proof goes through the explicit computation of the shape derivative (that seems to be completely new in the fractional context), and a refined adaptation of the moving plane method