On shape optimization problems involving the fractional laplacian

الأصل · EN

Our concern is the computation of optimal shapes in problems involving -Δ)¹/². We focus on the energy J(Ω) associated to the solution u_Ω of the basic Dirichlet problem (-Δ)¹/² u_Ω= 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.