On shape optimization problems involving the fractional laplacian

Our concern is the computation of optimal shapes in problems involving $\(-\Delta)^{1/2}$. We focus on the energy $J(\Omega)$ associated to the solution $u_\Omega$ of the basic Dirichlet problem $(-\Delta)^{1/2} u_\Omega = 1$ in $\Omega$, $ u = 0$ in $\Omega^c$. We show that regular minimizers $\Omega$ 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