On shape optimization problems involving the fractional laplacian

Abstract

Our concern is the computation of optimal shapes in problems involving $\(-Δ)^{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.