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 $J(Ω)$ associated to the solution $u\_Ω$ of the basic Dirichlet problem $(-Δ)^{1/2} u\_Ω= 1$ in $Ω$, $ u = 0$ in $Ω^c$. 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.