On shape optimization problems involving the fractional laplacian

¹ DMA/CNRS, Ecole Normale Supérieure, 45 rue d’Ulm, 75005 Paris, France
² IMJ and University Paris 7, 175 rue du Chevaleret, 75013 Paris France
gerard-varet@math.jussieu.fr

Received: 22 February 2012
Revised: 12 September 2012

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)^1/2u_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.