Volume 19, Number 4, October-December 2013
|Page(s)||976 - 1013|
|Published online||01 August 2013|
On shape optimization problems involving the fractional laplacian
1 DMA/CNRS, Ecole Normale
Supérieure, 45 rue
2 IMJ and University Paris 7, 175 rue du Chevaleret, 75013 Paris France
Revised: 12 September 2012
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 uΩ of the basic Dirichlet problem ( − Δ)1/2uΩ = 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.
Mathematics Subject Classification: 35J05 / 35Q35
Key words: Fractional laplacian / fhape optimization / shape derivative / moving plane method
© EDP Sciences, SMAI, 2013
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