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Article Dans Une Revue ESAIM: Control, Optimisation and Calculus of Variations Année : 2013

On shape optimization problems involving the fractional laplacian

Résumé

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.
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Dates et versions

hal-00673035 , version 1 (22-02-2012)
hal-00673035 , version 2 (18-02-2015)

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Anne-Laure Dalibard, David Gérard-Varet. On shape optimization problems involving the fractional laplacian. ESAIM: Control, Optimisation and Calculus of Variations, 2013, 19 (4), pp.976-1013. ⟨hal-00673035v2⟩
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