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On shape optimization problems involving the fractionallaplacian

Published online by Cambridge University Press:  01 August 2013

Anne-Laure Dalibard  and
David Gérard-Varet
Anne-Laure Dalibard
Affiliation:
DMA/CNRS, Ecole Normale Supérieure, 45 rue d’Ulm, 75005 Paris, France
David Gérard-Varet
Affiliation:
IMJ and University Paris 7, 175 rue du Chevaleret, 75013 Paris France. gerard-varet@math.jussieu.fr
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Abstract

Our concern is the computation of optimal shapes in problems involving(−Δ)1/2. We focus on the energyJ(Ω) associated to the solution uΩ of thebasic Dirichlet problem( − Δ)1/2uΩ = 1in Ω, u = 0 in Ωc. We show that regularminimizers Ω of this energy under a volume constraint are disks. Our proof goes throughthe explicit computation of the shape derivative (that seems to be completely new in thefractional context), and a refined adaptation of the moving plane method.

Keywords

Fractional laplacianfhape optimizationshape derivativemoving plane method
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 4 , October 2013 , pp. 976 - 1013
Copyright
© EDP Sciences, SMAI, 2013

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