Issue |
ESAIM: COCV
Volume 19, Number 4, October-December 2013
|
|
---|---|---|
Page(s) | 976 - 1013 | |
DOI | https://doi.org/10.1051/cocv/2012041 | |
Published online | 01 August 2013 |
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
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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