On shape optimization problems involving the fractional laplacian
ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 4, p. 976-1013

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.

DOI : https://doi.org/10.1051/cocv/2012041
Classification:  35J05,  35Q35
Keywords: fractional laplacian, fhape optimization, shape derivative, moving plane method
@article{COCV_2013__19_4_976_0,
     author = {Dalibard, Anne-Laure and G\'erard-Varet, David},
     title = {On shape optimization problems involving the fractional laplacian},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {4},
     year = {2013},
     pages = {976-1013},
     doi = {10.1051/cocv/2012041},
     zbl = {1283.49049},
     mrnumber = {3182677},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2013__19_4_976_0}
}
Dalibard, Anne-Laure; Gérard-Varet, David. On shape optimization problems involving the fractional laplacian. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 4, pp. 976-1013. doi : 10.1051/cocv/2012041. http://www.numdam.org/item/COCV_2013__19_4_976_0/

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