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.

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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