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 u_{Ω} of the basic Dirichlet problem ( - Δ)^{1/2}u_{Ω} = 1 in Ω, u = 0 in Ω^{c}. 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.

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}, pages = {976--1013}, publisher = {EDP-Sciences}, volume = {19}, number = {4}, year = {2013}, doi = {10.1051/cocv/2012041}, mrnumber = {3182677}, zbl = {1283.49049}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2012041/} }

TY - JOUR AU - Dalibard, Anne-Laure AU - Gérard-Varet, David TI - On shape optimization problems involving the fractional laplacian JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 976 EP - 1013 VL - 19 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2012041/ DO - 10.1051/cocv/2012041 LA - en ID - COCV_2013__19_4_976_0 ER -

%0 Journal Article %A Dalibard, Anne-Laure %A Gérard-Varet, David %T On shape optimization problems involving the fractional laplacian %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 976-1013 %V 19 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2012041/ %R 10.1051/cocv/2012041 %G en %F 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://archive.numdam.org/articles/10.1051/cocv/2012041/

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