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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.
@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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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, Tome 19 (2013) no. 4, pp. 976-1013. doi: 10.1051/cocv/2012041
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