# On shape optimization problems involving the fractional laplacian

Anne-Laure Dalibard; David Gérard-Varet

ESAIM: Control, Optimisation and Calculus of Variations (2013)

- Volume: 19, Issue: 4, page 976-1013
- ISSN: 1292-8119

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topDalibard, Anne-Laure, and Gérard-Varet, David. "On shape optimization problems involving the fractional laplacian." ESAIM: Control, Optimisation and Calculus of Variations 19.4 (2013): 976-1013. <http://eudml.org/doc/272782>.

@article{Dalibard2013,

abstract = {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.},

author = {Dalibard, Anne-Laure, Gérard-Varet, David},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {fractional laplacian; fhape optimization; shape derivative; moving plane method; fractional Laplacian; shape optimization},

language = {eng},

number = {4},

pages = {976-1013},

publisher = {EDP-Sciences},

title = {On shape optimization problems involving the fractional laplacian},

url = {http://eudml.org/doc/272782},

volume = {19},

year = {2013},

}

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

PB - EDP-Sciences

VL - 19

IS - 4

SP - 976

EP - 1013

AB - 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.

LA - eng

KW - fractional laplacian; fhape optimization; shape derivative; moving plane method; fractional Laplacian; shape optimization

UR - http://eudml.org/doc/272782

ER -

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