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

Abstract

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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.

How to cite

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Dalibard, 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 -

References

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