On shape optimization problems involving the fractional laplacian
Year: 2012
Type: preprint
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 $u\_Ω$ 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.
Source: arXiv (Cornell University)
Authors: Dalibard, Anne-Laure, Gérard-Varet, David
Institution: Département de mathématiques et applications
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Agence Nationale de la Recherche
OpenAlex ID: W4297569555