Dalibard, Anne-Laure; Gérard-Varet, David On shape optimization problems involving the fractional Laplacian. (English) Zbl 1283.49049 ESAIM, Control Optim. Calc. Var. 19, No. 4, 976-1013 (2013). Summary: Our concern is the computation of optimal shapes in problems involving \((-{\Delta})^{1/2}\). We focus on the energy \(J({\Omega})\) associated to the solution \(u_{{\Omega}}\) of the basic Dirichlet problem \((-{\Delta})^{1/2}u_{{\Omega}} = 1\) in \({\Omega}\), \(u = 0\) in \({\Omega}^{c}\). We show that regular minimizers \({\Omega}\) 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. Cited in 2 ReviewsCited in 18 Documents MSC: 49Q10 Optimization of shapes other than minimal surfaces 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35Q35 PDEs in connection with fluid mechanics Keywords:fractional Laplacian; shape optimization; shape derivative; moving plane method PDFBibTeX XMLCite \textit{A.-L. Dalibard} and \textit{D. Gérard-Varet}, ESAIM, Control Optim. Calc. Var. 19, No. 4, 976--1013 (2013; Zbl 1283.49049) Full Text: DOI arXiv