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On shape optimization problems involving the fractional Laplacian. (English) Zbl 1283.49049

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.

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
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