On shape optimization problems involving the fractional laplacian

TL;DR

This paper investigates the shape optimization problem involving the fractional Laplacian, demonstrating that regular minimizers under volume constraints are disks through explicit shape derivative computation and a refined moving plane method.

Contribution

It introduces a novel explicit computation of the shape derivative in the fractional setting and applies a refined moving plane method to prove disk optimality.

Findings

Regular minimizers are disks under volume constraints.

Explicit shape derivative computation in fractional context.

Application of a refined moving plane method.

Abstract

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.