On shape optimization problems involving the fractional laplacian

Details On shape optimization problems involving the fractional laplacian On shape optimization problems involving the fractional laplacian

2012-02-22

arxiv.org/abs/1202.4920v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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