First Eigenvalue and Torsional Rigidity: Isoperimetric Inequalities for the Fractional Laplacian

Abstract: We present a fractional counterpart of a generalized Kohler-Jobin inequality, showing that, among all bounded, open sets $Ω\subset \mathbb{R}<sup>N$ with Lipschitz boundary, having the same fractional torsional rigidity, the first Dirichlet eigenvalue $λ_1(Ω)$ of the fractional Laplacian attains its minimum on balls. With the same arguments we also establish a reverse Hölder inequality for an eigenfunction corresponding to $λ_1(Ω)$.