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

We present a fractional counterpart of a generalized Kohler-Jobin inequality, showing that, among all bounded, open sets $\mathrm{\Omega }\subset {\mathbb{R}}^N$ with Lipschitz boundary, having the same fractional torsional rigidity, the first Dirichlet eigenvalue ${\lambda }_1(\mathrm{\Omega })$ 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 ${\lambda }_1(\mathrm{\Omega })$.