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 $\Omega\subset \mathbb{R}^N$ with Lipschitz boundary, having the same fractional torsional rigidity, the first Dirichlet eigenvalue $\lambda_1(\Omega)$ of the fractional Laplacian attains its minimum on balls. With the same arguments we also establish a reverse H\"older inequality for an eigenfunction corresponding to $\lambda_1(\Omega)$.