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

TL;DR

This paper extends classical isoperimetric inequalities to the fractional Laplacian, demonstrating that among sets with equal fractional torsional rigidity, balls minimize the first eigenvalue, and establishing related inequalities for eigenfunctions.

Contribution

It introduces a fractional version of the Kohler-Jobin inequality and proves new isoperimetric inequalities for the fractional Laplacian.

Findings

Balls minimize the first eigenvalue for fixed fractional torsional rigidity.

Established a reverse H"older inequality for eigenfunctions.

Extended classical inequalities to fractional Laplacian setting.

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 $\lambda_1(\Omega)$.