Kohler-Jobin inequality for $p$-Laplace operator in the Gauss space

TL;DR

This paper extends the Kohler-Jobin inequality to the $p$-Laplace operator in Gaussian space, providing sharp bounds for eigenvalues based on torsional rigidity and establishing related inequalities.

Contribution

It introduces a generalized spectral inequality for the $p$-Laplace operator in Gaussian space, extending classical results and establishing new bounds and inequalities.

Findings

Derived a sharp lower bound for the first Dirichlet eigenvalue in Gaussian space.

Extended the classical Kohler-Jobin spectral inequality to the $p$-Laplace operator.

Established a Payne-Rayner type inequality in this context.

Abstract

A sharp lower bound for the first Dirichlet eigenvalue of the $p$-laplacian in Gaussian space is derived for sets with prescribed generalized torsional rigidity. The result provides an extension of the classical spectral inequality due to Kohler-Jobin. The proof is based on a careful analysis of the generalized torsional rigidity and on a sharp mass comparison result. Furthermore, a Payne-Rayner type inequality is established.