Isoperimetric estimates for the first Neumann eigenvalue of Hermite differential equations

TL;DR

This paper establishes isoperimetric inequalities for the first Neumann eigenvalue in Gaussian space, demonstrating that symmetric Euclidean balls maximize this eigenvalue among sets with fixed Gaussian measure.

Contribution

It proves that among symmetric sets with fixed Gaussian measure, the Euclidean ball maximizes the first Neumann eigenvalue in Gauss space, extending classical isoperimetric results.

Findings

Euclidean balls maximize the first Neumann eigenvalue in Gaussian space.

The result applies to possibly unbounded symmetric domains.

Provides isoperimetric inequalities for eigenvalues in Gauss space.

Abstract

We provide isoperimetric Szeg\"{o}-Weinberger type inequalities for the first nontrivial Neumann eigenvalue $\mu_{1}(\Omega)$ in Gauss space, where $\Omega$ is a possibly unbounded domain of $\mathbb{R}^{N}$. Our main result consists in showing that among all sets of $\mathbb{R}^{N}$ symmetric about the origin, having prescribed Gaussian measure, $\mu_{1}(\Omega)$ is maximum if and only if $\Omega$ is the euclidean ball centered at the origin.