Isoperimetric inequalities for the first Neumann eigenvalue in Gauss space

Abstract

We provide isoperimetric Szegö–Weinberger type inequalities for the first nontrivial Neumann eigenvalue ${\mu }_1(\Omega )$ in Gauss space, where Ω is a possibly unbounded domain of ${\mathbb{R}}^N$. Our main result consists in showing that among all sets Ω of ${\mathbb{R}}^N$ symmetric about the origin, having prescribed Gaussian measure, ${\mu }_1(\Omega )$ is maximum if and only if Ω is the Euclidean ball centered at the origin.