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.