Isoperimetric estimates for the first Neumann eigenvalue of Hermite differential equations

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.