Optimal Szeg\"o-Weinberger type inequalities

Denote with $\mu_{1}(\Omega;e^{h\left(|x|\right)})$ the first nontrivial eigenvalue of the Neumann problem \begin{equation*} \left\{\begin{array}{lll} -\text{div}\left(e^{h\left(|x|\right)}\nabla u\right) =\mu e^{h\left(|x|\right)}u & \text{in} & \Omega & & \frac{\partial u}{\partial \nu}=0 & \text{on} & \partial \Omega , \end{array} \right. \end{equation*} where $\Omega $ is a bounded and Lipschitz domain in $\mathbb{R}^{N}$. Under suitable assumption on $h$ we prove that the ball centered at the origin is the unique set maximizing $\mu_{1}(\Omega;e^{h\left(|x|\right)})$ among all Lipschitz bounded domains $\Omega $ of $\mathbb{R}^{N}$ of prescribed $e^{h\left(|x|\right)}dx$-measure and symmetric about the origin. Moreover, an example in the model case $h\left(|x|\right) =|x|^{2},$ shows that, in general, the assumption on the symmetry of the domain cannot be dropped. In the one-dimensional case, i.e. when $\Omega$ reduces to an interval $(a,b),$ we consider a wide class of weights (including both Gaussian and anti-Gaussian). We then describe the behavior of the eigenvalue as the interval $(a,b)$ slides along the $x$-axis keeping fixed its weighted length.