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Optimal Szegö-Weinberger type inequalities

Denote with $μ_{1}(Ω;e^{h\left(|x|\right)})$ 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) =μe^{h\left(|x|\right)}u & \text{in} & Ω& & \frac{\partial u}{\partial ν}=0 & \text{on} & \partial Ω, \end{array} \right. \end{equation*} where $Ω$ 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 $μ_{1}(Ω;e^{h\left(|x|\right)})$ among all Lipschitz bounded domains $Ω$ 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 $Ω$ 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.