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Optimal Szegö-Weinberger type inequalities

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  • Denote with $\mu _{1}(\Omega ;e^{h( |x|) })$ the first nontrivial eigenvalue of the Neumann problem \begin{eqnarray} &-div( e^{h( |x|) }\nabla u) =\mu e^{h(|x|) }u \quad in \ \Omega \\ &\frac{\partial u}{\partial \nu }=0 \quad on \ \partial \Omega, \end{eqnarray} 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 $\mu _{1}(\Omega ;e^{h( |x|)})$ among all Lipschitz bounded domains $\Omega $ of $\mathbb{R}^{N}$ of prescribed $ e^{h( |x|) }dx$-measure and symmetric about the origin. Moreover, an example in the model case $h( |x|) =|x|^{2},$ 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.
    Mathematics Subject Classification: Primary: 35B45; Secondary: 35P15.

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