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Optimal Szeg\"o-Weinberger type inequalities
Analysis of PDEs · Submitted 21 Nov 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 0.93 | 0.24 | 0.03 | asserted 0.71 |
| s1 | \end{equation*} where $\Omega $ is a bounded and Lipschitz domain in $\mathbb{R}^{N}$. | background 0.99 | 0.12 | 0.02 | asserted 0.71 |
| s2 | 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. | result 0.99 | 0.96 | 0.91 | asserted 1.00 |
| s3 | 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. | limitation 0.61 | 0.92 | 0.07 | asserted 0.99 |
| s4 | 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). | contribution 0.93 | 0.82 | 0.02 | asserted 0.88 |
| s5 | We then describe the behavior of the eigenvalue as the interval $(a,b)$ slides along the $x$-axis keeping fixed its weighted length. | contribution 0.91 | 0.92 | 0.04 | asserted 0.92 |
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Original abstract
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.
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For related results see also [ref] and [ref] .
The equality case in a Poincar\'e-Wirtinger type inequality · Introduction · 0.88
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