Optimal Szeg\"o-Weinberger type inequalities

TL;DR

This paper establishes optimal inequalities for the first nontrivial Neumann eigenvalue under weighted measures, proving that centered balls maximize this eigenvalue among symmetric domains, with extensions to one-dimensional weighted intervals.

Contribution

It proves that under certain conditions, centered balls uniquely maximize the weighted Neumann eigenvalue among symmetric domains with fixed measure, extending classical inequalities to weighted settings.

Findings

Centered balls maximize the eigenvalue among symmetric domains.

The eigenvalue behavior is characterized for weighted intervals in 1D.

Symmetry assumptions are essential for the optimality result.

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.