Optimal Szegö-Weinberger type inequalities
Gabriella Di Blasio, Francesco Chiacchio, Friedemann Brock · Communications on Pure & Applied Analysis · 2016
Denote with $\mu _{1}(\Omega ;e^{h( |x|) })$ the first nontrivialeigenvalue of the Neumann problem\begin{eqnarray}&-div( e^{h( |x|) } abla u) =\mu e^{h(|x|) }u \quad in \ \Omega \\&\frac{\partial u}{\partial u }=0 \quad on \ \partial \Omega,\end{eqnarray}where $\Omega $ is a bounded and Lipschitz domain in $\mathbb{R}^{N}$. Undersuitable assumption on $h$ we prove that the ball centered at the origin isthe unique set maximizing $\mu _{1}(\Omega ;e^{h( |x|)})$ amongall 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, ingeneral, the assumption on the symmetry of the domain cannot be dropped. Inthe one-dimensional case, i.e. when $\Omega $ reduces to an interval $(a,b),$ we consider a wide class of weights (including both Gaussian andanti-Gaussian). We then describe the behavior of the eigenvalue as theinterval $(a,b)$ slides along the $x$-axis keeping fixed its weighted length.