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- W2964069692 abstract "Denote with $mu _{1}(Omega ;e^{h( |x|) })$ the first nontrivialeigenvalue of the Neumann problembegin{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 $mathbb{R}^{N}$. 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." @default.
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- W2964069692 date "2016-01-01" @default.
- W2964069692 modified "2023-09-23" @default.
- W2964069692 title "Optimal Szegö-Weinberger type inequalities" @default.
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- W2964069692 doi "https://doi.org/10.3934/cpaa.2016.15.367" @default.
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