Optimal Szegö-Weinberger type inequalities

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Denote with μ₁(Ω;eh(|x|)) the first nontrivial eigenvalue of the Neumann problem equation* {arraylll -div(eh(|x|)∇ u) =μeh(|x|)u & in & Ω& & ∂ u/∂ ν=0 & on & ∂ Ω, array. equation* where Ω is a bounded and Lipschitz domain in Rⁿ. Under suitable assumption on h we prove that the ball centered at the origin is the unique set maximizing μ₁(Ω;eh(|x|)) among all Lipschitz bounded domains Ω of Rⁿ of prescribed eh(|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 Ω 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.