论文标题
Neumann特征值的最佳界限
Optimal bounds for Neumann eigenvalues in terms of the diameter
论文作者
论文摘要
在本文中，我们在两种情况下（密切相关）获得了所有Neumann特征值的最佳上限。首先，我们考虑一个一维sturm-liouville特征值问题，其中密度是$ h（x）$的函数，其功能是凹的。我们证明存在$μ_k（h）$的最大化器，我们将其完全表征。 Then we consider the Neumann eigenvalues (for the Laplacian) of a domain $Ω\subset \mathbb{R}^d$ of given diameter and we assume that its profile function (defined as the $d-1$ dimensional measure of the slices orthogonal to a diameter) has also some power that is concave.这包括$ \ mathbb {r}^d $中的凸域的情况，其中包含P.Kröger的先前结果。另一方面，在最后一节中，我们给出了上限未能为真实的域的示例，表明通常是$ \ sup d^2（ω）μ_k（ω）= +\ \ infty $。
In this paper, we obtain optimal upper bounds for all the Neumann eigenvalues in two situations (that are closely related). First we consider a one-dimensional Sturm-Liouville eigenvalue problem where the density is a function $h(x)$ whose some power is concave. We prove existence of a maximizer for $μ_k(h)$ and we completely characterize it. Then we consider the Neumann eigenvalues (for the Laplacian) of a domain $Ω\subset \mathbb{R}^d$ of given diameter and we assume that its profile function (defined as the $d-1$ dimensional measure of the slices orthogonal to a diameter) has also some power that is concave. This includes the case of convex domains in $\mathbb{R}^d$, containing and generalizing previous results by P. Kröger. On the other hand, in the last section, we give examples of domains for which the upper bound fails to be true, showing that, in general, $\sup D^2(Ω)μ_k(Ω)= +\infty$.
