论文标题
球体上的Neumann特征值的急剧不平等
Sharp inequalities for Neumann eigenvalues on the sphere
论文作者
论文摘要
We prove that the second nontrivial Neumann eigenvalue of the Laplace-Beltrami operator on the unit sphere $\mathbb{S}^n \subseteq \mathbb{R}^{n+1}$ is maximized by the union of two disjoint, equal, geodesic balls among all subsets of $\mathbb{S}^n$ of规定的数量。实际上，结果具有更强大的版本，涉及订单$ 2 $至$ n $的特征值的谐波平均值，并扩展到密度。 A（令人惊讶的）后果发生在第一个非平凡特征值下在体积约束下的地理球的最大性：Ashbaugh-Benguria结果的半球包含条件可以放松到较弱的较弱的情况下，即与规定的体积的地理球相交。尽管我们没有证明最后一个纳入结果是尖锐的，因为质量小于球的一半，但我们从数值上识别出比相应的测量球更高的密度，并且具有等于全球$ \ mathbb {s s}^2 $的支撑。
We prove that the second nontrivial Neumann eigenvalue of the Laplace-Beltrami operator on the unit sphere $\mathbb{S}^n \subseteq \mathbb{R}^{n+1}$ is maximized by the union of two disjoint, equal, geodesic balls among all subsets of $\mathbb{S}^n$ of prescribed volume. In fact, the result holds in a stronger version, involving the harmonic mean of the eigenvalues of order $2$ to $n$, and extends to densities. A (surprising) consequence occurs on the maximality of a geodesic ball for the first nontrivial eigenvalue under the volume constraint: the hemisphere inclusion condition of the Ashbaugh-Benguria result can be relaxed to a weaker one, namely empty intersection with a geodesic ball of the prescribed volume. Although we do not prove that this last inclusion result is sharp, for a mass less than the half of the sphere, we numerically identify a density with higher first eigenvalue than the corresponding geodesic ball and with support equal to the full sphere $\mathbb{S}^2$.
