论文标题
pólya型的不平等现象和半球的不平等现象
Pólya-type inequalities on spheres and hemispheres
论文作者
论文摘要
给定$ n- $ spheres或$ - $ hemispheres的laplace-beltrami操作员的特征值$λ$，带有多重性$ m $，因此$λ=λ_{k} = \ dots =λ_ $ \ left \ {k，\ dots，k+m-1 \ right \} $，pólya的猜想持有和失败。特别是，我们表明，在Neumann案中，Pólya的猜想适用于半球，但在$ n $大于两个大的情况下，在Dirichlet案件中没有。我们通过添加一个校正项，为所有特征值提供尖锐的下限和上限，进一步得出了Pólya-type的不平等。这使我们能够测量球体和半球特征值Weyl渐近学中的主要项的偏差。直接结果，我们为瓷砖半球的域获得了相似的结果。我们还分别获得了$ \ mathbb {s}^2 $和$ \ mathbb {s}^4 $的直接和反转li-yau不等式。
Given an eigenvalue $λ$ of the Laplace-Beltrami operator on $n-$spheres or $-$hemispheres, with multiplicity $m$ such that $λ=λ_{k}=\dots = λ_{k+m-1}$, we characterise the lowest and highest orders in the set $\left\{k,\dots,k+m-1\right\}$ for which Pólya's conjecture holds and fails. In particular, we show that Pólya's conjecture holds for hemispheres in the Neumann case, but not in the Dirichlet case when $n$ is greater than two. We further derive Pólya-type inequalities by adding a correction term providing sharp lower and upper bounds for all eigenvalues. This allows us to measure the deviation from the leading term in the Weyl asymptotics for eigenvalues on spheres and hemispheres. As a direct consequence, we obtain similar results for domains which tile hemispheres. We also obtain direct and reversed Li-Yau inequalities for $\mathbb{S}^2$ and $\mathbb{S}^4$, respectively.
