论文标题
在Pólya的猜想上，圆形部门和球
On the Pólya conjecture for circular sectors and for balls
论文作者
论文摘要
1954年，G。Polya猜想，在有限的$ω\ subset r^d $中，计数功能$ n（ω，λ）$ $ n（ω，λ）$ n dirichlet（neumann）的dirichlet（neumann）边界价值问题问题的特征值$ω\ subset r^d $比$（resp。 λ^{d/2} $。这里$λ$是光谱参数，$ω_d$是单位球的体积。我们证明了对任何圆形部门的Dirichlet和Neumann边界问题的猜想，以及用于任意维度的Dirichlet问题。我们大量使用\ cite {lps}的想法。
In 1954, G. Polya conjectured that the counting function $N(Ω,Λ)$ of the eigenvalues of the Laplace operator of the Dirichlet (resp. Neumann) boundary value problem in a bounded set $Ω\subset R^d$ is lesser (resp. greater) than $(2π)^{-d} ω_d |Ω| Λ^{d/2}$. Here $Λ$ is the spectral parameter, and $ω_d$ is the volume of the unit ball. We prove this conjecture for both Dirichlet and Neumann boundary problems for any circular sector, and for the Dirichlet problem for a ball of arbitrary dimension. We heavily use the ideas from \cite{LPS}.
