AI 中文总结
研究欧几里得球上诺伊曼拉普拉斯算子的波利亚猜想下界，通过严格罗宾比较、变分界等方法，在不同频率范围进行估计，最终证明该猜想，有限有理计算辅助证明。
AI 中文摘要
我们证明了欧几里得球的诺伊曼计数函数的波利亚猜想下界。若\(d\geq2\)，\(R>0\)且\(E\geq0\)，则有相应不等式。在\(d\geq3\)时径向边界条件为迪尼条件而非导数为零条件。通过严格的罗宾比较将其简化为贝塞尔相位估计，变分界处理低频，基于有限多个径向水平和贝塔矩的估计统一覆盖中间范围，高频估计明确，有限有理计算在证明附录中。
英文摘要
We prove Pólya's conjectured lower bound for the Neumann eigenvalue counting function of Euclidean balls. If $B_R^d\subset\mathbb R^d$ is the ball of radius $R$, then, for every $d\ge2$, $R>0$, and $E\ge0$, $$ N_{B_R^d}^{<}(E) \ge \frac{ω_d}{(2π)^d}|B_R^d|E^{d/2} = \frac{(R\sqrt E)^d}{2^dΓ(\frac d2+1)^2}, $$ where $ω_d$ is the volume of the unit $d$-ball and $N_{B_R^d}^{<}(E)$ counts Neumann eigenvalues strictly below $E$. Combined with the Dirichlet theorem for balls, this settles both Pólya inequalities for Euclidean balls in every dimension $d\ge2$. In the disk case, the proof replaces a computer-assisted finite-frequency step by explicit Rayleigh--Ritz estimates. In dimensions $d\ge3$, the radial Neumann condition is a Dini condition rather than a derivative-zero Bessel condition. A strict comparison with an auxiliary Robin problem transfers a derivative-zero Bessel phase estimate to the physical Neumann spectrum. The problem then becomes a comparison between a multiplicity-weighted phase staircase and an integral equal to the Weyl term. Variational trial spaces control low frequencies; finitely many radial levels and beta-integral estimates cover the intermediate range; and a uniform phase estimate treats high frequencies. All finite computations for $2\le d\le6$ are printed in the paper. For $d\ge7$, one compact two-parameter estimate is verified in exact rational arithmetic by the ancillary program.
Comments72 pages
