AI 中文总结
研究 \(\mathbb{R}^n\) 中有界凸域 \(\Omega\) 的第一个 \(p -\) 斯捷克洛夫特征值，通过证明相关不等式并结合等周不等式得出明确上界，还得到了 \(p -\) 拉普拉斯算子在凸域上第一个温茨尔特征值的上界。
AI 中文摘要
对于 \(p\in(1,n]\) 以及任何有界凸域 \(\Omega\subset\mathbb{R}^n\)，我们证明了尖锐不等式 \(\Lambda_p(\Omega):=\frac{W_p(\Omega)}{P(\Omega)V(\Omega)^{p/n}}\geq\omega_n^{-p/n}\)，其中 \(W_p(\Omega)=\int_{\partial\Omega}|x|^p\ dS\)，等式恰好在中心球处成立。结合等周不等式得到明确上界 \(\sigma_{1,p}(\Omega)\leq \frac{A(n,p)}{r(\Omega^*)^{p - 1}}\)，并给出了 \(p -\) 拉普拉斯算子在凸域上第一个温茨尔特征值的明确上界。
英文摘要
For $p\in(1,n]$ and any bounded convex domain $Ω\subset\mathbb{R}^n$, we prove the sharp inequality \[ Λ_p(Ω):=\frac{W_p(Ω)}{P(Ω)V(Ω)^{p/n}}\geqω_n^{-p/n}, \qquad W_p(Ω)=\int_{\partialΩ}|x|^p\ dS, \] with equality holding exactly at centered balls. Combining this with the isoperimetric inequality yields the explicit upper bound \[ σ_{1,p}(Ω)\leq \frac{A(n,p)}{r(Ω^*)^{p-1}}, \] where $Ω^*$ is a ball having the same perimeter as $Ω$, and $A(n,p)=1$ for $1<p\leq 2$, $A(n,p)=n^{p/2-1}$ for $2<p\leq n$. When $p=2$, the result recovers the higher-dimensional Weinstock inequality of Bucur et al. [J. Differential Geom. 2021]. We also obtain an explicit upper bound for the first Wentzell eigenvalue of the $p$-Laplacian on convex domains.
Comments22 pages
