非钝多面体区域中的尖锐Neumann特征值估计与$C^2$椭圆正则性
Sharp Neumann eigenvalue estimates and $C^2$ elliptic regularity in non-obtuse polyhedral domains
AI总结：
该研究证明了非钝多面体区域中关于Neumann谱的断言$\mathbf{(P_n)}$与椭圆正则性的断言$\mathbf{(Q_n)}$相互蕴含，进而得到其在所有维度均成立，给出了最优Neumann特征值下界与$C^2$椭圆正则性。
AI中文摘要：
对于整数$n\ge1$，考虑两个断言：$\mathbf{(P_n)}$：设$\Omega \subset S^n$是由全测地$S^{n-1}$围成、具有非钝二面角的球域，则在区间$[0,2(n+1)]$内，其Neumann谱仅能取$\{0,n,2(n+1)\}$中的值；此外，$n$是Neumann特征值当且仅当对应的特征函数是$\mathbb{R}^{n+1}$中线性函数的限制，而$2(n+2)$是Neumann特征值当且仅当对应的特征函数是$\mathbb{R}^{n+1}$中二次多项式的限制。$\mathbf{(Q_n)}$：在$\mathbb{R}^n$中具有非钝二面角的锥形多面体区域$\Omega$上，满足Neumann边界条件的$\Delta u = f$的弱解$u$，若$f$是Hölder连续的，则$u$属于$C^{2,\alpha}_{loc}(\overline{\Omega})$。我们证明蕴含关系$\mathbf{(Q_n)} \Rightarrow \mathbf{(P_n)}$和$\mathbf{(P_n)} \Rightarrow \mathbf{(Q_{n+1})}$，因此两个断言在所有维度均成立，这给出了非钝Riemannian多面体区域中的最优Neumann特征值下界与$C^2$椭圆正则性。
英文摘要：
For integers $n\ge 1$, consider assertions: $\mathbf{(P_n)}$: Let $Ω\subset S^n$ be a spherical domain enclosed by totally geodesic $S^{n-1}$'s with non-obtuse dihedral angles. Then in $[0,2(n+1)]$, its Neumann spectrum can only take values among $\{0,n,2(n+1)\}$. Moreover, $n$ is a Neumann eigenvalue if and only if the corresponding eigenfunction is the restriction of a linear function in $\mathbb{R}^{n+1}$, while $2(n+2)$ is a Neumann eigenvalue if and only if the corresponding eigenfunction is restriction of a quadratic polynomial in $\mathbb{R}^{n+1}$. $\mathbf{(Q_n)}$: A weak solution $u$ to $Δu = f$ with the Neumann boundary condition, with $f$ Hölder continuous, in a conical polyhedral domain $Ω$ in $\mathbb{R}^n$ with non-obtuse dihedral angles, is in $C^{2,α}_{loc}(\overlineΩ)$. We prove the implications \[\mathbf{(Q_n)} \Rightarrow \mathbf{(P_n)},\qquad \mathbf{(P_n)}\Rightarrow \mathbf{(Q_{n+1})}.\] Consequently, both assertions hold in all dimensions. These give the optimal Neumann eigenvalue lower bound and $C^2$ elliptic regularity in non-obtuse Riemannian polyhedral domains.
