E. Aubry, J. Bertrand, B. Colbois
In this paper, we show that the convex domains of the hyperbolic space which are almost extremal for the Faber-Krahn or the Payne-Polya-Weinberger inequalities are close to geodesic balls. Our proof is also valid in other space forms and allows us to recover known results in Euclidean space and on the sphere.
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We derive various pinching results for small Dirac eigenvalues using the classification of $\text{spin}^c$ and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for $\text{spin}^c$ manifolds which involves a general study on convergence of Riemannian manifolds with a principal $\mathbb{S}^1$-bundle. We also analyze the relation between the regularity of the Riemannian metric and the regularity of the curvature of the associated principal $\mathbb{S}^1$-bundle on $\text{spin}^c$ manifolds with Killing spinors.
Lower bounds for the eigenvalues of the Hodge Laplacian on certain non-convex domains
研究非凸域上Hodge拉普拉斯算子最小正特征值的几何下界，推导凸域多孔情况下的高阶精确特征值下界，并改进Cheeger不等式。
Some universal inequalities for Dirichlet eigenvalues of the Laplacian on a Euclidean convex domain
本文建立了欧几里得凸域上拉普拉斯算子狄利克雷本征值的两个通用不等式，探讨了相关数学性质。
A note on the distribution of Neumann eigenvalues of the Laplacian on a Euclidean convex domain
本文研究了欧几里得凸域上拉普拉斯算子Neumann特征值的分布，建立了两个通用不等式。
A universal inequality for Neumann eigenvalues of the Laplacian on a convex domain in Euclidean space
该研究针对欧氏空间有界凸域上的拉普拉斯算子Neumann特征值，得到了新的上界，并借此推导出相应的普适不等式。
Hot spots in convex hyperbolic planar domains with small eigenvalues
本文证明了在具有小Neumann或混合Dirichlet-Neumann特征值的双曲平面域中，Rauch热点猜想的一个变种，得出结论：在双曲平面上足够大的面积有界凸域中，第二Neumann拉普拉斯特征函数没有内部临界点。
The first Steklov eigenvalue on manifolds with nonnegative Ricci curvature and convex boundary
本文为具有非负 Ricci 曲率与凸边界的紧致 Riemannian 流形的第一非零 Steklov 特征值建立了新的下界，并在更弱的几何假设下获得了相关结果。
Two new universal inequalities for Neumann eigenvalues of the Laplacian on a planar convex domain
本文提出两个新通用不等式，用于平面凸域中拉普拉斯算子的Neumann特征值，扩展了相关理论的应用范围。
A quadratic comparison of Neumann eigenvalues on thin convex domains in arbitrary dimenstion
本文研究任意维数薄凸域的Neumann特征值，证明其与加权线段谱的$O(\varepsilon^2)$比较，并指出该二次精度最优。
Multiple eigenvalues and the width
该研究探讨了凸薄域在不同维度下的Neumann特征值简单性，揭示了特征值数量与宽度与直径比值的关系。
Optimal convex domains for the first curl eigenvalue
本文研究了在固定体积下最小化第一curl特征值的最优凸域，证明了该域无法为解析或稳定凸结构，并探讨了在限定区域内的最优域存在性。
On domain monotonicity of Neumann eigenvalues of convex domains
本文研究有界凸区域包含关系下 Neumann Laplacian 特征值的定量单调性，给出第一非平凡特征值的尖锐比较因子及高阶特征值界，并讨论非凸情形与开放问题。
We show that as the ratio between the first Dirichlet eigenvalues of a convex domain and of the ball with the same volume becomes large, the same must happen to the corresponding ratio of isoperimetric constants. The proof is based on the generalization to arbitrary dimensions of Polya and Szego's 1951 upper bound for the first eigenvalue of the Dirichlet Laplacian on planar star-shaped domains which depends on the support function of the domain. As a by-product, we also obtain a sharp upper bound for the spectral gap of convex domains.
A note on domain monotonicity for the Neumann eigenvalues of the Laplacian
本文证明了凸区域及其子区域上拉普拉斯算子诺伊曼特征值的区域单调性变体，并基于该方法给出了凸区域诺伊曼特征值的上界。
Let $Ω\subset \mathbb{R}^2$ be the exterior of a convex polygon whose side lengths are $\ell_1,...,\ell_M$. For $α>0$, let $H^Ω_α$ denote the Laplacian in $Ω$, $u\mapsto -Δu$, with the Robin boundary conditions $\partial u/\partialν=αu$, where $ν$ is the exterior unit normal at the boundary of $Ω$. We show that, for any fixed $m\in\mathbb{N}$, the $m$th eigenvalue $E^Ω_m(α)$ of $H^Ω_α$ behaves as \[ E^Ω_m(α)=-α^2+μ^D_m +\mathcal{O}\Big(\dfrac{1}{\sqrtα}\Big) \quad {as $α$ tends to $+\infty$}, \] where $μ^D_m$ stands for the $m$th eigenvalue of the operator $D_1\oplus...\oplus D_M$ and $D_n$ de…
Sharp ratios for low-index Neumann eigenvalues on convex domains
研究凸域上拉普拉斯算子诺伊曼特征值，证明了\(\mu_2(\Omega)\leq4\mu_1(\Omega)\)和\(\mu_3(\Omega)\leq9\mu_1(\Omega)\)的尖锐界，解决了相关问题并给出一维模型预测的下一个尖锐情形，常数在各维度最优。
We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact set in any dimension, subject to attractive Robin boundary conditions. As an improvement upon our previous work (arXiv:1608.04896, to appear in J. Convex Anal.), we show that under either a constraint of fixed perimeter or area, the maximiser within the class of exteriors of simply connected planar sets is always the exterior of a disk, without the need of convexity assumption. Moreover, we generalise the result to disconnected compact planar sets. Namely, we prove that unde…
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