图上Dirichlet、Neumann与Laplacian特征值的比较及其应用

Gradient and Hessian Estimates for Dirichlet and Neumann Eigenfunctions

We establish integral formulas and sharp two-sided bounds for the Ricci curvature, mean curvature and second fundamental form on a Riemannian manifold with boundary. As applications, sharp gradient and Hessian estimates are derived for the Dirichlet and Neumann eigenfunctions.

带位势的Witten-Laplace算子相关Dirichlet-to-Neumann映射的谱不变量

Spectral invariants of the Dirichlet-to-Neumann map associated to the Witten-Laplacian with potential

本研究针对带光滑边界的紧连通黎曼流形，提出计算带位势的Witten-Laplace算子相关Dirichlet-to-Neumann映射谱渐近公式全系数的有效方法，显式给出前四个谱不变量，可提供流形体积、曲率等信息。

共形平坦黎曼流形上Dirichlet Laplacian特征值的普遍不等式

Universal inequalities for eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds

本文研究了共形平坦黎曼流形上Dirichlet Laplacian的特征值问题，并在双曲空间$\mathbb{H}^n(-1)$上建立了特征值的普遍不等式。

On the spectrum of the Dirichlet-to-Neumann operator acting on forms of a Euclidean domain

We compute the whole spectrum of the Dirichlet-to-Neumann operator acting on differential p-forms on the unit Euclidean ball. Then, we prove a new upper bound for its first eigenvalue on a domain $Ω$ in Euclidean space in terms of the isoperimetric ratio ${\rm Vol}(\bdΩ)/{\rm Vol}(Ω)$.

On the Dirichlet-to-Neumann operator for the Connection Laplacian

We study the relationship between the symbol of the Dirichlet-to-Neumann operator associated with a connection Laplacian, and the geometry on and near the boundary. As a consequence, we show that the geometric data on the boundary, and when the dimension of the base is greater than two all corresponding normal derivatives, are determined by the symbol.

Isoperimetric relations between Dirichlet and Neumann eigenvalues

Inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian have received much attention in the literature, but open problems abound. Here, we study the number of Neumann eigenvalues no greater than the first Dirichlet eigenvalue. Based on a combination of analytical and numerical results, we conjecture that this number is controlled by the isoperimetric ratio of the domain. This has applications to the nodal deficiency of eigenfunctions and is closely related to a long-standing conjecture of Yau on the Hausdorff measure of nodal sets.

Non-compactness of the Neumann operator for the Kohn Laplacian on the Heisenberg ball

For $1\leq q \leq n-2$, we provide explicit examples to demonstrate non-compactness of the Neumann operator for the Kohn Laplacian acting on $L^2$ $(0,q)$-forms on the unit ball in $(2n+1)$-dimensional Heisenberg space.

通过弹性 Dirichlet-to-Neumann 映射确定 Riemannian 度量的边界

Boundary determination of the Riemannian metric by the elastic Dirichlet-to-Neumann map

本文研究弹性 Dirichlet-to-Neumann 映射的边界逆问题，通过计算其全象征，证明它可唯一确定边界度量的任意阶偏导数。

关于非紧流形上 $p$-拉普拉斯算子第一个狄利克雷特征值的估计

Estimate for the first Dirichlet eigenvalue of $p-$Laplacian on non-compact manifolds

本文研究了非紧黎曼流形上 $p$-拉普拉斯算子第一个狄利克雷特征值的估计问题，得出了具有非负里奇曲率的完整非紧流形上的精确下界结果。

超几何空间上狄利克雷拉普拉斯算子特征值的新不等式

New inequalities for eigenvalues of the Dirichlet Laplacian on the hyperbolic space

本文基于对狄利克雷拉普拉斯算子特征值通用不等式的研究，证明了超几何空间上相关特征值的新不等式，并验证了陈猜想在两种特殊有界域中的成立，允许ε损失。

Dirichlet-to-Neumann Map for Poincaré-Einstein Metrics in Even Dimensions

We study the linearization of the Dirichlet-to-Neumann map for Poincaré-Einstein metrics in even dimensions on an arbitrary compact manifold with boundary. By fixing a suitable gauge, we make the linearized Einstein equation elliptic. In this gauge the linearization of the Dirichlet-to-Neumann map appears as the scattering matrix for an elliptic operator of 0-type, modified by some differential operators. We study the scattering matrix by using the 0-calculus and generalize a result of Graham for the case of the standard hyperbolic metric on a ball.

非线性Neumann边界问题的n-拉普拉斯李乌利维尔方程在半空间上的解

Nonlinear Neumann boundary problems for $n$-Laplacian Liouville equation on a half space

本文研究了n-拉普拉斯李乌利维尔方程在半空间上正非线性Neumann边界条件下的解的分类，将二次阶结果推广到一般n≥2，并将临界p-拉普拉斯方程结果推广到p=n。

各向异性弹性 Dirichlet-to-Neumann 映射中的规范自由度

Gauge freedoms in the anisotropic elastic Dirichlet-to-Neumann map

本文研究从 Dirichlet-to-Neumann 映射恢复弹性参数的反问题，通过分析坐标变换下方程的不变性，揭示了黎曼弹性波方程及其映射中的规范自由度。

The complete Dirichlet-to-Neumann map for differential forms

The Dirichlet-to-Neumann map for differential forms on a Riemannian manifold with boundary is a generalization of the classical Dirichlet-to-Neumann map which arises in the problem of Electrical Impedance Tomography. We synthesize the two different approaches to defining this operator by giving an invariant definition of the complete Dirichlet-to-Neumann map for differential forms in terms of two linear operators Φ and Ψ. The pair (Φ, Ψ) is equivalent to Joshi and Lionheart's operator Π and determines Belishev and Sharafutdinov's operator Λ. We show that the Betti numbers of the manifold are d…

Dirichlet p-Laplacian eigenvalues and Cheeger constants on symmetric graphs

In this paper, we study eigenvalues and eigenfunctions of $p$-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of $p$-Laplacian, as $p\to 1,$ we identify the Cheeger constant of a symmetric graph with that of the quotient graph. By this approach, we calculate various Cheeger constants of spherically symmetric graphs.