On the fundamental eigenvalue ratio of the p-Laplacian

Small volume expansion of the splitting of multiple Neumann Laplacian eigenvalues due to a grounded inclusion in two dimensions

The first terms of the small volume asymptotic expansion for the splitting of Neumann boundary condition Laplacian eigenvalues due to a grounded inclusion of size ε are derived. An explicit formula to compute the first term from the eigenvalues and eigenfunctions of the unperturbed domain, the inclusion size and position is given. As a consequence, when an eigenvalue of double multiplicity splits in two distinct eigenvalues, one decays like O(1/log(ε)), the other like O(ε^2).

Nonlocal Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities

The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $Θ$ which give rise to self-adjoint Laplacians $-Δ_{Θ, Ω}$ in $L^2(Ω; d^n x)$ with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains $Ω\subset\bbR^n$, $n\in\bbN$, $n\geq 2$. Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains $Ω$, following an approach introduced by Filonov for this type of problems.

关于加权拉普拉斯算子第二罗宾特征值的等周不等式

An isoperimetric inequality for the second Robin eigenvalue of the Weighted Laplacian

本文证明在特定条件下，球体是最大化加权拉普拉斯算子第二罗宾特征值的最优形状。

关于欧几里得凸域上拉普拉斯算子狄利克雷本征值的一些通用不等式

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。

H$^+$-Eigenvalues of Laplacian and Signless Laplacian Tensors

We propose a simple and natural definition for the Laplacian and the signless Laplacian tensors of a uniform hypergraph. We study their H$^+$-eigenvalues, i.e., H-eigenvalues with nonnegative H-eigenvectors, and H$^{++}$-eigenvalues, i.e., H-eigenvalues with positive H-eigenvectors. We show that each of the Laplacian tensor, the signless Laplacian tensor and the adjacency tensor has at most one H$^{++}$-eigenvalue, but has several other H$^+$-eigenvalues. We identify their largest and smallest H$^+$-eigenvalues, and establish some maximum and minimum properties of these H$^+$-eigenvalues. We t…

关于欧几里得凸域上拉普拉斯算子Neumann特征值分布的注记

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。

Lower bounds for the first eigenvalue of $p$-Laplacian on Kähler manifolds

We study the eigenvalue problem for the $p$-Laplacian on Kähler manifolds. Our first result is a lower bound for the first nonzero eigenvalue of the $p$-Laplacian on compact Kähler manifolds in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature for $p\in (1, 2]$. Our second result is a sharp lower bound for the first Dirichlet eigenvalue of the $p$-Laplacian on compact Kähler manifolds with smooth boundary for $p\in (1, \infty)$. Our results generalize corresponding results for the Laplace eigenvalues on Kähler manifolds proved in […

On the honeycomb conjecture for Robin Laplacian eigenvalues

We prove that the optimal cluster problem for the sum of the first Robin eigenvalue of the Laplacian, in the limit of a large number of convex cells, is asymptotically solved by (the Cheeger sets of) the honeycomb of regular hexagons. The same result is established for the Robin torsional rigidity.

Asymptotic behavior for the radial eigenvalues of p-Laplacian in certain annular domains

In this paper we prove an asymptotic behavior for the radial eigenvalues to the Dirichlet $p$-Laplacian problem $-Δ_p\,u = λ\,|u|^{p-2}u$ in $Ω$, $u=0$ on $\partialΩ$, where $Ω$ is an annular domain $Ω=Ω_{R,\overline{R}}$ in $\mathbb{R}^N$.

The Largest Laplacian and Signless Laplacian H-Eigenvalues of a Uniform Hypergraph

In this paper, we show that the largest Laplacian H-eigenvalue of a $k$-uniform nontrivial hypergraph is strictly larger than the maximum degree when $k$ is even. A tight lower bound for this eigenvalue is given. For a connected even-uniform hypergraph, this lower bound is achieved if and only if it is a hyperstar. However, when $k$ is odd, it happens that the largest Laplacian H-eigenvalue is equal to the maximum degree, which is a tight lower bound. On the other hand, tight upper and lower bounds for the largest signless Laplacian H-eigenvalue of a $k$-uniform connected hypergraph are given.…

Lower bounds for the first eigenvalue of the Laplacian on Kähler manifolds

We establish lower bound for the first nonzero eigenvalue of the Laplacian on a closed Kähler manifold in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature. On compact Kähler manifolds with boundary, we prove lower bounds for the first nonzero Neumann or Dirichlet eigenvalue in terms of geometric data. Our results are Kähler analogues of well-known results for Riemannian manifolds.

Eigenvalue bounds for the fractional Laplacian: A review

We review some recent results on eigenvalues of fractional Laplacians and fractional Schrödinger operators. We discuss, in particular, Lieb-Thirring inequalities and their generalizations, as well as semi-classical asymptotics.

Mixed Eigenvalues of {\LARGE $\pmb p$\,}-Laplacian

The mixed principal eigenvalue of $p\,$-Laplacian (equivalently, the optimal constant of weighted Hardy inequality in $L^p$ space) is studied in this paper. Several variational formulas for the eigenvalue are presented. As applications of the formulas, a criterion for the positivity of the eigenvalue is obtained. Furthermore, an approximating procedure and some explicit estimates are presented case by case. An example is included to illustrate the power of the results of the paper.

On the second Robin eigenvalue of the Laplacian

We study the Robin eigenvalue problem for the Laplace-Beltrami operator on Riemannian manifolds. Our first result is a comparison theorem for the second Robin eigenvalue on geodesic balls in manifolds whose sectional curvatures are bounded from above. Our second result asserts that geodesic balls in nonpositively curved space forms maximize the second Robin eigenvalue among bounded domains of the same volume.

平面凸域上拉普拉斯算子Neumann特征值的两个新通用不等式

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。

First Robin Eigenvalue of the $p$-Laplacian on Riemannian Manifolds

We consider the first Robin eigenvalue $ł_p(M,\a)$ for the $p$-Laplacian on a compact Riemannian manifold $M$ with nonempty smooth boundary, with $\a \in \R$ being the Robin parameter. Firstly, we prove eigenvalue comparison theorems of Cheng type for $ł_p(M,\a)$. Secondly, when $\a>0$ we establish sharp lower bound of $ł_p(M,\a)$ in terms of dimension, inradius, Ricci curvature lower bound and boundary mean curvature lower bound, via comparison with an associated one-dimensional eigenvalue problem. The lower bound becomes an upper bound when $\a<0$. Our results cover corresponding comparison …

关于拉普拉斯算子诺伊曼特征值区域单调性的一个注记

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。

Comparison of Steklov eigenvalues and Laplacian eigenvalues on graphs

In this paper, we obtain a comparison of Steklov eigenvalues and Laplacian eigenvalues on graphs and discuss its rigidity. As applications of the comparison of eigenvalues, we obtain Lichnerowicz-type estimates and some combinatorial estimates for Steklov eigenvalues on graphs.

A new class of eigenvalue problems for perturbed p-Laplacians

This paper is devoted to a dispersion analysis of a class of perturbed p-Laplacians. Besides the p-Laplacian-like eigenvalue problems we also deal with new and non-standard eigenvalue problems, which can not be solved by the methods used in nonlinear eigenvalue problems for p-Laplacians and similar operators. Original techniques are suggested for solving these new problems (see Section 3). In addition, dispersion relations between the eigen-parameters, quantitative analysis of eigenvectors and variational principles for eigenvalues of perturbed p-Laplacians are also studied in this paper. The …

Absence of embedded eigenvalues for translationally invariant magnetic Laplacians

Translationnally invariant bidimensional magnetic Laplacians are considered. Using an improved version of the harmonic approximation, we establish the absence of point spectrum under various assumptions on the behavior of the magnetic field.

Zero and negative eigenvalues of the conformal Laplacian

We show that zero is not an eigenvalue of the conformal Laplacian for generic Riemannian metrics. We also discuss non-compactness for sequences of metrics with growing number of negative eigenvalues of the conformal Laplacian.

Eigenvalue inequalities for the Laplacian with mixed boundary conditions

Inequalities for the eigenvalues of the (negative) Laplacian subject to mixed boundary conditions on polyhedral and more general bounded domains are established. The eigenvalues subject to a Dirichlet boundary condition on a part of the boundary and a Neumann boundary condition on the remainder of the boundary are estimated in terms of either Dirichlet or Neumann eigenvalues. The results complement several classical inequalities between Dirichlet and Neumann eigenvalues due to Pólya, Payne, Levine and Weinberger, Friedlander, and others.

Estimates for the Lowest Eigenvalue of Magnetic Laplacians

We prove various estimates for the first eigenvalue of the magnetic Dirichlet Laplacian on a bounded domain in two dimensions. When the magnetic field is constant, we give lower and upper bounds in terms of geometric quantities of the domain. We furthermore prove a lower bound for the first magnetic Neumann eigenvalue in the case of constant field.

Computing all Laplacian H-eigenvalues for a k-uniform loose path of length three

The spectral theory of Laplacian tensor is an important tool for revealing some important properties of a hypergraph. It is meaningful to compute all Laplacian H-eigenvalues for some special $k$-uniform hypergraphs. For an odd-uniform loose path of length three, the Laplacian H-spectrum has been studied. However, all Laplacian H-eigenvalues of the class of loose paths have not been found out. In this paper, we compute all Laplacian H-eigenvalues for the class of loose paths. We show that the number of Laplacian H-eigenvalues of an odd(even)-uniform loose path with length three is $7$($14$). So…

On the multiplicity of the second eigenvalue of the Laplacian in non simply connected domains--with some numerics--

We revisit an interesting example proposed by Maria Hoffmann-Ostenhof, the second author and Nikolai Nadirashvili of a bounded domain in R2 for which the second eigenvalue of the Dirichlet Laplacian has multiplicity three. We also analyze carefully the first eigenvalues of the Laplacian in the case of the disk with two symmetric cracks placed on a smaller concentric disk in function of their size.

Inequalities between the lowest eigenvalues of Laplacians with mixed boundary conditions

The eigenvalue problem for the Laplacian on bounded, planar, convex domains with mixed boundary conditions is considered, where a Dirichlet boundary condition is imposed on a part of the boundary and a Neumann boundary condition on its complement. Given two different such choices of boundary conditions for the same domain, we prove inequalities between their lowest eigenvalues. As a special case, we prove parts of a conjecture on the order of mixed eigenvalues of triangles.

Eigenvalues of Dirichlet Laplacian within the class of open sets with constant diameter

This paper is about a shape optimization problem related to the Dirichlet Laplacian eingevalues in the Euclidean plane. More precisely we study the shape of the minimizer in the class of open sets of constant width. We prove that the disk is not a local minimizer except for a limited number of eigenvalues.

Complementary asymptotically sharp estimates for eigenvalue means of Laplacians

We present asymptotically sharp inequalities, containing a second term, for the Dirichlet and Neumann eigenvalues of the Laplacian on a domain, which are complementary to the familiar Berezin-Li-Yau and Kröger inequalities in the limit as the eigenvalues tend to infinity. We accomplish this in the framework of the Riesz mean $R_1(z)$ of the eigenvalues by applying the averaged variational principle with families of test functions that have been corrected for boundary behaviour.

伪拉普拉斯算子的小特征值

Small eigenvalues of pseudo-Laplacians

本文扩展了Otal-Rosas对超曲面小特征值的界限到伪拉普拉斯算子，并将Colin de Verdière的谱理论扩展到具有多个孔的超曲面。

Remarks on Bounds of Normalized Laplacian Eigenvalues of Graphs

Let $G$ be a connected undirected graph with $n$, $n\ge 3$, vertices and $m$ edges. Denote by $ρ_1 \ge ρ_2 \ge \cdots > ρ_n =0$ the normalized Laplacian eigenvalues of $G$. Upper and lower bounds of $ρ_i$, $i=1,2,\ldots , n-1$, are determined in terms of $n$ and general Randi\' c index, $R_{-1}$.

The Eigenvectors of the Zero Laplacian and Signless Laplacian Eigenvalues of a Uniform Hypergraph

In this paper, we show that the eigenvectors of the zero Laplacian and signless Lapacian eigenvalues of a $k$-uniform hypergraph are closely related to some configured components of that hypergraph. We show that the components of an eigenvector of the zero Laplacian or signless Lapacian eigenvalue have the same modulus. Moreover, under a {\em canonical} regularization, the phases of the components of these eigenvectors only can take some uniformly distributed values in $\{\{exp}(\frac{2jπ}{k})\;|\;j\in [k]\}$. These eigenvectors are divided into H-eigenvectors and N-eigenvectors. Eigenvectors …

Conformal mappings and first eigenvalue of Laplacian on surfaces

In this note we give a simple relation between conformal mapping and the first eigenvalue of Laplacian for surfaces in Euclidean spaces.

Mixed eigenvalues of p-Laplacian on trees

The purpose of the paper is to present quantitative estimates for the principal eigenvalue of discrete p-Laplacian on the set of rooted trees. Alternatively, it is studying the optimal constant of a class of weighted Hardy inequality. Three kinds of variational formulas in different formulation for the mixed principal eigenvalue of p-Laplacian on the set of trees with unique root as Dirichlet boundary are presented. As their applications, we obtain a basic estimate of the eigenvalue on trees.

Lower order eigenvalues of the poly-Laplacian with any order on spherical domains

We consider the lower order eigenvalues of poly-Laplacian with any order on spherical domains. We obtain universal inequalities for them and show that our results are optimal.

Sub-Laplacian eigenvalue bounds on CR manifolds

We prove upper bounds for sub-Laplacian eigenvalues independent of a pseudo-Hermitian structure on a CR manifold. These bounds are compatible with the Menikoff-Sjoestrand asymptotic law, and can be viewed as a CR version of Korevaar's bounds for Laplace eigenvalues of conformal metrics.

Metrics on a closed surface of genus two which maximize the first eigenvalue of the Laplacian

In this paper, we settle in the affirmative the Jakobson-Levitin-Nadirashvili-Nigam-Polterovich conjecture, stating that a certain singular metric on the Bolza surface, with area normalized, should maximize the first eigenvalue of the Laplacian.

An estimate of the first non-zero eigenvalue of the Laplacian by the Ricci curvature on edges of graphs

We define the distance between edges of graphs and study the coarse Ricci curvature on edges. We consider the Laplacian on edges based on the Jost-Horak's definition of the Laplacian on simplicial complexes. As one of our main results, we obtain an estimate of the first non-zero eigenvalue of the Laplacian by the Ricci curvature for a regular graph.

欧氏空间凸域上拉普拉斯算子Neumann特征值的一个普适不等式

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。

Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians

In this paper we study in detail some spectral properties of the magnetic discrete Laplacian. We identify its form-domain, characterize the absence of essential spectrum and provide the asymptotic eigenvalue distribution.

与聚集特征值相对应的 Laplace 特征函数的稳定计算

Stable Computation of Laplacian Eigenfunctions Corresponding to Clustered Eigenvalues

针对区域扰动导致 Laplace 特征值紧密聚集时特征函数难以精确计算的问题，本文利用特征值的形状差商提出一种稳定计算方法。

Sub-Laplacian eigenvalue bounds on sub-Riemannian manifolds

We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues $λ_k$ of conformal sub-Riemannian metrics that are asymptotically sharp as $k\to +\infty$. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for contact metric manifolds conformal to such Sasakian manifolds, we obtain eigenvalue inequalities that can be viewed as versions of the classical results by Korevaar and Buser in Riemannian geometry.

三维曲面的图拉普拉斯算子特征值灵活性

Flexibility of eigenvalues for graph Laplacians arising from genus 3 surfaces

本文研究三维曲面图拉普拉斯算子的小特征值近似，通过分析四顶点图的特征值集，揭示了特征值在不同曲面分解下的灵活性。

An Interlacing Approach for Bounding the Sum of Laplacian Eigenvalues of Graphs

We apply eigenvalue interlacing techniques for obtaining lower and upper bounds for the sums of Laplacian eigenvalues of graphs, and characterize equality. This leads to generalizations of, and variations on theorems by Grone, and Grone and Merris. As a consequence we obtain inequalities involving bounds for some well-known parameters of a graph, such as edge-connectivity, and the isoperimetric number.

Upper Eigenvalue Bounds for the Kirchhoff Laplacian on Embbeded Metric Graphs

We derive upper bounds for the eigenvalues of the Kirchhoff Laplacian on a compact metric graph depending on the graph's genus g. These bounds can be further improved if $g = 0$, i.e. if the metric graph is planar. Our results are based on a spectral correspondence between the Kirchhoff Laplacian and a particular a certain combinatorial weighted Laplacian. In order to take advantage of this correspondence, we also prove new estimates for the eigenvalues of the weighted combinatorial Laplacians that were previously known only in the weighted case.

The multiplicity of eigenvalues of the Hodge Laplacian on 5-dimensional compact manifolds

We study multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact two-forms, has nonzero eigenvalues of multiplicity two. The proof is based on the fact that Hodge Laplacian restricted to the subspace of co-exact two-forms is minus the square of the Beltrami operator, a first-order operator. We prove that for generic metrics the spectrum of the Beltrami operator is simple. Because the Beltrami operator in this setting is …

Comparison of Steklov eigenvalues on a domain and Laplacian eigenvalues on its boundary in Riemannian manifolds

We prove that in Riemannian manifolds the $k$-th Steklov eigenvalue on a domain and the square root of the $k$-th Laplacian eigenvalue on its boundary can be mutually controlled in terms of the maximum principal curvature of the boundary under sectional curvature conditions. As an application, we derive a Weyl-type upper bound for Steklov eigenvalues. A Pohozaev-type identity for harmonic functions on the domain and the min-max variational characterization of both eigenvalues are important ingredients.

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.

Sparse estimation of Laplacian eigenvalues in multiagent networks

We propose a method to efficiently estimate the Laplacian eigenvalues of an arbitrary, unknown network of interacting dynamical agents. The inputs to our estimation algorithm are measurements about the evolution of a collection of agents (potentially one) during a finite time horizon; notably, we do not require knowledge of which agents are contributing to our measurements. We propose a scalable algorithm to exactly recover a subset of the Laplacian eigenvalues from these measurements. These eigenvalues correspond directly to those Laplacian modes that are observable from our measurements. We …

Eigenvalues of the Laplacian and extrinsic geometry

We extend the results given by Colbois, Dryden and El Soufi on the relationships between the eigenvalues of the Laplacian and an extrinsic invariant called intersection index, in two directions. First, we replace this intersection index by invariants of the same nature which are stable under small perturbations. Second, we consider complex submanifolds of the complex projective space $\mathbb{C} P^N$ instead of submanifolds of $\mathbb{R}^N$ and we obtain an eigenvalue upper bound depending only on the dimension of the submanifold which is sharp for the first non-zero eigenvalue.

Eigenvalues of collapsing domains and drift Laplacians

By introducing a weight function to the Laplace operator, Bakry and Émery defined the "drift Laplacian" to study diffusion processes. Our first main result is that, given a Bakry-Émery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new relationship between Dirichlet eigenvalues of domains in $\R^n$ and Neumann eigenvalues of domains in $\R^{n+1}$ and a new maximum principle. Using our main result and maximum principle, we are able to gen…

Applications of the `Ham Sandwich Theorem' to eigenvalues of the Laplacian

We apply Gromov's ham sandwich method to get (1) domain monotonicity (up to a multiplicative constant factor); (2) reverse domain monotonicity (up to a multiplicative constant factor); and (3) universal inequalities for Neumann eigenvalues of the Laplacian on bounded convex domains in a Euclidean space.

On extremal eigenvalues of the graph Laplacian

Upper and lower estimates of eigenvalues of the Laplacian on a metric graph have been established in 2017 by G. Berkolaiko, J.B. Kennedy, P. Kurasov and D. Mugnolo. Both these estimates can be achieved at the same time only by highly degenerate eigenvalues which we call maximally degenerate. By comparison with the maximal eigenvalue multiplicity proved by I. Kac and V. Pivovarchik in 2011 we characterize the family of graphs exhibiting maximally degenerate eigenvalues which we call lasso trees, namely graphs constructed from trees by attaching lasso graphs to some of the vertices.

Bounds on the negative eigenvalues of Laplacians on finite metric graphs

For a self--adjoint Laplace operator on a finite, not necessarily compact, metric graph lower and upper bounds on each of the negative eigenvalues are derived. For compact finite metric graphs Poincaré type inequalities are given.

Eigenvalues for the Robin Laplacian in domains with variable curvature

We determine accurate asymptotics for the low-lying eigenvalues of the Robin Laplacian when the Robin parameter goes to $-\infty$. The two first terms in the expansion have been obtained by K. Pankrashkin in the $2D$-case and by K. Pankrashkin and N. Popoff in higher dimensions. The asymptotics display the influence of the scalar curvature and the splitting between every two consecutive eigenvalues. The analysis is based on the approach developed by Fournais-Helffer for the semi-classical magnetic Laplacian. We also propose a WKB construction as a candidate for the ground state energy.