Stability and Perturbations of the Domain for the First Eigenvalue of the 1-Laplacian

Explicit lower bound of the first eigenvalue of the Laplacian on Kähler manifolds

We establish an explicit lower bound of the first eigenvalue of the Laplacian on Kähler manifolds based off the comparison results of Li and Wang. The lower bound will depend on the diameter, dimension, holomorphic sectional curvature and orthogonal Ricci curvature.

四元数 Kähler 流形上 $p$-Laplacian 第一特征值的下界

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

本文研究紧致四元数 Kähler 流形上 $p$-Laplacian 的特征值，分别建立闭流形首个非零 Neumann 特征值及带光滑边界情形第一 Dirichlet 特征值的下界，推广了已有 Laplacian 结果。

First eigenvalue for p-Laplacian with mixed boundary conditions on manifolds

In this paper, we mainly study eigenvalue problems of p-Laplacian on domains with an interior hole. Firstly we prove Faber-Krahn-type inequalities, and Cheng-type eigenvalue comparison theorems on manifolds. Secondly, we prove a comparison theorem for eigenvalues with inner Dirichlet and outer Neumann boundary in minimal submanifolds in Euclidean space. Lastly we give a sharp estimate of the eigenvalue (with outer Dirichlet and inner Neumann boundaries) in terms of the Dircihlet eigenvalue, and also we give an upper bound of the eigenvalue with inner Dirichlet and outer Neumann problems by the…

The Gap of the Consecutive Eigenvalues of the Drifting Laplacian on Metric Measure Spaces

In this paper, we investigate eigenvalues of the Dirichlet problem and the closed eigenvalue problem of drifting Laplacian on the complete metric measure spaces and establish the corresponding general formulas. By using those general formulas, we give some upper bounds of consecutive gap of the eigenvalues of the eigenvalue problems, which is sharp in the sense of the order of the eigenvalues. As some interesting applications, we study the eigenvalue of drifting Laplacian on Ricci solitons, self-shrinkers and product Riemannian manifolds. We give the explicit upper bounds of the gap of the con…

带边流形上拉普拉斯算子特征值的极值度量

Extremal metrics for the eigenvalues of the Laplacian on manifolds with boundary

本文研究带边流形上拉普拉斯算子特征值的极值度量问题，证明诺伊曼拉普拉斯算子在任意紧致流形上不存在极值度量，并在环域中构造共形极值度量，同时证明狄利克雷拉普拉斯算子在紧致曲面上也不存在极值度量。

Lower bounds for the first Laplacian eigenvalue of geodesic balls of spherically symmetric manifolds

We obtain lower bounds for the first Laplacian eigenvalues of geodesic balls of spherically symmetric manifolds. These lower bounds are only $C^{0}$ dependent on the metric coefficients.

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。

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。

Principal eigenvalue problem for infinity Laplacian in metric spaces

This paper is concerned with the Dirichlet eigenvalue problem associated to the $\infty$-Laplacian in metric spaces. We establish a direct PDE approach to find the principal eigenvalue and eigenfunctions in a proper geodesic space without assuming any measure structure. We provide an appropriate notion of solutions to the $\infty$-eigenvalue problem and show the existence of solutions by adapting Perron's method. Our method is different from the standard limit process via the variational eigenvalue formulation for $p$-Laplacian in the Euclidean space.

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 […

Some Remarks on Robin-Laplacian Eigenvalues

We study some properties of Laplacian eigenvalues with negative Robin boundary conditions. We will show some monotonicity properties on annuli of the first eigenvalue by means of shape optimization techniques.

Eigenvalue gaps for the Laplacian on hypersurfaces of the sphere

We provide lower estimates for the eigenvalues of the laplacian for hypersurfaces of the round sphere.

Asymptotic behavior of the eigenvalues of the p(x)-Laplacian

We obtain asymptotic estimates for the eigenvalues of the p(x)-Laplacian defined consistently with a homogeneous notion of first eigenvalue recently introduced in the literature.

Shape derivative of the first eigenvalue of the 1-Laplacian

We compute the shape derivative of the first eigenvalue of the 1-Laplacian. As an application, we find that a ball is critical among all volume-preserving deformations.

The eigenvalue problem for the regional fractional Laplacian in the small order limit

In this note, we study the asymptotic behavior of eigenvalues and eigenfunctions of the regional fractional Laplacian $(-Δ)^s$ as $ s \to 0^+$. Our analysis leads to a study of the regional logarithmic Laplacian, which arises as a formal derivative of regional fractional Laplacians at $s = 0$.

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.

Remarks on Eigenvalue Problems for Fractional $p(\cdot)$-Laplacian

In this paper, we give some properties and remarks of the new fractional Sobolev spaces with variable exponents. We also study the eigenvalue problem involving the new fractional $p(\cdot)$-Laplacian.

The Neumann eigenvalue problem for the $\infty$-Laplacian

The first nontrivial eigenfunction of the Neumann eigenvalue problem for the $p$-Laplacian, suitable normalized, converges as $p$ goes to $\infty$ to a viscosity solution of an eigenvalue problem for the $\infty$-Laplacian. We show among other things that the limit of the eigenvalue, at least for convex sets, is in fact the first nonzero eigenvalue of the limiting problem. We then derive a number of consequences, which are nonlinear analogues of well-known inequalities for the linear (2-)Laplacian.

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。

A Variational Characterisation of the Second Eigenvalue of the p-Laplacian on Quasi Open Sets

In this article, we prove a minimax characterization of the second eigenvalue of the p-Laplacian operator on p-quasi-open sets, using a construction based on minimizing movements. This leads also to an existence theorem for spectral functionals depending on the first two eigenvalues of the p-Laplacian.

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 …

离散p-Laplacian的特征值：通过图手术方法

Eigenvalues of the discrete p-Laplacian via graph surgery

本文通过图手术和Hellmann-Feynman型微扰理论，将离散带符号p-Laplacian的特征值表征为更简单图参数依赖特征值的临界值。

On growth monotonicity estimates of the principal Dirichlet-Laplacian eigenvalue

In the present paper we obtain growth monotonicity estimates of the principal Dirichlet-Laplacian eigenvalue in bounded non-Lipschitz domains. The proposed method is based on composition operators generated by quasiconformal mappings and their applications to weighted Sobolev inequalities.

Monotonicity of eigenvalues of the fractional p-Laplacian with singular weights

We study a nonlinear, nonlocal eigenvalue problem driven by the fractional p-Laplacian with an indefinite, singular weight chosen in an optimal class. We prove the existence of an unbounded sequence of positive variational eigenvalues and alternative characterizations of the first and second eigenvalues. Then, by means of such characterizations, we prove strict decreasing monotonicity of such eigenvalues with respect to the weight function.

Kroger's upper bound types for the Dirichlet eigenvalues of the fractional Laplacian

We establish an upper bound of the sum of the eigenvalues for the Dirichlet problem of the fractional Laplacian. Our result is obtained by a subtle computation of the Rayleigh quotient for specific functions.

On the Second Eigenvalue of Combination Between Local and Nonlocal $p$-Laplacian

In this paper, we study Mountain Pass Characterization of the second eigenvalue of the operator $-\De_p u -\De_{J,p}u$ and study shape optimization problems related to these eigenvalues.

关于Robin边界条件下无穷拉普拉斯算子第二特征值的研究

On the second eigenvalue of the infinity Laplacian with Robin boundary conditions

研究Robin边界条件下p-拉普拉斯算子第二特征值在p趋于无穷时的极限，证明其收敛于无穷拉普拉斯算子的第二特征值。

Eigenvalues homogenization for the fractional $p-$Laplacian operator

In this work we study the homogenization for eigenvalues of the fractional $p-$Laplace in a bounded domain both with Dirichlet and Neumann conditions. We obtain the convergence of eigenvalues and the explicit order of the convergence rates.

m阶对数拉普拉斯算子的Dirichlet特征值计数函数与下界

Counting function and lower bound for Dirichlet eigenvalues of the m-order logarithmic Laplacian

本文研究有界Lipschitz域上m阶对数拉普拉斯算子Dirichlet特征值的计数函数极限及下界。

An optimal mass transport approach for limits of eigenvalue problems for the fractional $p$-Laplacian

We find interpretation using optimal mass transport theory for eigenvalue problems obtained as limits of the eigenvalue problems for the fractional $p-$Laplacian operators as $p\to +\infty$. We deal both with Dirichlet and Neumann boundary conditions.

旋度旋度与Dirichlet拉普拉斯算子特征值的比较

Curl curl versus Dirichlet Laplacian eigenvalues

该研究给出有界三维区域上curl curl算子特征值的Dirichlet拉普拉斯特征值上界估计，补充了其与Neumann拉普拉斯特征值的不等式，对应常参数Maxwell特征值问题。

Estimates for Robin $p$-Laplacian eigenvalues of convex sets with prescribed perimeter

In this paper, we prove an upper bound for the first Robin eigenvalue of the $p$-Laplacian with a positive boundary parameter and a quantitative version of the reverse Faber-Krahn type inequality for the first Robin eigenvalue of the $p$-Laplacian with negative boundary parameter, among convex sets with prescribed perimeter. The proofs are based on a comparison argument obtained by means of inner sets, introduced by Payne, Weimberger and Polya.

Eigenvalues for a nonlocal pseudo $p-$Laplacian

In this paper we study the eigenvalue problems for a nonlocal operator of order $s$ that is analogous to the local pseudo $p-$Laplacian. We show that there is a sequence of eigenvalues $λ_n \to \infty$ and that the first one is positive, simple, isolated and has a positive and bounded associated eigenfunction. For the first eigenvalue we also analyze the limits as $p\to \infty$ (obtaining a limit nonlocal eigenvalue problem analogous to the pseudo infinity Laplacian) and as $s\to 1^-$ (obtaining the first eigenvalue for a local operator of $p-$Laplacian type). To perform this study we have to …

Eigenvalues of the negative $(p,q)$-Laplacian under a Steklov-like boundary condition

In this paper we consider in a bounded domain $Ω\subset \mathbb{R}^N$ with smooth boundary an eigenvalue problem for the negative $(p,q)$-Laplacian with a Steklov type boundary condition, where $p\in (1,\infty)$, $q\in (2,\infty)$ and $p\neq q$. A full description of the set of eigenvalues of this problem is provided, thus essentially extending a recent result by Abreu and Madeira [1] related to the $(p,2)$-Laplacian.

On the first eigenvalue for a (p(x),q(x))-Laplacian elliptic system

In this article, we deal about the first eigenvalue for a nonlinear gradient type elliptic system involving variable exponents growth conditions. Positivity, boundedness and regularity of associated eigenfunctions for auxiliaries systems are established.

次椭圆 $p$-Laplacian 的变分特征值

Variational eigenvalues of the subelliptic $p$-Laplacian

本文利用 Lusternik-Schnirelman 理论，证明了次椭圆 $p$-Laplacian 在 Dirichlet 边界条件下存在非递减的变分特征值序列。

About small eigenvalues of Witten Laplacian

We study the eigenvalues of the semiclassical Witten Laplacian $Δ_ϕ$ associated to a potential $ϕ$. We consider the case where the sequence of Arrhenius numbers $S_1\leq \ldots\leq S_n$ associated to $ϕ$ is degenerated, that is the preceding inequality are not necessarily strict.

Optimal partitions for Robin Laplacian eigenvalues

We prove the existence of an optimal partition for the multiphase shape optimization problem which consists in minimizing the sum of the first Robin Laplacian eigenvalue of $k$ mutually disjoint {\it open} sets which have a $\mathcal H ^ {d-1}$-countably rectifiable boundary and are contained into a given box $D$ in $R^d$

The principal eigenvalue of the $\infty$-Laplacian with the Neumann boundary condition

We prove the existence of a principal eigenvalue associated to the $\infty$-Laplacian plus lower order terms and the Neumann boundary condition in a bounded smooth domain. As an application we get uniqueness and existence results for the Neumann problem and a decay estimate for viscosity solutions of the Neumann evolution problem.

带球形障碍物的Steklov拉普拉斯特征值问题的一个稳定性结果

A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle

该文研究带内部固定球形障碍物的第一Steklov-拉普拉斯特征值问题，证明内球和体积固定时球壳在近球形集合中局部最大化第一特征值。

Stability of variational eigenvalues for the fractional p-Laplacian

By virtue of $Γ-$convergence arguments, we investigate the stability of variational eigenvalues associated with a given topological index for the fractional $p$-Laplacian operator, in the singular limit as the nonlocal operator converges to the $p$-Laplacian. We also obtain the convergence of the corresponding normalized eigenfunctions in a suitable fractional norm.

Small order asymptotics of the Dirichlet eigenvalue problem for the fractional Laplacian

In this article, we study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian $(-Δ)^s$ in bounded open Lipschitz sets in the small order limit $s \to 0^+$. While it is easy to see that all eigenvalues converge to $1$ as $s \to 0^+$, we show that the first order correction in these asymptotics is given by the eigenvalues of the logarithmic Laplacian operator, i.e., the singular integral operator with symbol $2\log|ξ|$. By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that $L^2$-nor…

On Ashbaugh-Benguria's Conjecture about Lower Order Dirichlet Eigenvalues of the Laplacian

In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the Dirichlet Laplacian on bounded domains of a Euclidean space which strengthens the well-known Ashbaugh-Beguria inequality conjectured by Payne-Pólya-Weinberger on the ratio of the first two Dirichlet eigenvalues and makes an important step toward the proof of a conjecture by Ashbaugh-Benguria.

分数阶拉普拉斯算子特征值的通有性质

Generic properties of eigenvalues of the fractional Laplacian

该研究针对分数阶拉普拉斯算子的Dirichlet特征值问题，证明光滑有界域经任意小扰动后其所有特征值均为单的，推得区间上特征值全为单，并得出非局部算子特征值在参数通有选择下也为单的结论。

Variational Eigenvalues of the fractional $g$-Laplacian

In the present work we study existence of sequences of variational eigenvalues to non-local non-standard growth problems ruled by the fractional $g-$Laplacian operator with different boundary conditions (Dirichlet, Neumann and Robin). Due to the non-homogeneous nature of the operator several drawbacks must be overcome, leading to some results that contrast with the case of power functions.

Superlinear perturbations of the eigenvalue problem for the Robin Laplacian plus an indefinite and unbounded potential

We consider a superlinear perturbation of the eigenvalue problem for the Robin Laplacian plus an indefinite and unbounded potential. Using variational tools and critical groups, we show that when $λ$ is close to a nonprincipal eigenvalue, then the problem has seven nontrivial solutions. We provide sign information for six of them.

Two estimates for the first Robin eigenvalue of the Finsler Laplacian with negative boundary parameter

We prove two bounds for the first Robin eigenvalue of the Finsler Laplacian with negative boundary parameter in the planar case. In the constant area problem, we show that the Wulff shape is the maximizer only for values which are close to 0 of the boundary parameter and, in the fixed perimeter case, that the Wulff shape maximizes the first eigenvalue for all values of the parameter.

Properties of eigenvalues and some regularities on fractional $p$-Laplacian with singular weights

We provide fundamental properties of the first eigenpair for fractional $p$-Laplacian eigenvalue problems under singular weights, which is related to Hardy type inequality, and also show that the second eigenvalue is well-defined. We obtain a-priori bounds and the continuity of solutions to problems with such singular weights with some additional assumptions. Moreover, applying the above results, we show a global bifurcation emenating from the first eigenvalue, the Fredholm alternative for non-resonant problems, and obtain the existence of infinitely many solutions for some nonlinear problems …

On the fundamental eigenvalue ratio of the p-Laplacian

It is shown that the fundamental eigenvalue ratio λ_2 / λ_1 of the p-Laplacian is bounded by a quantity depending only on the dimension N and p.

Neumann fractional $p-$Laplacian: eigenvalues and existence results

We develop some properties of the $p-$Neumann derivative for the fractional $p-$Laplacian in bounded domains with general $p>1$. In particular, we prove the existence of a diverging sequence of eigenvalues and we introduce the evolution problem associated to such operators, studying the basic properties of solutions. Finally, we study a nonlinear problem with source in absence of the Ambrosetti-Rabinowitz condition.

Non-standard eigenvalue problems for perturbed $p$-Laplacians

This paper is devoted to multi-parameter eigenvalue problems for perturbed $p$-Laplacians, modelling travelling waves for a class of non-linear evolution PDE. Dispersion relations between the eigen-para-meters, the existence of eigenvectors and positive eigenvectors, variational principles for eigenvalues of perturbed $p$-Laplacians and constructing analytical solutions are the main subject of this paper. 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…

格子子图上分数阶 Laplace 算子的特征值估计

Eigenvalue estimates for the fractional Laplacian on lattice subgraphs

本文研究图子图上带 Dirichlet 边界条件的分数阶 Laplace 算子，并针对格子图建立前 k 个特征值之和的上下界估计，推广了 Li-Yau 与 Kröger 的经典结论。

Characterization of Vibrating Plates by Bi-Laplacian Eigenvalue Problems

In this paper we derive boundary integral identities for the bi-Laplacian eigenvalue problems under Dirichlet, Navier and simply-supported boundary conditions. By using these identities, we first obtain the uniqueness criteria for the solutions of the bi-Laplacian eigenvalue problems, and then prove that each eigenvalue of the problem with simply-supported boundary condition increases strictly with Poisson's ratio, thereby showing that each natural frequency of a simply-supported vibrating plate increases strictly with Poisson's ratio. In addition, we obtain boundary integral representations f…