arXiv research

This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problems which yield to the eigenvalue and generalized eigenvalue problems. We also prov…

In this paper, two interesting eigenvalue comparison theorems for the first non-zero Steklov eigenvalue of the Laplacian have been established for manifolds with radial sectional curvature bounded from above. Besides, sharper bounds for the first non-zero eigenvalue of the Wentzell eigenvalue problem of the weighted La…

The paper sets lower bounds for Laplacian eigenvalues and clamped plate problem eigenvalues.

Improved lower bounds for poly-Laplacian eigenvalues in arbitrary dimensions.

Paper finds inequalities for eigenvalues of buckling problems on special metric spaces.

Numerical methods solve Steklov eigenvalue problems to generate free boundary minimal surfaces.

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 eigen…

For a bounded domain $Ω$ with a piecewise smooth boundary in an $n$-dimensional Euclidean space $\mathbf{R}^{n}$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. First we give a general inequality for eigenvalues of the Laplacian. As an application, we study lower order eigenvalues of the Lap…

This paper studies eigenvalues of the buckling problem of arbitrary order on bounded domains in Euclidean spaces and spheres. We prove universal bounds for the k-th eigenvalue in terms of the lower ones independent of the domains. Our results strengthen the recent work in [28] and generalize Cheng-Yang's recent estimat…

Study eigenvalues and shapes, proving sharp inequalities for Steklov eigenvalues.

The paper studies eigenvalue inequalities for a clamped plate problem involving a generalized elliptic differential operator.

The paper studies eigenvalues of Xin-Laplacian on Riemannian manifolds.

In this paper we study eigenvalues of the closed eigenvalue problem of the Witten-Laplacian on an $n$-dimensional compact Riemannian manifold. Estimates for eigenvalues are given. As applications, we give a sharp upper bound for the $k^{\text{th}}$ eigenvalue and for isoparametric minimal hypersurfaces in the unit sphe…

Study eigenvalue problems on Riemannian manifolds with specific curvature conditions.

New method generalizes eigenvalue inequality to surfaces with boundaries.

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Steklov problem on functions. We derive a number of upper and lower bounds for the first eigenvalue in several contexts: many of thes…

Paper finds how Steklov eigenvalues change on graphs and trees.

Machine learning models solve inverse eigenvalue problems for symmetric potentials and refractive indices.

New biharmonic Steklov problem on forms yields eigenvalue estimates.

Paper studies eigenvalues of a specific operator on Riemannian manifolds.

Study proves inequalities for eigenvalues of fourth-order elliptic operators on Riemannian manifolds.

The paper proves inequalities for Steklov eigenvalues on finite graphs.

The paper provides estimates for eigenvalues of elliptic differential problems.

In this paper, we study estimates for eigenvalues of the clamped plate problem. A sharp upper bound for eigenvalues is given and the lower bound for eigenvalues in [10] is improved.

Let $\om $ be a bounded domain in an $n$-dimensional Euclidean space $\Bbb R^n$. We study eigenvalues of an eigenvalue problem of a system of elliptic equations: $$ \{\aligned &Δ{\mathbf u}+ α{\rm grad}(\text{div}{\mathbf u})=-σ{\mathbf u}, \ \text{in $Ω$}, &{\mathbf u}|_{\partial Ω}={\mathbf 0}. \aligned . $$ Estimate…

Study eigenvalues of conformal Laplacian under Sire-Xu normalization.

In study of eigenvalue problems, a classical problem is the Stekloff eigenvalue problem. There are many estimates of the first non- zero Stekloff eigenvalue, including a sharp estimate on surfaces, obtained by Escobar in "The geometry of the first non-zero Stekloff eigenvalue, J. Funct. Anal. 150 (1997)". In this paper…

Sharp bounds found for Steklov-type eigenvalues on surfaces.

Study on biharmonic Steklov problem on differential forms.

Let $Ω$ be a star-shaped bounded domain in $(\mathbb{S}^{n}, ds^{2})$ with smooth boundary. In this article, we give a sharp lower bound for the first non-zero eigenvalue of the Steklov eigenvalue problem in $Ω.$ This result is the generalization of a result given by Kuttler and Sigillito for a star-shaped bounded doma…

Paper finds principal eigenvalue for infinity Laplacian in metric spaces.

Study eigenvalues of p-Laplacian on Kähler manifolds, proving lower bounds.

Study on biharmonic Steklov problems with Neumann boundary conditions and eigenvalue estimates.

We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem and its higher eigenvalue versions via the direct method of calculus of variations. The principal results include the gen…

In this paper, we investigate the Dirichlet problem of Laplacian on complete Riemannian manifolds. By constructing new trial functions, we obtain a sharp upper bound of the gap of the consecutive eigenvalues in the sense of the order, which affirmatively answers to a conjecture proposed by Chen-Zheng-Yang. In addition,…

Paper develops IFTRR to solve sparse generalized eigenvalue problems efficiently.

We investigate the eigenvalues of the buckling problem of arbitrary order on compact domains in Euclidean spaces and spheres. We obtain universal bounds for the $k$th eigenvalue in terms of the lower eigenvalues independently of the particular geometry of the domain.

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…

Eigenvalue bounds for forms on warped manifolds studied.

Study on eigenvalues of complex Hessian operator on pseudoconvex manifolds.

The paper studies eigenvalue problems on manifolds and recovers known inequalities.

Upper bounds found for eigenvalues of weighted Steklov and (p,q)-Laplacian problems.

Extends plate problems to differential forms on manifolds.

The ball maximizes the first biharmonic Steklov eigenvalue.

Study on shape optimization for specific eigenvalue problems on domains.

Physics-informed GP regression solves eigenvalue problems by identifying non-trivial eigenspaces.

The paper finds minimum Steklov eigenvalues on combinatorial graphs.

We consider the classification problem for compact Lie groups $G\subset U(n)$ which are generated by a single conjugacy class with a fixed number $N$ of distinct eigenvalues. We give an explicit classification when N=3, and apply this to extract information about Galois representations and braid group representations.