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…
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48 results for “eigenvalues”
The paper compares Steklov and Laplacian eigenvalues on graphs.
problem Understanding the relationship between Steklov and Laplacian eigenvalues on graphs.
method Analyzing eigenvalues and discussing rigidity.
result Obtained Lichnerowicz-type estimates and combinatorial estimates for Steklov eigenvalues.
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 explores inequalities between eigenvalues on Riemannian manifolds.
problem Investigating relationships between eigenvalues on Riemannian manifolds.
method Constructing gradient estimates for a first eigenfunction to derive inequalities.
result Obtained some relationships between weighted -Laplacian first eigenvalues.
Eigenvalues of Steklov eigenproblems change predictably with boundary tweaks.
problem Understanding how Steklov eigenvalues respond to boundary changes.
method Analyzing smooth boundary perturbations of Steklov eigenvalues.
result Steklov eigenvalues are generically simple under such perturbations.
Study compares eigenvalues on spherically symmetric manifolds to Euclidean balls.
problem Comparing eigenvalues on spherically symmetric manifolds to Euclidean balls.
method Examines Dirichlet Laplace eigenvalues on balls of spherically symmetric manifolds and Euclidean space.
result Eigenvalues on spherically symmetric manifolds are smaller for small radii, but larger for hyperbolic spaces.
The paper sets lower bounds for Laplacian eigenvalues and clamped plate problem eigenvalues.
problem Eigenvalues of the Laplace operator and clamped plate problem.
method Sharp lower bounds for Laplacian eigenvalues and clamped plate problem eigenvalues.
result Sharp lower bounds for Laplacian eigenvalues and clamped plate problem eigenvalues.
Paper finds how Steklov eigenvalues change on graphs and trees.
problem Understanding how Steklov eigenvalues vary on graphs and trees.
method Analyzes monotonicity of Steklov eigenvalues on graphs and trees.
result Extends Steklov eigenvalue results to higher eigenvalues and trees.
Eigenvalue estimate for shrinkers in mean curvature flow.
problem Eigenvalue estimates on shrinkers for mean curvature flow.
method Generalized earlier work of Ding and Xin to noncompact cases.
result Eigenvalue estimate holds on every properly embedded shrinker.
In this paper we study eigenvalues of the closed eigenvalue problem of the Witten-Laplacian on an -dimensional compact Riemannian manifold. Estimates for eigenvalues are given. As applications, we give a sharp upper bound for the eigenvalue and for isoparametric minimal hypersurfaces in the unit sphe…
For a bounded domain with a piecewise smooth boundary in an -dimensional Euclidean space , 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…
The paper provides estimates for eigenvalues of elliptic differential problems.
problem Computing eigenvalue estimates for elliptic differential problems.
method Analytical computation of eigenvalues for specific types of elliptic differential equations.
result Universal estimates of eigenvalues and gaps between consecutive eigenvalues are derived.
The paper compares eigenvalues of Dirichlet, Neumann, and Laplacian on graphs.
problem Eigenvalue comparisons on graphs.
method Analytical comparisons and discussions of eigenvalues and their applications.
result Extensions of eigenvalue estimates for Dirichlet and Neumann eigenvalues.
Sharp bounds derived for the first two Steklov eigenvalues of exterior domains.
problem Finding bounds for the first two eigenvalues of Steklov eigenvalue problems on exterior domains.
method Sharp lower and upper bounds derived using the support function and distance function to the origin of the boundary.
result Sharp bounds for the first two eigenvalues of Steklov eigenvalue problems on exterior domains.
Estimates for eigenvalues of a system of of elliptic equations and of the biharmonic operatormath.DG
Let $\om $ be a bounded domain in an -dimensional Euclidean space . 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…
Improved lower bounds for poly-Laplacian eigenvalues in arbitrary dimensions.
problem Lower bounds for higher eigenvalues of the poly-Laplacian operator.
method Sharp inequalities and eigenvalue bounds in low and arbitrary dimensions.
result Improved lower bounds for eigenvalues of the poly-Laplacian in arbitrary dimensions.
Study eigenvalues of p-Laplacian on quaternionic Kähler manifolds.
problem Finding lower bounds for eigenvalues of p-Laplacian on quaternionic Kähler manifolds.
method Analytical proofs for both Neumann and Dirichlet boundary conditions.
result Established lower bounds for eigenvalues on compact quaternionic Kähler manifolds.
We study the eigenvalue problem for the Riemannian Pucci operator on geodesic balls. We establish upper and lower bounds for the principal Pucci eigenvalues depending on the curvature, extending Cheng's eigenvalue comparison theorem for the Laplace-Beltrami operator. For manifolds with bounded sectional curvature, we p…
The paper studies eigenvalues of Xin-Laplacian on Riemannian manifolds.
problem Eigenvalue problems related to Xin-Laplacian on Riemannian manifolds.
method Establishing general formulas and applying Chen-Cheng type results.
result Sharp estimates for the upper bound of the second nonzero eigenvalue of the Laplace-Beltrami operator.
The paper finds new inequalities for Laplacian and biharmonic eigenvalues on manifolds.
problem Eigenvalue inequalities for Laplacian and biharmonic operators on submanifolds.
method Using Sobolev inequalities to establish new eigenvalue inequalities.
result Established new inequalities for Laplacian and biharmonic eigenvalues.
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 minim…
Study on eigenvalues of complex Hessian operator on pseudoconvex manifolds.
problem Eigenvalue problem for complex Hessian operator on pseudoconvex manifolds.
method Established -regularity and uniqueness of the first eigenfunction, derived variational formula for the first eigenvalue.
result Derivation of a bifurcation-type theorem and geometric bounds for the eigenvalue.
Paper finds principal eigenvalue for infinity Laplacian in metric spaces.
problem Finding the principal eigenvalue of the infinity Laplacian in metric spaces.
method Direct PDE approach and Perron's method to establish existence of solutions.
result Existence of solutions to the infinity eigenvalue problem in metric spaces.
Estimates for eigenvalues on Riemannian manifolds using classical inequalities.
problem Estimating eigenvalues of the Dirichlet Laplacian on Riemannian manifolds.
method Building on Li-Yau's and Yang's inequalities, deriving upper and lower bounds.
result Explicit estimates on lower bounds for eigenvalues of the Dirichlet Laplacian on projective spaces and their minimal submanifolds.
Lower bounds for eigenvalues on Bakry-Emery manifolds proven.
problem Eigenvalue estimates on Bakry-Emery manifolds.
method Generalised maximum principle and heat kernel estimates.
result Lower bounds for all eigenvalues proven.
Study proves inequalities for eigenvalues of fourth-order elliptic operators on Riemannian manifolds.
problem Eigenvalue inequalities for fourth-order elliptic operators on Riemannian manifolds.
method Proves inequalities using Payne-Pólya-Weinberger-Yang type for eigenvalues of fourth-order elliptic operators in divergence form on complete Riemannian manifolds.
result Generalizes eigenvalue inequalities for the clamped plate problem to complete Riemannian manifolds.
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.
The paper proves stability of eigenvalue inequalities on surfaces.
problem Stability of isoperimetric inequalities for Laplace eigenvalues on surfaces.
method Employing eigenvalues of measures and Sobolev space , the paper proves stability estimates for the first and second nonzero Laplace eigenvalues on surfaces.
result Metrics almost maximizing the normalized eigenvalue are -close to a maximal metric.
Study on Kähler manifolds shows rigidity of eigenvalues with positive Ricci bound.
problem Optimal rigidity results for eigenvalues on Kähler manifolds with positive Ricci lower bound.
method Established optimal rigidity results for eigenvalues on Kähler manifolds with positive Ricci lower bound.
result Complex projective space is the only Kähler manifold with the largest multiplicity of the first eigenvalue.
Estimates small eigenvalues for geometrically finite manifolds.
problem Estimating small eigenvalues of Schrödinger operators.
method Geometrically finite manifolds, Riemannian vector bundles.
result Estimates the number of small eigenvalues.
The paper finds universal inequalities for eigenvalues on hyperbolic spaces.
problem Eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds.
method Establishes universal inequalities for eigenvalues of the Dirichlet Laplacian on hyperbolic spaces.
result Establishes universal inequalities for eigenvalues of the Dirichlet Laplacian on hyperbolic spaces.
Upper bound found for Steklov eigenvalue of a surface of revolution.
problem Finding an upper limit for Steklov eigenvalues of a specific surface.
method Analyzing a surface of revolution with boundary conditions of two spheres.
result An upper bound for the first Steklov eigenvalue is derived and shown to be sharp in some cases.
The paper examines the rigidity of eigenvalues in shrinking Ricci solitons.
problem Rigidity of eigenvalues in shrinking Ricci solitons.
method Analysis of the drifted Laplacian on shrinking Ricci solitons, showing eigenvalue bounds and rigidity results.
result If the eigenvalue is close to a lower bound, the -soliton must be the trivial Gaussian soliton.
Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surfacemath.DG
Sharp bounds found for Steklov-type eigenvalues on surfaces.
problem Finding bounds for the first eigenvalue of Steklov-type problems on compact surfaces.
method Proved bounds using Gaussian curvature constraints and properties of geodesic curvature.
result Sharp lower bounds for the first eigenvalue of Steklov-type problems on compact surfaces.
New biharmonic Steklov problem on forms yields eigenvalue estimates.
problem Eigenvalue estimates for differential forms with curvature quantities.
method Introduced a new biharmonic Steklov problem and proved existence of a discrete spectrum.
result Established Kuttler-Sigillito inequalities connecting eigenvalues of differential forms.
Study eigenvalues of ellipsoids near a sphere, comparing to sphere's.
problem Analyzing changes in Laplacian eigenvalues for ellipsoids near a sphere.
method Comparison with standard Euclidean unit sphere, under Gaussian curvature condition.
result Eigenvalues of ellipsoids near a sphere, with comparison to sphere's.
Variational method for eigenvalues on manifolds.
problem Optimizing functionals involving eigenvalues of Riemannian manifolds.
method New Palais-Smale sequences and min-max methods for locally-Lipschitz functionals.
result Convergence of Palais-Smale sequences in Laplace and Steklov eigenvalues.
Conformal and extrinsic upper bounds for the harmonic mean of Neumann and Steklov eigenvaluesmath.DG
Study bounds Neumann and Steklov eigenvalues on manifolds and submanifolds.
problem Bounding Neumann and Steklov eigenvalues on manifolds and submanifolds.
method Using conformal and extrinsic volumes, the paper derives upper bounds for eigenvalues.
result Upper bounds for harmonic mean of Neumann and Steklov eigenvalues.
We study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). In particular, we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a Euclidean space. This inequality controls the th eigenvalue by the lower eigenvalues,…
We prove that in Riemannian manifolds the -th Steklov eigenvalue on a domain and the square root of the -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 upp…
Machine learning models solve inverse eigenvalue problems for symmetric potentials and refractive indices.
problem Solving inverse eigenvalue problems for symmetric potentials and refractive indices.
method Supervised regression models (k-Nearest Neighbours, Random Forests, Multi-Layer Perceptron) trained on eigenvalue datasets.
result Machine learning methods can numerically solve inverse eigenvalue problems under appropriate parameter tuning.
Some universal inequalities for Dirichlet eigenvalues of the Laplacian on a Euclidean convex domainmath.SP
Universal inequalities found for Laplacian eigenvalues on convex domains.
problem Finding bounds for Laplacian eigenvalues on convex domains.
method Established two universal inequalities.
result Found new bounds for Laplacian eigenvalues.
A note on the distribution of Neumann eigenvalues of the Laplacian on a Euclidean convex domainmath.SP
Universal inequalities for Laplacian eigenvalues on convex domains.
problem Eigenvalue distribution of the Laplacian on convex domains.
method Established two universal inequalities.
result Two new inequalities for Laplacian eigenvalues.
We prove inequalities for Laplace eigenvalues of Kaehler manifolds generalising to higher eigenvalues the classical inequality for the first Laplace eigenvalue due to Bourguignon, Li, and Yau in 1994. We also obtain similar inequalities for analytic varieties in Kaehler manifolds.
New upper bound for Neumann Laplacian eigenvalues on convex domains.
problem Bounding Neumann eigenvalues on convex domains.
method Deriving a new upper bound for eigenvalues.
result Universal inequalities for Neumann eigenvalues derived from the upper bound.
Two new universal inequalities for Neumann eigenvalues of the Laplacian on a planar convex domainmath.SP
New inequalities for planar convex domains' Laplacian eigenvalues.
problem Neumann eigenvalues of the Laplacian on planar convex domains.
method Established two new universal inequalities.
result New inequalities for Laplacian eigenvalues on convex domains.
Paper finds a graph Steklov eigenvalue estimate with rigidity results.
problem Estimating Steklov eigenvalues on graphs.
method Lichnerowicz-type estimate for the first Steklov eigenvalues.
result Rigidity results for the Steklov eigenvalues on graphs.
Researchers investigate extremal eigenvalues of GJMS operators in fixed conformal classes.
problem Investigating extremal eigenvalues of GJMS operators in fixed conformal classes.
method Developed a general framework for existence theory of extremals, defined and investigated generalised eigenvalues, and established semi-continuity results and Euler-Lagrange equations.
result Proved several new (non)-existence results for extremals of renormalised eigenvalues over the conformal class.