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 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.
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.
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.
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.
Estimates eigenvalues of poly-Laplace operator on lattice subgraphs.
problem Estimating eigenvalues of poly-Laplace operator on subgraphs of lattice graphs.
method Introduced discrete poly-Laplace operator, derived upper and lower bounds for eigenvalues.
result Poly-Laplace eigenvalues are at least squares of lower-order poly-Laplace eigenvalues.
Geodesic balls with non-negative Ricci curvature have a sharp lower bound on their first Dirichlet eigenvalue.
problem Finding a sharp lower bound for the first Dirichlet eigenvalue of geodesic balls.
method Quantitative explicit inequality linking the width of geodesic balls to the spectral gap.
result A quantitative inequality relating the width of geodesic balls to the spectral gap between the first Dirichlet eigenvalue and its lower bound.
Paper proves new inequalities for hyperbolic space Laplacian eigenvalues.
problem Universal inequalities for eigenvalues of the Dirichlet Laplacian.
method Proves new inequalities for eigenvalues on hyperbolic space.
result Verifies Cheng's conjecture up to a small loss.
Paper calculates eigenvalues of a specific triangle on a sphere.
problem Computing eigenvalues of a specific triangle on a sphere.
method Computed first two Dirichlet eigenvalues and eigenfunctions of the equilateral Schwarz triangle (3/2 3/2 3/2) on the sphere.
result Computed the first two Dirichlet eigenvalues and eigenfunctions of the equilateral Schwarz triangle (3/2 3/2 3/2).
Study on Neumann eigenvalues controlled by domain isoperimetric ratio.
problem Control the number of Neumann eigenvalues no greater than the first Dirichlet eigenvalue.
method Combination of analytical and numerical results, related to Yau's conjecture.
result Neumann eigenvalues are controlled by the isoperimetric ratio of the domain.
The paper studies higher order Dirichlet-to-Neumann maps on graphs and their eigenvalues.
problem Analyzing eigenvalues of higher order Dirichlet-to-Neumann maps on graphs.
method Introducing and studying higher order Dirichlet-to-Neumann maps on graphs, deriving estimates on eigenvalues.
result Raulot-Savo-type estimates on the eigenvalues of the DtN maps.
The paper proves inequalities for lattice eigenvalues, answering a question.
problem Finding bounds for Dirichlet Laplace eigenvalues on integer lattices.
method Proving analogues of existing inequalities for eigenvalues.
result Answers a question posed by Chung and Oden about eigenvalues on integer lattices.
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…
Eigenvalues of manifolds with cylindrical boundaries approximated by graph Laplacians.
problem Approximating eigenvalues of manifolds with cylindrical boundaries.
method Using truncated graph Laplacians constructed from (ε,ρ)-proximity graphs. result Eigenvalues of truncated graph Laplacians converge to Dirichlet eigenvalues of the Laplace-Beltrami operator.
Introduces differential forms to study inequalities between eigenvalues.
problem Eigenvalue inequalities for Laplacian and other operators.
method Uses differential forms and the de Rham complex.
result Shows differential forms are central to Rohleder's work.
Study spectral properties of modified Dirichlet-to-Neumann map on differential forms.
problem Spectral properties of modified Dirichlet-to-Neumann map on differential forms.
method Investigation of self-adjointness and purely discrete spectrum of the operator Λ on coclosed forms.
result Hersch-Payne-Schiffer type inequality relating eigenvalues of Λ to eigenvalues of Hodge Laplacian on the boundary.
Study estimates eigenvalues for concave Hessian operators on convex domains.
problem Estimating eigenvalues for concave elliptic Hessian operators.
method Investigates Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators.
result Existence and properties of the first nonzero eigenvalue and eigenfunction.
Paper derives inequalities for eigenvalues of Witten-Laplacian under fixed volume constraint.
problem Eigenvalue inequalities of Witten-Laplacian on bounded domains.
method Rearrangement technique and trial functions under fixed weighted volume constraint.
result Several isoperimetric inequalities for eigenvalues of Witten-Laplacian.
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 p-Laplacian first eigenvalues. The paper examines how the first Steklov-Dirichlet eigenvalue changes with the distance between two concentric circles.
problem Investigating the monotonicity of the first Steklov-Dirichlet eigenvalue on eccentric annuli.
method The approach involves showing differentiability, deriving integral expressions for the derivative, and using variational formulations to find upper and lower bounds.
result The paper proves the monotonicity of the first Steklov-Dirichlet eigenvalue on eccentric annuli with respect to the distance between the centers of the inner and outer boundaries.
Sharp lower bound for p-Laplacian eigenvalue on non-compact manifolds.
problem Estimating eigenvalues of p-Laplacian on non-compact manifolds. method Sharp lower bound established through domain properties and curvature conditions.
result Sharp lower bound for the first Dirichlet eigenvalue of p-Laplacian. Liu's paper contains an error regarding eigenvalues.
problem Eigenvalues of Dirichlet and buckling problems.
method Review and identification of an error.
result Error in the Payne conjecture for eigenvalues.
Optimizing shapes for a specific eigenvalue problem involving two balls.
problem Optimizing shapes for the first mixed Steklov-Dirichlet eigenvalue.
method Geometric proof based on Newton's shell theorem.
result Geometric insight into eigenvalue optimization.
Optimal bounds for Laplacian eigenvalues on weighted graphs.
problem Finding lower bounds for Laplacian eigenvalues in weighted graphs.
method Formulating bounds in terms of graph geometry, specifically inradius of subsets.
result Optimal lower bounds for the first non-zero eigenvalue in finite volume and Dirichlet Laplacian on subsets with geometric conditions.
The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.
problem Eigenvalue problems and heat trace asymptotics for different operators.
method Establishes connections and inequalities for eigenvalues and heat traces.
result Eigenvalue inequalities and three-term asymptotic formulas for heat traces of various operators.
In this paper, we derive the CR Reilly's formula and its applications to studying of the first eigenvalue estimate for CR Dirichlet eigenvalue problem and embedded p-minimal hypersurfaces. In particular, we obtain the first Dirichlet eigenvalue estimate in a compact pseudohermitian (2n+1)-manifold with boundary and the…
Sharp lower bound found for geodesic ball eigenvalues.
problem Finding the minimum eigenvalue for geodesic balls.
method Applied Li-Schoen's uniform Poincare inequality for non-negative Ricci curvature manifolds.
result Sharp lower bound of the first Dirichlet eigenvalue for geodesic balls.
Lower bounds for Steklov eigenvalues derived from Markov operators and mass concentration.
problem Finding bounds for Steklov eigenvalues in various geometric settings.
method Developed techniques involving accelerated Markov operators and mass concentration deformations.
result Proved higher order Cheeger type inequalities for Steklov eigenvalues.
Study on eigenvalue rate of geodesic balls in asymptotically hyperbolic Einstein manifolds.
problem Rate of decrease of the first Dirichlet eigenvalue of geodesic balls.
method Investigation of eigenvalues in asymptotically hyperbolic Einstein manifolds with nonnegative Yamabe type conformal infinity.
result Two-term asymptotic of eigenvalues is the same as in hyperbolic space for nonnegative Yamabe type conformal infinity.
Researchers found the first and second eigenvalues are Courant-sharp on a Möbius strip.
problem Determining Courant-sharp eigenvalues on a Möbius strip.
method Analyzing the eigenvalues and nodal patterns of the Möbius strip.
result Only the first and second eigenvalues are Courant-sharp on the Möbius strip.
In this paper we consider a domain in a space of negative constant sectional curvature. Such assumption about the sectional curvature let us develop a new technique and improve existing lower bounds of eigenvalues from Dirichlet eigenvalue problem, obtained by Alessandro Savo in 2009.
We give a new estimate on the lower bound of the first Dirichlet eigenvalue of a compact Riemannian manifold with negative lower bound of Ricci curvature and provide a solution for a conjecture of H. C. Yang.
We compute the first Dirichlet eigenvalue of a geodesic ball in a rotationally symmetric model space in terms of the moment spectrum for the Brownian motion exit times from the ball. This expression implies an estimate as exact as you want for the first Dirichlet eigenvalue of a geodesic ball in these rotationally symm…
Estimates eigenvalues and spectrum for graph substructures using isocapacitary constants.
problem Estimating eigenvalues and spectrum for graph substructures.
method Introducing Cheeger type constants via isocapacitary constants to estimate eigenvalues and spectrum.
result Estimates for first Dirichlet, Neumann, and Steklov eigenvalues, as well as the bottom of the spectrum of the Laplace operator and Dirichlet-to-Neumann operator.
The paper finds inequalities for eigenvalues of fourth order elliptic operators on Riemannian manifolds.
problem Eigenvalue inequalities for fourth order elliptic operators on Riemannian manifolds.
method Analyzes eigenvalues of fourth order elliptic operators in divergence form with Dirichlet boundary conditions on bounded domains in compact Riemannian manifolds.
result General inequalities for eigenvalues are derived.
The paper finds metrics for surfaces with boundaries that match specific eigenvalues and areas.
problem Finding metrics for surfaces with boundaries that match specific eigenvalues and areas.
method The approach involves constructing a metric on a compact surface with boundary that satisfies given eigenvalues and area constraints.
result A metric can be constructed on a compact surface with boundary that matches a given sequence of eigenvalues and area.
The paper compares Dirichlet and Neumann eigenvalues on various curved surfaces.
problem Comparing eigenvalues on curved surfaces.
method Variational principle of the Hodge Laplacian on 1-forms.
result Strict inequalities between Dirichlet and Neumann eigenvalues on specific surfaces.
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…
The study examines metrics that extremize eigenvalues of a specific map on manifolds with boundary.
problem Variational properties of the spectrum of the Dirichlet-to-Robin map on manifolds with boundary.
method Analysis of the extremal metrics for the first and second normalized eigenvalues of the Dirichlet-to-Robin map.
result Existence and characterization of extremal metrics for the first and second eigenvalues of the Dirichlet-to-Robin map.
The paper studies eigenvalue problems on manifolds and recovers known inequalities.
problem Eigenvalue problems on complete compact Riemannian manifolds with Dirichlet boundary conditions.
method Cheng comparison estimates, Faber-Krahn inequality, Cheeger estimates.
result Eigenvalue bounds and convergence to Cheeger's constant as p,qo1,1. Proves Weyl's law for metric spaces with Ricci curvature.
problem Proving Weyl's law for metric measure spaces with bounded Ricci curvature.
method Analyzes RCD∗(K,N) spaces to prove asymptotic eigenvalue formula. result Establishes Weyl's law for Dirichlet eigenvalues in metric measure spaces.
We give a new estimate on the lower bound for the first Dirichlet eigenvalue for a compact manifold with positive Ricci curvature in terms of the in-diameter and the lower bound of the Ricci curvature. The result improves the previous estimates.
The paper finds lower bounds for the first eigenvalue of p-Laplacian in specific manifolds.
problem Finding lower bounds for the first eigenvalue of p-Laplacian in Riemannian manifolds.
method Established and enhanced lower bounds for the eigenvalue under specific conditions.
result Provided an estimation for the first Dirichlet eigenvalue in asymptotically hyperbolic Einstein manifolds.
Study on biharmonic Steklov problems with Neumann boundary conditions and eigenvalue estimates.
problem Biharmonic Steklov problems with Neumann boundary conditions.
method Introduced a biharmonic Steklov problem and proved its well-posedness. Established eigenvalue estimates using Kuttler-Sigillito inequalities.
result Eigenvalue estimates for the biharmonic Steklov problem with Neumann boundary conditions.
Universal inequalities for Laplacian eigenvalues on discrete groups.
problem Proving inequalities for Laplacian eigenvalues on discrete groups.
method Analyzing Laplacian eigenvalues with Dirichlet boundary conditions on subsets of discrete groups.
result Yang-type universal inequalities for Cayley graphs of amenable groups and the d-regular tree.
The paper finds metrics for manifolds with prescribed volumes and eigenvalues.
problem Finding Riemannian metrics with specific volumes and eigenvalues.
method Prescribes Dirichlet eigenvalues for a compact manifold with a non-empty boundary.
result Existence of metrics with prescribed volume and eigenvalues.
The paper studies eigenvalues and Cheeger constants on symmetric graphs.
problem Characterizing eigenvalues and Cheeger constants on symmetric graphs.
method Characterization of the first eigenfunction via sign condition, and calculation of Cheeger constants using the limit of p-Laplacian eigenvalues. result Identifies Cheeger constants of symmetric graphs and their quotients.
The Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on an annulus than on any other surface of revolution in R3 with the same boundary. This is established by defining a sequence of shrinking cylinders about the axis of symmetry and proving that flattening a surface outside of each cylinde…