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.
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.
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.
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.
Graph Laplacians adapt to different manifold dimensions, while Dirichlet energies converge to a tensorized Dirichlet energy.
problem Understanding machine learning methods for data with varying intrinsic dimensions.
method Γ-convergence of graph Dirichlet energies and spectral convergence of graph Laplacians on intersecting manifolds of varying dimensions.
result Normalized Dirichlet energy converges to a tensorized Dirichlet energy that adapts to all dimensions simultaneously.
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.
Researchers calculate spectral invariants from Dirichlet-to-Neumann map for Witten-Laplacian with potential.
problem Calculating spectral invariants from Dirichlet-to-Neumann map for Witten-Laplacian with potential.
method Established an effective procedure to calculate all coefficients of the spectral asymptotic formula of the Dirichlet-to-Neumann map.
result Explicitly provided the first four coefficients of the spectral asymptotic formula.
Study Hodge Laplacians for manifold data, improving error bounds.
problem Approximating Laplace-Beltrami operator on differential forms.
method Higher-order graph Laplacians (Hodge Laplacians) as approximations.
result High-probability error bound for Dirichlet forms.
The paper finds a family of 2D surfaces where Laplacian and Dirichlet-to-Neumann commute.
problem Finding surfaces with commuting boundary Laplacian and Dirichlet-to-Neumann map.
method Investigated 2D submanifolds of R^2, classified surfaces of genus 0 or with k≥3 boundary components.
result One-parameter family of 2D submanifolds with commuting Laplacian and Dirichlet-to-Neumann map.
Study connects boundary geometry to symbol of Dirichlet-to-Neumann operator.
problem Determining geometric data from boundary symbol of connection Laplacian.
method Analyze symbol of Dirichlet-to-Neumann operator associated with connection Laplacian.
result Geometric data on boundary and normal derivatives are determined by symbol.
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.
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. 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.
Researchers express spectral determinants on hyperbolic cones.
problem Express spectral determinants on 2D hyperbolic cones.
method Explicitly expressed spectral determinants in terms of cone angle and geodesic radius.
result Results in recent paper by Freixas i Montplet and von Pippich were incorrect.
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. New inequality for eigenfunctions on curved spaces.
problem Eigenfunctions on non-smooth spaces with Ricci curvature.
method Sharp reverse-Hölder inequality for Dirichlet Laplacian eigenfunctions.
result Generalizes classical comparison theorem to curved spaces.
Recent research connects Hörmander's old work to modern boundary Laplacian analysis.
problem How close is the Dirichlet-to-Neumann map to the boundary Laplacian?
method Investigates techniques from Hörmander's 1950s manuscript to solve modern boundary Laplacian problems.
result Obtained results for DtN maps on non-smooth boundaries, Helmholtz equation, and differential forms.
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→1, we ident…
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.
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.
We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between…
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…
For a bounded domain Ω with a piecewise smooth boundary in an n-dimensional Euclidean space Rn, 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…
We study the Dirichlet problem at infinity on a Cartan-Hadamard manifold for a large class of operators containing in particular the p-Laplacian and the minimal graph operator.
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.
The first eigenvalue of the Laplacian on a unique Hurwitz surface has a sevenfold multiplicity and specific numerical values.
problem Identifying the spectrum of the Laplacian on a specific Hurwitz surface.
method Analytical proof for the multiplicity and numerical identification of the first eigenvalue; numerical identification of the eigenspace representation; determination of Dirichlet domain.
result The first eigenvalue of the Laplacian on the Fricke-Macbeath surface has a sevenfold multiplicity and is contained in the interval [1.23, 1.26].
Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.
problem Geometric and topological properties of Finsler metric measure manifolds with integral weighted Ricci curvature bounds.
method Establish Laplacian comparison theorem, volume comparison theorems, volume growth estimate, Gromov pre-compactness, local Dirichlet isoperimetric constant estimate.
result First Dirichlet eigenvalue estimate and gradient estimate for harmonic functions.
Abstract: Expresses zeta-determinant of Dirichlet-to-Neumann operator on forms.
problem Express zeta-determinant of Dirichlet-to-Neumann operator on forms.
method Expresses zeta-determinant as difference of Laplacian determinants with boundary conditions.
result Computes terms explicitly for dimensions 2 and 3.
The paper proves spectral convergence rates for graph Laplacian to manifold Laplace-Beltrami operator.
problem Spectral convergence of graph Laplacian to manifold Laplace-Beltrami operator.
method Analysis of Dirichlet form convergence and construction of approximate eigenfunctions via manifold heat kernel.
result Proves spectral convergence rates for Gaussian kernelized graph Laplacian.
Paper derives a formula for the determinant of Dirichlet-to-Neumann operator on Riemann surfaces.
problem Bounding asymptotics of a conformal invariant under degeneration of Riemann surfaces.
method Meyer-Vietoris formula, gluing, height function on moduli space, properness of height function, Steklov isospectral metrics, Laplacian with Dirichlet/Neumann boundary conditions.
result Properness of height function on moduli space of genus zero hyperbolic surfaces implies compactness theorem for Steklov isospectral metrics.
The paper sets up eigenvalue comparison theorems for specific Laplacians on manifolds.
problem Eigenvalue comparison theorems for Witten-Laplacian and weighted p-Laplacian on manifolds with modified Ricci curvature. method Established Cheng-type eigenvalue comparison theorems for the first Dirichlet eigenvalues of the Witten-Laplacian and weighted p-Laplacian on geodesic balls. result Successfully set up eigenvalue comparison theorems for the Witten-Laplacian and weighted p-Laplacian. DiAL uses Bayesian Dirichlet random fields for active learning with sparse labels.
problem Active learning with limited labeled data.
method Bayesian Dirichlet random field for feature-conditional class probabilities, calibrating with graph Laplacian.
result Competitive performance in low-label rate graph learning tasks.
The paper studies the zeta-regularized determinant of a pseudo-Laplacian on a cuspidal end with flat unitary line bundle.
problem Analyzing the asymptotic behavior of the zeta-regularized determinant of a pseudo-Laplacian.
method Investigates the asymptotic behavior of the zeta-regularized determinant of a pseudo-Laplacian ΔL,0+μ as μ and a vary. result Determines the asymptotic behavior of the zeta-regularized determinant for various values of μ and a. 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. For negatively curved manifolds, a condition is found for intrinsic ultracontractivity of heat semigroups.
problem Investigating intrinsic ultracontractivity for domains in negatively curved manifolds.
method Using volume doubling property, Poincaré inequality, and Li-Yau Gaussian estimate for the Dirichlet heat kernel.
result The reciprocal of the bottom of the spectrum and the supremum of the torsion function are comparable with the square of the capacitary width for small capacitary width.
We discuss optimal lower bounds for eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In particular, we study the first non-zero eigenvalue in the finite volume case and the first eigenvalue of the Dirichlet L…
New method generalizes eigenvalue inequality to surfaces with boundaries.
problem Eigenvalue inequality for surfaces with boundaries.
method Generalized Rohleder's approach to differential forms, presenting Hodge-Laplacian spectrum.
result Obtained inequality for eigenvalues of Hodge-Laplacian and Dirichlet problems.
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.
The paper studies eigenfunctions and nodal sets of the Witten-Laplacian.
problem Characterizing nodal lines and eigenfunctions of the Witten-Laplacian.
method Courant-type nodal domain theorem for Dirichlet and closed eigenvalue problems.
result Upper bound for the multiplicity of closed eigenvalues of the Witten-Laplacian.
We study the asymptotic Dirichlet problem for A-harmonic functions on a Cartan-Hadamard manifold whose radial sectional curvatures outside a compact set satisfy an upper bound K(P)≤−r(x)2logr(x)1+ε and a pointwise pinching condition ∣K(P)∣≤CK∣K(P′)∣ for some const…
The article proves Talenti's comparison theorem for Poisson equations on Riemannian manifolds with nonnegative Ricci curvature.
problem Proving Talenti's comparison theorem for Poisson equations on Riemannian manifolds with nonnegative Ricci curvature.
method Analyzing complete noncompact Riemannian manifolds with nonnegative Ricci curvature, applying Talenti's comparison theorem to Poisson equations.
result Obtained the Faber-Krahn inequality for the first eigenvalue of Dirichlet Laplacian, L1- and L∞-moment spectrum, and a reverse Hölder inequality for eigenfunctions of Dirichlet Laplacian. Formula for Laplacian determinants on polygonal domains with slits.
problem Determining the ζ-regularized determinant of the Laplacian on polygonal domains with slits. method Patchwork method for heat trace asymptotics, comparison formula for smooth conformal metrics.
result Polyakov-Alvarez type formula for Laplacian determinants on polygonal domains with slits.
We present a riemannian structure on the disk that has a remarkably rich structure. Geodesics are hypocycloids and the (negative of the) laplacian has integer spectrum with multiplicity the Dirichlet divisor function. Eigenfunctions of the laplacian are orthogonal polynomials naturally suited to the analysis of acousti…
Estimates gaps between eigenvalues for elliptic operators on manifolds.
problem Estimating the gaps between consecutive eigenvalues for elliptic differential operators.
method Analyzes a class of second-order elliptic differential operators in divergence form with Dirichlet boundary conditions.
result Estimates for the upper bound of gaps between eigenvalues, with results matching known best estimates for specific cases.
Study sets lower bounds for Kähler manifolds' Laplacian eigenvalues.
problem Finding bounds for eigenvalues on Kähler manifolds.
method Establishes lower bounds using geometric data like dimension, diameter, and curvature.
result Proves bounds for Laplacian eigenvalues on Kähler manifolds.
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.
For a bounded domain Ω with a piecewise smooth boundary in a complete Riemannian manifold M, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of L2(Ω) in place of the Rayleigh-Ritz formula, we obtain inequalities for …