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.
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.
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.
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 prove rigidity of first conformal Steklov eigenvalue on specific shapes.
problem Rigidity of the first conformal Steklov eigenvalue on annuli and Möbius bands.
method Proof relies on uniqueness results, compactness theorem, and asymptotic control of Steklov eigenvalues.
result Rigidity of the first conformal Steklov eigenvalue on annuli and Möbius bands proved.
Upper bounds found for eigenvalues of weighted Steklov and (p,q)-Laplacian problems.
problem Finding upper bounds for eigenvalues of weighted Steklov and (p,q)-Laplacian problems.
method Proving upper bounds using the weighted p-Laplace operator and (p,q)-Laplacian on submanifolds. result Reilly-type upper bounds for the first eigenvalues of Steklov and (p,q)-Laplacian problems.
Proves inequality for Steklov eigenvalues in hyperbolic space.
problem Finding bounds for Steklov eigenvalues in hyperbolic geometry.
method Proves isoperimetric inequality for harmonic mean of eigenvalues.
result Establishes inequality for hyperbolic Steklov eigenvalues.
The ball maximizes the first biharmonic Steklov eigenvalue.
problem Maximizing the first biharmonic Steklov eigenvalue for bounded domains.
method Comparing domains with fixed measure to find the maximum eigenvalue.
result The ball maximizes the first positive biharmonic Steklov eigenvalue.
Paper extends Steklov eigenvalue estimate to weighted graphs.
problem Steklov eigenvalue estimation on weighted graphs.
method Extended Perrin's estimate to general weighted graphs.
result Characterized rigidity of the extended estimate.
Study on biharmonic Steklov problem on differential forms.
problem Characterize and estimate eigenvalues of biharmonic Steklov problem.
method Introduce boundary conditions, prove properties, derive inequalities.
result Characterize smallest eigenvalue and prove spectrum properties.
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.
Upper bound found for Steklov eigenvalues counting function.
problem Counting Steklov eigenvalues on compact manifolds with boundary.
method Used Weyl's law and Pólya's Conjecture in the Steklov case.
result Obtained an upper bound for the counting function.
Sharp upper bounds found for Steklov eigenvalues of warped products.
problem Finding bounds for Steklov eigenvalues of specific metric configurations.
method Investigation of Steklov spectrum for warped products with a fiber of dimension 2.
result Sharp upper bounds for Steklov eigenvalues in terms of the eigenvalues of the Laplacian on the fiber.
Optimal constants for isoperimetric inequalities involving Steklov eigenvalues on surfaces are determined.
problem Determining optimal constants for isoperimetric inequalities involving Steklov eigenvalues on surfaces.
method Analyzing Riemannian surfaces with boundary, considering both given topology and conformal class, and proving inequalities relating conformal invariants and eigenvalues.
result New examples of topological disks realizing optimal constants and inequalities relating conformal invariants of Steklov eigenvalues on surfaces and disks are provided.
The paper finds minimum Steklov eigenvalues on combinatorial graphs.
problem Finding the minimum Steklov eigenvalues on combinatorial graphs.
method Extending Friedman's nodal domain theory for Laplacian eigenfunctions to Steklov eigenfunctions.
result The minimum of the imth Steklov eigenvalue on a connected combinatorial graph is essentially attained by a star or a regular comb with minimal brooms. In this paper, motivated by the work of Raulot and Savo, we generalize Raulot-Savo's estimate for the first Steklov eigenvalues of Euclidean domains to higher Steklov eigenvalues.
The paper bounds higher Steklov eigenvalues of graphs on surfaces.
problem Bounding higher Steklov eigenvalues of graphs on surfaces.
method Using metrical deformation via probability flows, the upper bound is derived.
result The upper bound of higher Steklov eigenvalues is established.
Lower bound found for Steklov eigenvalue on curved manifolds.
problem Finding bounds for Steklov eigenvalues on curved spaces.
method Established a new lower bound using geometric curvature conditions.
result Found a new lower bound for the first non-zero Steklov eigenvalue.
We prove a lower bound for the k-th Steklov eigenvalues in terms of an isoperimetric constant called the k-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Ste…
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.
Recent developments link Steklov eigenvalues to manifold geometry.
problem Steklov eigenvalues and eigenfunctions on compact Riemannian manifolds.
method Analytical and geometric approaches, including isoperimetric bounds, stability analysis, optimisation, and discretization.
result Connections between Steklov eigenvalues and manifold geometry, including optimisation and isospectrality.
New metrics on 3D manifolds with large Steklov eigenvalues.
problem Finding metrics with large Steklov eigenvalues on compact manifolds.
method Expressed Steklov spectrum of warped products and applied to metrics with fixed volume.
result Examples of metrics on 3D manifolds with arbitrarily large first non-zero Steklov eigenvalue.
This is a continuation of our previous work arXiv:1601.05617 on trace and inverse trace of Steklov eigenvalues. More new inequalities for the trace and inverse trace of Steklov eigenvalues are obtained.
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 proves inequalities for Steklov eigenvalues on finite graphs.
problem Eigenvalues of Laplacians for reversible Markov chains and Steklov eigenvalues.
method Generalized Cheeger inequalities, convergence results, and resolvent convergence.
result Sharp estimate for the first non-trivial Steklov eigenvalue.
Unified approach to Laplace and Steklov eigenvalues via n-harmonic maps.
problem Eigenvalue problems on manifolds of arbitrary dimension.
method Unified description using n-harmonic maps. result Uncovering two new features of Steklov eigenvalues.
Upper bounds for Steklov eigenvalues on curved submanifolds.
problem Eigenvalue bounds for Steklov problem on submanifolds.
method Reilly-type upper bounds for p-Steklov eigenvalues. result Proved upper bounds for the first non-zero eigenvalue.
Higher surgeries preserve Steklov spectra in 3D and above.
problem Effect of topology changes on Steklov eigenvalues in higher dimensions.
method Perform surgeries of codimension 2 or higher on compact manifolds.
result Topology changes do not affect Steklov spectra in dimensions 3 and above.
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.
Study eigenvalues and shapes, proving sharp inequalities for Steklov eigenvalues.
problem Eigenvalue continuity and shape optimization for Laplace and Steklov problems.
method Variational eigenvalue analysis, Sobolev space convergence, shape optimization techniques.
result Sharp isoperimetric inequalities for Steklov eigenvalues, upper bound 8πk for k-th perimeter-normalized eigenvalue. Let N be a complete Riemannian manifold of dimension n+1 whose Riemannian metric g is conformally equivalent to a metric with non-negative Ricci curvature. The normalized Steklov eigenvalues of a bounded domain in N are bounded above in terms of the isoperimetric ratio of the domain. Consequently, the normalized Steklo…
Sharp Steklov eigenvalue estimates for differential forms on manifolds.
problem Estimating the first positive eigenvalue of the Steklov eigenvalue problem for differential forms.
method Established a weighted Reilly formula for differential forms and applied it to geometric conditions.
result Sharp lower bound for the first positive eigenvalue of the Steklov eigenvalue problem on differential forms.
The paper sets geometric lower bounds for low Steklov eigenvalues on manifolds.
problem Finding geometric lower bounds for low Steklov eigenvalues on manifolds.
method Using trace inequalities relating Steklov eigenvalues to Neumann eigenvalues of subdomains containing boundary collars.
result Geometric lower bounds for low Steklov eigenvalues, complementing earlier results.
Study magnetic Steklov eigenvalues on manifolds with boundary.
problem Eigenvalue problem for magnetic Steklov operators on compact manifolds.
method Equivalent characterizations, bounds, comparison results.
result Established bounds for the smallest eigenvalue of magnetic Steklov operators.
The paper studies Steklov eigenvalues in space forms and warped product manifolds, deriving bounds and monotonicity results.
problem Estimating Steklov eigenvalues in space forms and warped product manifolds.
method Monotonicity results for Steklov eigenvalues in geodesic disks and warped product manifolds with non-negative Ricci curvature.
result Sharp bounds and monotonicity results for Steklov eigenvalues on warped product manifolds.
Paper proves a new isoperimetric inequality for Steklov eigenvalues.
problem Finding a new isoperimetric inequality for Steklov eigenvalues.
method Proving a Brock-type inequality under specific conditions.
result Extension of Brock's classical result to Witten-Laplacian.
Paper proves surfaces with specific symmetries have the first Steklov eigenvalue.
problem Proving surfaces with certain symmetries have the first Steklov eigenvalue.
method Analyzing surfaces with reflection planes and genus zero.
result Surfaces with n distinct reflection planes have the first Steklov eigenvalue. Lower bound for Steklov eigenvalues on negatively curved manifolds.
problem Finding a geometric lower bound for the first nonzero Steklov eigenvalue.
method Combining a uniform lower bound for the first eigenvalue of the Steklov-Dirichlet problem and a tubular neighborhood theorem for totally geodesic hypersurfaces.
result A geometric lower bound for the first nonzero Steklov eigenvalue in terms of total and boundary volumes.
The paper finds large Steklov eigenvalues on manifolds using homogenization.
problem Finding large Steklov eigenvalues on manifolds.
method Using homogenization theory, the paper constructs manifolds with large Steklov eigenvalues.
result The paper proves that Kokarev's upper bound for the first nonzero normalised Steklov eigenvalue on orientable surfaces of genus 0 is saturated.
The paper provides estimates for Steklov eigenvalues of surfaces with boundary.
problem Estimating Steklov eigenvalues of surfaces with boundary components.
method Computable lower bounds for the first non-zero Steklov eigenvalue using geometric quantities specific to manifolds with boundary.
result The geometry of the manifold away from the boundary affects the Steklov eigenvalue.
Study proves inequality for eigenvalues in symmetric spaces.
problem Proving an inequality for Steklov eigenvalues in symmetric spaces.
method Analyzes eigenvalues on bounded domains in noncompact rank-1 symmetric spaces.
result Extends previous results to non-Euclidean spaces.
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.
Upper bounds for Steklov eigenvalues derived from intersection indices.
problem Finding upper bounds for Steklov eigenvalues of submanifolds in Euclidean space.
method Using intersection indices of submanifolds and their boundaries.
result Explicit upper bounds involving intersection index, volume, and dimensional constants.
Study Steklov eigenvalues on hyperbolic triangle-tiling graphs.
problem Analyzing Steklov eigenvalues on specific hyperbolic graph structures.
method Introduced a graph roughly isometric to hyperbolic plane, used discretization to transfer bounds.
result Steklov eigenvalues tend to zero proportionally to the inverse of the domain size.
Consider a compact Riemannian manifold with boundary. In this short note we prove that under certain positive curvature assumptions on the manifold and its boundary the Steklov eigenvalues of the manifold are controlled by the Laplace eigenvalues of the boundary. Additionally, in two dimensions we obtain an upper bound…
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 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.
Study Cheeger inequalities for Riemannian manifolds with boundary.
problem Estimating Steklov eigenvalues on Riemannian manifolds with boundary.
method Establish Cheeger-type inequalities using isocapacitary constants.
result Cheeger inequalities for Steklov eigenvalues on compact and non-compact manifolds.