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.
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.
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.
We use layer potential to establish that the boundary biharmonic Steklov operators are elliptic pseudo-differential operators. Thus we are able to establish lower bounds on both the measure of boundary nodal sets and interior nodal sets for biharmonic Steklov eigenfunctions.
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.
We prove Li-Yau-Kröger type bounds for Neumann-type eigenvalues of the poly-harmonic operator and of the biharmonic operator on bounded domains in a Euclidean space. We also prove sharp estimates for lower order eigenvalues of a biharmonic Steklov problem and of the Laplacian, which directly implies two sharp Reilly-ty…
Let Ω be a bounded domain with C∞ boundary in an n-dimensional C∞ Riemannian manifold, and let ϱ be a non-negative bounded function defined on ∂Ω. It is well-known that for the biharmonic equation Δ2u=0 in Ω with the 0-Dirichlet boundary condition, there exists an infinite se…
In this paper, we prove some isoperimetric bounds for lower order eigenvalues of the Wentzell-Laplace operator on bounded domains of a Euclidean space or a Hadamard manifold, of the Laplacian on closed hypersurfaces of a Euclidean space or a Hadamard manifold, and of a biharmonic Steklov problem on bounded domains of a…
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…
We study the biharmonic Steklov eigenvalue problem on a compact Riemannian manifold Ω with smooth boundary. We give a computable, sharp lower bound of the first eigenvalue of this problem, which depends only on the dimension, a lower bound of the Ricci curvature of the domain, a lower bound of the mean curvature of i…
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 finds finitely many non-congruent polygonal domains with same Steklov spectrum.
problem Inverse Steklov problem on convex polygons.
method Analysis of Steklov eigenvalues and isoperimetric bounds.
result For almost all convex polygonal domains, there exist at most finitely many non-congruent domains with the same Steklov spectrum.
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.
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.
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.
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.
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.
Sharp upper bound found for Steklov spectrum on revolution submanifolds.
problem Finding bounds for Steklov spectrum on specific submanifolds.
method Analyzing submanifolds of revolution in Euclidean space.
result Sharp upper bound established for Steklov spectrum.
Generic metrics on manifolds yield simple Steklov eigenvalues and Morse boundary functions.
problem Understanding the properties of Steklov eigenfunctions under generic metrics.
method Analyzing smooth compact manifolds with smooth boundaries and generic metrics of Ck type. result Nonzero Steklov eigenvalues are simple and non-constant eigenfunctions are Morse functions on the boundary.
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.
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.
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 study counts critical points of Steklov eigenfunctions on manifolds.
problem Counting critical points of Steklov eigenfunctions on manifolds.
method Established an identity relating indexes of eigenfunctions and their restrictions to the boundary, and used it to count critical points.
result A precise count of interior critical points of Steklov eigenfunctions in terms of manifold's Euler characteristic and boundary sign changes.
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.
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.
Paper introduces magnetic Steklov operator on differential forms and its properties.
problem Analyzing the boundary value problem of magnetic Steklov operator.
method Introduced magnetic Steklov operator and proved its well-posedness. Also, computed spectral properties.
result An analogue of Diamagnetic Inequality does not always hold for magnetic Steklov operators.
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.
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.
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 study the heat trace asymptotics associated with the Steklov eigenvalue problem on a Riemannian manifold with boundary. In particular, we describe the structure of the Steklov heat invariants and compute the first few of them explicitly in terms of the scalar and mean curvatures. This is done by applying the Seeley …
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.
f-Biharmonic maps are the extrema of the f-bienergy functional. f-biharmonic submanifolds are submanifolds whose defining isometric immersions are f-biharmonic maps. In this paper, we prove that an f-biharmonic map from a compact Riemannian manifold into a non-positively curved manifold with constant f-bienergy density…
We continue our study [Ou4] of f-biharmonic maps and f-biharmonic submanifolds by exploring the applications of f-biharmonic maps and the relationships among biharmonicity, f-biharmonicity and conformality of maps between Riemannian manifolds. We are able to characterize harmonic maps and minimal submanifolds by using …
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.
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.
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.
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.
We study biharmonic maps and f-biharmonic maps from a round sphere (S2,g0), the latter maps are equivalent to biharmonic maps from Riemann spheres (S2,f−1g0). We proved that for rotationally symmetric maps between rotationally symmetric spaces, both biharmonicity and f-biharmonicity reduce to a 2nd order l…
The paper solves the Steklov spectral inverse problem for conformal metrics.
problem Recovering a metric from its Steklov spectrum in dimension n≥3.
method Combines wave trace formula techniques with geodesic X-ray transform.
result Steklov isospectral metrics must coincide under real-analyticity assumption.
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.
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.
The article explores constructing biharmonic and conformal biharmonic maps to spheres.
problem Constructing biharmonic and conformal biharmonic maps to spheres.
method Geometric algorithm to render harmonic maps biharmonic or conformally biharmonic.
result Explicit critical points for conformal-biharmonic maps between spheres are found.
Survey on biharmonic Riemannian submersions, a dual concept of biharmonic submanifolds.
problem Understanding biharmonic Riemannian submersions as a dual concept.
method Short survey focusing on biharmonic Riemannian submersions.
result Survey provides insights into the dual concept of biharmonic submanifolds.
The paper studies f-biharmonic maps and submersions in space forms.
problem Characterizing f-biharmonic maps and submersions in space forms. method Analyzing f-biharmonic curves, developing classifications, and using integrability data. result Proper f-biharmonic developable surfaces exist only in the case of cylinders. Study of p-biharmonic curves and their properties.
problem Generalizing biharmonic curves to p-biharmonic curves. method Classification and analysis of p-biharmonic curves on surfaces and space forms. result Existence and stability of p-biharmonic curves on closed surfaces.