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…
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…
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…
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…
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.
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.
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. 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.
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.
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 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.
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.
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 …
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.
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.
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.
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.
This paper is devoted to an inverse Steklov problem for a particular class of n-dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. We prove that the knowledge of the Steklov spectrum determines uniquely the associated warping function up to a natural invariance.
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. 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.
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.
The bienergy of smooth maps between Riemannian manifolds, when restricted to unit vector fields, yields two different variational problems depending on whether one takes the full functional or just the vertical contribution. Their critical points, called biharmonic unit vector fields and biharmonic unit sections, form …
We use two of the most fruitful methods for constructing isospectral manifolds, the Sunada method and the torus action method, to construct manifolds whose Dirichlet-to-Neumann operators are isospectral at all frequencies. The manifolds are also isospectral for the Robin boundary value problem for all choices of Robin …
The paper explores p-biharmonic hypersurfaces in Einstein and conformally flat spaces.
problem Characterizing p-biharmonic submanifolds in Einstein spaces.
method Analyzing properties and constructing examples of p-biharmonic hypersurfaces.
result New examples of proper p-biharmonic hypersurfaces constructed.
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.
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.
Study on biharmonic heat equation on manifolds with curvature constraints.
problem Analyzing entire solutions of biharmonic heat equation on manifolds.
method Exponential decay estimates for biharmonic heat kernel under Ricci curvature and noncollapsing conditions. Proving uniqueness criteria for Cauchy problem.
result Conservation law for biharmonic heat kernel and uniform L-infinity estimate for entire solutions.
In this note, we give a brief survey on some recent developments of biharmonic submanifolds. After reviewing some recent progress on Chen's biharmonic conjecture, the Generalized Chen's conjecture on biharmonic submanifolds of non-positively curved manifolds, and some classifications of biharmonic submanifolds of spher…
Numerical methods solve Steklov eigenvalue problems to generate free boundary minimal surfaces.
problem Generating free boundary minimal surfaces using Steklov eigenvalue problems.
method Maximizing Steklov eigenvalues over a class of metrics, using conformal uniformization and gradient-based optimization.
result Numerical solutions for free boundary minimal surfaces with various boundary components.
We consider a shape optimization problem for the first mixed Steklov-Dirichlet eigenvalues of domains bounded by two balls in two-point homogeneous space. We give a geometric proof which is motivated by Newton's shell theorem
Upper bounds for Steklov eigenvalues in subgraphs of polynomial growth Cayley graphs.
problem Finding upper bounds for Steklov eigenvalues in subgraphs of polynomial growth Cayley graphs.
method Discretizing a bounded domain and using comparison theorems.
result The $k^{\mbox{th}}$ eigenvalue tends to 0 proportionally to 1/∣B∣d−11. 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.
Proves Chen's conjecture on biharmonic submanifolds in Euclidean space and space forms.
problem Chen's conjecture on biharmonic submanifolds in Euclidean space.
method Derived a fundamental identity involving the mean curvature vector field and used it to prove the conjecture.
result Proved Chen's conjecture on biharmonic submanifolds in a Euclidean space and space forms.
Study proves spectral determination of triangles and quadrilaterals, with restrictions on higher-order polygons.
problem Determining the geometry of convex polygons from their Steklov spectra.
method Analysis of characteristic polynomial and spectral properties of Steklov spectrum.
result Almost all triangles and certain quadrilaterals are uniquely determined by their Steklov spectra.
Study on shape optimization for specific eigenvalue problems on domains.
problem Shape optimization of eigenvalue problems for fourth order Steklov.
method Asymptotic expansion and sharp upper bound derivation.
result Derivation of eigenvalue spectra and shape optimization conclusions.
The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been a growing interest in the Steklov problem from the viewpoint of spectral geometr…
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.
New statistical biharmonic maps derived from a variation problem.
problem Variation problem for mappings between statistical manifolds.
method Statistical biharmonic maps derived from the Euler-Lagrange equation.
result Improper affine hyperspheres induce examples of statistical biharmonic maps.
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…
We show that the ball does not maximize the first nonzero Steklov eigenvalue among all contractible domains of fixed boundary volume in Rn when n≥3. This is in contrast to the situation when n=2, where a result of Weinstock from 1954 shows that the disk uniquely maximizes the first Steklov eigenval…