Yongjie Shi, Chengjie Yu
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.
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In this paper, we obtain a comparison of Steklov eigenvalues and Laplacian eigenvalues on graphs and discuss its rigidity. As applications of the comparison of eigenvalues, we obtain Lichnerowicz-type estimates and some combinatorial estimates for Steklov eigenvalues on graphs.
Boundary Perturbations of Steklov Eigenvalues
研究Steklov特征值在边界扰动下的行为，证明其在光滑扰动下通常简单，补充了度量扰动的相关结论。
In this paper, we obtain a Lichnerowicz-type estimate for the first Steklov eigenvalues on graphs and discuss its rigidity.
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 for these Steklov eigenvalues in terms of topology of the surface without any curvature restrictions.
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 bound of the counting function of Steklov eigenvalues
本文研究了带边界紧流形上Steklov特征值计数函数，获得了涉及Weyl定律主项的上界估计，可视为一般流形Steklov情形下Pólya猜想的弱化版本，并描述了特征函数在边界附近的衰减行为。
We prove two explicit bounds for the multiplicities of Steklov eigenvalues $σ_k$ on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index $k$ of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given smooth Riemannian surface with boundary, the multiplicities of Steklov eigenvalues $σ_k$ are uniformly bounded in $k$.
We prove that in Riemannian manifolds the $k$-th Steklov eigenvalue on a domain and the square root of the $k$-th Laplacian eigenvalue on its boundary can be mutually controlled in terms of the maximum principal curvature of the boundary under sectional curvature conditions. As an application, we derive a Weyl-type upper bound for Steklov eigenvalues. A Pohozaev-type identity for harmonic functions on the domain and the min-max variational characterization of both eigenvalues are important ingredients.
We show that for compact Riemannian manifolds of dimension at least $3$ with nonempty boundary, we can modify the manifold by performing surgeries of codimension $2$ or higher, while keeping the Steklov spectrum nearly unchanged. This shows that certain changes in the topology of a domain do not have an effect when considering shape optimization questions for Steklov eigenvalues in dimensions $3$ and higher. Our result generalizes the 1-dimensional surgery in [FS2] to higher dimensional surgeries and to higher eigenvalues. It is proved in [FS2] that the unit ball does not maximize the first no…
Using expander graphs, we construct a sequence of smooth compact surfaces with boundary of perimeter N, and with the first non-zero Steklov eigenvalue uniformly bounded away from zero. This answers a question which was raised in [9]. The genus grows linearly with N, this is the optimal growth rate.
We give explicit isoperimetric upper bounds for all Steklov eigenvalues of a compact orientable surface with boundary, in terms of the genus, the length of the boundary, and the number of boundary components. Our estimates generalize a recent result of Fraser-Schoen, as well as the classical inequalites obtained by Hersch-Payne-Schiffer, whose approach is used in the present paper.
Large Steklov eigenvalues under volume constraints
本文通过辅助 Steklov 问题给出乘积流形 Steklov 谱的表达式，并证明在固定体积和边界条件下，三维流形可具有任意大的第一个非零 Steklov 特征值。
We obtain upper bounds for the Steklov eigenvalues $σ_k(M)$ of a smooth, compact, connected, $n$-dimensional submanifold $M$ of Euclidean space with boundary $Σ$ that involve the intersection indices of $M$ and of $Σ$. One of our main results is an explicit upper bound in terms of the intersection index of $Σ$, the volume of $Σ$ and the volume of $M$ as well as dimensional constants. By also taking the injectivity radius of $Σ$ into account, we obtain an upper bound that has the optimal exponent of $k$ with respect to the asymptotics of the Steklov eigenvalues as $k \to \infty$.
We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with connected boundary: we prove that each eigenvalue is uniquely minimized by the ball. We also observe that each surface of revolution with connected boundary is isospectral to the disk.
We prove that the isoperimetric inequality due to Hersch-Payne-Schiffer for the n-th nonzero Steklov eigenvalue of a bounded simply-connected planar domain is sharp for all n=1,2,... The equality is attained in the limit by a sequence of simply-connected domains degenerating to the disjoint union of n identical disks. We give a new proof of this inequality for n=2 and show that it is strict in this case. Related results are also obtained for the product of two consecutive Steklov eigenvalues.
Some recent developments on the Steklov eigenvalue problem
本文综述Steklov特征值问题近期进展，涵盖等周型界、度量变形下的稳定性、优化与自由边界极小曲面、反问题及特征函数几何，并对比Laplace谱，提出开放问题。
We present a unified description of extremal metrics for the Laplace and Steklov eigenvalues on manifolds of arbitrary dimension using the notion of $n$-harmonic maps. Our approach extends the well-known results linking extremal metrics for eigenvalues on surfaces with minimal immersions and harmonic maps. In the process, we uncover two previously unknown features of the Steklov eigenvalues. First, we show that in higher dimensions there is a unique normalization involving both the volume of the boundary and of the manifold itself, which leads to meaningful extremal eigenvalue problems. Second…
Given a smooth compact hypersurface $M$ with boundary $Σ=\partial M$, we prove the existence of a sequence $M_j$ of hypersurfaces with the same boundary as $M$, such that each Steklov eigenvalue $σ_k(M_j)$ tends to zero as $j$ tends to infinity. The hypersurfaces $M_j$ are obtained from $M$ by a local perturbation near a point of its boundary. Their volumes and diameters are arbitrarily close to those of $M$, while the principal curvatures of the boundary remain unchanged.
In recent years, eigenvalue optimization problems have received a lot of attention, in particular, due to their connection with the theory of minimal surfaces. In the present paper we prove that on any orientable surface there exists a smooth metric maximizing the first normalized Steklov eigenvalue. For surfaces of genus zero, this has been earlier proved by A. Fraser and R. Schoen. Our approach builds upon their ideas and further developments due to R. Petrides. As a corollary, we show that there exist free boundary branched minimal immersions of an arbitrary compact orientable surface with …
We show rigidity of the first conformal Steklov eigenvalue on annuli and Möbius bands. The proof relies, among others, on uniqueness results due to Fraser--Schoen, a compactness theorem of the second named author, and recent work of the authors on asymptotic control of Steklov eigenvalues in glueing constructions.
Upper bound for Steklov eigenvalues of warped products with fiber of dimension 2
本文针对纤维为紧致曲面的扭曲乘积，利用纤维上Laplacian特征值给出Steklov特征值的尖锐上界，并应用于三维球体旋转度量，得到相邻Steklov特征值谱间隙的尖锐估计。
We consider the Steklov problem associated with the weighted p-Laplace operator and $(p,q)$-Laplacian on submanifolds with the boundary of Euclidean spaces and prove Reilly-type upper bounds for their first eigenvalues.
In this paper we construct compact manifolds with fixed boundary geometry which admit Riemannian metrics of unit volume with arbitrarily large Steklov spectral gap. We also study the effect of localized conformal deformations that fix the boundary geometry. For instance, we prove that it is possible to make the spectral gap arbitrarily large using conformal deformations which are localized on domains of small measure, as long as the support of the deformations contains and connects each component of the boundary.
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.
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 Steklov eigenvalues. In particular it extends the Cheeger type inequality for the first nonzero Steklov eigenvalue previously studied by Escobar in 1997 and by Jammes in 2015 to higher order Steklov eigenvalues. The technique we develop to get this lower bound is based on considerin…
Extension and rigidity of Perrin's lower bound estimate for Steklov eigenvalues on graphs
本文将Perrin关于单位权图Steklov特征值的下界估计推广到一般加权图，并刻画了该估计的刚性条件。
We show that the ball does not maximize the first nonzero Steklov eigenvalue among all contractible domains of fixed boundary volume in $\mathbb{R}^n$ when $n \geq 3$. 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 eigenvalue among all simply connected domains in the plane having the same boundary length. When $n \geq 3$, we show that increasing the number of boundary components does not increase the normalized (by boundary volume) first Steklov eigenvalue. This is in contrast to recent results whi…
Lower bounds for the low Steklov eigenvalues
针对具有 b 个边界分支的紧致连通可定向黎曼流形，通过迹不等式和边界环的电阻解释，获得了低 Steklov 特征值（σ_k, 1≤k≤b-1）的几何下界，揭示了内部几何对低特征值的影响。
Let M be a compact Riemannian manifold with boundary. Let b>0 be the number of connected components of its boundary. For manifolds of dimension at least 3, we prove that it is possible to obtain an arbitrarily large (b+1)-th Steklov eigenvalue using a smooth conformal perturbation which is supported in a thin neighbourhood of the boundary, identically equal to 1 on the boundary. For j<b+1, it is also possible to obtain arbitrarily large j-th eigenvalue, but this require the conformal factor to spread throughout the interior of the manifold M. This is in stark contrast with the situation for th…
In this paper we study spectral properties of Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator $Λ$ is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there.We investigate properies of eigenvalues of $Λ$ and prove a Hersch-Payne-Schiffer type inequality relating products of those eigenvalues to eigenvalues of Hodge Laplacian on the boundary. Moreover, non-trivial eigenvalues of $Λ$ are always at least as large as eigenvalues of Dirichlet-to-…
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 Laplacian, which can be seen as a natural generalization of the classical Steklov eigenvalue problem, have been obtained.
Eigenvalue bounds for the Steklov problem on differential forms in warped product manifolds
本文研究了扭曲积流形上微分p-形式的Steklov问题，导出了多种几何特征值界，包括Escobar型下界和旋转超曲面的精确界，并揭示了底层拓扑对p-形式谱的影响。
We introduce the biharmonic Steklov problem on differential forms by considering suitable boundary conditions. We characterize its smallest eigenvalue and prove elementary properties of the spectrum. We obtain various estimates for the first eigenvalue, some of which involve eigenvalues of other problems such as the Dirichlet, Neumann, Robin and Steklov ones. Independently, new inequalities relating the eigenvalues of the latter problems are proved.
Sharp upper bounds for Steklov eigenvalues of a hypersurface of revolution with two boundary components in Euclidean space
本文研究了欧氏空间中具有两个边界分量的旋转超曲面的 Steklov 特征值精确上界问题，给出了首个非零特征值的精确上界 $B_n(L)$ 和 $B_n$，并证明了这些上界的稳定性性质。
In this paper, we obtain some new estimates for the trace and inverse trace of Steklov eigenvalues. The estimates generalize some previous results of Hersch-Payne-Schiffer , Brock}, Raulot-Savo and Dittmar.
Scaling inequalities for Steklov eigenvalues in space forms and sharp eigenvalue estimates on warped product manifolds
本文在空间形式中推导了Steklov特征值的尺度不等式，并在扭曲乘积流形上获得了精确特征值估计，确认了关于四阶Steklov问题的猜想。
Isoperimetric inequalities involving Steklov eigenvalues on surfaces
本文研究了带边界曲面上涉及 Steklov 特征值的等周不等式最优常数，在 Riemann 曲面拓扑和共形类层面证明了相关不等式，提供了实现最优常数的拓扑圆盘新实例，并证明了环面和 Möbius 带上第一共形 Steklov 特征值的刚性。
Geometric bounds for low Steklov eigenvalues of finite volume hyperbolic surfaces
针对带测地边界的有限体积双曲曲面，研究得到低Steklov特征值的几何下界，明确了其与最短分离多重测地线长度的依赖关系及紧性，拓展改进了紧情形的已有结果。
In this article, we prove an isoperimetric inequality for the harmonic mean of the first $(n-1)$ nonzero Steklov eigenvalues on bounded domains in $n$-dimensional Hyperbolic space. Our approach to prove this result also gives a similar inequality for the first $n$ nonzero Steklov eigenvalues on bounded domains in $n$-dimensional Euclidean space.
In this paper we prove that given a volume, among all domains with smooth boundary in rank-1 symmetric spaces of noncompact type, geodesic balls maximizes the first nonzero Steklov eigenvalue. We also prove a comparison result for the first nonzero Steklov eigenvalue for domains in simply connected Riemannian manifolds with certain curvature bounds.
The first Steklov eigenvalue on manifolds with nonnegative Ricci curvature and convex boundary
本文为具有非负 Ricci 曲率与凸边界的紧致 Riemannian 流形的第一非零 Steklov 特征值建立了新的下界，并在更弱的几何假设下获得了相关结果。
In this paper, we generalize the Hersch-Payne-Schiffer inequality for Steklov eigenvalues to higher dimensional case by extending the trick used by Hersch, Payne and Schiffer to higher dimensional manifolds.
Upper bounds for the Steklov eigenvalues of warped products
本文研究了 warped products 的 Steklov 特征值上界，结合体积、拉普拉斯特征值和挠动函数的 L^p-范数，得出了不同维度和 p 值下的上界结果，并在某些情况下给出了最优估计。
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 Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniformly bounded above. On a compact surface with boundary, the normalized Steklov eigenvalues are uniformly bounded above in terms of the genus. We also obtain a relationship bet…
Upper bounds for Steklov eigenvalues of a hypersurface of revolution
本文为边界由两个不同半径球面组成的旋转曲面建立了第一Steklov特征值的上界，并证明了该上界在某些情形下是尖锐的。
We prove several results about the multiplicity of the first Steklov eigenvalues on compact surfaces with boundary. We improve some bounds on the multiplicity, especially for the first eigenvalue, and we prove they are sharp on some surfaces of small genus. In a previous article, we defined a new chromatic invariant of surfaces with boundary and conjectured that this invariant is related to the bound on the first eigenvalue. In the present article, we study this invariant, and prove that the conjecture is true when the known bound is sharp.
In this paper, we obtain monotonicity of Steklov eigenvalues on graphs which as a special case on trees extends the results of He-Hua [Calc. Var. Partial Differential Equations 61 (2022), no. 3, Paper No. 101, arXiv: 2103.07696] to higher Steklov eigenvalues and gives affirmative answers to two problems proposed in He-Hua [arXiv: 2103.07696]. As applications of the monotonicity of Steklov eigenvalues, we obtain some estimates for Steklov eigenvalues on trees generalizing the isodiametric estimate for the first positive Steklov eigenvalues on trees in He-Hua [arXiv:2011.11014].
The Steklov spectrum of convex polygonal domains I: spectral finiteness
本文研究凸多边形的 Steklov 反问题，证明几乎所有凸多边形至多有有限多个等谱但不全等的区域，并给出数量上界及特征值等周界。
Metric upper bounds for Steklov and Laplace eigenvalues
本文利用度量-测度技术与集中不等式，建立带边界紧 Riemann 流形 Steklov 特征值的两个上界，并推广得到 Laplace 特征值界及边界几何推论。
A note on the magnetic Steklov operator on functions
本文研究了紧致黎曼流形上一般磁性势的磁性 Steklov 特征值问题，通过引入磁性 Cheeger 常数证明了第一特征值的下界，计算了特定欧氏球的完整谱，并推广了非磁性情况下的特征值差值几何上界。
In this note, an isoperimetric inequality for the harmonic mean of lower order Steklov eigenvalues is proved on bounded domains in noncompact rank-$1$ symmetric spaces. This work extends result of \cite{BR.01} and \cite{V.21} proved for bounded domains in Euclidean space and Hyperbolic space, respectively.
Let $Ω$ be a star-shaped bounded domain in $(\mathbb{S}^{n}, ds^{2})$ with smooth boundary. In this article, we give a sharp lower bound for the first non-zero eigenvalue of the Steklov eigenvalue problem in $Ω.$ This result is the generalization of a result given by Kuttler and Sigillito for a star-shaped bounded domain in $\mathbb{R}^2.$ Further we also obtain a two sided bound for the first non-zero eigenvalue of the Steklov problem on the ball in $\mathbb{R}^n$ with rotationally invariant metric and with bounded radial curvature.
This work is an extension of a result given by Kuttler and Sigillito (SIAM Rev $10$:$368-370$, $1968$) on a star-shaped bounded domain in $\mathbb{R}^2$. Let $Ω$ be a star-shaped bounded domain in a hypersurface of revolution, having smooth boundary. In this article, we obtain a sharp lower bound for all Steklov eigenvalues on $Ω$ in terms of the Steklov eigenvalues of the largest geodesic ball contained in $Ω$ with the same center as $Ω$. We also obtain similar bounds for all Steklov eigenvalues on star-shaped bounded domain in paraboloid, $P = \left\lbrace (x, y, z) \in \mathbb{R}^{3} : z = …
We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying shar…
Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian manifold to weighted Laplace eigenvalues of that manifold. The domains are obtained by removing small geodesic balls that are asymptotically densely uniformly distributed as their radius tends to zero. We use this relationship to construct manifolds that have large Steklov eigenvalues. In dimension two, and with constant weight equal to 1, we prove that Kokarev's upper bound of $8π$ for the first nonzero normalised Steklov eigenvalue on or…
Conformal and extrinsic upper bounds for the harmonic mean of Neumann and Steklov eigenvalues
本文研究了Neumann和Steklov特征值调和平均值的共形与内蕴上界，并给出了闭子流形在空间形式中的最优上界。
In this short note, we show the rigidity of a trace estimate for Steklov eigenvalues with respect to functions in our previous work (Trace and inverse trace of Steklov eigenvalues. J. Differential Equations 261 (2016), no. 3, 2026--2040.). Namely, we show that equality of the estimate holds if and only if the manifold is a direct product of a round ball and a closed manifold. The key ingredient in the proof is a decomposition theorem for flat and totally geodesic Riemannian submersions which may be of independent interests.
We obtain supremum of the k-th normalized Steklov eigenvalues of all rotational symmetric conformal metrics on the cylinder with k>1. The case k=1 for all conformal metrics has been completely solved by Fraser and Schoen. We give geometric description in terms of minimal surfaces for metrics attaining the supremum. We also obtain some partial results on the comparison of the normalized Stekov eigenvalues of rotationally symmetric metrics and general conformal metrics on the cylinder. A counter example is constructed to show that for that the first normalized Steklov eigenvalue of rotationally …
A weighted Reilly formula for differential forms and sharp Steklov eigenvalue estimates
本文建立带边黎曼流形上微分形式的加权Reilly公式，并利用它得到微分形式Steklov特征值第一正特征值的尖锐下界及与边界Hodge Laplacian谱的比较结果，同时提出一个类比Escobar猜想的公开问题。
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-type inequalities for the corresponding first nonzero eigenvalue.
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