Zunwu He, Bobo Hua
In this paper, we study the upper bounds for discrete Steklov eigenvalues on trees via geometric quantities. For a finite tree, we prove sharp upper bounds for the first nonzero Steklov eigenvalue by the reciprocal of the size of the boundary and the diameter respectively. We also prove similar estimates for higher order Steklov eigenvalues.
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We consider Steklov eigenvalues of three-dimensional, nearly-spherical domains. In previous work, we have shown that the Steklov eigenvalues are analytic functions of the domain perturbation parameter. Here, we compute the first-order term of the asymptotic expansion, which can explicitly be written in terms of the Wigner 3-jsymbols. We analyze the asymptotic expansion and prove the isoperimetric result that, if l is a square integer, the volume-normalized l-th Steklov eigenvalue is stationary for a ball.
Weyl asymptotics for Poincaré-Steklov eigenvalues in a domain with Lipschitz boundary
本文证明了在Lipschitz边界区域中，Poincaré-Steklov谱问题特征值的Weyl渐近公式成立。
We extend some classical inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian to the context of mixed Steklov--Dirichlet and Steklov--Neumann eigenvalue problems. The latter one is also known as the sloshing problem, and has been actively studied for more than a century due to its importance in hydrodynamics. The main results of the paper are applied to obtain certain geometric information about nodal sets of sloshing eigenfunctions. The key ideas of the proofs include domain monotonicity for eigenvalues of mixed Steklov problems, as well as an adaptation of Filonov's me…
On the Eigenvalues of the Biharmonic Steklov Problem on a Thin Set
本文研究了薄集上双调和Steklov问题的特征值渐进行为，证明其随厚度趋于零而收敛于零。
Recently, D. Bucur and M. Nahon used boundary homogenisation to show the remarkable flexibility of Steklov eigenvalues of planar domains. In the present paper we extend their result to higher dimensions and to arbitrary manifolds with boundary, even though in those cases the boundary does not generally exhibit any periodic structure. Our arguments use framework of variational eigenvalues and provides a different proof of the original results. Furthermore, we present an application of this flexibility to the optimisation of Steklov eigenvalues under perimeter constraint. It is proved that the b…
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.
We obtain asymptotic formulae for the Steklov eigenvalues and eigenfunctions of curvilinear polygons in terms of their side lengths and angles. These formulae are quite precise: the errors tend to zero as the spectral parameter tends to infinity. The Steklov problem on planar domains with corners is closely linked to the classical sloshing and sloping beach problems in hydrodynamics; as we show it is also related to quantum graphs. Somewhat surprisingly, the arithmetic properties of the angles of a curvilinear polygon have a significant effect on the boundary behaviour of the Steklov eigenfunc…
We introduce the Steklov flows on finite trees, i.e. the flows (or currents) associated with the Steklov problem. By constructing appropriate Steklov flows, we prove the monotonicity and rigidity of the first nonzero Steklov eigenvalues on trees: for finite trees $\g_1$ and $\g_2,$ the first nonzero Steklov eigenvalue of $\g_1$ is greater than or equal to that of $\g_2$, provided that $\g_1$ is a subgraph of $\g_2.$ Moreover, we give the sufficient and necessary condition in which the equality holds.
Sharp bounds for higher Steklov-Dirichlet eigenvalues on domains with spherical holes
本文研究了带有球形孔的领域中高阶Steklov-Dirichlet特征值的精确界，通过等周不等式和空间形式领域的扩展，获得了星形领域中特征值的精确上下界。
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 consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of boundary mass concentration. We discuss the asymptotic behavior of the Neumann eigenvalues in a ball and we deduce that the Steklov eigenvalues minimize the Neumann eigenvalues. Moreover, we study the dependence of the eigenvalues of the Steklov problem upon perturbation of the mass density and show that the Steklov eigenvalues violates a maximum principle in spectral optimization problems.
We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar domains. The normalization involves the perimeter and scale-invariant geometric factors which measure deviation of the domain from roundness. We prove sharp upper bounds for both starlike and simply connected domains, for a large collection of spectral functionals including partial sums of the zeta function and heat trace. The proofs rely on a special class of quasiconformal mappings.
We give an overview of results on shape optimization for low eigenvalues of the Laplacian on bounded planar domains with Neumann and Steklov boundary conditions. These results share a common feature: they are proved using methods of complex analysis. In particular, we present modernized proofs of the classical inequalities due to Szego and Weinstock for the first nonzero Neumann and Steklov eigenvalues. We also extend the inequality for the second nonzero Neumann eigenvalue, obtained recently by Nadirashvili and the authors, to non-homogeneous membranes with log-subharmonic densities. In the h…
Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues
本文提出方法，证明平面角域上晃动问题（混合Steklov-Neumann边值问题）特征值的二项渐近公式，证实Fox-Kuttler猜想，并推广至相关混合问题及多边形Steklov谱研究。
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.
We consider a variant of the classic Steklov eigenvalue problem, which arises in the study of the best trace constant for functions in Sobolev space. We prove that the elementary symmetric functions of the eigenvalues depend real-analytically upon variation of the underlying domain and we compute the corresponding Hadamard-type formulas for the shape derivatives. We also consider isovolumetric and isoperimetric domain perturbations and we characterize the corresponding critical domains in terms of appropriate overdetermined systems. Finally, we prove that balls are critical domains for the ele…
Steklov Eigenvalue Problems on Nearly Spherical and Nearly Annular Domains
本文研究近球形与近环形区域上的 Steklov 特征值问题，利用球谐函数恒等式和加法公式给出区域为临界点的解析条件，并通过数值方法验证了二维和三维结果。
Geometric bounds for Steklov and weighted Neumann eigenvalues on Euclidean domains
研究在欧几里得空间中维数d≥3的有界域的前两个非零Steklov特征值的上界，并证明当d≥7时这些界是最佳的。
Bounds for higher Steklov and mixed Steklov Neumann eigenvalues on domains with holes
本文研究了带球形孔洞区域上的Steklov与混合Steklov-Neumann特征值，推导了同心环域上第二非零特征值的显式表达式，并给出了4阶对称区域前n个非零特征值的精确上界，最后通过数值模拟提出了相关猜想。
Isoperimetric inequalities for the lowest magnetic Steklov eigenvalue
本文研究了平面域上磁斯托克罗夫特问题的最低特征值优化，证明圆盘在给定面积下最大化该特征值，并建立了外域下的等周不等式。
We study a new link between the Steklov and Neumann eigenvalues of domains in Euclidean space. This is obtained through an homogenisation limit of the Steklov problem on a periodically perforated domain, converging to a family of eigenvalue problems with dynamical boundary conditions. For this problem, the spectral parameter appears both in the interior of the domain and on its boundary. This intermediary problem interpolates between Steklov and Neumann eigenvalues of the domain. As a corollary, we recover some isoperimetric type bounds for Neumann eigenvalues from known isoperimetric bounds f…
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.
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.
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。
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 note we present upper bounds for the variational eigenvalues of the Steklov $p$-Laplacian on domains of $\mathbb R^n$, $n\geq 2$. We show that for $1<p\leq n$ the variational eigenvalues $σ_{p,k}$ are bounded above in terms of $k,p,n$ and $|\partialΩ|$ only. In the case $p>n$ upper bounds depend on a geometric constant $D(Ω)$, the $(n-1)$-distortion of $Ω$ which quantifies the concentration of the boundary measure. We prove that the presence of this constant is necessary in the upper estimates for $p>n$ and that the corresponding inequality is sharp, providing examples of domains with …
Steklov eigenvalues of nearly hyperspherical domains
本文研究近超球面区域的Steklov特征值，计算其一阶渐近展开并用Pochhammer符号和Wigner 3j-符号显式表示，进而证明在固定体积下球对某些特征值非最优、对另一些特征值为驻点。
We present upper and lower bounds for Steklov eigenvalues for domains in $\mathbb{R}^{N+1}$ with $C^2$ boundary compatible with the Weyl asymptotics. In particular, we obtain sharp upper bounds on Riesz-means and the trace of corresponding Steklov heat kernel. The key result is a comparison of Steklov eigenvalues and Laplacian eigenvalues on the boundary of the domain by applying Pohozaev-type identities on an appropriate tubular neigborhood of the boundary and the min-max principle. Asymptotically sharp bounds then follow from bounds for Riesz-means of Laplacian eigenvalues.
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.
The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalues on cuboids in dimension d greater or equal to 3. Apart from the standard Weyl term, we calculate explicitly the second term in the asymptotics, capturing the contribution of the (d-2)-dimensional facets of a cuboid. Our approach is based on lattice counting techniques. While this strategy is similar to the one used for the Dirichlet Laplacian, the Steklov case carries additional complications. In particular, it is …
On domain monotonicity of Steklov eigenvalues
本文证明了严格星形有界域上Laplacian的Steklov特征值的一种域单调性变体，并应用于凸域得到上下界。
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 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…
In recent decades qualitative inverse scattering methods with eigenvalues as target signatures received much attention. To understand those methods a knowledge on the properties of the related eigenvalue problems is essential. However, even the existence of eigenvalues for such (nonselfadjoint) problems is a challenging question and existing results for absorbing media are usually established under unrealistic assumptions or a smoothing of the eigenvalue problem. We present a technique to prove the existence of infinitely many eigenvalues for such problems under realistic assumptions. In parti…
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…
Let $Ω$ be a bounded domain with $C^2$-smooth boundary in an $n$-dimensional oriented Riemannian manifold. It is well-known that for the bi-harmonic equation $Δ^2 u=0$ in $Ω$ with the $0$-Dirichlet boundary condition, there exists an infinite set $\{u_k\}$ of biharmonic functions in $Ω$ with positive eigenvalues $\{λ_k\}$ satisfying $Δu_k+ λ_k \varrho \frac{\partial u_k}{\partial ν}=0$ on the boundary $\partial Ω$. In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of the biharmonic Stekloff eigenvalues $λ_k$.
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。
The second eigenvalue of the Robin Laplacian is shown to be maximal for the ball among domains of fixed volume, for negative values of the Robin parameter $α$ in the regime connecting the first nontrivial Neumann and Steklov eigenvalues, and even somewhat beyond the Steklov regime. The result is close to optimal, since the ball is not maximal when $α$ is sufficiently large negative, and the problem admits no maximiser when $α$ is positive.
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…
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。
In this paper, we address the problem of maximizing the Steklov eigenvalues with a diameter constraint. We provide an estimate of the Steklov eigenvalues for a convex domain in terms of its diameter and volume and we show the existence of an optimal convex domain. We establish that balls are never maximizers, even for the first non-trivial eigenvalue that contrasts with the case of volume or perimeter constraints. Under an additional regularity assumption, we are able to prove that the Steklov eigenvalue is multiple for the optimal domain. We illustrate our theoretical results by giving some o…
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 consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of mass concentration at the boundary of a ball. We discuss the asymptotic behavior of the Neumann eigenvalues and find explicit formulas for their derivatives at the limiting problem. We deduce that the Neumann eigenvalues have a monotone behavior in the limit and that Steklov eigenvalues locally minimize the Neumann eigenvalues.
We consider three different questions related to the Steklov and mixed Steklov problems on surfaces. These questions are connected by the techniques that we use to study them, which exploit symmetry in various ways even though the surfaces we study do not necessarily have inherent symmetry. In the spirit of the celebrated Hersch-Payne-Schiffer and Weinstock inequalities for Steklov eigenvalues, we obtain a sharp isoperimetric inequality for the mixed Steklov eigenvalues considering the interplay between the eigenvalues of the mixed Steklov-Neumann and Steklov-Dirichlet eigenvalues. In 1980, Ba…
Biharmonic Steklov problems with Neumann boundary conditions and spectral inequalities on differential forms
引入微分形式上的纽曼型边界条件双调和Steklov问题，证明其适定性、离散谱存在性及特征值变分刻画，并建立Kuttler-Sigillito型不等式联系其他问题特征值。
Sharp bounds for the first two eigenvalues of an exterior Steklov eigenvalue problem
针对n≥3的欧氏空间外域上的Steklov特征值问题，推导了第一个特征值的精确上下界，还得到n=3时连通边界的容量精确上界及第二个特征值的上界。
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…
Spectral ratios and gaps for Steklov eigenvalues of balls with revolution-type metrics
本文研究了具有旋转类型度量的球的Steklov特征值谱比与谱隙，在不依赖Ricci曲率或边界凸性条件下，获得了3维及以上谱比和3维谱隙的最优上界，并在4维及以上谱隙取得上界。
The second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form $α/L(Ω)$, and $α$ lies between $-2π$ and $2π$. Corollaries include Szegő's sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock's inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter. The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization o…
The exterior Steklov problem for Euclidean domains
研究欧几里得外部区域上的Steklov特征值问题，给出多种等价表述，并证明凸域上第一外部Steklov特征值的Escobar型下界，该下界由边界主曲率表示且对球达到最优。
We study the eigenvalues of the Dirichlet-to-Neumann operator on a finite subgraph of the integer lattice Zn. We estimate the first n+1 eigenvalues using the number of vertices of the subgraph. As a corollary, we prove that the first non-trivial eigenvalue of the Dirichlet-to-Neumann operator tends to zero as the number of vertices of the subgraph tends to infinity.
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.
Estimates for the first and second Steklov-Dirichlet eigenvalues
本文研究环形区域中第一和第二 Steklov-Dirichlet 特征值的渐近行为，证明其收敛性质，并由此在小孔条件下建立等周不等式。
We study bounds on the Riesz means of the mixed Steklov-Neumann and Steklov-Dirichlet eigenvalue problem on a bounded domain $Ω$ in $\mathbb{R}^n$. The Steklov-Neumann eigenvalue problem is also called the sloshing problem. We obtain two-term asymptotically sharp lower bounds on the Riesz means of the sloshing problem and also provide an asymptotically sharp upper bound for the Riesz means of mixed Steklov-Dirichlet problem. The proof of our results for the sloshing problem uses the average variational principle and monotonicity of sloshing eigenvalues. In the case of Steklov-Dirichlet eigenva…
Sharp bounds for higher mixed Steklov-Robin eigenvalues on domains with holes
本文针对含球形孔的4阶对称有界域，证明了每个k∈{2,3,…,n+1}的混合Steklov-Robin特征值的等周不等式，并强调对称性假设的关键性。
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.
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