Bounds for the Steklov eigenvalues

Let \(\Omega \) 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 \(\Omega .\) This result extends a result given by Kuttler and Sigillito (SIAM Rev 10:368–370, 1968) for a star-shaped bounded domain in \(\mathbb {R}^2\). Further we also obtain a two sided bound for the eigenvalues of the Steklov problem on a ball in \(\mathbb {R}^n\) with rotationally invariant metric and with bounded radial curvature.     

On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues

We prove that the Hersch-Payne-Schiffer isoperimetric inequality for the nth nonzero Steklov eigenvalue of a bounded simply connected planar domain is sharp for all n ⩾ 1. The equality is attained in the limit by a sequence of simply connected domains degenerating into a disjoint union of n identical disks. Similar results are obtained for the product of two consecutive Steklov eigenvalues. We also give a new proof of the Hersch-Payne-Schiffer inequality for n = 2 and show that it is strict in this case.     

An inequality for Steklov eigenvalues for planar domains

Study of the zeta function associated to the Neumann operator on planar domains yields an inequality for Steklov eigenvalues for planar domains.This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.     

Large deviations of the extreme eigenvalues of random deformations of matrices

Consider a real diagonal deterministic matrix X _n of size n with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We show that the joint law of the extreme eigenvalues of the perturbed model satisfies a large deviation principle in the scale n, with a good rate function given by a variational formula. We tackle both cases when the extreme eigenvalues of X _n converge to the edges of the support of the limiting measure and when we allow some eigenvalues of X _n , that we call outliers, to converge out of the bulk. We can also generalise our results to the case when X _n is random, with law proportional to e ^{?n Tr V(X)} dX, for V growing fast enough at infinity and any perturbation of finite rank.     

Nonresonance between the first two eigenvalues for a Steklov problem

In this paper, we study the solvability of the Steklov problem Δ_pu=|u|^p−2u in Ω, on ∂Ω, under assumptions on the asymptotic behaviour of the quotients f(x,s)/|s|^p−2s and pF(x,s)/|s|^p which extends the classical results with Dirichlet boundary conditions that for a.e. x∈∂Ω, the limits at the infinity of these quotients lie between the first two eigenvalues.     

Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem

In this paper, we find upper bounds for the eigenvalues of the Laplacian in the conformal class of a compact Riemannian manifold (M,g). These upper bounds depend only on the dimension and a conformal invariant that we call “min-conformal volume”. Asymptotically, these bounds are consistent with the Weyl law and improve previous results by Korevaar and Yang and Yau. The proof relies on the construction of a suitable family of disjoint domains providing supports for a family of test functions. This method is interesting for itself and powerful. As a further application of the method we obtain an upper bound for the eigenvalues of the Steklov problem in a domain with C¹ boundary in a complete Riemannian manifold in terms of the isoperimetric ratio of the domain and the conformal invariant that we introduce.     

The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues on Riemannian manifolds

Let Ω be a bounded domain with C²-smooth boundary in an n-dimensional oriented Riemannian manifold. It is well known that for the biharmonic equation Δ²u=0 in Ω with the condition u=0 on ∂Ω, there exists an infinite set {uk} of biharmonic functions in Ω with positive eigenvalues {λk} satisfying on ∂Ω. In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of the biharmonic Steklov eigenvalues λk.     

An optimization problem for nonlinear Steklov eigenvalues with a boundary potential

In this paper, we analyze an optimization problem for the first (nonlinear) Steklov eigenvalue plus a boundary potential with respect to the potential function which is assumed to be uniformly bounded and with fixed L¹$L^1$-norm.     

Superalgebra of conformal deformations for twistor-like superspaces

A twistor-like extension of the N=2 conformal superspace (SSp) with nontrivial central charges is presented. Parameters of new (anti)commutative generators carrying an index of the spin SU (2)-symmetry are used to construct massless harmonics, dramatically reducing the number of essential coordinates. Under this approach, the usual vector coordinates become complex, while the Grassmann coordinates are reduced to a single variable which is scalar under the Lorentz transformation. The choice of the appropriate analytical subspace in the initial SSp is rather difficult, but our closed algebra gives the possibility of covariantly reducing the theory from the initial SSp to an SSp of a smaller Grassmann dimension. A special definition of the coordinates and the flat space for this structure allows one to obtain the Penrose twistor equation and the Ogievetsky harmonics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 114–121, January, 1997.     

Steklov特征值问题的一种有效的Legendre-Galerkin谱逼近

本文给出Steklov特征值问题基于Legendre-Galerkin逼近的一种有效的谱方法.首先利用Legendre多项式构造了一组适当的基函数使得离散变分形式中的矩阵是稀疏的,然后推导了2维及3维情形下离散变分形式基于张量积的矩阵形式,由此可以快速地计算出离散的特征值和特征向量.文章还给出了误差分析和数值试验,数值结果表明本文提出的方法是稳定和有效的.     

The first Steklov eigenvalue, conformal geometry, and minimal surfaces

We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first nonzero eigenvalue σ₁ of the Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces Σ with genus γ and k boundary components we obtain the upper bound σ₁L(∂Σ)?2(γ+k)π. For γ=0 and k=1 this result was obtained by Weinstock in 1954, and is sharp. We attempt to find the best constant in this inequality for annular surfaces (γ=0 and k=2). For rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a solution of the free boundary problem for the area functional in the ball. For a general class of (not necessarily rotationally symmetric) metrics on the annulus, which we call supercritical, we prove that σ₁(Σ)L(∂Σ) is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the boundary. Motivated by the annulus case, we show that a proper submanifold of the ball is immersed by Steklov eigenfunctions if and only if it is a free boundary solution. We then prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities. We show that these bounds are only achieved when the manifold is minimally immersed by first Steklov eigenfunctions. We also use these ideas to show that any free boundary solution in two dimensions has area at least π, and we observe that this implies the sharp isoperimetric inequality for free boundary solutions in the two-dimensional case.     

The Neumann problem for the Hénon equation,trace inequalities and Steklov eigenvalues

We consider the Neumann problem for the Hénon equation. We obtain existence results and we analyze the symmetry properties of the ground state solutions. We prove that some symmetry and variational properties can be expressed in terms of eigenvalues of a Steklov problem. Applications are also given to extremals of certain trace inequalities.     

Upper bounds for Steklov eigenvalues of submanifolds in Euclidean space via the intersection index

We obtain upper bounds for the Steklov eigenvalues ${\sigma }_k(M)$ of a smooth, compact, 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 $k\to \infty$.     

A spectral projection method for transmission eigenvalues

We consider a nonlinear integral eigenvalue problem, which is a reformulation of the transmission eigenvalue problem arising in the inverse scattering theory. The boundary element method is employed for discretization, which leads to a generalized matrix eigenvalue problem. We propose a novel method based on the spectral projection. The method probes a given region on the complex plane using contour integrals and decides whether the region contains eigenvalue(s) or not. It is particularly suitable to test whether zero is an eigenvalue of the generalized eigenvalue problem, which in turn implies that the associated wavenumber is a transmission eigenvalue. Effectiveness and efficiency of the new method are demonstrated by numerical examples.     

Radially increasing minimizing surfaces or deformations under pointwise constraints on positions and gradients

In this paper, we prove existence of radially symmetric minimizersu_A(x)=U_A(|x|), having U_A(⋅)AC monotone and increasing, for the convex scalar multiple integral(∗ ) among those u(⋅) in the Sobolev space. Here, |∇u(x)| is the Euclidean norm of the gradient vector and B_R is the ball ; while A is the boundary data.Besides being e.g. superlinear (but no growth needed if (∗) is known to have minimum), our Lagrangian?^∗∗:R×R→[0,∞] is just convex lsc and and ?^∗∗(s,⋅) is even; while ρ₁(⋅) and ρ₂(⋅) are Borel bounded away from .Remarkably, (∗) may also be seen as the calculus of variations reformulation of a distributed-parameter scalar optimal control problem. Indeed, state and gradient pointwise constraints are, in a sense, built-in, since ?^∗∗(s,v)=∞ is freely allowed.     

Biharmonic maps, conformal deformations and the Hopf maps

We consider harmonic semi-conformal maps between two Riemannian manifolds. By deforming conformally the codomain metric, we construct new examples of non-harmonic biharmonic maps.