Steklov Eigenvalue Research Articles

Estimates of the first Steklov eigenvalue of properly embedded minimal hypersurfaces with free boundary

We consider a properly embedded minimal hypersurfacewith free boundary in a compact n-dimensional Riemannian manifold M be with nonnegative Ricci curvature and strictly convex boundary. Here, we obtain a new estimate from below for the first nonzero Steklov eigenvalue.

Estimates of the first Steklov eigenvalue of properly embedded minimal hypersurfaces with free boundary

Estimates of the first Steklov eigenvalue of properly embedded minimal hypersurfaces with free boundary

An adaptive algorithm based on the shifted inverse iteration for the Steklov eigenvalue problem

This paper proposes and analyzes an a posteriori error estimator for the finite element multi-scale discretization approximation of the Steklov eigenvalue problem. Based on the a posteriori error estimates, an adaptive algorithm of shifted inverse iteration type is designed. Finally, numerical experiments comparing the performances of three kinds of different adaptive algorithms are provided, which illustrate the efficiency of the adaptive algorithm proposed here.

A Type of Multigrid Method Based on the Fixed-Shift Inverse Iteration for the Steklov Eigenvalue Problem

For the Steklov eigenvalue problem, we establish a type of multigrid discretizations based on the fixed-shift inverse iteration and study in depth its a priori/a posteriori error estimates. In addition, we also propose an adaptive algorithm on the basis of the a posteriori error estimates. Finally, we present some numerical examples to validate the efficiency of our method.

The shape of the fundamental sloshing mode in axisymmetric containers

We numerically study positions of high spots (extrema) of the fundamental sloshing mode of a liquid in an axisymmetric tank. Our approach is based on a linear model and reduces the problem to an appropriate Steklov eigenvalue problem. We propose a numerical scheme for calculating sloshing modes and a novel method for making images of an oscillating fluid.

Second Domain Variation for Problems with Robin Boundary Conditions

In this paper, the first and second domain variations for functionals related to elliptic boundary and eigenvalue problems with Robin boundary conditions are computed. Extremal properties of the ball among nearly spherical domains of given volume are derived. The discussion leads to a Steklov eigenvalue problem. As a by-product, a general characterization of the optimal shapes is obtained.

A Multilevel Correction Method for Steklov Eigenvalue Problem by Nonconforming Finite Element Methods

AbstractIn this paper, a multilevel correction scheme is proposed to solve the Steklov eigenvalue problem by nonconforming finite element methods. With this new scheme, the accuracy of eigenpair approximations can be improved after each correction step which only needs to solve a source problem on finer finite element space and an Steklov eigenvalue problem on the coarsest finite element space. This correction scheme can increase the overall efficiency of solving eigenvalue problems by the nonconforming finite element method. Furthermore, as same as the direct eigenvalue solving by nonconforming finite element methods, this multilevel correction method can also produce the lower-bound approximations of the eigenvalues.

Steklov Eigenvalues and Quasiconformal Maps of Simply Connected Planar Domains

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.

Local and parallel finite element algorithms for the Steklov eigenvalue problem

Based on the work of Xu and Zhou [Math Comput 69 (2000) 881–909], we propose and analyze in this article local and parallel finite element algorithms for the Steklov eigenvalue problem. We also prove a local error estimate which is suitable for the case that the locally refined region contains singular points lying on the boundary of domain, which is an improvement of the existing results. Numerical experiments are reported finally to validate our theoretical analysis. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 399–417, 2016

Sharp eigenvalue bounds and minimal surfaces in the ball

We prove existence and regularity of metrics on a surface with boundary which maximize sigma_1 L where sigma_1 is the first nonzero Steklov eigenvalue and L the boundary length. We show that such metrics arise as the induced metrics on free boundary minimal surfaces in the unit ball B^n for some n. In the case of the annulus we prove that the unique solution to this problem is the induced metric on the critical catenoid, the unique free boundary surface of revolution in B^3. We also show that the unique solution on the Mobius band is achieved by an explicit S^1 invariant embedding in B^4 as a free boundary surface, the critical Mobius band. For oriented surfaces of genus 0 with arbitrarily many boundary components we prove the existence of maximizers which are given by minimal embeddings in B^3. We characterize the limit as the number of boundary components tends to infinity to give the asymptotically sharp upper bound of 4pi. We also prove multiplicity bounds on sigma_1 in terms of the topology, and we give a lower bound on the Morse index for the area functional for free boundary surfaces in the ball.

A virtual element method for the Steklov eigenvalue problem

The aim of this paper is to develop a virtual element method for the two-dimensional Steklov eigenvalue problem. We propose a discretization by means of the virtual elements presented in [L. Beirão da Veiga et al., Basic principles of virtual element methods, Math. Models Methods Appl. Sci.23 (2013) 199–214]. Under standard assumptions on the computational domain, we establish that the resulting scheme provides a correct approximation of the spectrum and prove optimal-order error estimates for the eigenfunctions and a double order for the eigenvalues. We also prove higher-order error estimates for the computation of the eigensolutions on the boundary, which in some Steklov problems (computing sloshing modes, for instance) provides the quantity of main interest (the free surface of the liquid). Finally, we report some numerical tests supporting the theoretical results.

Doubling Property and Vanishing Order of Steklov Eigenfunctions

The paper is concerned with the doubling estimates and vanishing order of the Steklov eigenfunctions on the boundary of a smooth domain in ℝ n . The eigenfunction is given by a Dirichlet-to-Neumann map. We improve the doubling property shown by Bellova and Lin. Furthermore, we show that the optimal vanishing order of Steklov eigenfunction is everywhere less than Cλ where λ is the Steklov eigenvalue and C depends only on Ω.

Boundary Integrals and Approximations of Harmonic Functions

Steklov expansions for a harmonic function on a rectangle are derived and studied with a view to determining an analog of the mean value theorem for harmonic functions. It is found that the value of a harmonic function at the center of a rectangle is well approximated by the mean value of the function on the boundary plus a very small number (often 3 or fewer) of specific further boundary integrals. These integrals are coefficients in the Steklov representation of the function. Similar approximations are found for the central values of solutions of Robin and Neumann boundary value problems. The results follow from analyses of the explicit expressions for the Steklov eigenvalues and eigenfunctions.

An efficient Legendre-Galerkin spectral approximation for the Steklov eigenvalue problem

In this paper, we give an effective spectral method based on the Legendre-Galerkin approximation for the Steklov eigenvalue problem. We first construct an appropriate set of basis functions such that the matrices in the discrete variational form are sparse. Then we deduce the matrix formulations based on the tensor-product for the discrete variational form in two- and three-dimensional cases, respectively. By using these tensor-product forms we can compute the discrete eigenvalues and eigenvectors rapidly. We also provide an error analysis and some numerical experiments. The numerical results indicate that our method is very stable and effective.

A Multilevel Correction Scheme for the Steklov Eigenvalue Problem

Combining the correction technique proposed by Lin and Xie and the shifted inverse iteration, a multilevel correction scheme for the Steklov eigenvalue problem is proposed in this paper. The theoretical analysis and numerical experiments indicate that the scheme proposed in this paper is efficient for both simple and multiple eigenvalues of the Steklov eigenvalue problem.

Une inégalité de Cheeger pour le spectre de Steklov

We prove a Cheeger inequality for the first positive Steklov eigenvalue. It involves two isoperimetric constants.

Extremal problems for Steklov eigenvalues on annuli

We obtain supremum of the \(k\)-th normalized Steklov eigenvalues of all rotationally symmetric conformal metrics on \([0,T]\times \mathbb {S}^1\), \(k>1\). This generalizes the corresponding result of Fraser and Schoen for the case \(k=1\). We give geometric description in terms of minimal surfaces for metrics attaining the supremum. We obtain some results on the comparison of the normalized Steklov eigenvalues of rotationally symmetric metrics and general conformal metrics on \([0,T]\times \mathbb {S}^1\). We also construct an example of a conformal metric on \([0,T]\times \mathbb {S}^1\) whose first normalized Steklov eigenvalue is larger than that of the corresponding rotationally symmetric conformal metric.

An improved two-grid finite element method for the Steklov eigenvalue problem

A combination method of finite element method and accelerated two-grid scheme is constructed for solving numerically the two-dimensional Steklov eigenvalue problem. This algorithm involves solving a small Steklov eigenvalue problem on the coarse mesh with mesh size H and a linear algebraic system on the fine mesh with mesh size h. By solving a slightly different linear problem on the fine grid, the computation will be more effective and convenient and the scaling between H and h becomes h=O(H4), which can save a large amount of computational time. Finally, numerical tests confirm the theoretical results of the presented method.

An extremal eigenvalue problem for the Wentzell–Laplace operator

We consider the question of giving an upper bound for the first nontrivial eigenvalue of the Wentzell–Laplace operator of a domain Ω, involving only geometrical information. We provide such an upper bound, by generalizing Brock's inequality concerning Steklov eigenvalues, and we conjecture that balls maximize the Wentzell eigenvalue, in a suitable class of domains, which would improve our bound. To support this conjecture, we prove that balls are critical domains for the Wentzell eigenvalue, in any dimension, and that they are local maximizers in dimension 2 and 3, using an order two sensitivity analysis. We also provide some numerical evidence.

Spectre et géométrie conforme des variétés compactes à bord

AbstractWe prove that on any compact manifold $M^{n}$ with boundary, there exists a conformal class $C$ such that for any Riemannian metric $g\in C$ of unit volume, the first positive eigenvalue of the Neumann Laplacian satisfies ${\it\lambda}_{1}(M^{n},g)<n\,\text{Vol}(S^{n},g_{\text{can}})^{2/n}$. We also prove a similar inequality for the first positive Steklov eigenvalue. The proof relies on a handle decomposition of the manifold. We also prove that the conformal volume of $(M,C)$ is $\text{Vol}(S^{n},g_{\text{can}})$, and that the Friedlander–Nadirashvili invariant and the Möbius volume of $M$ are equal to those of the sphere. If $M$ is a domain in a space form, $C$ is the conformal class of the canonical metric.