Steklov Eigenvalue Research Articles

The Steklov spectrum of surfaces: asymptotics and invariants

AbstractWe obtain precise asymptotics for the Steklov eigenvalues on a compact Riemannian surface with boundary. It is shown that the number of connected components of the boundary, as well as their lengths, are invariants of the Steklov spectrum. The proofs are based on pseudodifferential techniques for the Dirichlet-to-Neumann operator and on a number–theoretic argument.

Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces

We investigate in this paper the existence of a metric which maximizes the first eigenvalue of the Laplacian on Riemannian surfaces. We first prove that, in a given conformal class, there always exists such a maximizing metric which is smooth except at a finite set of conical singularities. This result is similar to the beautiful result concerning Steklov eigenvalues recently obtained by Fraser and Schoen (Sharp eigenvalue bounds and minimal surfaces in the ball, 2013). Then we get existence results among all metrics on surfaces of a given genus, leading to the existence of minimal isometric immersions of smooth compact Riemannian manifold (M, g) of dimension 2 into some k-sphere by first eigenfunctions. At last, we also answer a conjecture of Friedlander and Nadirashvili (Int Math Res Not 17:939–952, 1999) which asserts that the supremum of the first eigenvalue of the Laplacian on a conformal class can be taken as close as we want of its value on the sphere on any orientable surface.

Sharp Bounds for the First Eigenvalue of a Fourth-Order Steklov Problem

We study the biharmonic Steklov eigenvalue problem on a compact Riemannian manifold with smooth boundary. We give a 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 its boundary and the inner radius. The proof is obtained by estimating the isoperimetric ratio of non-negative subharmonic functions on $$\Omega $$ , which is of independent interest. We also give a comparison theorem for geodesic balls.

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-norm.

Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary

We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact $n$-dimensional Riemannian manifold which has nonnegative Ricci curvature and strictly convex boundary. When $n=3$, this implies an apriori curvature estimate for these minimal surfaces in terms of the geometry of the ambient manifold and the topology of the minimal surface. An important consequence of the estimate is a smooth compactness theorem for embedded minimal surfaces with free boundary when the topological type of these minimal surfaces is fixed.

The spectral gap of graphs and Steklov eigenvalues on surfaces

Using expander graphs, we construct a sequence $\{\Omega_N\}_{N\in\mathbb{N}}$ of smooth compact surfaces with boundary of perimeter $N$, and with the first non-zero Steklov eigenvalue $\sigma_1(\Omega_N)$ uniformly bounded away from zero. This answers a question which was raised in [10]. The sequence $\sigma_1(\Omega_N) L(\partial\Omega_n)$ grows linearly with the genus of $\Omega_N$, which is the optimal growth rate.

A Posteriori Error Estimates with Computable Upper Bound for the Nonconforming RotatedQ1Finite Element Approximation of the Eigenvalue Problems

This paper discusses the nonconforming rotatedQ1finite element computable upper bound a posteriori error estimate of the boundary value problem established by M. Ainsworth and obtains efficient computable upper bound a posteriori error indicators for the eigenvalue problem associated with the boundary value problem. We extend the a posteriori error estimate to the Steklov eigenvalue problem and also derive efficient computable upper bound a posteriori error indicators. Finally, through numerical experiments, we verify the validity of the a posteriori error estimate of the boundary value problem; meanwhile, the numerical results show that the a posteriori error indicators of the eigenvalue problem and the Steklov eigenvalue problem are effective.

Multiple Solutions for a Class of Semilinear Elliptic Equations with Nonlinear Boundary Conditions

In this paper, using Local Linking Theorem, we obtain the existence of multiple solutions for a class of semilinear elliptic equations with nonlinear boundary conditions, in which the nonlinearites are compared with higher Neumann eigenvalue and the first Steklov eigenvalue.

The Spectral Element Method for the Steklov Eigenvalue Problem

This paper discusses the spectral element approximation with LGL node basis for the Steklov eigenvalue problem, and analyzes the a priori error estimates. Finally, numerical experi-ments on the square and the L-shaped domain are carried out to get very accurate approximations by the spectral element method.

Multiplicity of solutions to a nonlinear elliptic problem with nonlinear boundary conditions

We study the problem $$\left\{\begin{array}{ll}\Delta_p u = |u|^{q-2}u, & \quad x \in \Omega ,\\ |\nabla u|^{p-2} \frac{\partial u}{\partial \nu}= \lambda |u|^{p-2}u, &\quad x \in \partial \Omega, \end{array}\right.$$ where \({\Omega \subset \mathbb{R}^N}\) is a bounded smooth domain, \({\nu}\) is the outward unit normal at \({\partial \Omega}\) and \({\lambda > 0}\) is regarded as a bifurcation parameter. When p = 2 and in the superlinear regime q > 2, we show existence of n nontrivial solutions for all \({\lambda > \lambda_n}\) , \({\lambda_n}\) being the n-th Steklov eigenvalue. It is proved in addition that bifurcation from the trivial solution takes place at all \({\lambda_n}\) ’s. Similar results are obtained in the sublinear case 1 < q < 2. In this case, bifurcation from infinity takes place in those \({\lambda_n}\) with odd multiplicity. Partial extensions of these features are shown in the nonlinear diffusion case \({p \neq 2}\) and related problems under spatially heterogeneous reactions are also addressed.

Heat Invariants of the Steklov Problem

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 calculus to the Dirichlet-to-Neumann operator, whose spectrum coincides with the Steklov eigenvalues. As an application, it is proved that a three-dimensional ball is uniquely defined by its Steklov spectrum among all Euclidean domains with smooth connected boundary.

Local a priori/a posteriori error estimates of conforming finite elements approximation for Steklov eigenvalue problems

Based on the work of Xu and Zhou (2000), this paper makes a further discussion on conforming finite elements approximation for Steklov eigenvalue problems, and proves a local a priori error estimate and a new local a posteriori error estimate in \(\left\| \cdot \right\|_{1,\Omega _0 }\) norm for conforming elements eigenfunction, which has not been studied in existing literatures.

A type of multilevel method for the Steklov eigenvalue problem

A new type of iteration method is proposed in this paper to solve the Steklov eigenvalue problem by the finite element method. In this scheme, solving the Steklov eigenvalue problem is transformed into a series of solutions of boundary value problems on multilevel meshes by the multigrid method and solutions of the Steklov eigenvalue problem on the coarsest mesh. Besides the multigrid scheme, all other efficient iteration methods can also serve as the linear algebraic solver for the associated boundary value problems. The computational work of this new scheme for the Steklov eigenvalue problem can reach the same optimal order as the solution of the corresponding boundary value problem. Therefore, an improvement of efficiency for the Steklov eigenvalue solving method can be achieved. Some numerical experiments are presented to validate the efficiency of the new method.

A Two-Grid Method of Nonconforming Element Based on the Shifted-Inverse Power Method for the Steklov Eigenvalue Problem

This paper establishes a new kind of two-grid discretization scheme of nonconforming Crouzeix-Raviart element based on the shifted-inverse power method for the Steklov eigenvalue problem. The error estimates are provided from the work of Yang and Bi (SIAM J. Numer. Anal., 49, pp.1602-1624, 2011). Finally, numerical experiments are reported to illustrate the high efficiency of the two-grid discretization scheme proposed in this paper.

Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations

The paper deals with error estimates and lower bound approximations of the Steklov eigenvalue problems on convex or concave domains by nonconforming finite element methods. We consider four types of nonconforming finite elements: Crouzeix-Raviart, Q 1 rot , EQ 1 rot and enriched Crouzeix-Raviart. We first derive error estimates for the nonconforming finite element approximations of the Steklov eigenvalue problem and then give the analysis of lower bound approximations. Some numerical results are presented to validate our theoretical results.

Convex shape optimization for the least biharmonic Steklov eigenvalue

The least Steklov eigenvalue d 1 for the biharmonic operator in bounded domains gives a bound for the positivity preserving property for the hinged plate problem, appears as a norm of a suitable trace operator, and gives the optimal constant to estimate the L 2 -norm of harmonic functions. These applications suggest to address the problem of minimizing d 1 in suitable classes of domains. We survey the existing results and conjectures about this topic; in particular, the existence of a convex domain of fixed measure minimizing d 1 is known, although the optimal shape is still unknown. We perform several numerical experiments which strongly suggest that the optimal planar shape is the regular pentagon. We prove the existence of a domain minimizing d 1 also among convex domains having fixed perimeter and present some numerical results supporting the conjecture that, among planar domains, the disk is the minimizer.

Spectral Method with the Tensor-Product Nodal Basis for the Steklov Eigenvalue Problem

This paper discusses spectral method with the tensor-product nodal basis at the Legendre-Gauss-Lobatto points for solving the Steklov eigenvalue problem. A priori error estimates of spectral method are discussed, and based on the work of Melenk and Wohlmuth (2001), a posterior error estimator of the residual type is given and analyzed. In addition, this paper combines the shifted-inverse iterative method and spectral method to establish an efficient scheme. Finally, numerical experiments with MATLAB program are reported.

Multiscale Asymptotic Method for Steklov Eigenvalue Equations in Composite Media

In this paper we consider the multiscale analysis of a Steklov eigenvalue equation with rapidly oscillating coefficients arising from the modeling of a composite media with a periodic microstructure. There are mainly two new results in the present paper. First, we obtain the convergence rate with $ \varepsilon^{1/2} $ for the multiscale asymptotic expansions of the eigenvalues and the eigenfunctions of the Steklov eigenvalue problem. Second, the boundary layer solution is defined. Numerical simulations are then carried out to validate the above theoretical results.

A three dimensional Steklov eigenvalue problem with exponential nonlinearity on the boundary

We investigate the existence of pairs (λ,u), with λ>0 and u a harmonic function in a bounded domain Ω⊂R3, such that the nonlinear boundary condition ∂νu=λμsinhu holds on ∂Ω, where μ is a non negative weight function. This type of exponential boundary condition arises in corrosion modeling (Butler–Volmer condition).

A new procedure for the construction of hierarchical high order Hdiv and Hcurl finite element spaces

This paper considers a systematic procedure for the construction of a hierarchy of high order finite element approximation for Hdiv and Hcurl spaces based on triangular and quadrilateral partitions of bidimensional domains. The principle is to choose an appropriate set of vectors, based on the geometry of each element, which are multiplied by an available set of H1 hierarchical scalar basic functions. This strategy produces vector basis functions with continuous normal or tangent components on the elements interfaces, properties that characterise functions in Hdiv or Hcurl, respectively. We also present a numerical study to evaluate the correct balancedness of the resulting Hdiv spaces of degree k and L2 spaces of degree k−1 on the resolution of the mixed formulation for a Steklov eigenvalue problem.