A transfer principle and applications to eigenvalue estimates for graphs

In this paper, we prove a variant of the Burger-Brooks transfer principle which, combined with recent eigenvalue bounds for surfaces, allows to obtain upper bounds on the eigenvalues of graphs as a function of their genus. More precisely, we show the existence of a universal constants $C$ such that the $k$-th eigenvalue $\lambda_k^{nr}$ of the normalized Laplacian of a graph $G$ of (geometric) genus $g$ on $n$ vertices satisfies $$\lambda_k^{nr}(G) \leq C \frac{d_{\max}(g+k)}{n},$$ where $d_{\max}$ denotes the maximum valence of vertices of the graph. This result is tight up to a change in the value of the constant $C$, and improves recent results of Kelner, Lee, Price and Teng on bounded genus graphs. To show that the transfer theorem might be of independent interest, we relate eigenvalues of the Laplacian on a metric graph to the eigenvalues of its simple graph models, and discuss an application to the mesh partitioning problem, extending pioneering results of Miller-Teng-Thurston-Vavasis and Spielman-Tang to arbitrary meshes.Comment: Major revision, 16 page

Flux reconstructions in the Lehmann-Goerisch method for lower bounds on eigenvalues

The standard application of the Lehmann-Goerisch method for lower bounds on eigenvalues of symmetric elliptic second-order partial differential operators relies on determination of fluxes $\sigma_i$ that approximate co-gradients of exact eigenfunctions scaled by corresponding eigenvalues. Fluxes $\sigma_i$ are usually computed by a global saddle point problem solved by mixed finite element methods. In this paper we propose a simpler global problem that yields fluxes $\sigma_i$ of the same quality. The simplified problem is smaller, it is positive definite, and any $H(\mathrm{div},\Omega)$ conforming finite elements, such as Raviart-Thomas elements, can be used for its solution. In addition, these global problems can be split into a number of independent local problems on patches, which allows for trivial parallelization. The computational performance of these approaches is illustrated by numerical examples for Laplace and Steklov type eigenvalue problems. These examples also show that local flux reconstructions enable to compute lower bounds on eigenvalues on considerably finer meshes than the traditional global reconstructions

Range of the first three eigenvalues of the planar Dirichlet Laplacian

We conduct extensive numerical experiments aimed at finding the admissible range of the ratios of the first three eigenvalues of a planar Dirichlet Laplacian. The results improve the previously known theoretical estimates of M Ashbaugh and R Benguria. We also prove some properties of a maximizer of the ratio $\lambda_3/\lambda_1$.Comment: 21 page; 9 figure

Isoperimetric and Universal Inequalities for Eigenvalues

This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and bi-Laplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet and Neumann Laplacians and of the vibrating clamped plate problem (i.e., the biharmonic operator with ``Dirichlet'' boundary conditions). We also discuss the known universal inequalities for the eigenvalues of the Dirichlet Laplacian and the vibrating clamped plate and buckling problems and go on to present some new ones. Some of the names associated with these inequalities are Rayleigh, Faber-Krahn, Szego-Weinberger, Payne-Polya-Weinberger, Sperner, Hile-Protter, and H. C. Yang. Occasionally, we will also comment on extensions of some of our inequalities to bounded domains in other spaces, specifically, S^n or H^n.Comment: 45 pages. This is my contribution to the proceedings of the Instructional Conference on Spectral Theory and Geometry held in Edinburgh in March-April 199

Eigenvalue gaps for the Cauchy process and a Poincar\'e inequality

A connection between the semigroup of the Cauchy process killed upon exiting a domain $D$ and a mixed boundary value problem for the Laplacian in one dimension higher known as the "mixed Steklov problem," was established in a previous paper of the authors. From this, a variational characterization for the eigenvalues $\lambda_n$, $n\geq 1$, of the Cauchy process in $D$ was obtained. In this paper we obtain a variational characterization of the difference between $\lambda_n$ and $\lambda_1$. We study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound estimates for $\lambda_* - \lambda_1$ where $\lambda_*$ is the eigenvalue corresponding to the "first" antisymmetric eigenfunction for $D$. The proof is based on a variational characterization of $\lambda_* - \lambda_1$ and on a weighted Poincar\'e--type inequality. The Poincar\'e inequality is valid for all $\alpha$ symmetric stable processes, $0<\alpha\leq 2$, and any other process obtained from Brownian motion by subordination. We also prove upper bound estimates for the spectral gap $\lambda_2-\lambda_1$ in bounded convex domains

Three methods for two-sided bounds of eigenvalues - a comparison

We compare three finite element based methods designed for two-sided bounds of eigenvalues of symmetric elliptic second order operators. The first method is known as the Lehmann-Goerisch method. The second method is based on Crouzeix-Raviart nonconforming finite element method. The third one is a combination of generalized Weinstein and Kato bounds with complementarity based estimators. We concisely describe these methods and use them to solve three numerical examples. We compare their accuracy, computational performance, and generality in both the lowest and higher order case

Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations

We consider three problems for the Helmholtz equation in interior and exterior domains in R^d (d=2,3): the exterior Dirichlet-to-Neumann and Neumann-to-Dirichlet problems for outgoing solutions, and the interior impedance problem. We derive sharp estimates for solutions to these problems that, in combination, give bounds on the inverses of the combined-field boundary integral operators for exterior Helmholtz problems.Comment: Version 3: 42 pages; improved exposition in response to referee comments and added several reference

On strongly indefinite systems involving fractional elliptic operators

In this paper we discuss the existence and regularity of solutions of strongly indefinite systems involving fractional elliptic operators on a smooth bounded domain $\Omega$ in $\R^n$.Comment: arXiv admin note: text overlap with arXiv:1705.0633

Eigenvalue asymptotic of Robin Laplace operators on two-dimensional domains with cusps

We consider Robin Laplace operators on a class of two-dimensional domains with cusps. Our main results include the formula for the asymptotic distribution of the eigenvalues of such operators. In particular, we show how the eigenvalue asymptotic depends on the geometry of the cusp and on the boundary conditions

On positive viscosity solutions of fractional Lane-Emden systems

We discuss on existence, nonexistence and uniqueness of positive viscosity solutions for Lane-Emden systems involving the fractional Laplacian on bounded domains. As a byproduct, we obtain the critical hyperbole associated to the these systems