Related papers: Courant-sharp eigenvalues of Neumann 2-rep-tiles

This paper concerns the behavior of the eigenfunctions and eigenvalues of the round sphere's Laplacian acting on the space of sections of a real line bundle which is defined on the complement of an even numbers of points in $S^2$. Of…

We address the problem of splitting of eigenvalues of the Neumann Laplacian under singular domain perturbations. We consider a domain perturbed by the excision of a small spherical hole shrinking to an interior point. Our main result…

We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We…

In this paper we prove discreteness of the spectrum of the Neu\-mann-Lap\-la\-ci\-an (the free membrane problem) in a large class of non-convex space domains. The lower estimates of the first non-trivial eigenvalue are obtained in terms of…

In this article, we illustrate and draw connections between the geometry of zero sets of eigenfunctions, graph theory and the vanishing order of eigenfunctions. We identify the nodal set of an eigenfunction of the Laplacian (with smooth…

We consider the nonlinear eigenvalue problem, with Dirichlet boundary condition, for a class of very degenerate elliptic operators, with the aim to show that, at least for square type domains having fixed volume, the symmetry of the domain…

We prove that every nodal domain of an eigenfunction of the Laplacian of eigenvalue $\lambda$ on a $d$-dimensional closed Riemannian manifold contains a ball of radius $c\lambda^{-1/2}(\log\lambda)^{-(d-2)/2}$. This ball is centered at a…

We extend the framework of submanifolds in Riemannian geometry to Riemann-Cartan geometry, which addresses connections with torsion. This procedure naturally introduces a 2-form on submanifolds associated with the nontrivial ambient…

We prove sharp criteria on the behavior of radial curvature for the existence of asymptotically flat or hyperbolic Riemannian manifolds with prescribed sets of eigenvalues embedded in the spectrum of the Laplacian. In particular, we…

We provide an upper estimate for the eigenvalues of the curl curl operator on a bounded, three-dimensional Euclidean domain in terms of eigenvalues of the Dirichlet Laplacian. The result complements recent inequalities between curl curl and…

We give an example of a domain in dimension $N \geq 3$, homeomorphic to a ball and with analytic boundary, for which the second eigenvalue of the Dirichlet Laplacian has an eigenfunction with a closed nodal surface. The domain is…

There exists a planar domain with piecewise smooth boundary and one hole such that the second eigenfunction for the Laplacian with Neumann boundary conditions attains its maximum and minimum inside the domain.

Recent work of the authors and their collaborators has uncovered fundamental connections between the Dirichlet-to-Neumann map, the spectral flow of a certain family of self-adjoint operators, and the nodal deficiency of a Laplacian…

We discuss several properties of eigenvalues and eigenfunctions of the $p$-Laplacian on a ball subject to zero Dirichlet boundary conditions. Among main results, in two dimensions, we show the existence of nonradial eigenfunctions which…

We consider eigenfunctions of the Laplace-Beltrami operator on special surfaces of revolution. For this separable system, the nodal domains of the (real) eigenfunctions form a checker-board pattern, and their number $\nu_n$ is proportional…

For smooth bounded domains in $\mathbb{R}$, we prove upper and lower $L^2$ bounds on the boundary data of Neumann eigenfunctions, and prove quasi-orthogonality of this boundary data in a spectral window. The bounds are tight in the sense…

We study the nodal set of eigenfunctions of the Laplace operator on the right angled isosceles triangle. A local analysis of the nodal pattern provides an algorithm for computing the number of nodal domains for any eigenfunction. In…

In this paper we establish new quantitative stability estimates with respect to domain perturbations for all the eigenvalues of both the Neumann and the Dirichlet Laplacian. Our main results follow from an abstract lemma stating that it is…

The elastic Neumann--Poincar\'e operator is a boundary integral operator associated with the Lam\'e system of linear elasticity. It is known that if the boundary of a planar domain is smooth enough, it has eigenvalues converging to two…

We initiate a systematic study of eigenvectors of random graphs. Whereas much is known about eigenvalues of graphs and how they reflect properties of the underlying graph, relatively little is known about the corresponding eigenvectors. Our…