We prove the “hot spots” conjecture of J. Rauch in the case that the domain Ω is a planar convex domain satisfying diam(Ω)2/|Ω| < 1.378. Specifically, we show that an eigenfunction corresponding to the lowest nonzero eigenvalue of the Neumann Laplacian on Ω attains its maximum (minimum) at points on ∂Ω. When Ω is a disk, diam(Ω)2/|Ω| t 1.273. Hence, the above condition indicates that Ω is a nea...

Abstract. The largest eigenvalue of a matrix is always larger or equal than its largest diagonal entry. We show that for a large class of random Laplacian matrices, this bound is essentially tight: the largest eigenvalue is, up to lower order terms, often the size of the largest diagonal entry. Besides being a simple tool to obtain precise estimates on the largest eigenvalue of a large class of...

We prove that among all triangles of given diameter, the equilateral triangle minimizes the sum of the first n eigenvalues of the Neumann Laplacian, when n 3 . The result fails for n = 2 , because the second eigenvalue is known to be minimal for the degenerate acute isosceles triangle (rather than for the equilateral) while the first eigenvalue is 0 for every triangle. We show the third eigenva...

In this paper, a rigorous convergence and error analysis of a Galerkin boundary element method for the Dirichlet Laplacian eigenvalue problem is presented. The formulation of the eigenvalue problem in terms of a boundary integral equation yields a nonlinear boundary integral operator eigenvalue problem. This nonlinear eigenvalue problem and its Galerkin approximation are analyzed in the framewo...

We consider the problem of minimising the kth eigenvalue, k ≥ 2, of the (p-)Laplacian with Robin boundary conditions with respect to all domains in R of given volume M . When k = 2, we prove that the second eigenvalue of the p-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For p = 2 and k...

Let G be a nonsingular connected mixed graph. We determine the mixed graphs G on at least seven vertices with exactly two Laplacian eigenvalues greater than 2. In addition, all mixed graphs G with exactly one Laplacian eigenvalue greater than 2 are also characterized. c © 2006 Elsevier Ltd. All rights reserved.

We consider Γ = (X, E) a dual polar graph and we give a tight frame on each eigenspace of the Laplacian operator associated to Γ. We compute the constants associated to each tight frame and as an application we give a formula for the product in the Norton algebra attached to the eigenspace corresponding to the second largest eigenvalue of the Laplacian.

In an important paper, Alon [2] derived a Cheeger–type inequality [8], by bounding from below the second smallest eigenvalue of the Laplacian of a finite undirected graph by a function of a (vertex) isoperimetric constant. More precisely, let G=(V,E) be a finite, undirected, connected graph, and let λ2(G) denote twice (for reasons explained below) the smallest non-zero eigenvalue of the Laplaci...

Let π : Z → Y be a Riemannian V -submersion of compact V manifolds. We study when the pull-back of an eigenform of the Laplacian on Y is an eigenform of the Laplacian on Z, and when the associated eigenvalue can change.

We prove an asymptotic estimate for the growth of variational eigenvalues of fractional p-Laplacian eigenvalue problems on a smooth bounded domain.