Isoperimetric Inequalities and Eigenvalues

Weighted Graph Laplacians and Isoperimetric Inequalities

We consider a weighted Cheeger’s constant for a graph and we examine the gap between the first two eigenvalues of Laplacian. We establish several isoperimetric inequalities concerning the unweighted Cheeger’s constant, weighted Cheeger’s constants and eigenvalues for Neumann and Dirichlet conditions .

Isoperimetric Inequalities and Eigenvalues of the Laplacian

Given a domain D in euclidean space or on a Riemannian manifold, one naturally associates with it such fundamental geometric quantities as its volume, the measure of its boundary, various curvature functions and their extremes or integrals. One can also associate with the domain D the set of eigenfunctions and eigenvalues of the Laplace operator, subject to given boundary conditions. Recent wor...

Isoperimetric Inequalities and Eigenvalues

Laplacians of graphs and Cheeger inequalities

We define the Laplacian for a general graph and then examine several isoperimetric inequalities which relate the eigenvalues of the Laplacian to a number of graphs invariants such as vertex or edge expansions and the isoperimetric dimension of a graph.

Maximizing Neumann Fundamental Tones of Triangles

We prove sharp isoperimetric inequalities for Neumann eigenvalues of the Laplacian on triangular domains. The first nonzero Neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter, and hence among all triangles of given area. Similar results are proved for the harmonic and arithmetic means of the first two nonzero eigenvalues.

Isoperimetric Inequalities for the Eigenvalues of Natural Schrödinger Operators on Surfaces

This paper deals with eigenvalue optimization problems for a family of natural Schrödinger operators arising in some geometrical or physical contexts. These operators, whose potentials are quadratic in curvature, are considered on closed surfaces immersed in space forms and we look for geometries that maximize the eigenvalues. We show that under suitable assumptions on the potential, the first ...