This paper develops the necessary tools to understand the relationship between eigenvalues of the Laplacian matrix of a graph and the connectedness of the graph. First we prove that a graph has k connected components if and only if the algebraic multiplicity of eigenvalue 0 for the graph’s Laplacian matrix is k. We then prove Cheeger’s inequality (for dregular graphs) which bounds the number of...

In this article, we prove the existence of multiple solutions for following fractional Schrödinger-Poisson system with sign-changing potential (−∆)u+ V (x)u+ λφu = f(x, u), x ∈ R, (−∆)φ = u, x ∈ R, where (−∆)α denotes the fractional Laplacian of order α ∈ (0, 1), and the potential V is allowed to be sign-changing. Under certain assumptions on f , we obtain infinitely many solutions for this sys...

For every k ∈ N we prove the existence of a quasi-open set minimizing the k-th eigenvalue of the Dirichlet Laplacian among all sets of prescribed Lebesgue measure. Moreover, we prove that every minimizer is bounded and has finite perimeter. The key point is the observation that such quasi-open sets are shape subsolutions for an energy minimizing free boundary problem.

The purpose of this paper is to prove the L∞ gradient estimates and L∞ gradient estimates for the unit spectral projection operators of the Dirichlet Laplacian and Neumann (or more general, Ψ1-Robin) Laplacian on compact Riemannian manifolds (M, g) of dimension n ≥ 2 with C2 boundary . And we also get an upper bounds for normal derivatives of the unit spectral projection operators of the Dirich...

In recent years, improvements in various scientific image acquisition techniques gave rise to the need for adaptive processing methods aimed for large data-sets corrupted by noise and deformations. In this work, we consider data-sets of images sampled from an underlying low-dimensional manifold (i.e. an image-valued manifold), where the images are obtained through arbitrary planar rotations. To...

We compute the Green’s function for the Hodge Laplacian on the symmetric spaces M × Σ, where M is a simply connected n-dimensional Riemannian or Lorentzian manifold of constant curvature and Σ is a simply connected Riemannian surface of constant curvature. Our approach is based on a generalization to the case of differential forms of the method of spherical means and on the use of Riesz distrib...

In this paper we deal with two nonlocal operators, that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. Precisely, for a fixed s ∈ (0, 1) we consider the integral definition of the fractional Laplacian given by (−∆)u(x) := c(n, s) 2 ∫ Rn 2u(x)− u(x+ y)− u(x− y) |y|n+2s dy , x ∈ R n , where c(n, s) is a positive normalizing consta...

Spectral Graph Theory is the study of the spectra of certain matrices defined from a given graph, including the adjacency matrix, the Laplacian matrix and other related matrices. Graph spectra have been studied extensively for more than fifty years. In the last fifteen years, interest has developed in the study of generalized Laplacian matrices of a graph, that is, real symmetric matrices with ...

The fractional Laplacian and the fractional derivative are two different mathematical concepts (Samko et al, 1987). Both are defined through a singular convolution integral, but the former is guaranteed to be the positive definition via the Riesz potential as the standard Laplace operator, while the latter via the Riemann-Liouville integral is not. It is noted that the fractional Laplacian can ...