CR invariant differential operators on densities with leading part a power of the sub-Laplacian are derived. One family of such operators is constructed from the " conformally invariant powers of the Laplacian " via the Fefferman metric; the powers which arise for these operators are bounded in terms of the dimension. A second family is derived from a CR tractor calculus which is developed here...

In this thesis we discuss some new results concerning the combinatorial Laplace operator of a simplicial complex. The combinatorial Laplacian of a simplicial complex encodes information about the relationships between adjacent simplices in the complex. This thesis is divided into two relatively disjoint parts. In the first portion of the thesis, we derive a relationship between the Laplacian sp...

We consider a multi-dimensional continuum Schrödinger operator H, which is given by perturbation of the negative Laplacian compactly supported bounded potential. show that for fairly large class test functions, second-order Szegő-type asymptotics spatially truncated Fermi projection H independent potential and, thus, identical to known Laplacian.

Abstract We consider a convolution-type operator on vector bundles over metric-measure spaces. This extends the analogous convolution Laplacian functions in our earlier work to bundles, and is natural extension of graph connection Laplacian. prove that for Euclidean or Hermitian connections closed Riemannian manifolds, spectrum this both approximate

We show that the action of conformal vector fields on functions on the sphere determines the spectrum of the Laplacian (or the conformal Laplacian), without further input of information. The spectra of intertwining operators (both differential and non-local) with principal part a power of the Laplacian follows as a corollary. An application of the method is the sharp form of Gross’ entropy ineq...

The aim of the book is to provide an analysis of the boundary element method for the numerical solution of Laplacian eigenvalue problems. The representation of Laplacian eigenvalue problems in the form of boundary integral equations leads to nonlinear eigenvalue problems for related boundary integral operators. The solution of boundary element discretizations of such eigenvalue problems require...

In this paper, we study the nonlinear heat equation ∂ ∂t △u(x, t) − c♦u(x, t) = f(x, t, u(x, t)), where△k is the Laplacian operator iterated k− times and is defined by (1.4)and ♦k is the Diamond operator iterated k− times and is defined by (1.2). We obtain an interesting kernel related to the nonlinear heat equation.

We construct an anisotropic, degenerate, fractional operator that nevertheless satisfies a strong form of the maximum principle. By applying such an operator to the concavity function associated to the solution of an equation involving the usual fractional Laplacian, we obtain a fractional form of the celebrated convexity maximum principle devised by Korevaar in the 80’s. Some applications are ...

We prove a Liouville-type theorem for bounded stable solutions v ∈ C(R) of elliptic equations of the type (−∆)v = f(v) in R, where s ∈ (0, 1) and f is any nonnegative function. The operator (−∆) stands for the fractional Laplacian, a pseudo-differential operator of symbol |ξ|.

If Ω is a ball in Rn (n ≥ 2), then the boundary integral operator of the double layer potential for the Laplacian is self-adjoint on L2(∂Ω). In this paper we prove that the ball is the only bounded Lipschitz domain on which the integral operator is self-adjoint.