Related papers: Laplace contour integrals and linear differential …

- The purpose of this article is to study possible concentrations of eigenfunc-tions of Laplace operators (or more generally quasi-modes) on product manifolds. We show that the approach of the first author and Zworski [10, 11] applies…

Let $(\Sigma,g)$ be a compact Riemannian surface without boundary and $\lambda_1(\Sigma)$ be the first eigenvalue of the Laplace-Beltrami operator $\Delta_g$. Let $h$ be a positive smooth function on $\Sigma$. Define a functional…

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold without boundary and $e_\lambda$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator with respect to the metric $g$, i.e \[ -\Delta_g e_\lambda = \lambda^2…

We first strictly expressed the basic notions and research methods of abstract operators, which systematically expounded the main results of abstract operator theory. By combining abstract operators with the Laplace transform, we can easily…

A new method for finding first integrals of discrete equations is presented. It can be used for discrete equations which do not possess a variational (Lagrangian or Hamiltonian) formulation. The method is based on a newly established…

Let $\{e_j\}$ be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold $(M,g)$. Let $H \subset M$ be a submanifold and let $\{\psi_k\}$ be an orthonormal basis of Laplace eigenfunctions of $H$ with the induced…

The relaxation functions introduced empirically by Debye, Cole-Cole, Cole-Davidson and Havriliak-Negami are, each of them, solutions to their respective kinetic equations. In this work, we propose a generalization of such equations by…

We begin with a treatment of the Caputo time-fractional diffusion equation, by using the Laplace transform, to obtain a Volterra intego-differential equation where we may examine the weakly singular nature of this convolution…

We consider the question of when the Laplace eigenfunctions on an arbitrary flat torus $\mathbf{T}_\Gamma:=\mathbf{R}^d/\Gamma$ are flexible enough to approximate, over the natural length scale of order $1/\sqrt\lambda$, where $\lambda\gg1$…

Let $\Delta_M$ be the Laplace operator on a compact $n$-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions $u:\Delta u + \lambda u =0$. In dimension $n=2$ we refine the Donnelly-Fefferman estimate…

The convergence problem of the Laplace-Beltrami operators plays an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve the operator. In this note…

The paper deals with the approximate solution of integro-differential equations of Prandtl's type. Quadrature methods involving ``optimal'' Lagrange interpolation processes are proposed and conditions under which they are stable and…

Under the lack of variational structure and nondegeneracy, we investigate three notions of \textit{generalized principal eigenvalue} for a general infinity Laplacian operator with gradient and homogeneous term. A Harnack inequality and…

In this paper we study the existence and the analytic dependence upon domain perturbation of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation. The problem is defined in a pair of sets consisting of a…

Boundary integral equations are an efficient and accurate tool for the numerical solution of elliptic boundary value problems. The solution is expressed as a layer potential; however, the error in its evaluation grows large near the…

We study the analytical properties of the Laplace transform of the lognormal distribution. Two integral expressions for the analytic continuation of the Laplace transform of the lognormal distribution are provided, one of which takes the…

In this paper we introduce the p-Laplace equations for the intermediate Schouten curvature in conformal geometry. These p-Laplace equations provide more tools for the study of geometry and topology of manifolds. First, the positivity of the…

Two-term Weyl-type asymptotic law for the eigenvalues of one-dimensional fractional Laplace operator (-d^2/dx^2)^(alpha/2) (0 < alpha < 2) in the interval (-1,1) is given: the n-th eigenvalue is equal to (n pi/2 - (2 - alpha) pi/8)^alpha +…

From a suitable integral representation of the Laplace transform of a positive semi-definite quadratic form of independent real random variables with not necessarily identical densities a univariate integral representation is derived for…

A fundamental solution of Laplace's equation in three dimensions is expanded in harmonic functions that are separated in parabolic or elliptic cylinder coordinates. There are two expansions in each case which reduce to expansions of the…