Abstract. Let Q be a first-order differential operator on a compact, smooth oriented Riemannian manifold with smooth boundary. Then, Friedrichs’ Extension Lemma states that the minimal closed extension Qmin (the closure of the graph) and the maximal closed extension Qmax (in the sense of distributions) of Q in L-spaces (1 ≤ p < ∞) coincide. In the present paper, we show that the same is true fo...

In this paper, we provide stability and convergence analysis for a class of finite difference schemes for unsteady incompressibleNavier-Stokes equations in vorticity-stream function formulation. The no-slip boundary condition for the velocity is converted into local vorticity boundary conditions. Thom’s formula, Wilkes’ formula, or other local formulas in the earlier literature can be used in t...

The XXZ spin chain with a boundary magnetic field h is considered, using the vertex operator approach to diagonalize the Hamiltonian. We find explicit bosonic formulas for the two vacuum vectors with zero particle content. There are three distinct regions when h ≥ 0, in which the structure of the vacuum states is different. Excited states are given by the action of vertex operators on the vacuu...

A static mixed boundary value problem of physically nonlinear elasticity for a continuously inhomogeneous body is considered. Using the two-operator Green-Betti formula and the fundamental solution of an auxiliary linear operator, a non-standard boundary-domain integro-differential formulation of the problem is presented, with respect to the displacements and their gradients. Using a cut-off fu...

We consider the numerical solution for the Helmholtz equation in R with mixed boundary conditions. The solvability of this mixed boundary value problem is established by the boundary integral equation method. Based on the Green formula, we express the solution in terms of the boundary data. The key to the numerical realization of this method is the computation of weakly singular integrals. Nume...

context of finite difference schemes in vorticity formulation has a long history, going back at least to the 1930s when This paper discusses three basic issues related to the design of finite difference schemes for unsteady viscous incompressible Thom’s formula (see (2.4)) was derived [20]. Thom’s forflows using vorticity formulations: the boundary condition for mula is generally referred to as...

This paper deals with the boundary value problem involving the differential equation begin{equation*}     ell y:=-y''+qy=lambda y,  end{equation*}  subject to the standard boundary conditions along with the following discontinuity  conditions at a point $ain (0,pi)$  begin{equation*}     y(a+0)=a_1 y(a-0),quad y'(a+0)=a_1^{-1}y'(a-0)+a_2 y(a-0), end{equation*} where $q(x),  a_1 , a_2$ are  rea...

In any reaction-diffusion system of predator-prey models, the population densities of species are determined by the interactions between them, together with the influences from the spatial environments surrounding them. Generally, the prey species would die out when their birth rate is too low, the habitat size is too small, the predator grows too fast, or the predation pressure is too high. To...

We show the general formula of the one loop effective potential of the 5D SU(N) gauge theory compactified on an orbifold, S/Z2. The formula shows the case when there are fundamental, (anti-)symmetric tensor and adjoint representational bulk fields. Our calculation method is also applicable when there are bulk fields belonging to higher dimensional representations. The supersymmetric version of ...

We consider solutions to the wave equation in 3+1 spacetime dimensions whose data is compactly supported at some initial time. For points outside a ball containing the initial support, we develop an outgoing wave condition, and associated one-way propagation formula, for the partial waves in the spherical-harmonic decomposition of the solution. The propagation formula expresses the l-th partial...