Boundary Value Problems for Harmonic Functions on the Heisenberg Group

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

Orthogonal polynomials and a Dirichlet problem related to the Hilbert transform

Various kinds of Riemann boundary value problems (BVPs) for analytic functions on closed curves or on open arc, doubly periodic Riemann BVPs, doubly quasi-periodic Riemann BVPs, and BVPs for polyanalytic functions have been widely investigated in [1-8]. The main ap- proach is to use the decomposition of polyanalytic functions and their generalization to transform the boundary value problems to their corresponding boundary value problems for analytic functions. Recently, inverse Riemann BVPs for generalized analytic functions or bianalytic functions have been investigated in [9-12]. In this paper, we consider a kind of Riemann BVP of non-normal type on the infinite straight line and discuss the solvable conditions and the general solution for it.

Let A/gn be the Kohn Laplacian on the Heisenberg group HI and let Q be a halfspace of HI whose boundary is parallel to the center of HI. In this paper we prove that if u is a non-negative A/n-superharmonic function such that u E L () then u _ 0 in Q.

Let L = −1/4 The purpose of this note is to present a simplified calculation of the Fourier transform of fundmental solution of theb-heat equation on the Heisenberg group. The Fourier transform of the fundamental solution has been computed by a number of authors (Gav77, Hul76, CT00, Tie06). We use the approach of (CT00, Tie06) and compute the heat kernel using Hermite functions but differ from the earlier approaches by working on a different, though biholomorphically equivalent, version of the Heisenberg group. The simplification in the computation occurs because the differential operators on this equivalent Heisenberg group take on a simpler form. Moreover, in the proof of Theorem 1.2, we reduce the n-dimensional heat equation to a 1-dimensional heat equation, and this technique would also be useful when analyzing the heat equation on the nonisotropic Heisenberg group (e.g., see (CT00)). We actually use the same version of the Heisenberg group as Hulanicki (Hul76), but he computes the fundamental solution of the heat equation associated to the sub-Laplacian and not the Kohn Laplacian acting on (0,q)-forms.

This paper is concerned with a PDE approach to horizontally quasiconvex (h-quasiconvex) functions in the Heisenberg group based on a nonlinear second-order elliptic operator. We discuss sufficient and necessary conditions for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to the associated elliptic equation. Since the notion of h-quasiconvexity is equivalent to the horizontal convexity (h-convexity) of the function's sublevel sets, we further adopt these conditions to study the h-convexity preserving property for horizontal mean curvature flow in the Heisenberg group. Under the comparison principle, we show that the curvature flow starting from a star-shaped h-convex set preserves the h-convexity during the evolution.

The paper considers a boundary value problem for a singularly perturbed linear differential equation with constant third-order coefficients. In this problem, a small parameter is indicated before the highest derivatives that are part of the differential equation and the boundary condition at t = 0.The fundamental system of solutions of a homogeneous singularly perturbed differential equation is constructed on the basis of asymptotic representations obtained for the roots of the corresponding characteristic equation. This system was used to construct the Cauchy function, special functions of boundary value problems, and also the Green function. With the help of these functions, an analytical formula is obtained for solving a singularly perturbed boundary value problem and it turns out that this solution has an initial zero-order jump at t = 0. It is proved that the solution to the considered singularly perturbed boundary value problem tends to the corresponding unperturbed problem obtained from it under .

CHAPTER V - VARIOUS GENERALIZED BOUNDARY VALUE PROBLEMS

In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group. The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk. Here the non commutative nature of the increments produces a new gradient which naturally involves a Brownian bridge on the Heisenberg group. This new inequality contains the optimal logarithmic Sobolev inequality for the Gaussian distribution in two dimensions. We compare this new inequality with the sub-elliptic logarithmic Sobolev inequality of Hong-Quan Li and with the more recent inequality of Fabrice Baudoin and Nicola Garofalo obtained using a generalized curvature criterion. Finally, we extend this inequality to the case of homogeneous Carnot groups of rank two.

It is natural to begin the study of those aspects of the theory of boundary value problems that are of interest here with a rather detailed discussion of a number of properties of ordinary differential equations. In the first place, ordinary differential Operations are the simplest entities in the theory in which we are interested that can, in many respects, be studied exhaustively. In the second place, the fundamental kind of Operators that are of interest in what follows will be Operators that act on tensor products ℍ0ℍ1⨂…⨂ℍ (Chapter IV) and are constructed from mutually commuting operators L k : ℍ k → ℍ k Here the material for the construction of L k will be ordinary differential Operators that have properties depending on the boundary conditions that enter into their definitions.

This paper presents results on some boundary value problems for holomorphic functions of several complex variables in polydomains. The Cauchy kernel is one of the significant tools for solving the Riemann and the Riemann-Hilbert boundary value problems for holomorphic functions as well as for establishment of the connection between them. For polydomains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in polydomains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. The general integral representation formulas for the functions, holomorphic in polydomains, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holomorphic function for arbitrary polydomains is given and an exact, yet compact way of notation for holomorphic functions in arbitrary polydomains is introduced and applied. The Riemann jump problem and the Riemann-Hilbert problem are solved for holomorphic functions of several complex variables with the unit torus as the jump manifold. The higher-dimensional Plemelj-Sokhotzki formula for holomorphic functions in polydomains is established. The canonical functions of the Riemann problem for polydomains are represented and applied in order to construct solutions for both of the homogeneous and inhomogeneous problems. For all three boundary value problems, well-posed formulations are given which does not demand more solvability conditions than in the one variable case. The connection between the Riemann and the Riemann-Hilbert problem for polydomains is proven. Thus contrary to earlier research the results are similar to the respective ones for just one variable.

The existence and uniqueness of the solution to the boundary value problem for a third-order mixed-type equation with variable coefficients in the lower terms is proven, with conditions for the gluing of the function itself and its first- and second-order derivatives on the line y=0, where the type of equation changes when a second-order mixed parabolic-hyperbolic operator is applied to a first-order linear differential operator with constant coefficients. By reducing the order of the equation, the problem was reduced to the Tricomi problem for a second-order mixed parabolic-hyperbolic equation with continuous conditions for the function itself and its first-order derivative with respect to y on the line of change of the equation type. By the method of elimination of the system of equations obtained from the parabolic and hyperbolic parts of the domains, the solvability of the problem was reduced to the solvability of the Fredholm integral equation of the second kind. A sufficient condition for the solvability of the integral equation was obtained through estimates of the kernel of the equation. The solution of the problem was split into two problems in the regions under consideration: in the parabolic part of the region, the first boundary value problem for the heat conduction equation was solved using the Green's function method, and in the hyperbolic part of the region, bounded by the characteristics of the equation and the line y=0, the solution of the problem using the Riemann function construction method was determined as the solution of the Cauchy problem. By applying a curvilinear integral, the solution to the problem in the areas under consideration was found. The necessity of the requirement of continuity of the function itself and its first two derivatives with respect to y on the line of change of the equation type was justified. Sufficient conditions for the unique classical solvability of the boundary value problem were established. The obtained conditions for the solvability of the boundary value problem provided a theoretical basis for the development of numerical methods for solving applied problems in aerohydrodynamics, geophysics, and engineering thermodynamics

We develop a modulus method for surface families inside a domain in the Heisenberg group and we prove that the stretch map between two Heisenberg spherical rings is a minimiser for the mean distortion among the class of contact quasiconformal maps between these rings which satisfy certain boundary conditions.

On Schur multiplier and projective representations of Heisenberg groups

We prove Cordes type estimates for subelliptic linear partial differential operators in non-divergence form with measurable coefficients in the Heisenberg group. As an application we establish interior horizontal W 2 , 2 W^{2,2} -regularity for p-harmonic functions in the Heisenberg group H 1 {\mathbb H}^1 for the range 17 − 1 2 ≤ p > 5 + 5 2 \frac {\sqrt {17}-1}{2} \leq p > \frac {5+\sqrt {5}}{2} .