Neumann boundary value problem in domains of the Heisenberg Group $\mathbb H_n$

TL;DR

This paper investigates the Neumann boundary value problem for the Kohn-Laplacian on the Heisenberg group, providing explicit Green's functions and solutions for circular data in specific domains.

Contribution

It derives explicit Green's functions for the Neumann problem on the Korányi ball and half-space in the Heisenberg group, enabling solutions for inhomogeneous boundary data.

Findings

Explicit Green's functions for the Neumann problem in the Korányi ball and half-space.

Solutions for inhomogeneous Neumann boundary value problems with circular data.

Uniqueness and existence results for the Neumann problem in these domains.

Abstract

Existence and uniqueness of the solution of the Neumann problem for the Kohn-Laplacian on the Kor\'anyi ball of the Heisenberg group $\mathbb{H}_n$ are discussed. Explicit representations of Green's type function (Neumann function) for the half space and Kor\'anyi ball in $\mathbb{H}_n$ for circular functions have been obtained. These functions are then used on above regions in $\mathbb{H}_n$ to solve the inhomogeneous Neumann boundary value problem for circular data.