Well-posedness of a Neumann-type problem on a gauge ball in H-type groups

Mukund Madhav Mishra,Ashutosh Pandey

doi:10.1186/s13661-020-01389-2

Abstract

We discuss the existence and uniqueness of solution for the second boundary value problem of potential theory often referred to as the Neumann problem, on a gauge ball for the canonical sub-Laplacian in H-type groups. In this way we extend the classical results of the problem as well as its generalization to the Heisenberg group.

Highlights

1 Introduction In the study of partial differential equations, two boundary value problems associated with the Laplace equation occupy a special place, namely the Dirichlet and Neumann problems
While the Dirichlet problem asks to obtain a harmonic function in a domain whose value agrees with a prescribed function on the boundary, the Neumann problem requires the normal derivative of the solution function to agree with a prescribed function [17]
For domains having C1,α boundary where α is in

Summary

Introduction

In the study of partial differential equations, two boundary value problems associated with the Laplace equation occupy a special place, namely the Dirichlet and Neumann problems. While the Dirichlet problem asks to obtain a harmonic function in a domain whose value agrees with a prescribed (continuous) function on the boundary, the Neumann problem requires the normal derivative of the solution function to agree with a prescribed function [17]. For domains having C1,α boundary where α is in (0, 1], for example, a unit ball, the Dirichlet problem is solvable for any continuous boundary value (see, for details, [10]) whereas the Neumann problem is solvable under the essential condition that the integral of the values assigned to the normal derivative vanishes over the boundary surface (see, for details, [14]), i.e., f ds = 0

Mishra and Pandey Boundary Value Problems

Zj for every

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