Neumann Problem for a Smooth Bounded Domain in the Heisenberg Group <img src=image/13443204_01.gif>

Boundary value problems arise while studying differential equations and play a fundamental role in diverse areas of scientific, medical, and engineering disciplines, such as medical sciences involving diffusion processes of drugs, neuroscience, environmental studies, modelling in economics and finance, and simulations for computer graphics. Consequently, their study becomes essential in real-world applications. Two boundary value problems, namely the Dirichlet and Neumann problems associated with the Laplace equation, are of substantial significance in the discipline of partial differential equations. The Dirichlet problem involves finding a harmonic function within a domain, subject to the condition that its values coincide with a given continuous function on the boundary. On the other hand, the Neumann problem demands a solution in the form of a harmonic function whose normal derivative equals a specified function on the boundary of the domain. These problems acquire increased significance when the regularity of the associated differential operator is degraded. The Heisenberg group, a non-abelian and a non-compact Lie group, becomes a nice object to study these boundary value problems as being the simplest example having said properties in association with a subelliptic Laplace like operator called the Kohn-Laplacian. Gaveau was the first to discuss the Dirichlet problem for the Kohn–Laplacian on the Heisenberg groups in 1977. Later, Jerison further discussed it by calculating estimates in the Dirichlet problem in a smooth domain D, along with the regularity of the solution. The Neumann problem for the Kohn-Laplacian on the Koranyi ball in the Heisenberg group was initially addressed by Kumar, Dubey and Mishra in 2016, which was further generalized to H-type groups by Pandey and Mishra for certain gauge balls in H-type groups. We further generalize the existence and uniqueness results of the Neumann problem for the Kohn-Laplacian for bounded domains with smooth boundary that have no characteristic points in the Heisenberg group. We have established certain estimates of the derivatives of the fundamental solution and obtained the necessary and sufficient condition for the solvability of the interior Neumann problem for the same.