Existence and uniqueness of $t$-graphs of prescribed mean curvature in Heisenberg groups

TL;DR

This paper investigates the existence and uniqueness of solutions to the prescribed mean curvature equation for t-graphs in Riemannian Heisenberg groups, extending results to non-constant mean curvature and sub-Riemannian cases.

Contribution

It characterizes conditions for existence and uniqueness of classical solutions without boundary data and extends to non-constant mean curvature and sub-Riemannian equations.

Findings

Characterized existence of solutions in bounded domains.

Provided conditions for solution uniqueness.

Extended results to non-constant mean curvature and sub-Riemannian equations.

Abstract

We study the prescribed mean curvature equation for $t$-graphs in a Riemannian Heisenberg group of arbitrary dimension. We characterize the existence of classical solutions in a bounded domain without imposing Dirichlet boundary data, and we provide conditions that guarantee uniqueness. Moreover, we extend previous results to solve the Dirichlet problem when the mean curvature is non-constant. Finally, by an approximation technique, we obtain solutions to the sub-Riemannian prescribed mean curvature equation.