This talk is devoted to the prescribed mean curvature equation for t-graphs in a Riemannian Heisenberg group. Following results due to Giusti, we will prove existence of BV solutions in
the so-called non-extremal domains without imposing Dirichlet conditions. Then the regularity will be improved in consecutive steps to obtain classical solutions in these domains. The main
goals of the talk are the characterization of the existence of classical solutions to the equation and the study of several conditions that imply the uniqueness of them. Finally, by an approximation technique, we obtain solutions BVloc (Ω) ∩ ∞ Lloc (Ω) to the sub-Riemannian prescribed mean curvature equation. These results are obtained in collaboration with S. Verzellesi.