Minimal graphs in $$Nil_3:$$ N i l 3 : existence and non-existence results

Abstract

We study the minimal surface equation in the Heisenberg space, \(Nil_3.\) A geometric proof of non existence of minimal graphs over non convex, bounded and unbounded domains is achieved for some prescribed boundary data (our proof holds in the Euclidean space as well). We solve the Dirichlet problem for the minimal surface equation over bounded and unbounded convex domains, taking bounded, piecewise continuous boundary value. We are able to construct a Scherk type minimal surface and we use it as a barrier to construct non trivial minimal graphs over a wedge of angle \(\theta \in [\frac{\pi }{2}, \pi [,\) taking non negative continuous boundary data, having at least quadratic growth. In the case of an half-plane, we are also able to give solutions (with either linear or quadratic growth), provided some geometric hypothesis on the boundary data are satisfied. Finally, some open problems arising from our work, are posed.