H-minimal graphs of low regularity in ${\mathbb{H}}^1$

Abstract

In this paper we investigate H-minimal graphs of lower regularity. We show that noncharacteristic $C^1$ H-minimal graphs whose components of the unit horizontal Gauss map are in $W^{1,1}$ are ruled surfaces with $C^2$ seed curves. Moreover, in light of a structure theorem of Franchi, Serapioni and Serra Cassano, we see that any H-minimal graph is, up to a set of perimeter zero, composed of such pieces. Along these lines, we investigate ways in which patches of $C^1$ H-minimal graphs can be glued together to form continuous piecewise $C^1$ H-minimal surfaces. We apply this description of H-minimal graphs to the question of the existence of smooth solutions to the Dirichlet problem with smooth data. We find a necessary and sufficient condition for the existence of smooth solutions and produce examples where the conditions are satisfied and where they fail. In particular we illustrate the failure of the smoothness of the data to force smoothness of the solution to the Dirichlet problem by producing a class of curves whose H-minimal spanning graphs cannot be $C^2$.