JournalscmhVol. 81 , No. 2DOI 10.4171/cmh/55

H-minimal graphs of low regularity in <em><strong>H</strong><sup>1</sup></em>

  • Scott D. Pauls

    Dartmouth College, Hanover, United States
H-minimal graphs of low regularity in <em><strong>H</strong><sup>1</sup></em> cover

Abstract

In this paper we investigate H-minimal graphs of lower regularity. We show that noncharacteristic C1C^1 H-minimal graphs whose components of the unit horizontal Gauss map are in W1,1W^{1,1} are ruled surfaces with C2C^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 C1C^1 H-minimal graphs can be glued together to form continuous piecewise C1C^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 C2C^2.