Lower bound for energies of harmonic tangent unit-vector fields on   convex polyhedra
  We derive a lower bound for energies of harmonic maps of convex polyhedra in $ \R^3 $ to the unit sphere $S^2,$ with tangent boundary conditions on the faces. We also establish that $C^\infty$ maps, satisfying tangent boundary conditions, are dense with respect to the Sobolev norm, in the space of continuous tangent maps of finite energy. 
A. Majumdar, J.M. Robbins, M. Zyskin