Lower bound for energies of harmonic tangent unit-vector fields on convex polyhedra

Abstract

We derive a lower bound for energies of harmonic maps of convex polyhedra in ${\mathbb{R}}^3$ 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.

Acknowledgment added, typos removed, minor corrections