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

TL;DR

This paper establishes a lower energy bound for harmonic tangent unit-vector fields on convex polyhedra and proves the density of smooth tangent maps in the space of finite-energy tangent maps.

Contribution

It provides a fundamental lower bound for energies of harmonic tangent maps on convex polyhedra and shows smooth maps are dense in the space of finite-energy tangent maps.

Findings

Derived a lower energy bound for harmonic tangent maps.

Proved density of smooth tangent maps in Sobolev space.

Established properties of tangent boundary conditions on convex polyhedra.

Abstract

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.