# Energies of $S^{2}$-valued harmonic maps on polyhedra with tangent boundary conditions

### A. Majumdar

School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK, Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS12 6QZ, UK### J.M. Robbins

School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK### M. Zyskin

School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK

## Abstract

A unit-vector field $\mathbf{n}\:\text{:}P\rightarrow S^{2}$ on a convex polyhedron $P \subset \mathbb{R}^{3}$ satisfies tangent boundary conditions if, on each face of *P*, **n** takes values tangent to that face. Tangent unit-vector fields are necessarily discontinuous at the vertices of *P*. We consider fields which are continuous elsewhere. We derive a lower bound $E_{P}^{−}(h)$ for the infimum Dirichlet energy $E_{P}^{\mathrm{\inf }}(h)$ for such tangent unit-vector fields of arbitrary homotopy type *h*. $E_{P}^{−}(h)$ is expressed as a weighted sum of minimal connections, one for each sector of a natural partition of $S^{2}$ induced by *P*. For *P* a rectangular prism, we derive an upper bound for $E_{P}^{\mathrm{\inf }}(h)$ whose ratio to the lower bound may be bounded independently of *h*. The problem is motivated by models of nematic liquid crystals in polyhedral geometries. Our results improve and extend several previous results.

## Cite this article

A. Majumdar, J.M. Robbins, M. Zyskin, Energies of $S^{2}$-valued harmonic maps on polyhedra with tangent boundary conditions. Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), no. 1, pp. 77–103

DOI 10.1016/J.ANIHPC.2006.11.003