JournalsaihpcVol. 25, No. 1pp. 77–103

Energies of S2 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
Energies of \( S^{2} \)-valued harmonic maps on polyhedra with tangent boundary conditions cover
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Abstract

A unit-vector field n:PS2\mathbf{n}\:\text{:}P\rightarrow S^{2} on a convex polyhedron PR3P \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 EP(h)E_{P}^{−}(h) for the infimum Dirichlet energy EPinf(h)E_{P}^{\mathrm{\inf }}(h) for such tangent unit-vector fields of arbitrary homotopy type h. EP(h)E_{P}^{−}(h) is expressed as a weighted sum of minimal connections, one for each sector of a natural partition of S2S^{2} induced by P. For P a rectangular prism, we derive an upper bound for EPinf(h)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 S2 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