We consider an energy functional motivated by the celebrated K 13 problem in the Oseen-Frank theory of nematic liquid crystals. It is defined for spherevalued functions and appears as the usual Dirichlet energy with an additional surface term.It is known that this energy is unbounded from below and our aim has been to study the local minimizers. We show that even having a critical point in a suitable energy space imposes severe restrictions on the boundary conditions. Having suitable boundary conditions makes the energy functional bounded and in this case we study the partial regularity of the minimizers.for functions n with |n(x)| = 1 a.e. inΩ where

Article (Accepted Version) http://sro.sussex.ac.uk Day, Stuart and Taheri, Ali (2018) Geodesics on SO(n) and a class of spherically symmetric maps as solutions to a nonlinear generalised harmonic map problem. Topological Methods in Nonlinear Analysis, 51 (2). pp. 637-662.

In this paper we consider an energy functional depending on the norm of the gradient and seek to extremise it over an admissible class of Sobolev maps defined on an annulus and taking values on the unit sphere whilst satisfying suitable boundary conditions. We establish the existence of an infinite family of solutions with certain symmetries to the associated nonlinear Euler-Lagrange system in even dimensions and discuss the stability of such extremisers by way of examining the positivity of the second variation of the energy at these solutions.

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