Far-Field Expansions for Harmonic Maps and the Electrostatics Analogy in Nematic Suspensions

For a smooth bounded domain G⊂R3$$G\subset {{\mathbb {R}}}^3$$, we consider maps n:R3\G→S2$$n:{\mathbb {R}}^3\setminus G\rightarrow {\mathbb {S}}^2$$ minimizing the energy E(n)=∫R3\G|∇n|2+Fs(n⌊∂G)$$E(n)=\int _{{\mathbb {R}}^3{\setminus } G}|\nabla n|^2 +F_s(n_{\lfloor \partial G})$$ among S2$${\mathbb {S}}^2$$-valued map such that n(x)≈n0$$n(x)\approx n_0$$ as |x|→∞$$|x|\rightarrow \infty $$. This is a model for a particle G immersed in nematic liquid crystal. The surface energy Fs$$F_s$$ describes the anchoring properties of the particle and can be quite general. We prove that such minimizing map n has an asymptotic expansion in powers of 1/r. Further, we show that the leading order 1/r term is uniquely determined by the far-field condition n0$$n_0$$ for almost all n0∈S2$$n_0\in {\mathbb {S}}^2$$, by relating it to the gradient of the minimal energy with respect to n0$$n_0$$. We derive various consequences of this relation in physically motivated situations: when the orientation of the particle G is stable relative to a prescribed far-field alignment n0$$n_0$$; and when the particle G has some rotational symmetries. In particular, these corollaries justify some approximations that can be found in the physics literature to describe nematic suspensions via a so-called electrostatics analogy.