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

Abstract

For a smooth bounded domain $$G\subset {{\mathbb {R}}}^3$$ , we consider maps $$n:{\mathbb {R}}^3\setminus G\rightarrow {\mathbb {S}}^2$$ minimizing the energy $$E(n)=\int _{{\mathbb {R}}^3{\setminus } G}|\nabla n|^2 +F_s(n_{\lfloor \partial G})$$ among $${\mathbb {S}}^2$$ -valued map such that $$n(x)\approx n_0$$ as $$|x|\rightarrow \infty $$ . This is a model for a particle G immersed in nematic liquid crystal. The surface energy $$F_s$$ 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 $$n_0$$ for almost all $$n_0\in {\mathbb {S}}^2$$ , by relating it to the gradient of the minimal energy with respect to $$n_0$$ . 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 $$n_0$$ ; 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.