Abstract We establish, as 0, an asymptotic expansion for the minimal Dirichlet energy of S² -valued maps outside a finite number of particles of size with fixed centers xⱼ R³, under general anchoring conditions at the particle boundaries. Up to a scaling factor, this expansion is of the form align* E_ = ⱼ ⱼ -4 ₈ ₉ vᵢ, vⱼ|xᵢ-xⱼ| +o () \, , align* where ⱼ is the minimal energy after zooming in at scale around each particle, and vⱼ R³ is determined by the far-field behavior of the corresponding single-particle minimizer. The Coulomb-like interaction in this expansion agrees with the electrostatic analogy: a linearized approximation commonly used in the physics literature for colloid interactions in nematic liquid crystal. We obtain here for the first time a precise estimate of the energy error introduced by that linearization, by developing new tools that address the lack of convergence rate when zooming in at scale.
Bronsard et al. (Thu,) studied this question.