Asymptotics for Minimizers of Landau-de Gennes with a Magnetic Field and Tangential Anchoring

In this article we prove existence of minimizers of the Landau-de Gennes energy for liquid crystals with homogeneous external magnetic field and strong uniaxial planar anchoring. Next we consider the asymptotics of solutions to the joint minimization of the energy w.r.t. the function and its boundary condition. This constitutes a generalization to arbitrary regular particle shapes of the results obtained in [BLS, arXiv:2403.20274] in a particular setting. Moreover, we show the absence of line singularities in some asymptotic parameter regime. Finally we characterize the optimal orientation of particles vis-\`a-vis the magnetic field direction and compute it explicitly for different particle shapes.