Biaxiality in the asymptotic analysis of a 2-D Landau-de Gennes model for liquid crystals

Abstract: We consider the Landau-de Gennes variational problem on a bound-ed, two dimensional domain, subject to Dirichlet smooth boundary conditions. We prove that minimizers are maximally biaxial near the singularities, that is, their biaxiality parameter reaches the maximum value $1$. Moreover, we discuss the convergence of minimizers in the vanishing elastic constant limit. Our asymptotic analysis is performed in a general setting, which recovers the Landau-de Gennes problem as a specific case.