Landau-de Gennes model with sextic potentials: asymptotic behavior of minimizers

We study a class of Landau-de Gennes energy functionals with a sextic bulk energy density in a three-dimensional domain. We examine the asymptotic behavior of uniformly bounded minimizers in two distinct scenarios: one where their energy remains uniformly bounded, and another where it logarithmically diverges as a function of the elastic constant. In the first case, we show that up to a subsequence, the minimizers converge to a locally minimizing harmonic map in both the $ H_{\loc}^1 $ and $ C_{\loc}^j, j\in \Z_+ $ norms within compact subsets that are distant from the singularities of the limit. For the second case, we establish the existence of a closed set denoted as $ \cS_{\op{line}} $. This set has finite length and consists of finite segments of lines locally such that the energy of minimizers are locally uniformly bounded away from it. This work solves an open question raised by Canevari (ARMA, 223 (2017), 591-676), specifically concerning point and line defects in the Landau-de Gennes model with sextic potentials.