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Weak Anchoring for a Two-Dimensional Liquid Crystal

We study the weak anchoring condition for nematic liquid crystals in the context of the Landau-De Gennes model. We restrict our attention to two dimensional samples and to nematic director fields lying in the plane, for which the Landau-De Gennes energy reduces to the Ginzburg--Landau functional, and the weak anchoring condition is realized via a penalized boundary term in the energy. We study the singular limit as the length scale parameter $\varepsilon\to 0$, assuming the weak anchoring parameter $λ=λ(\varepsilon)\to\infty$ at a prescribed rate. We also consider a specific example of a bulk nematic liquid crystal with an included oil droplet and derive a precise description of the defect locations for this situation, for $λ(\varepsilon)=K\varepsilon^{-α}$ with $α\in (0,1]$. We show that defects lie on the weak anchoring boundary for $α\in (0,\frac12)$, or for $α=\frac12$ and $K$ small, but they occur inside the bulk domain $Ω$ for $α>\frac12$ or $α=\frac12$ with $K$ large.