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

Weak Anchoring for a Two-Dimensional Liquid Crystal