Inside the light boojums: a journey to the land of boundary defects

We consider minimizers of the energy \begin{equation*} E_\varepsilon (u) : = \int_\Omega \left[ \frac12 |\nabla u|^2 + {1\over 4\varepsilon^2} (|u|^2-1)^2\right]dx + \frac1{2\varepsilon^{s}}\int_{\partial\Omega} W(u,g) \, ds,\ u:\Omega\to{\mathbb C}, \end{equation*} in a two-dimensional domain $\Omega$, with $0$$<$$s$$<$$1$ and weak anchoring potential \begin{equation*} W(u,g):= \frac 12 (|u|^2-1)^2 +\left( \langle u, g\rangle -\cos\alpha\right)^2,\ 0<\alpha<\frac\pi 2. \end{equation*} This functional was previously derived as a thin-film limit of the Landau-de Gennes energy, assuming weak anchoring on the boundary favoring a nematic director lying along a cone of fixed aperture, centered at the normal vector to the boundary. In the regime where $s\, [\alpha^2+(\pi-\alpha)^2]$$<$$\pi^2/2$, any limiting map $u_\ast:\Omega\to{\mathbb S}^1$ has only boundary vortices, where its phase jumps by either $2\alpha$ (light boojums) or $2(\pi-\alpha)$ (heavy boojums). Our main result is the fine-scale description of the light boojums.