Inside the Light Boojums: a Journey to the Land of Boundary Defects

We consider minimizers of the energy Eε(u)=:∫Ω[1/2|■u|^2+1/4ε2(|u|^2-1)^2]dx+1/2ε^s∫■ΩW(u,g)ds,u:Ω→■,0<s<1,in a two-dimensional domain W,with weak anchoring potential W(u,g)=:1/2(|u|^2-1)^2+(<u,g>-cosα)^2,0<a<π/2.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[α^2+(π-α)^2]<π^2/2,any limiting map u*:Ω→S^1 has only boundary vortices,where its phase jumps by either 2α(light boojums)or 2(π-α)(heavy boojums).Our main result is the fine-scale description of the light boojums.