Lower Bounds for the Energy of Unit Vector Fields and Applications

We prove lower bounds for the Dirichlet energy of a unit vector field defined in a perforated domain of R2with nonzero degree on the outer boundary in terms of the total diameter of the holes. We use this to derive lower bounds, and then compactness results for sequences (uε) of minimizers or almost-minimizers of Ginzburg–Landau functionals with coupling constant 1/ε2tending to +∞.