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 (ue) of minimizers or almost-minimizers of Ginzburg–Landau functionals with coupling constant 1/e2tending to +∞.
We prove a new inequality for the Jacobian (or vorticity) associated to the Ginzburg–Landau energy in any dimension. It allows to retrieve existing lower bounds on the energy, to extend them to the case of unbounded vorticity, and to get a few other corollaries. It also provides a new estimate on the time-variation for time-dependent families, which has applications for the study of Ginzburg–Landau dynamics.
This chapter considers the limitations of traditional faculty assessment systems in the context of Boyer’s four domains of scholarship and suggests a new organizing template for realigning assessment systems to accommodate work in all four domains.