Maps into projective spaces: Liquid crystal and conformal energies

Variational problems for the liquid crystal energy of mappings
from three-dimensional domains into the real projective plane are studied.
More generally, we study the dipole problem, the relaxed energy, and density
properties concerning the conformal p-energy of mappings from n-dimensional
domains that are constrained to take values into the p-dimensional real projective
space, for any positive integer p. Furthermore, a notion of optimally
connecting measure for the singular set of such class of maps is given.