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.