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. A liquid crystal is a state of a matter, called mesomorphic, intermediate between a crystalline solid and a normal isotropic liquid, in which long rod-shaped molecules display orientational order. According to the continuum description in the Ericksen-Leslie theory [10, 29], a configuration of a liquid crystal which occupies a domain Ω in R is described mathematically as a unitary vector field u(x) in Ω. The bulk energy associated to the configuration u is given by