Sobolev maps into the projective line with bounded total variation

Variational problems for Sobolev maps with bounded total variation that take values into the 1-dimensional projective space are studied. We focus on the different features from the case of Sobolev maps with bounded conformal p-energy that take values into the p-dimensional projective space, for p ≥ 2 integer, recently studied in [18]. In the last decades there has been a growing interest in the study of variational problems for maps defined between manifolds. The most relevant problem is perhaps the one concerned with harmonic maps defined in three dimensional domains Ω that are constrained to take values into the two-dimensional unit sphere S. In this framework, one considers the Dirichlet energy