Variational problems for maps of bounded variation with values in \(S^ 1\)
The authors' main aim is to characterize the weak limits of smooth maps from an open subset \(\Omega\) of a Riemannian manifold into \(S^ 1\) (the unit circle). This is achieved in the class of Cartesian currents in \(\Omega\times S^ 1\), \(\text{cart}(\Omega\times S^ 1)\), introduced earlier by these authors [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 16, No. 3, 393-485 (1989; Zbl 0713.49014)]. In the present paper it is shown, among others, that each element of \(\text{cart}(\Omega\times S^ 1)\) is the weak limit of a sequence of smooth maps (from \(\Omega\) into \(S^ 1\)) with bounded variations. This result is then applied to prove the existence of minimizers of certain integral functionals in a subclass of \(\text{cart}(\Omega\times S^ 1)\). In this spirit, the dipole problem (minimizing the total variation under prescribed homological singularities) is solved.
scientific article; zbMATH DE number 509971
scientific article