The BV-energy of maps into a manifold: Relaxation and density results

Abstract

Let Y be a smooth compact oriented Riemannian manifold without boundary, and assume that its 1-homology group has no torsion. Weak limits of graphs of smooth maps uk : Bn → Y with equibounded total variation give rise to equivalence classes of Cartesian currents in cart1,1 (Bn × Y) for which we introduce a natural BV-energy. Assume moreover that the first homotopy group of Y is commutative. In any dimension n we prove that every element T in cart1,1 (Bn × Y) can be approximated weakly in the sense of currents by a sequence of graphs of smooth maps uk : Bn → Y with total variation converging to the BV-energy of T. As a consequence, we characterize the lower semicontinuous envelope of functions of bounded variations from Bn into Y.