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 u_k:B^n\to Y with equibounded total variation give rise to equivalence classes of Cartesian currents in cart^{1,1}(B^n\times Y) for which we introduce a natural BV-energy. Assume moreover that the first homotopy group of lY is commutative. In any dimension n we prove that every element T in cart^{1,1}(B^n\times Y can be approximated weakly in the sense of currents by a sequence of graphs of smooth maps u_k:B^n\to 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 B^n into Y.

The BV-energy of maps into a manifold:relaxation and density results / Giaquinta, M; Mucci, Domenico. - In: ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE. - ISSN 0391-173X. - 5:4(2006), pp. 483-548.

### 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 u_k:B^n\to Y with equibounded total variation give rise to equivalence classes of Cartesian currents in cart^{1,1}(B^n\times Y) for which we introduce a natural BV-energy. Assume moreover that the first homotopy group of lY is commutative. In any dimension n we prove that every element T in cart^{1,1}(B^n\times Y can be approximated weakly in the sense of currents by a sequence of graphs of smooth maps u_k:B^n\to 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 B^n into Y.
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The BV-energy of maps into a manifold:relaxation and density results / Giaquinta, M; Mucci, Domenico. - In: ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE. - ISSN 0391-173X. - 5:4(2006), pp. 483-548.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11381/1652005
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